INTERMITTENT DEMAND FORECASTING
WITH INTEGER AUTOREGRESSIVE
MOVING AVERAGE MODELS
A Thesis submitted for the degree of Doctor of Philosophy
By
Maryam Mohammadipour
School of Business and Management
Faculty of Enterprise and Innovation
Buckinghamshire New University
Brunel University
April, 2009
ii
Abstract
This PhD thesis focuses on using time series models for counts in modelling and
forecasting a special type of count series called intermittent series. An intermittent
series is a series of nonnegative integer values with some zero values. Such series
occur in many areas including inventory control of spare parts. Various methods
have been developed for intermittent demand forecasting with Croston?s method
being the most widely used.
Some studies focus on finding a model underlying Croston?s method. With none of
these studies being successful in demonstrating an underlying model for which
Croston?s method is optimal, the focus should now shift towards stationary models
for intermittent demand forecasting.
This thesis explores the application of a class of models for count data called the
Integer Autoregressive Moving Average (INARMA) models. INARMA models have
had applications in different areas such as medical science and economics, but this is
the first attempt to use such a modelbased method to forecast intermittent demand.
In this PhD research, we first fill some gaps in the INARMA literature by finding the
unconditional variance and the autocorrelation function of the general INARMA(p,q)
model. The conditional expected value of the aggregated process over lead time is
also obtained to be used as a lead time forecast. The accuracy of hstepahead and
lead time INARMA forecasts are then compared to those obtained by benchmark
methods of Croston, SyntetosBoylan Approximation (SBA) and ShaleBoylan
Johnston (SBJ).
The results of the simulation suggest that in the presence of a high autocorrelation in
data, INARMA yields much more accurate onestep ahead forecasts than benchmark
methods. The degree of improvement increases for longer data histories. It has been
shown that instead of identification of the autoregressive and moving average order
of the INARMA model, the most general model among the possible models can be
used for forecasting. This is especially useful for short history and high
autocorrelation in data.
The findings of the thesis have been tested on two real data sets: (i) Royal Air Force
(RAF) demand history of 16,000 SKUs and (ii) 3,000 series of intermittent demand
from the automotive industry. The results show that for sparse data with long history,
there is a substantial improvement in using INARMA over the benchmarks in terms
of Mean Square Error (MSE) and Mean Absolute Scaled Error (MASE) for the one
step ahead forecasts. However, for series with short history the improvement is
narrower. The improvement is greater for hstep ahead forecasts. The results also
confirm the superiority of INARMA over the benchmark methods for lead time
forecasts.
iii
Acknowledgements
Completing a PhD is by far the most challenging yet rewarding task I have ever
undertaken. Fulfilling this task was not possible without the support of numerous
people, to whom I am grateful.
Most importantly, I would like to express my deep and sincere gratitude to my
supervisor Prof. John Boylan, Head of Research, Faculty of Enterprise and
Innovation, Buckinghamshire New University. His indepth knowledge, constant
encouragement, support and particularly his advice have been invaluable in ensuring
I was on the right track throughout the PhD studies. I appreciate all his contributions
of time and ideas.
I also wish to express my appreciation to Dr Aris Syntetos of Salford Business
School, University of Salford, for providing me with the empirical data for this
thesis. I would also like to thank Ms Laura Bray, Faculty Research Officer,
Buckinghamshire New University for her sympathetic help she has provided during
the last three years.
I am grateful to the Buckinghamshire New University for the scholarship they
provided for this PhD study and also the support to enable me to attend and present
my academic papers at international conferences.
My special appreciation goes to my parents, Mina Moradkhan and Gholamhossein
Mohammadipour, who always encouraged me to concentrate on my studies. Their
nonstop support has made my dreams come true. I also want to thank my brother, Dr
Mani Mohammadipour, and my uncle, Dr Shahryar Mohammadipour, for their love
and friendship.
Last but not least, words cannot express my special thanks to my beloved husband,
Soroosh Saghiri, for his endless love, faith and encouragement in fulfilling this
aspiration.
iv
To Soroosh
my husband
my best friend
v
Contents
Chapter 1 Introduction ............................................................................................ 1
1.1 Introduction ....................................................................................................... 2
1.2 Research Overview ........................................................................................... 3
1.3 Business Context ............................................................................................... 3
1.4 Research Background ....................................................................................... 4
1.5 Research Problem ............................................................................................. 6
1.5.1 Initial Problem .............................................................................................. 6
1.5.2 Research Questions ....................................................................................... 6
1.6 Research Methodology ..................................................................................... 8
1.6.1 Research Philosophy ..................................................................................... 9
1.6.2 Research Approach ....................................................................................... 9
1.6.3 Research Strategy ......................................................................................... 9
1.6.3.1 Mathematical Analysis ......................................................................... 9
1.6.3.2 Simulation .......................................................................................... 10
1.6.3.3 Empirical Analysis ............................................................................. 11
1.7 Thesis Structure .............................................................................................. 11
Chapter 2 Forecasting Intermittent Demand ....................................................... 13
2.1 Introduction ..................................................................................................... 13
2.2 Definition of Intermittent Demand ................................................................. 14
2.3 Methods of Forecasting Intermittent Demand ................................................ 17
2.3.1 Croston?s Method ....................................................................................... 18
2.3.1.1 Bias Correction for Bernoulli Demand Incidence .............................. 20
2.3.1.2 Bias Correction for Poisson Demand Incidence ................................. 21
2.3.1.3 Modified Croston (MC) ...................................................................... 22
2.3.1.4 Models underlying Croston?s Method ................................................ 23
2.3.2 Bootstrapping .............................................................................................. 25
2.3.2.1 Snyder (2002) ..................................................................................... 25
2.3.2.2 Willemain et al. (2004) ....................................................................... 26
2.3.2.3 Porras and Dekker (2007) ................................................................... 27
2.3.3 Causal Models ............................................................................................ 28
2.3.4 Conclusions on IDF Methods ..................................................................... 30
2.4 Assessing Forecast Accuracy .......................................................................... 30
2.4.1 Absolute Accuracy Measures ..................................................................... 32
vi
2.4.2 Accuracy Measures Relative to another Method ........................................ 32
2.4.3 Conclusions on Accuracy Measures ........................................................... 34
2.5 Comparative Studies ....................................................................................... 36
2.6 INARMA Models ........................................................................................... 37
2.7 Conclusions ..................................................................................................... 38
Chapter 3 Integer Autoregressive Moving Average Models............................... 41
3.1 Introduction ..................................................................................................... 41
3.2 DARMA Models ............................................................................................. 43
3.2.1 DAR(1) Model ............................................................................................ 44
3.2.2 Applications of DARMA Models ............................................................... 45
3.3 INARMA Models ........................................................................................... 46
3.3.1 INAR(1) Model .......................................................................................... 47
3.3.2 INAR(2) Model .......................................................................................... 50
3.3.3 INAR(p) Model .......................................................................................... 52
3.3.4 INMA(1) Model .......................................................................................... 54
3.3.5 INMA(2) Model .......................................................................................... 56
3.3.6 INMA(q) Model .......................................................................................... 57
3.3.7 INARMA(1,1) Model ................................................................................. 58
3.3.8 INARMA(p,q) Model ................................................................................. 60
3.3.8.1 First and Second Unconditional Moments ......................................... 61
3.3.8.2 Autocorrelation Function (ACF) ........................................................ 62
3.3.9 Applications of INARMA Models ............................................................. 67
3.3.10 Aggregation in INARMA Models .......................................................... 70
3.3.10.1 Overlapping Temporal Aggregation .............................................. 72
3.3.10.2 Nonoverlapping Temporal Aggregation ....................................... 73
3.3.10.3 Crosssectional Aggregation .......................................................... 76
3.3.10.4 Over a Forecast Horizon Aggregation ........................................... 80
3.4 Summary of Literature Review ....................................................................... 81
3.5 Conclusions ..................................................................................................... 85
Chapter 4 Identification in INARMA Models ..................................................... 86
4.1 Introduction ..................................................................................................... 86
4.2 Testing Serial Dependence ............................................................................. 87
4.2.1 Runs Test .................................................................................................... 87
4.2.2 The Score Test ............................................................................................ 88
4.2.3 Portmanteautype Tests .............................................................................. 88
4.3 Identification based on ACF and PACF ......................................................... 90
vii
4.3.1 Autocorrelation Function (ACF) ................................................................ 90
4.3.2 Partial Correlation Function (PACF) .......................................................... 91
4.3.2.1 PACF of an INAR(p) Model .............................................................. 91
4.3.2.2 PACF of an INMA(q) Model ............................................................. 94
4.3.2.3 PACF of an INARMA(p,q) Model ..................................................... 95
4.4 Residual Analysis ............................................................................................ 96
4.5 Identification based on Penalty Functions ...................................................... 99
4.6 The Identification Procedure ......................................................................... 101
4.7 Conclusions ................................................................................................... 102
Chapter 5 Estimation in INARMA Models ........................................................ 104
5.1 Introduction ................................................................................................... 104
5.2 Estimation in an INARMA(0,0) Model ........................................................ 106
5.3 Estimation in an INAR(1) Model.................................................................. 107
5.3.1 YW for INAR(1) ....................................................................................... 107
5.3.2 CLS for INAR(1) ...................................................................................... 108
5.3.3 CML for INAR(1) ..................................................................................... 109
5.3.4 Conditional GMM for INAR(1) ............................................................... 111
5.4 Estimation in an INAR(p) Model.................................................................. 112
5.4.1 YW for INAR(p) ....................................................................................... 112
5.4.2 CLS for INAR(p) ...................................................................................... 113
5.4.3 CML for INAR(p) ..................................................................................... 113
5.5 Estimation in an INMA(1) Model ................................................................. 115
5.5.1 YW for INMA(1) ...................................................................................... 115
5.5.2 CLS for INMA(1) ..................................................................................... 115
5.6 Estimation in an INMA(q) Model ................................................................. 116
5.6.1 YW for INMA(q) ...................................................................................... 116
5.6.2 CLS for INMA(q) ..................................................................................... 117
5.6.3 GMM based on Probability Generation Functions for INMA(q) ............. 117
5.7 Estimation in an INARMA(1,1) Model ........................................................ 119
5.7.1 YW for INARMA(1,1) ............................................................................. 119
5.7.2 CLS for INARMA(1,1) ............................................................................ 120
5.8 YW Estimators of an INARMA(2,2) Model ................................................ 121
5.9 Conclusions ................................................................................................... 122
Chapter 6 Forecasting in INARMA Models ....................................................... 124
6.1 Introduction ................................................................................................... 124
6.2 MMSE Forecasts ........................................................................................... 125
viii
6.2.1 MMSE Forecasts for an INAR(p) Model ................................................. 125
6.2.2 MMSE Forecasts for an INMA(q) Model ................................................ 127
6.2.3 MMSE Forecasts for an INARMA(p,q) Model ........................................ 128
6.3 Forecasting over Lead Time ......................................................................... 129
6.3.1 Lead Time Forecasting for an INAR(1) Model ........................................ 129
6.3.2 Lead Time Forecasting for an INMA(1) Model ....................................... 132
6.3.3 Lead Time Forecasting for an INARMA(1,1) Model ............................... 133
6.3.4 Lead Time Forecasting for an INARMA(p,q) Model ............................... 134
6.4 Conclusions ................................................................................................... 136
Chapter 7 Simulation Design ............................................................................... 137
7.1 Introduction ................................................................................................... 137
7.2 Rationale for Simulation ............................................................................... 138
7.3 Simulation Design ......................................................................................... 139
7.3.1 The Range of Series .................................................................................. 139
7.3.2 Producing INARMA(p,q) Series .............................................................. 140
7.3.3 Control Parameters ................................................................................... 141
7.3.3.1 INARMA Parameters ....................................................................... 141
7.3.3.2 Length of Series ................................................................................ 143
7.3.3.3 Forecast Horizon and Lead Time ..................................................... 143
7.3.3.4 Benchmark Methods? Parameters..................................................... 144
7.3.4 Identification Procedure ............................................................................ 144
7.3.5 Estimation of Parameters .......................................................................... 145
7.3.6 Forecasting Method .................................................................................. 146
7.3.7 Performance Metrics ................................................................................. 147
7.4 Verification ................................................................................................... 147
7.5 Conclusions ................................................................................................... 148
Chapter 8 Simulation Results .............................................................................. 149
8.1 Introduction ................................................................................................... 149
8.2 Details of Simulation .................................................................................... 150
8.3 Accuracy of INARMA Parameter Estimates ................................................ 151
8.4 Forecasting Accuracy of INARMA Estimation Methods ............................. 154
8.5 CrostonSBA Categorization ........................................................................ 161
8.6 INARMA vs Benchmark Methods ............................................................... 166
8.6.1 INARMA with Known Order ................................................................... 166
8.6.2 INARMA with Unknown Order ............................................................... 173
8.6.2.1 Identification among Two Processes ................................................ 174
ix
8.6.2.2 AllINAR(1) ..................................................................................... 178
8.6.2.3 Identification among Four Processes................................................ 180
8.6.2.4 AllINARMA(1,1) ............................................................................ 188
8.6.2.5 AllINAR(1) vs Benchmark Methods .............................................. 191
8.6.3 Lead Time Forecasts ................................................................................. 193
8.7 Conclusions ................................................................................................... 199
Chapter 9 Empirical Analysis .............................................................................. 202
9.1 Introduction ................................................................................................... 202
9.2 Rationale for Empirical Analysis .................................................................. 203
9.3 Demand Data Series ...................................................................................... 204
9.4 Design of Empirical Analysis ....................................................................... 206
9.5 INARMA vs Benchmark Methods ............................................................... 207
9.5.1 AllINAR(1) ............................................................................................. 208
9.5.2 AllINARMA(1,1) .................................................................................... 211
9.5.3 Identification among four Processes ......................................................... 214
9.5.4 INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1) Series .............. 217
9.5.5 hstepahead Forecasts for INAR(1) Series .............................................. 222
9.5.6 The Effect of Length of History ............................................................... 226
9.5.7 Lead Time Forecasting for INAR(1) Series ............................................. 228
9.5.8 Lead Time Forecasting for allINAR(1) ................................................... 234
9.6 Conclusions ................................................................................................... 238
Chapter 10 Conclusions and Further Research ................................................... 241
10.1 Introduction ................................................................................................... 241
10.2 Contributions ................................................................................................. 242
10.3 Conclusions from the Theoretical Part of the Thesis .................................... 243
10.3.1 The Unconditional Variance of an INARMA(p,q) Model ................... 244
10.3.2 The Autocorrelation Function of an INARMA(p,q) Model ................. 245
10.3.3 The YW Estimators of an INARMA(1,1) Model ................................. 246
10.3.4 Lead Time Forecasting of an INARMA(p,q) Model ............................ 246
10.4 Conclusions from the Simulation Part of the Thesis..................................... 247
10.4.1 The Performance of Different Estimation Methods ............................. 248
10.4.2 The CrostonSBA Categorization ......................................................... 248
10.4.3 Identification in INARMA Models ...................................................... 249
10.4.4 Comparing INARMA with the Benchmark Methods ........................... 250
10.5 Conclusions from the Empirical Part of the Thesis ...................................... 252
10.5.1 Identification in INARMA Models ...................................................... 252
x
10.5.2 The Performance of Different Estimation Methods ............................. 252
10.5.3 Comparing INARMA with the Benchmark Methods ........................... 253
10.5.4 The Problem with MASE ..................................................................... 253
10.6 Practical and Software implications .............................................................. 254
10.7 Limitations and Further Research ................................................................. 256
References..................................................................................................................... 260
Appendix 3.A Autocorrelation Function of an INARMA(1,1) Model ......................... 272
Appendix 3.B The Unconditional Variance of an INARMA(p,q) Model .................... 274
Appendix 3.C The CrossCovariance Function between Y and Z for an INARMA(p,q)
Model ............................................................................................................................. 280
Appendix 3.D Over Lead Time Aggregation of an INAR(1) Model ........................... 281
Appendix 4.A Infinite Autoregressive Representation of an INARMA(p,q) Model ... 286
Appendix 5.A The CLS Estimators of an INARMA(1,1) Model ................................. 289
Appendix 5.B The Unconditional Variance of an INARMA(2,2) Model .................... 291
Appendix 6.A Lead Time Forecasting for an INAR(2) Model .................................... 293
Appendix 6.B Lead Time Forecasting for an INARMA(1,2) Model ........................... 297
Appendix 6.C Lead Time Forecasting for an INARMA(p,q) Model ........................... 300
Appendix 8.A The MSE of YW and CLS Estimates for INAR(1), INMA(1) and
INARMA(1,1) Processes ............................................................................................... 303
Appendix 8.B Impact of YW and CLS Estimates on Accuracy of Forecasts using MASE
....................................................................................................................................... 306
Appendix 8.C CrostonSBA Categorization for INAR(1), INMA(1) and INARMA(1,1)
....................................................................................................................................... 308
Appendix 8.D Comparing the Accuracy of INARMA Forecasts for all points in time and
issue points .................................................................................................................... 311
Appendix 8.E Comparison of MASE of INARMA (known order) with Benchmarks 314
Appendix 8.F Comparison of sixstep ahead MSE of INARMA (known order) with
Benchmarks ................................................................................................................... 317
Appendix 8.G Comparison of allINAR(1) and allINARMA(1,1) ............................. 320
Appendix 8.H Comparison of MASE of INARMA (unknown order) with Benchmarks
....................................................................................................................................... 323
Appendix 8.I Comparison of MASE of INARMA with Benchmarks for Lead Time
Forecasts ........................................................................................................................ 325
Appendix 9.A INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1) Series of 16,000
Series ............................................................................................................................. 330
Appendix 9.B hstepahead Forecasts for the INARMA(0,0), INMA(1), and
INARMA(1,1) Series..................................................................................................... 334
Appendix 9.C Lead Time Forecasts for the INARMA(0,0), INMA(1), and
INARMA(1,1) Series..................................................................................................... 341
xi
List of Illustrations
Figure 11 The structure of Chapter 1 ............................................................................... 2
Figure 12 The structure of the research methodology ...................................................... 8
Figure 13 The structure of the thesis .............................................................................. 12
Figure 21 The categorization scheme for intermittent demand data (Syntetos et al. 2005)
......................................................................................................................................... 16
Figure 31 The covariance at lag , , when
qkpk ?? ,
........................................... 63
Figure 32 The covariance at lag at lag , , when
qk pk ?? ,
................................. 64
Figure 33 The covariance at lag , , when
qk pk ?? ,
........................................... 65
Figure 34 The covariance at lag , , when
qkpk ?? ,
........................................... 65
Figure 35 Nonoverlapping temporal aggregation for ....................................... 74
Figure 36 The correlation between the INMA parts of the nonoverlapping temporal
aggregation of INAR(1) process over two periods .......................................................... 75
Figure 41 Time series plot of one demand series among 16,000 series ......................... 98
Figure 42 Correlograms of the selected series among 16,000 series ............................. 98
Figure 43 Correlograms of the residuals of the INAR(1) model ................................... 99
Figure 61 hstepahead forecast for an INAR(p) model when
ph ?
........................... 126
Figure 62 hstepahead forecast for an INMA(q) model when .......................... 127
Figure 81 Cutoff values for Croston and SBA when (Syntetos et al., 2005) .. 162
Figure 3.B1 in an INARMA(p,q) process when .............................. 276
Figure 3.B2 in an INARMA(p,q) process when .............................. 277
Figure 3.B3 in an INARMA(p,q) process when .............................. 278
k
k?
k
k?
k
k?
k
k?
12?k
qh ?
20.??
),cov( ji ZY
qp ?
),cov( ji ZY qp ?
),cov( ji ZY qp ?
xii
List of Tables
Table 21 The categorization scheme for intermittent demand data (Eaves and
Kingsman, 2004) ............................................................................................................. 15
Table 22 Example data (Hyndman, 2006)...................................................................... 34
Table 23 Comparing ES with ZF based on different accuracy measures ...................... 34
Table 24 Accuracy measures for simulation and empirical studies ............................... 35
Table 25 The categorization of methods of intermittent demand forecasting based on
their assumptions ............................................................................................................. 38
Table 31 Review of count models in time series............................................................ 43
Table 32 Literature survey on Integer Autoregressive Moving Average models .......... 82
Table 51 Research papers on estimation of parameters of INARMA models ............. 105
Table 52 The relationship between the order of the model and the type of YW equation
....................................................................................................................................... 116
Table 61 Coefficients of in each of for an INAR(1) model ...................... 130
Table 62 Coefficients of in each of for an INAR(1) model ................. 131
Table 63 Coefficients of in each of for an INMA(1) model ................ 133
Table 64 Parameters of the overleadtimeaggregated INARMA(p,q) model ............ 135
Table 71 Range of autoregressive and moving average parameters ............................ 142
Table 72 Range of INARMA parameters studied in the literature ............................... 142
Table 73 Parameter space for the selected INARMA models ...................................... 143
Table 81 MSE of YW, CLS and CML estimates for INAR(1) series when ...... 152
Table 82 Accuracy of YW and CLS estimates for INAR(1) series ............................... 153
Table 83 Accuracy of YW and CLS estimates for INMA(1) series ............................... 154
Table 84 Accuracy of YW and CLS estimates for INARMA(1,1) series ...................... 155
Table 85 Onestep ahead forecast error comparison (YW, CLS and CML) for INAR(1)
series .............................................................................................................................. 156
Table 86 Onestep ahead forecast error comparison (YW and CLS) for INMA(1) series
....................................................................................................................................... 157
Table 87 Onestep ahead forecast error comparison (YW and CLS) for INARMA(1,1)
series .............................................................................................................................. 157
Table 88 Threestep ahead forecast error comparison (YW and CLS) for INAR(1) series
....................................................................................................................................... 158
Table 89 Sixstep ahead forecast error comparison (YW and CLS) for INAR(1) series 159
Table 810 Threestep ahead forecast error comparison (YW and CLS) for INMA(1) series
....................................................................................................................................... 159
tY 11
?
??
l
jjtY }{
ijkt
Z ?
1
1
?
??
l
jjtY }{
ijkt
Z ?
1
1
?
??
l
jjtY }{
24?n
xiii
Table 811 Sixstep ahead forecast error comparison (YW and CLS) for INMA(1) series
....................................................................................................................................... 159
Table 812 Threestep ahead forecast error comparison (YW and CLS) for INARMA(1,1)
series .............................................................................................................................. 160
Table 813 Sixstep ahead forecast error comparison (YW and CLS) for INARMA(1,1)
series .............................................................................................................................. 160
Table 814 The advantage of SBA over Croston for and all points in time ......... 163
Table 815 The advantage of SBA over Croston for and all points in time ......... 163
Table 816 Onestep ahead for INARMA(0,0) series (known
order) .............................................................................................................................. 167
Table 817 Onestep ahead for INMA(1) series (known order) 168
Table 818 Onestep ahead with smoothing parameter 0.2 for
INAR(1) series (known order) ........................................................................................ 169
Table 819 Onestep ahead with smoothing parameter 0.5 for
INAR(1) series (known order) ........................................................................................ 169
Table 820 Onestep ahead with smoothing parameter 0.2 for
INARMA(1,1) series (known order) ............................................................................. 170
Table 821 Onestep ahead with smoothing parameter 0.5 for
INARMA(1,1) series (known order) .............................................................................. 170
Table 822 Threestep ahead for INARMA(0,0) series (known
order) .............................................................................................................................. 171
Table 823 Threestep ahead for INMA(1) series (known order)
....................................................................................................................................... 171
Table 824 Threestep ahead with smoothing parameter 0.2 for
INAR(1) series (known order) ........................................................................................ 171
Table 825 Threestep ahead with smoothing parameter 0.5 for
INAR(1) series (known order) ........................................................................................ 172
Table 826 Threestep ahead with smoothing parameter 0.2 for
INARMA(1,1) series (known order) ............................................................................. 172
Table 827 Threestep ahead with smoothing parameter 0.5 for
INARMA(1,1) series (known order) .............................................................................. 173
Table 828 The percentage of correct identification for INARMA(0,0) series................ 175
Table 829 The percentage of correct identification for INAR(1) series ......................... 175
Table 830 Accuracy of INAR(1) forecasts for LjungBox and AIC identification
procedures ...................................................................................................................... 177
Table 831 Accuracy of INAR(1) forecasts when the order is known ............................ 178
Table 832 Accuracy of forecasts with identification and allINAR(1) for INARMA(0,0)
series .............................................................................................................................. 179
Table 833 The percentage of correct identification for INARMA(0,0) series................ 181
20.??
50.??
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
xiv
Table 834 The percentage of correct identification for INAR(1) series ......................... 181
Table 835 The percentage of correct identification for INMA(1) series ........................ 182
Table 836 The percentage of correct identification for INARMA(1,1) series................ 182
Table 837 Accuracy of INAR(1) forecasts for onestage and twostage identification
procedures ...................................................................................................................... 184
Table 838 Accuracy of INMA(1) forecasts for onestage and twostage identification
procedures ...................................................................................................................... 185
Table 839 Accuracy of INARMA(1,1) forecasts for onestage and twostage identification
procedures ...................................................................................................................... 186
Table 840 Accuracy of INAR(1) forecasts when the order in known ............................ 187
Table 841 Accuracy of INMA(1) forecasts when the order in known ........................... 187
Table 842 Accuracy of INARMA(1,1) forecasts when the order in known................... 187
Table 843 Accuracy of forecasts with identification and allINARMA(1,1) for
INARMA(0,0) series ...................................................................................................... 189
Table 844 Accuracy of forecasts with identification and allINARMA(1,1) for INAR(1)
series .............................................................................................................................. 189
Table 845 Accuracy of forecasts with identification and allINARMA(1,1) for INMA(1)
series .............................................................................................................................. 190
Table 846 for INARMA(0,0) series (unknown order) ........... 191
Table 847 for INMA(1) series (unknown order) .................... 191
Table 848 with smoothing parameter 0.2 for INARMA(1,1) series
(unknown order) ............................................................................................................. 192
Table 849 with smoothing parameter 0.5 for INARMA(1,1) series
(unknown order) ............................................................................................................. 192
Table 850
BenchmarkINARMA MSEMSE /
of leadtime forecasts for INARMA(0,0)
series .............................................................................................................................. 194
Table 851 of leadtime forecasts for INMA(1) series 194
Table 852 of leadtime forecasts with smoothing
parameter 0.2 for INAR(1) series .................................................................................... 194
Table 853
BenchmarkINARMA MSEMSE /
of leadtime forecasts with smoothing
parameter 0.5 for INAR(1) series .................................................................................... 195
Table 854 of leadtime forecasts with smoothing
parameter 0.2 for INARMA(1,1) series ........................................................................... 195
Table 855 of leadtime forecasts with smoothing
parameter 0.5 for INARMA(1,1) series ........................................................................... 196
Table 856 of leadtime forecasts for INARMA(0,0)
series .............................................................................................................................. 196
Table 857 of leadtime forecasts for INMA(1) series 197
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
)( 3?l
BenchmarkINARMA MSEMSE / )( 3?l
BenchmarkINARMA MSEMSE / )( 3?l
)( 3?l
BenchmarkINARMA MSEMSE / )( 3?l
BenchmarkINARMA MSEMSE / )( 3?l
BenchmarkINARMA MSEMSE / )( 6?l
BenchmarkINARMA MSEMSE / )( 6?l
xv
Table 858 of leadtime forecasts with smoothing
parameter 0.2 for INAR(1) series .................................................................................... 197
Table 859 of leadtime forecasts with smoothing
parameter 0.5 for INAR(1) series .................................................................................... 197
Table 860 of leadtime forecasts with smoothing
parameter 0.2 for INARMA(1,1) series ........................................................................... 198
Table 861 of leadtime forecasts with smoothing
parameter 0.5 for INARMA(1,1) series ........................................................................... 198
Table 91 Comparing INAR(1) with benchmarks for all points in time (16000 series) 208
Table 92 Comparing INAR(1) with benchmarks for issue points (16000 series) ........ 209
Table 93 Comparing INAR(1) with benchmarks for all points in time (3000 series) .. 210
Table 94 Comparing INAR(1) with benchmarks for issue points (3000 series) .......... 211
Table 95 Comparing INARMA(1,1) with benchmarks for all points in time (16000
series) ............................................................................................................................. 212
Table 96 Comparing INARMA(1,1) with benchmarks for issue points (16000 series)
....................................................................................................................................... 212
Table 97 Comparing INARMA(1,1) with benchmarks for all points in time (3000
series) ............................................................................................................................. 213
Table 98 Comparing INARMA(1,1) with benchmarks for issue points (3000 series) 213
Table 99 The effect of identification on INARMA forecasts for all points in time (16000
series) ............................................................................................................................. 214
Table 910 The effect of identification on INARMA forecasts for issue points (16000
series) ............................................................................................................................. 215
Table 911 The effect of identification on INARMA forecasts for all points in time (3000
series) ............................................................................................................................. 216
Table 912 The effect of identification on INARMA forecasts for issue points (3000
series) ............................................................................................................................. 216
Table 913 Only INARMA(0,0) series for all points in time (3000 series) .................. 218
Table 914 Only INARMA(0,0) series for issue points (3000 series) ........................... 218
Table 915 Only INAR(1) series for all points in time (3000 series) ............................ 219
Table 916 Only INAR(1) series for issue points (3000 series) .................................... 219
Table 917 Only INMA(1) series for all points in time (3000 series) ........................... 220
Table 918 Only INMA(1) series for issue points (3000 series) ................................... 220
Table 919 Only INARMA(1,1) series for all points in time (3000 series) .................. 221
Table 920 Only INARMA(1,1) series for issue points (3000 series) ........................... 222
Table 921 Threestepahead YWINAR(1) for all points in time (16000 series) ........ 223
Table 922 Threestepahead YWINAR(1) for issue points (16000 series)................. 223
Table 923 Sixstepahead YWINAR(1) for all points in time (16000 series) ............ 224
BenchmarkINARMA MSEMSE / )( 6?l
BenchmarkINARMA MSEMSE / )( 6?l
BenchmarkINARMA MSEMSE / )( 6?l
BenchmarkINARMA MSEMSE / )( 6?l
xvi
Table 924 Sixstepahead YWINAR(1) for issue points (16000 series) .................... 224
Table 925 Threestepahead YWINAR(1) for all points in time (3000 series) .......... 225
Table 926 Threestepahead YWINAR(1) for issue points (3000 series)................... 225
Table 927 Sixstepahead YWINAR(1) for all points in time (3000 series) .............. 226
Table 928 Sixstepahead YWINAR(1) for issue points (3000 series) ...................... 226
Table 929 The estimation and performance periods .................................................... 226
Table 930 The forecasting accuracy for all points in time for case 1 .......................... 227
Table 931 The forecasting accuracy for issue points for case 1 ................................... 227
Table 932 The forecasting accuracy for all points in time for case 2 .......................... 227
Table 933 The forecasting accuracy for issue points for case 2 ................................... 227
Table 934 The forecasting accuracy for all points in time for case 3 .......................... 228
Table 935 The forecasting accuracy for issue points for case 3 ................................... 228
Table 936 Leadtime forecasts for INAR(1) series for all points in time (16000
series) ............................................................................................................................. 230
Table 937 Leadtime forecasts for INAR(1) series for issue points (16000 series)
....................................................................................................................................... 230
Table 938 Leadtime forecasts for INAR(1) series for all points in time (16000
series) ............................................................................................................................. 230
Table 939 Leadtime forecasts for INAR(1) series for issue points (16000 series)
....................................................................................................................................... 231
Table 940 Leadtime forecasts for INAR(1) series for all points in time (3000
series) ............................................................................................................................. 231
Table 941 Leadtime forecasts for INAR(1) series for issue points (3000 series)
....................................................................................................................................... 232
Table 942 Leadtime forecasts for INAR(1) series for all points in time (3000
series) ............................................................................................................................. 232
Table 943 Leadtime forecasts for INAR(1) series for issue points (3000 series)
....................................................................................................................................... 233
Table 944 Leadtime forecasts for allINAR(1) series for all points in time
(16000 series) ................................................................................................................ 235
Table 945 Leadtime forecasts for allINAR(1) series for issue points (16000
series) ............................................................................................................................. 235
Table 946 Leadtime forecasts for allINAR(1) series for all points in time
(16000 series) ................................................................................................................ 235
Table 947 Leadtime forecasts for allINAR(1) series for issue points (16000
series) ............................................................................................................................. 236
Table 948 Leadtime forecasts for allINAR(1) series for all points in time (3000
series) ............................................................................................................................. 236
)( 3?l
)( 3?l
)( 6?l
)( 6?l
)( 3?l
)( 3?l
)( 6?l
)( 6?l
)( 3?l
)( 3?l
)( 6?l
)( 6?l
)( 3?l
xvii
Table 949 Leadtime forecasts for allINAR(1) series for issue points (3000
series) ............................................................................................................................. 237
Table 950 Leadtime forecasts for allINAR(1) series for all points in time (3000
series) ............................................................................................................................. 237
Table 951 Leadtime forecasts for allINAR(1) series for issue points (3000
series) ............................................................................................................................. 238
Table 6.A1 Coefficients of in each of for an INAR(2) model .................. 294
Table 6.A2 Coefficients of in each of for an INAR(2) model ................ 295
Table 6.A3 Coefficients of in each of for an INAR(2) model ............. 296
Table 6.B1 Coefficients of in each of for an INARMA(1,2) model ......... 298
Table 6.B2 Coefficients of in each of for an INARMA(1,2) model .... 299
Table 6.C1 Parameters of the overleadtimeaggregated INARMA(p,q) model ........ 302
Table 8.A1 MSE of YW and CLS estimates of for an INAR(1) process ................ 303
Table 8.A2 Comparison of YW and CLS estimates of for an INAR(1) process ..... 303
Table 8.A3 Comparison of YW and CLS estimates of for an INMA(1) process ... 304
Table 8.A4 Comparison of YW and CLS estimates of for an INMA(1) process .... 304
Table 8.A5 Comparison of YW and CLS estimates of for an INARMA(1,1) process
....................................................................................................................................... 305
Table 8.A6 Comparison of YW and CLS estimates of for an INARMA(1,1) process
....................................................................................................................................... 305
Table 8.A7 Comparison of YW and CLS estimates of for an INARMA(1,1) process
....................................................................................................................................... 305
Table 8.B1 Forecast error comparison (YW and CLS) for INAR(1) series ................... 306
Table 8.B2 Forecast error comparison (YW and CLS) for INMA(1) series .................. 306
Table 8.B3 Comparison error comparison (YW and CLS) for INARMA(1,1) series .... 307
Table 8.C1 MSE of Croston and SBA with smoothing parameter 0.2 for INAR(1) series
....................................................................................................................................... 308
Table 8.C2 MSE of Croston and SBA with smoothing parameter 0.5 for INAR(1) series
....................................................................................................................................... 308
Table 8.C3 MSE of Croston and SBA with smoothing parameter 0.2 for INMA(1)
series .............................................................................................................................. 309
Table 8.C4 MSE of Croston and SBA with smoothing parameter 0.5 for INMA(1)
series .............................................................................................................................. 309
Table 8.C5 MSE of Croston and SBA with smoothing parameter 0.2 for INARMA(1,1)
series .............................................................................................................................. 310
Table 8.C6 MSE of Croston and SBA with smoothing parameter 0.5 for INARMA(1,1)
series .............................................................................................................................. 310
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xviii
Table 8.D1 Forecast accuracy for all points in time and issue points for INAR(1) series
(known order) ................................................................................................................. 311
Table 8.D2 Forecast accuracy for all points in time and issue points for INMA(1) series
(known order) ................................................................................................................. 312
Table 8.D3 Forecast accuracy for all points in time and issue points for INARMA(1,1)
series (known order) ....................................................................................................... 313
Table 8.E1 for INARMA(0,0) series (known order) ......... 314
Table 8.E2 for INMA(1) series (known order) .................. 314
Table 8.E3 for INAR(1) series (known order)................... 315
Table 8.E4 for INARMA(1,1) series (known order) ......... 316
Table 8.F1 Sixstep ahead for INARMA(0,0) series (known
order) .............................................................................................................................. 317
Table 8.F2 Sixstep ahead for INMA(1) series (known order) 317
Table 8.F3 Sixstep ahead with smoothing parameter 0.2 for
INAR(1) series (known order) ........................................................................................ 318
Table 8.F4 Sixstep ahead with smoothing parameter 0.5 for
INAR(1) series (known order) ........................................................................................ 318
Table 8.F5 Sixstep ahead with smoothing parameter 0.2 for
INARMA(1,1) series (known order) ............................................................................. 319
Table 8.F6 Sixstep ahead with smoothing parameter 0.5 for
INARMA(1,1) series (known order) .............................................................................. 319
Table 8.G1 Accuracy of forecasts by allINAR(1) and allINARMA(1,1) approaches for
INAR(1) series ............................................................................................................... 320
Table 8.G2 Accuracy of forecasts by allINAR(1) and allINARMA(1,1) approaches for
INMA(1) series .............................................................................................................. 321
Table 8.G3 Accuracy of forecasts by allINAR(1) and allINARMA(1,1) approaches for
INARMA(1,1) series ...................................................................................................... 322
Table 8.H1 for INARMA(0,0) series (unknown order) ..... 323
Table 8.H2 for INMA(1) series (unknown order) ............. 323
Table 8.H3 for INARMA(1,1) series (unknown order) ..... 324
Table 8.I1 of leadtime forecasts for INARMA(0,0)
series .............................................................................................................................. 325
Table 8.I2 of leadtime forecasts for INMA(1) series
....................................................................................................................................... 325
Table 8.I3 of leadtime forecasts with smoothing
parameter 0.2 for INAR(1) series .................................................................................... 326
Table 8.I4 of leadtime forecasts with smoothing
parameter 0.5 for INAR(1) series .................................................................................... 326
BenchmarkINARMA MASEMASE /
BenchmarkINARMA MASEMASE /
BenchmarkINARMA MASEMASE /
BenchmarkINARMA MASEMASE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MASEMASE /
BenchmarkINARMA MASEMASE /
BenchmarkINARMA MASEMASE /
BenchmarkINARMA MASEMASE / )( 3?l
BenchmarkINARMA MASEMASE / )( 3?l
BenchmarkINARMA MASEMASE / )( 3?l
BenchmarkINARMA MASEMASE / )( 3?l
xix
Table 8.I5 of leadtime forecasts with smoothing
parameter 0.2 for INARMA(1,1) series ........................................................................... 326
Table 8.I6 of leadtime forecasts with smoothing
parameter 0.5 for INARMA(1,1) series ........................................................................... 327
Table 8.I7 of leadtime forecasts for INARMA(0,0)
series .............................................................................................................................. 327
Table 8.I8 of leadtime forecasts for INMA(1) series
....................................................................................................................................... 327
Table 8.I9 of leadtime forecasts with smoothing
parameter 0.2 for INAR(1) series .................................................................................... 328
Table 8.I10 of leadtime forecasts with smoothing
parameter 0.5 for INAR(1) series .................................................................................... 328
Table 8.I11 of leadtime forecasts with smoothing
parameter 0.2 for INARMA(1,1) series ........................................................................... 328
Table 8.I12 of leadtime forecasts with smoothing
parameter 0.5 for INARMA(1,1) series ........................................................................... 329
Table 9.A1 Only INARMA(0,0) series for all points in time (16000 series) ............... 330
Table 9.A2 Only INARMA(0,0) series for issue points (16000 series) ....................... 331
Table 9.A3 Only INAR(1) series for all points in time (16000 series) ........................ 331
Table 9.A4 Only INAR(1) series for issue points (16000 series) ................................ 331
Table 9.A5 Only INMA(1) series for all points in time (16000 series) ....................... 332
Table 9.A6 Only INMA(1) series for issue points (16000 series) ............................... 332
Table 9.A7 Only INARMA(1,1) series for all points in time (16000 series) ............... 332
Table 9.A8 Only INARMA(1,1) series for issue points (16000 series) ....................... 333
Table 9.B1 Threestepahead INARMA(0,0) for all points in time (16000 series) ..... 334
Table 9.B2 Threestepahead INARMA(0,0) for issue points (16000 series) ............. 334
Table 9.B3 Sixstepahead INARMA(0,0) for all points in time ( 16000 series) ........ 334
Table 9.B4 Sixstepahead INARMA(0,0) for issue points (16000 series) ................. 335
Table 9.B5 Threestepahead YWINMA(1) for all points in time (16000 series) ...... 335
Table 9.B6 Threestepahead YWINMA(1) for issue points (16000 series) .............. 335
Table 9.B7 Sixstepahead YWINMA(1) for all points in time ( 16000 series) ......... 335
Table 9.B8 Sixstepahead YWINMA(1) for issue points (16000 series) .................. 336
Table 9.B9 Threestepahead INARMA(1,1) for all points in time (16000 series) ..... 336
Table 9.B10 Threestepahead INARMA(1,1) for issue points (16000 series) ........... 336
Table 9.B11 Sixstepahead INARMA(1,1) for all points in time ( 16000 series) ...... 336
Table 9.B12 Sixstepahead INARMA(1,1) for issue points (16000 series) ............... 337
Table 9.B13 Threestepahead INARMA(0,0) for all points in time (3000 series) ..... 337
Table 9.B14 Threestepahead INARMA(0,0) for issue points (3000 series) ............. 337
BenchmarkINARMA MASEMASE / )( 3?l
BenchmarkINARMA MASEMASE / )( 3?l
BenchmarkINARMA MASEMASE / )( 6?l
BenchmarkINARMA MASEMASE / )( 6?l
BenchmarkINARMA MASEMASE / )( 6?l
BenchmarkINARMA MASEMASE / )( 6?l
BenchmarkINARMA MASEMASE / )( 6?l
BenchmarkINARMA MASEMASE / )( 6?l
xx
Table 9.B15 Sixstepahead INARMA(0,0) for all points in time ( 3000 series) ........ 337
Table 9.B16 Sixstepahead INARMA(0,0) for issue points (3000 series) ................. 338
Table 9.B17 Threestepahead YWINMA(1) for all points in time (3000 series) ...... 338
Table 9.B18 Threestepahead YWINMA(1) for issue points (3000 series) .............. 338
Table 9.B19 Sixstepahead YWINMA(1) for all points in time ( 3000 series) ......... 338
Table 9.B20 Sixstepahead YWINMA(1) for issue points (3000 series) .................. 339
Table 9.B21 Threestepahead CLSINARMA(1,1) for all points in time (3000 series)
....................................................................................................................................... 339
Table 9.B22 Threestepahead CLSINARMA(1,1) for issue points (3000 series) ..... 339
Table 9.B23 Sixstepahead CLSINARMA(1,1) for all points in time (3000 series) 339
Table 9.B24 Sixstepahead CLSINARMA(1,1) for issue points (3000 series) ......... 340
Table 9.C1 Leadtime forecasts for INARMA(0,0) series for all points in time
(16000 series) ................................................................................................................ 341
Table 9.C2 Leadtime forecasts for INARMA(0,0) series for issue points (16000
series) ............................................................................................................................. 341
Table 9.C3 Leadtime forecasts for INARMA(0,0) series for all points in time
(16000 series) ................................................................................................................ 341
Table 9.C4 Leadtime forecasts for INARMA(0,0) series for issue points (16000
series) ............................................................................................................................. 342
Table 9.C5 Leadtime forecasts for INMA(1) series for all points in time (16000
series) ............................................................................................................................. 342
Table 9.C6 Leadtime forecasts for INMA(1) series for issue points (16000
series) ............................................................................................................................. 342
Table 9.C7 Leadtime forecasts for INMA(1) series for all points in time (16000
series) ............................................................................................................................. 343
Table 9.C8 Leadtime forecasts for INMA(1) series for issue points (16000
series) ............................................................................................................................. 343
Table 9.C9 Leadtime forecasts for INARMA(1,1) series for all points in time
(16000 series) ................................................................................................................ 343
Table 9.C10 Leadtime forecasts for INARMA(1,1) series for issue points
(16000 series) ................................................................................................................ 344
Table 9.C11 Leadtime forecasts for INARMA(1,1) series for all points in time
(16000 series) ................................................................................................................ 344
Table 9.C12 Leadtime forecasts for INARMA(1,1) series for issue points
(16000 series) ................................................................................................................ 344
Table 9.C13 Leadtime forecasts for INARMA(0,0) series for all points in time
(3000 series) .................................................................................................................. 345
Table 9.C14 Leadtime forecasts for INARMA(0,0) series for issue points (3000
series) ............................................................................................................................. 345
)( 3?l
)( 3?l
)( 6?l
)( 6?l
)( 3?l
)( 3?l
)( 6?l
)( 6?l
)( 3?l
)( 3?l
)( 6?l
)( 6?l
)( 3?l
)( 3?l
xxi
Table 9.C15 Leadtime forecasts for INARMA(0,0) series for all points in time
(3000 series) .................................................................................................................. 345
Table 9.C16 Leadtime forecasts for INARMA(0,0) series for issue points (3000
series) ............................................................................................................................. 345
Table 9.C17 Leadtime forecasts for INMA(1) series for all points in time (3000
series) ............................................................................................................................. 346
Table 9.C18 Leadtime forecasts for INMA(1) series for issue points (3000
series) ............................................................................................................................. 346
Table 9.C19 Leadtime forecasts for INMA(1) series for all points in time (3000
series) ............................................................................................................................. 346
Table 9.C20 Leadtime forecasts for INMA(1) series for issue points (3000
series) ............................................................................................................................. 347
Table 9.C21 Leadtime forecasts for INARMA(1,1) series for all points in time
(3000 series) .................................................................................................................. 347
Table 9.C22 Leadtime forecasts for INARMA(1,1) series for issue points (3000
series) ............................................................................................................................. 347
Table 9.C23 Leadtime forecasts for INARMA(1,1) series for all points in time
(3000 series) .................................................................................................................. 348
Table 9.C24 Leadtime forecasts for INARMA(1,1) series for issue points (3000
series) ............................................................................................................................. 348
)( 6?l
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)( 3?l
)( 3?l
)( 6?l
)( 6?l
)( 3?l
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xxii
List of abbreviations
ACF Autocorrelation Function
AE Absolute Error
ARIMA Autoregressive Integrated Moving Average
CLS Conditional Least Squares
CML Conditional Maximum Likelihood
DARMA Discrete Autoregressive Moving Average
DSD Discrete SelfDecomposable
EWMA Exponentially Weighted Moving Average
GMM General Method of Moments
IDF Intermittent Demand Forecasting
INARMA Integer Autoregressive Moving Average
LTD Lead Time Demand
MAD Mean Absolute Deviation
MAPE Mean Absolute Percentage Error
MASE Mean Absolute Scaled Error
MC Modified Croston
MCMC Markov Chain Monte Carlo
ME Mean Error
MSE Mean Square Error
ML Maximum Likelihood
PACF Partial Autocorrelation Function
PB Percentage Better
RGRMSE Relative Geometric Root Mean Square Error
SBA SyntetosBoylan Approximation
SBJ ShaleBoylanJohnston
SES Single (or simple) Exponential Smoothing
SMA Simple Moving Average
YW Yule Walker
ZF Zero Forecast
M.Mohammadipour, 2009, Chapter 1 1
Chapter 1 INTRODUCTION
This chapter lays the foundations for this thesis and provides an overall perspective
for this PhD research. It introduces the research by means of an overview of the
project. The business context and research background are described. The research
problem and research questions are then defined and the methodology is briefly
outlined.
All the above topics are explained in detail by a stepbystep process through
chapters 2 to 9. However, the introductory chapter is intended to outline the research
through a summary of the research background and problems, expected results, and
designated methodology. The structure of chapter 1 and sequences of sections are
shown in Figure 11.
M.Mohammadipour, 2009, Chapter 1 2
Figure ?11 The structure of Chapter 1
1.1 Introduction
To attain a unified understanding of related concepts in this PhD thesis, a brief
description of the key terms and phrases used in the study is provided in this section.
These are the definitions that will be adopted in this thesis. More discussion will
follow in later chapters.
Time series
Bowerman et al. (2005) have defined a time series as ?a chronological sequence of
observations on a particular variable? (p.4). Demand of a product over time and
inventory level for a product over time are examples of time series.
One of the most general classes of models for forecasting a time series is the class of
autoregressive integrated moving average (ARIMA) models (Box et al., 1994).
ARIMA models are based on adding linear combinations of lags of the differenced
series and/or lags of the forecast errors to the prediction equation, as needed to
remove any last traces of autocorrelation from the forecast errors.
Time series of counts
Time series of counts is defined by McKenzie (2003) as ?counts of events, objects or
1.1 Introduction
1.2 Research Overview
1.5 Research Problem
1.3 Business Context
1.6 Research Methodology
1.7 Thesis Structure
1.4 Research Background
M.Mohammadipour, 2009, Chapter 1 3
individuals in consecutive intervals or at consecutive points in time?. Integer
autoregressive moving average (INARMA) models have been proposed for
forecasting time series of counts, and have received a certain amount of attention
over the last 25 years. These models are explained in detail in chapter 3.
Intermittent series
Intermittent series are time series of nonnegative integer values where some values
are zero (Shenstone and Hyndman, 2005). A common example is intermittent
demand. Intermittent demand should be distinguished from lumpy demand where the
nonzero values are highly variable. Methods for intermittent demand forecasting
(IDF) are reviewed in chapter 2.
1.2 Research Overview
This PhD concentrates on modelling intermittent demand by integer autoregressive
moving average (INARMA) processes and proposing a forecasting method based on
such processes. The study first focuses on stochastic characteristics of INARMA
processes, including finding the second unconditional moment and the
autocorrelation function (ACF) and partial autocorrelation function (PACF)
structure. Lead time forecasts are also developed for INARMA models.
INARMA models are then used for intermittent demand forecasting. The results are
compared to some benchmark methods in the literature, namely Croston's method
(Croston, 1972), SyntetosBoylan Approximation (SBA) (Syntetos and Boylan,
2005), and ShaleBoylanJohnston (SBJ) (Shale et al., 2006). The rationale for this
choice of benchmarks is given in chapter 2.
1.3 Business Context
Accurate demand forecasting is a significant concern for many organizations. It lays
a foundation for every part of inventory management. Various forecasting models
have been developed to incorporate different components of demand such as trend
and seasonality. A class of demand called intermittent demand exists in which some
M.Mohammadipour, 2009, Chapter 1 4
periods have no demand at all. This is especially the case for service parts and capital
goods (Willemain et al., 2004). Examples are commonly found in the aerospace,
automotive, computer components, electronics, and industrial machinery industries.
In general, all companies that must stock spare parts face intermittent demand.
Intermittent time series are also common in business and economic data. There are
many cases for which nonintermittent data is intermittent at lower levels of data
disaggregation (for example, greater frequency or smaller geography).
The items with intermittent demand are not rare and in fact they constitute the
majority of items held by many stockists (Johnston et al., 2003). These items are also
important from a financial point of view. In the U.S. alone, service parts management
grew to a $700 billion business sector in 2001 (Patton and Steele, 2003). The fact
that, in many cases, service parts face a high risk of obsolescence makes accurate
forecasting for such items even more important. Lower stockholding costs and
higher service levels are the outcomes of more accurate forecasts.
INARMA models have had applications in a wide area including medical science
(Franke and Seligmann, 1993; Cardinal et al., 1999), economics (B?ckenholt, 1999;
Br?nn?s and Hellstr?m, 2001; Freeland and McCabe, 2004b) and service industries
(Br?nn?s et al., 2002). However, the performance of these models in an intermittent
demand context is yet to be tested. This testing is to be conducted in this research as
a new application for INARMA models.
Although the empirical analysis of this thesis focuses on intermittent demand data,
the theoretical and simulation findings can generally be applied to any time series
with low nonnegative integervalued data.
1.4 Research Background
Many companies use single exponential smoothing (SES) for intermittent demand
forecasting (Teunter and Sani, 2009) but Croston (1972) shows that this can result in
biased forecasts and excessive stock levels. He proposes a method based on separate
exponential smoothing estimates of demand size and interval between demands,
M.Mohammadipour, 2009, Chapter 1 5
which he claims to be unbiased. However, Syntetos and Boylan (2001) show that
Croston?s method is positively biased. Some methods have been suggested to reduce
this bias (Lev?n and Segerstedt, 2004; Syntetos and Boylan, 2005). While it is
confirmed that the correction by Syntetos and Boylan does reduce the bias of
Croston?s method, the modification by Lev?n and Segerstedt is even more biased
than Croston?s method (Teunter and Sani, 2009).
In parallel with these studies, some authors focus on finding a model underlying
Croston?s method (Shenstone and Hyndman, 2005; Snyder and Beaumont, 2007).
This is especially useful to find the distribution and prediction intervals of the
forecasts. However, none of these studies has yet demonstrated an underlying model
for which Croston?s method is optimal. It has been suggested that the focus should
now be moved to stationary models for IDF such as time series models for counts
(Shenstone and Hyndman, 2005).
Time series models for counts occur as counts of individuals (e.g. the number of
people in a queue waiting to receive a service at a certain moment) or events (e.g. the
number of accidents in a firm each three months). If these discrete variates are large
numbers, they could be approximated by continuous variates; otherwise, special
models should be used.
A class of models for count data has been developed, namely the integer
autoregressive moving average (INARMA) models. These models were originally
introduced in the 1980s (McKenzie, 1985; AlOsh and Alzaid, 1987) and it will be
shown in chapter 3 that they are analogous to wellknown ARMA models (Box et al.,
1994).
Intermittent demand data belong to a broader class called count data. Although count
data frequently occur in many industries, not many stochastic models have been
developed for them (Gooijer and Hyndman, 2006). As one of these few models,
integer autoregressive moving average models have recently received much
attention. This PhD thesis aims to bridge the gap between models for counts and
intermittent demand forecasting, focusing specifically on INARMA models. In doing
so, some issues such as identification, estimation of parameters and lead time
forecasting need to be addressed. Identification of the autoregressive and moving
M.Mohammadipour, 2009, Chapter 1 6
average orders of INARMA models has been mainly conducted in the literature
using the autocorrelation function (ACF) and the partial autocorrelation function
(PACF) (Jung and Tremayne, 2006a; Zheng et al., 2006; Zhu and Joe, 2006; Bu and
McCabe, 2008). To our knowledge, the ACF of the general INARMA(p,q) process
has not been looked at before. Therefore, we establish the unconditional variance and
the ACF of such processes. The conditional expected value of the overleadtime
aggregated process are also obtained. The accuracy of hstepahead and lead time
INARMA forecasts are compared to those obtained by the benchmark methods
mentioned in section 1.2. In doing so, the difficulties of forecasting intermittent
demand are borne in mind, particularly if the length of data history is short or the
data is sparse.
1.5 Research Problem
1.5.1 Initial Problem
The main problem addressed in this research is as follows:
o In the context of intermittent demand, is there any benefit in modelling and
forecasting the demand using INARMA models, in terms of forecast
accuracy, compared to simpler methods?
1.5.2 Research Questions
Based upon the above initial research problem, the four detailed questions for the
research are as follows:
I. How can the appropriate integer autoregressive moving average (INARMA)
model be identified for a time series of counts?
Different methods have been suggested for identification of ARIMA processes
including: using the sample autocorrelation function (SACF) and the sample
partial autocorrelation function (SPACF), and using a penalty function such as
the Akaike information criterion (AIC) or the Bayesian information criterion
M.Mohammadipour, 2009, Chapter 1 7
(BIC). This research focuses on finding the ACF and PACF structure of
INARMA processes. But, using ACF and PACF needs a visual check.
Identification based on these functions has not been automated for INARMA
models as it has been for ARMA models (e.g. in AutoBox). Therefore automated
methods should be used. It has been suggested in the literature that data should be
first tested for any serial dependence. As a result, two identification procedures
are compared in this research: a twostage procedure first uses a test of serial
dependence to distinguish between an independent and identically distributed
(i.i.d.) process and other INARMA processes and then the AIC is used to select
among other possible INARMA processes. A onestage procedure does not have
the first step and only uses the AIC. However, the correct model might not be
identified at all times. In such cases, it is of interest to find the impact that
misidentification has on forecast accuracy.
II. How can the parameters of integer autoregressive moving average
(INARMA) models be estimated?
The performance of different estimation methods are to be tested in terms of both
the accuracy of parameter estimates and their impact on forecast accuracy.
III. How can an INARMA process be forecasted over a lead time?
One of the application areas of leadtime aggregation is in the inventory control
field where there is a lead time between placing an order by a manufacturer and
receiving it from its supplier. The manufacturer has to place an order to cover the
demand over the lead time and, therefore, the leadtime demand has to be
forecasted. The aggregated INARMA(p,q) process over a lead time and its
conditional expected value are found. The latter is then used as the lead time
forecast.
IV. Do INARMA models provide more accurate forecasts for intermittent
demand than nonoptimal smoothingbased methods?
This research has suggested using INARMA models to forecast intermittent
demand. The accuracy of forecasts provided by INARMA methods then has to be
compared with some of the methods that have been used in the literature of
M.Mohammadipour, 2009, Chapter 1 8
intermittent demand forecasting. As this research solely focuses on demand
forecasting and not inventory control, forecastaccuracy metrics will be used
rather than accuracyimplication metrics (measures that analyze the effect of
forecasting methods on inventory performance measures (Boylan and Syntetos,
2006)). As previously mentioned, the methods that will be used for comparison
purposes are Croston's method (Croston, 1972), the SyntetosBoylan
Approximation (Syntetos and Boylan, 2005), and the method of ShaleBoylan
Johnston (Shale et al., 2006).
1.6 Research Methodology
This section will try to sufficiently clarify the ?philosophy?, ?approach? and
?strategy? of this study to demonstrate how the expected results in this PhD may be
achieved. This research follows the ?positivism? philosophy where the research
approach will be ?deductive?. Accordingly, based upon the detailed hypotheses and
the required level of generalisation, ?simulation? and then ?empirical study? is
believed to be the most suitable strategy for this research.
To explain the research methodology of this study, the approach of the ?research
process onion? as termed by Saunders et al. (2003) is followed. On this basis, an
appropriate research philosophy, research approach, and research strategy are
proposed for the study. Consistent with the research strategy, source(s) of data are
identified for analysis. The structure of this section is shown in Figure 12.
Figure ?12 The structure of the research methodology
1.6.1 Research Philosophy
1.6.2 Research Approach
1.6.3 Research Strategy
1.6.3.1 Mathematical Analysis 1.6.3.2 Simulation 1.6.3.3 Empirical Study
M.Mohammadipour, 2009, Chapter 1 9
1.6.1 Research Philosophy
Taking into account the existing knowledge of the overall environment of forecasting
and time series analysis, the ground philosophy of this research tends towards
?positivism? (rather than pure ?realism? or ?interpretivism?), i.e. the end product of
the research can be lawlike generalisations. The nature of intermittent data makes it
difficult for humans to detect patterns for this type of data. This means that a
positivistic deductive approach is a natural starting point. An interpretivistic
approach may be more appropriate for investigating human adjustments to
intermittent demand forecasts.
1.6.2 Research Approach
Research approaches are classified into two main groups of ?deductive? and
?inductive?. While the former works on the basis of a clear understanding of the
research theory and questions, the latter tries to find these through investigations with
real world data (Saunders et al., 2003). Although some inductive studies have been
done in the area of forecasting (e.g. Collopy and Armstrong, 1992; Adya et al.,
2001), most of the studies in this field are deductive. A theory is developed first and
a research strategy to test it is designed. This research is also based on developing a
theoretical structure and testing the findings by simulation and empirical analysis.
Hence, a deductive approach is followed in this PhD thesis.
1.6.3 Research Strategy
A research strategy endeavours to plan the process of answering the research
questions (section 1.5.2). Our strategy consists of three steps: mathematical analysis,
simulation and empirical study.
1.6.3.1 Mathematical Analysis
Although the introduction of INARMA models dates back to the 1980s, there are still
M.Mohammadipour, 2009, Chapter 1 10
an increasing number of studies which investigate the statistical properties of these
models. The mathematical analysis of this research aims at extending the theory of
INARMA modelling making it more complete.
The mathematical results of this PhD are provided in chapters 3 to 6. Some stochastic
properties of the general INARMA(p,q) process have been obtained which are: the
unconditional variance and the autocorrelation function (ACF) and partial
autocorrelation function (PACF). The results of aggregation of an INARMA(p,q)
process over lead time along with the conditional expected value of the aggregated
process are established. These results are then used in chapters 8 and 9 to compare
the performance of the INARMA method with some benchmark methods in
forecasting intermittent demand.
1.6.3.2 Simulation
Simulation will be used for the following reasons:
? to assess the percentage of theoretically generated INARMA time series that
can be identified correctly
? to investigate the effect of misidentification on forecasting accuracy
? to compare the performance of different estimation methods
? to assess the sensitivity of the results (parameter estimates and forecasting
accuracy) to the sparsity of data and the length of history
? to compare the INARMA forecasts with those of the benchmark methods
It can be seen that simulation is essential for the first three objectives (and the part of
the fourth objective relating to parameter estimates) because only for theoretically
generated data, the order of the INARMA model and the parameters are known. For
the remaining objectives, simulation is useful to gain additional insights. The design
of the simulation study is explained in chapter 7.
M.Mohammadipour, 2009, Chapter 1 11
1.6.3.3 Empirical Analysis
The findings of this PhD thesis are to be tested on real empirical data to assess the
practical validity and applicability of the main results of the study. As discussed,
simulation helps us to test the accuracy of model identification, when we know that
the intermittent demand follows an INARMA process with known order and
parameters. However, this is not true for real data in that we do not have such
information from the beginning. Therefore, empirical analysis would help us to test
the applicability of the results in real situations.
Although many studies focus on the statistical aspects of INARMA models, there are
fewer studies regarding the application of these models (Jung and Tremayne, 2003).
There is especially a lack of empirical testing of INARMA models on intermittent
demand. The demand data series used in this PhD thesis are Royal Air Force (RAF)
individual demand histories of 16,000 SKUs over a period of 6 years (monthly
observations) and 3,000 series of intermittent demand for 24 periods (two years
monthly series) from the automotive industry.
1.7 Thesis Structure
Based on the position adumbrated in this chapter, this PhD thesis is structured as
follows. Chapter 2 discusses different definitions and categorizations of intermittent
demand, reviews methods for intermittent demand forecasting and the accuracy
measures that can be used in this context.
Chapter 3 briefly reviews different count models and introduces INARMA models.
The stochastic properties of these models and their application in the literature are
examined. The unconditional variance and ACF of the INARMA(p,q) process are
obtained.
Chapter 4 reviews different approaches for identification of the order of INARMA
models. The PACF structure of INARMA models is also derived.
Chapter 5 discusses methods for estimation of the parameters of these models.
Chapter 6 investigates the forecasting of an INARMA process over lead time.
Chapter 7 discusses the design of simulation, and the results of simulation experiments
M.Mohammadipour, 2009, Chapter 1 12
are presented in Chapter 8.
Chapter 9 assesses the validity of theoretical and simulation results on real intermittent
demand data.
Finally, the findings of this PhD research, the limitations and some potential future
studies are reviewed in chapter 10. The structure of the thesis is shown in Figure 13.
Figure ?13 The structure of the thesis
Chapter 1 Introduction
Chapter 2 Forecasting Intermittent
Demand
Chapter 5 Estimation in INARMA
Models
Chapter 3 Integer Autoregressive
Moving Average Models
Chapter 6 Forecasting in
INARMA Models
Chapter 7 Simulation Design
Chapter 4 Identification in
INARMA Models
Chapter 8 Simulation Results
Chapter 10 Conclusions and
Further Research
Chapter 9 Empirical Analysis
M.Mohammadipour, 2009, Chapter 2 13
Chapter 2 FORECASTING INTERMITTENT
DEMAND
2.1 Introduction
This chapter aims to provide an overview of the literature on forecasting intermittent
demand. This research focuses on integer autoregressive moving average (INARMA)
models to address intermittent demand modelling and forecasting. As the focus of
this research is on forecasting, we do not review the literature on inventory control
for slowmoving items.
The chapter is organized as follows. Intermittent demand is defined in detail in
section 2.2. Methods for forecasting intermittent demand are then discussed in
section 2.3. We start our review with Croston?s method (Croston, 1972) as the most
M.Mohammadipour, 2009, Chapter 2 14
widely used approach in this field. Some variants of Croston?s method are then
reviewed. Studies based on bootstrapping to forecast leadtime demand are also
discussed. As this research is based on comparing different forecasting methods, the
forecast accuracy measures need to be selected. Some accuracy measures cannot be
used in an intermittent demand context. Section 2.4 determines these measures and
classifies the measures that can be applied to intermittent series. A number of studies
have compared different methods of forecasting intermittent demand. These studies
are reviewed in section 2.5. The motives to use INARMA models for modelling and
forecasting intermittent demand are discussed in section 2.6 and, finally, section 2.7
concludes the chapter.
2.2 Definition of Intermittent Demand
Intermittent demand is defined by Silver et al. (1998) as ?infrequent in the sense that
the average time between transactions is considerably larger than the unit period,
the latter being the interval of forecast updating? (footnote, p.127). The main
disadvantage of this definition is its impracticality, i.e. it does not define how long
the average time between transactions should be for demand to be considered
intermittent.
The definition provided by Johnston and Boylan (1996) is more practical. They
suggest that demand is intermittent when the mean interarrival time between
demands is greater than 1.25 review intervals. This cutoff value is based on a
comparison of Croston's method and Single Exponential Smoothing (SES) using
simulated data. However, this definition also has its limitations: it is simulation
based, it depends on specific methods, and it does not take into account the
?lumpiness? of the demand. Intermittent demand is often lumpy, which means that
the variability among the nonzero values is high (Willemain et al., 2004). However,
these two concepts should be distinguished.
Shenstone and Hyndman (2005) introduce another practical definition: ?Data for
intermittent demand items consist of time series of nonnegative integer values where
some values are zero?. It can be seen that this does not take into account the
lumpiness of demand either.
M.Mohammadipour, 2009, Chapter 2 15
There are two main categorization schemes for intermittent demand in the literature.
The first approach, proposed by Williams (1984) and later revised by Eaves and
Kingsman (2004), is based on second moment variability (transaction variability,
demand size variability and leadtime variability). The other approach, suggested by
Syntetos et al. (2005) is based on frequency of transactions and demand size
variability.
The classification approach by Eaves and Kingsman (2004) is a revision of the
method of Williams (1984) who classified demand by decomposing the variance of
leadtime demand into transaction variability, demand size variability and leadtime
variability. This is shown in Table 21.
Table ?21 The categorization scheme for intermittent demand data (Eaves and Kingsman, 2004)
Leadtime demand component Demand pattern
classification Transaction variability Demand size variability Leadtime variability
Low Low Smooth
Low High Irregular
High Low Slow moving
High High Low Mildly intermittent
High High High Highly intermittent
The main disadvantage of Eaves and Kingsman?s classification is its lack of
practicality. Unlike Syntetos et al. (2005), they do not provide cutoff values for
different classes.
Syntetos et al. (2005) categorize demand based on the expected mean square error
(MSE) of each forecasting method. They propose four categories of demand shown
in Figure 21 which are: ?erratic?, ?lumpy?, ?smooth?, and ?intermittent?. Each of
these demand classes are uniquely specified by two parameters: ? which is the
average interdemand interval, and ? which is the squared coefficient of variation of
the demand when it occurs.
The classification by Johnston and Boylan (1996) is based on comparing Croston's
method with SES as methods widely used in forecasting software packages.
Therefore, although it has the benefit of being practical, one can argue that it is not a
comprehensive classification method that can be used for all forecasting methods.
The categorization scheme of Syntetos et al. (2005) also includes the Syntetos
M.Mohammadipour, 2009, Chapter 2 16
Boylan Approximation (SBA) (explained in section 2.3.1.1) in addition to the other
two methods. The main benefit of their scheme is again its practicality (because of
determining the cutoff values) and that it is empirically validated. However, as the
authors mentioned, the categories are based on only one forecast accuracy measure
(MSE) (see section 2.4 for more information on accuracy measures).
Figure ?21 The categorization scheme for intermittent demand data (Syntetos et al. 2005)
The classification by Syntetos et al. (2005) distinguishes between intermittent and
smooth demand to determine which method should be used in each situation. This is
because the SyntetosBoylan Approximation and SES methods are not universal, i.e.
these methods are not appropriate for both categories of demand.
This research aims at using integer autoregressive moving average (INARMA)
models to model and forecast intermittent demand. This method can be used for both
intermittent and nonintermittent data provided that the data is not lumpy. Therefore,
we do not need to focus on a specific classification method for intermittence to find
when the method should be used. We do need to follow a standard definition that is
practical and universal (not limited to specific methods).
Among the definitions that we reviewed before, only the one by Shenstone and
Hyndman (2005) met all the above criteria and therefore is used for the purpose of
this study. As a result, whenever a data series has at least one zero, it is considered as
an intermittent data series, and an INARMA forecasting method can be used.
However, the universality of the INARMA approach means that it can also be used
Lumpy
Erratic
Intermittent
Smooth
?= 0.49
?= 1.31
M.Mohammadipour, 2009, Chapter 2 17
when all the observations are positive.
As previously mentioned, although the classification by Syntetos et al. (2005) is
practical, it is only based on specific forecasting methods. We will test this
classification when the data is INARMA. INARMA forecasts will be compared to
the best benchmark method based on this classification.
The main reason for not using Eaves and Kingsman classification method is its lack
of practicality and also because leadtime variability will not be studied in this thesis.
As mentioned before, the other reason for the selection of Syntetos et al.?s
classification is that two methods used in their classification (Croston?s method and
SBA) will be used in this research for comparison reasons.
This research focuses on using INARMA models with Poisson marginal distributions
to model intermittent demand. Due to the properties of the Poisson distribution,
lumpy demand cannot be modelled by these models. As will be explained in chapter
9, a filtering mechanism will be used to eliminate the lumpy series when dealing with
empirical data.
2.3 Methods of Forecasting Intermittent Demand
Inventories with intermittent demands are common in practice (Shenstone and
Hyndman, 2005) and they create significant problems in the manufacturing and
supply environment (Syntetos and Boylan, 2001). The accurate forecasting of
demand is one of the most important issues of inventory management (Ghobbar and
Friend, 2003). This is more difficult when demand has an intermittent nature
(Willemain et al., 2004).
Willemain et al. (2004) divide intermittent demand forecasting (IDF) methods into:
? simple statistical smoothing methods
? Croston?s method and its variants
? bootstrap methods
from which we focus on the last two classes because, according to Croston (1972),
the first class results in biased forecasts when applied immediately after a demand
M.Mohammadipour, 2009, Chapter 2 18
occurrence. These methods have also been the subject of some comparison studies,
including Teunter and Duncan (2009). These studies are reviewed in section 2.5.
2.3.1 Croston?s?Method
The most widely used approach for forecasting intermittent demand is Croston?s
method (Shenstone and Hyndman, 2005). Croston (1972) suggested that conventional
forecasting methods like Simple Moving Averages (SMA) and Single Exponential
Smoothing (SES) may not be appropriate for slowmoving items. The bias associated
with SES is expressed in a quantitative form for the case in which forecasts are
updated after a demand occurrence. He then proposed a method based on separate
Single Exponential Smoothing (SES) forecasts on the size of a demand and on the time
period between observing two demands, both with the same smoothing constant ?.
Let ?? be the demand occurring during the time period ? and the indicator variable ??
be:
??=?
1 when demand occurs
0 when no demand occurs
?
Equation ?21
Furthermore, let ?? be the number of periods with nonzero demand during the interval
[0,?], ??=? ??
?
?=1 . The size of the ?th nonzero demand is then shown by ?? and the
interarrival time between ???1 and ?? is ??.
Croston (1972) adopts a stochastic model of arrival and size of demand, assuming
that demand sizes, ??, are normally distributed, ?(?,?
2), and that demand is random
and has a Bernoulli probability 1/? of occurring in every review period
(subsequently, the interarrival time, ??, follows the geometric distribution with a
mean ?). Both demand sizes and intervals are assumed to be stationary.
The demand at period ? is given by:
??=????
Equation ?22
Using Croston?s method, the demand size and interarrival time between demands
M.Mohammadipour, 2009, Chapter 2 19
are separately forecasted using Single Exponential Smoothing (SES), with forecasts
being updated only after demand occurrence. Let ?? and ?? be the forecasts of the
(?+ 1)th demand size and interarrival time respectively, based on demands up to
period j. Then, based on Croston?s method:
??=?1??????1 +???
Equation ?23
??=?1??????1 +???
Equation ?24
The forecast for the next time period is then given by the smoothed size of demand
divided by the smoothed interarrival time:
??=??/??
Equation 25
According to Croston (1972), the expected value and variance of the forecast for the
stochastic model are given by:
?????=?/?
Equation 26
var????=
?
2??
?
??1
?2
?2 +
?2
?
?
Equation 27
based upon which, he claims the method is unbiased. However, Syntetos and Boylan
(2001) showed that Croston?s method is biased due to the fact that
?(??)??(??)/?(??). Syntetos and Boylan (2005) suggest some modifications to
reduce the bias which will be discussed in section 2.3.1.1.
Rao (1973) made corrections to some of the expressions of the forecast variance.
However, these changes have no effect on the forecast of mean demand.
Croston?s method is based on using the same smoothing constant (?) for updating
demand sizes and demand intervals. Schultz (1987) suggests using different
smoothing constants, ? and ?, for size and interval between demands.
M.Mohammadipour, 2009, Chapter 2 20
Croston's method is also based on assumptions of independence and normality of
demand, independence of demand sizes and interarrival times, and independence of
interarrival times with a Geometric distribution. The validity of these assumptions
for realworld data has been discussed by several authors (e.g. Willemain et al.,
1994; Snyder, 2002; Shenstone and Hyndman, 2005). Willemain et al. (1994) argue
that not only might there be autocorrelation among demand sizes and interarrival
times, but it is also possible for sizes and intervals to be correlated.
Shenstone and Hyndman (2005) explore possible models underlying Croston?s
method. These models are reviewed in section 2.3.1.4.
2.3.1.1 Bias Correction for Bernoulli Demand Incidence
As explained in the previous section, Syntetos and Boylan (2001) showed that
Croston?s forecasts are biased. A new estimator is proposed later by Syntetos and
Boylan (2005) to reduce the bias associated with Croston?s method. It is shown that
the bias can be approximated by ?
?
2??
?
??1
?2
? as:
??
??
??
??
?
?
+
?
2??
?
??1
?2
Equation 28
and based on Equation 28 they suggest that the forecast should be multiplied by
(1?
?
2
) in order to reduce the bias.
???1?
?
2
?
??
??
??
?
?
Equation 29
Therefore, the new estimator of mean demand is:
??=?1?
?
2
?
??
??
Equation 210
All of Croston?s assumptions are maintained in the derivation of the new estimator
M.Mohammadipour, 2009, Chapter 2 21
(however, the assumption of normality of demand sizes is not necessary for the
derivation). Some of these assumptions have been changed in another study by Shale
et al. (2006) which will be discussed in the next section.
The new method, called the SyntetosBoylan Approximation (SBA) in the literature,
is then compared to three other forecasting methods: Simple Moving Average, Single
Exponential Smoothing, and Croston?s method on 3,000 real intermittent demand
data series from the automotive industry. The results suggest that the new method is
the most accurate estimator for the faster intermittent demand data.
Teunter and Sani (2009) also compared Croston's method, SBA and modified
Croston (see section 2.3.1.3) and show that SBA has the smallest average standard
deviations of all.
2.3.1.2 Bias Correction for Poisson Demand Incidence
Shale et al. (2006) consider the case of Poisson demand incidence instead of a
Bernoulli distribution as in Croston?s model. Also, they assume that the interarrival
time follows a negative exponential instead of a geometric distribution. They derive
the bias expected for this case and provide the correction factor for application. This
is done for the cases where either a simple or an exponentially weighted moving
average (EWMA) is used. Hereafter, this method is called the ShaleBoylan
Johnston (SBJ) method.
For the case of a simple moving average, when the ? most recent interarrival times
have a negative exponential distribution, the average of them has an Erlang
distribution. The correction factor for an adapted Croston?s estimate of mean demand
for this case is shown to be [(??1)/?]. The adaptation consists of a ratio of simple
moving average of demand size and demand interval. The estimate of demand would
then be:
?????=?
??1
?
?
?(??)
?(??)
Equation 211
M.Mohammadipour, 2009, Chapter 2 22
where ?(??) and ?(??) are the arithmetic averages rather than the exponentially
weighted averages of demand sizes and interval between demands. In the latter case,
it is shown that the smoothing parameter ??? of EWMA is linked to ? as follows:
?=
2??
?
Equation 212
Based on the above result, the estimate of demand (which will be used later in this
research) is:
?????=?1?
?
2??
?
?(??)
?(??)
Equation 213
When ? is not an integer, the probability distribution is the continuous form of the
Erlang distribution which is Gamma distribution. The correction factor for this case
is also [(??1)/?].
It has been shown through simulation that these correction factors are very close to
the observed average bias associated with Croston?s method (Shale et al., 2006).
2.3.1.3 Modified Croston (MC)
Lev?n and Segerstedt (2004) propose a Modified Croston procedure, based on an
earlier working paper by Segerstedt (2000), which they claim works for both fast
moving and slowmoving items. The procedure is given by:
??
??=???1
?? +??
??
??????1
????1
???
Equation 214
where ??
?? is the forecasted demand rate at the end of period ??, ?? is the time
period in which the demand ?? occurs, ?? is the measured demand quantity during
the nth period, and ? is the smoothing constant. Demand is assumed to follow the
Erlang distribution. This modified Croston procedure is then compared to Croston?s
method in which:
M.Mohammadipour, 2009, Chapter 2 23
??
?=
??
??
Equation 215
??=???1 +?(???1????1)
Equation 216
??=???1 +?(???1????1)
Equation 217
Where ??
? is the forecasted demand rate using Croston?s method and ?? is the
forecasted demand interval calculated at the end of period ?? with other notations as
before.
The simulation results suggest that an inventory control system based on the MC
procedure and the Erlang distribution yields fewer shortages than a system using
exponential smoothing and the Normal distribution. The authors believe that is due to
the MC procedure and not on the assumption of Erlang distribution.
Lev?n and Segerstedt (2004) claim that the MC estimator avoids bias. However,
Boylan and Syntetos (2007) prove that not only is it biased, but also its bias is
substantially greater than the original Croston procedure, especially for highly
intermittent series. The results of a comparison study by Teunter and Sani (2009)
confirm the bias of the MC method, showing that it has the highest bias when
compared to the original Croston?s method, SB approximation and the Syntetos
approximation (Syntetos, 2001). Therefore, the MC method will not be considered
further in this thesis.
2.3.1.4 Models?underlying?Croston?s?Method
Shenstone and Hyndman (2005) use autoregressive integrated moving average
(ARIMA) models for both size of demand and interarrival times. In an attempt to
find an underlying model for Croston?s method, they have looked at several models.
First, they assume that ??~ARIMA(0,1,1) and ??~ARIMA(0,1,1), where ?? is the
size of the jth nonzero demand, ?? is the interarrival time between ???1 and ??, and
?? and ?? are independent. They also restrict the sample space of the underlying
M.Mohammadipour, 2009, Chapter 2 24
model to be positive by defining log(??)~ARIMA(0,1,1) and
log(??)~ARIMA(0,1,1). The other two models are the ones proposed by Snyder
(2002) in which ??~ARIMA(0,1,1) and ??~ Geometric(?), and
log(??)~ARIMA(0,1,1) and ??~ Geometric(?) where ? is the average interarrival
time of the series.
Shenstone and Hyndman (2005) suggest that any model assumed to be underlying
Croston?s method must be nonstationary and defined on a continuous sample space
including negative values. However, they argue that none of the above mentioned
models is consistent with the properties of intermittent demand data. Finally, they
suggest that instead of focusing on models based on SES, it is worth considering
stationary models such as Poisson autoregressive models. These models have not
been used for intermittent demand data before, opening up a new area of study.
In a recent working paper, Snyder and Beaumont (2007) claim to find a logically
sound model underlying Croston?s method. They assume that the sizes of positive
demands follow a Poisson distribution. However, they assume that the Poisson
distribution is shifted by one to the right to avoid zero demands. This results in the
following probability mass function for a positive demand at period ?:
???=
(???1?1)
??1
???1?!
?????1+1
Equation 218
where ? is the value of the positive demand. The probability of a positive demand in
period ? is then defined by ?? which is:
??=
1
??
Equation 219
Finally, the probability of demand in period ? is given by:
????=??=?
1??? if ?= 0
????? if ?> 0
?
Equation 220
The above equation is claimed to be the underlying model for Croston?s method
i.i.d.
i.i.d.
M.Mohammadipour, 2009, Chapter 2 25
(Snyder and Beaumont, 2007). It can be considered as a model underlying Croston's
method. However, they have not performed any assessment of the optimality of the
proposed model for Croston's method. If a model exists for which Croston?s method
is optimal, then it has yet to be formally established and proven.
2.3.2 Bootstrapping
Bootstrapping has been proposed by Willemain et al. (2004) and Porras (2005). It is
based on repeatedly sampling ? demands from demand history to estimate the
distribution of lead time demand (lead time = ?). The main advantage of bootstrapping
is that it forecasts the whole lead time demand distribution.
There are many variants of the bootstrapping method (see for example Efron, 1979;
Bookbinder and Lordahl, 1989) many of them having the disadvantage of being
complex, a disadvantage that also holds for the bootstrapping method proposed by
Willemain et al. (2004). The bootstrapping method proposed by Porras and Dekker
(2007) is simpler. Both of these methods along with the method proposed by Snyder
(2002) are discussed in this section.
2.3.2.1 Snyder (2002)
Snyder (2002) proposes a parametric bootstrap method to generate an approximation
for the leadtime demand distribution from Croston's model. In each iteration, first,
the values for noise terms are generated from a normal distribution with mean 0 and
variance ?2. Then the values for indicator variables (??, Equation 21) are generated
from a Bernoulli distribution with probability
p
. A realisation of future demand
series is then produced based on Croston?s method (Equation 23 and Equation 24),
and finally, the leadtime demand is calculated from ? ??
?+?
?=?+1 .
In order to tackle the problem of having negative demands, Snyder (2002) suggests
two adaptations. The first is to apply exponential smoothing to the logarithm of the
data, which is called logspace adaptation. The other method, called the adaptive
variance version, differs from Croston?s method in two respects: (i) variability is
M.Mohammadipour, 2009, Chapter 2 26
measured in terms of variances instead of mean absolute deviations (MAD); (ii) a
second smoothing parameter, ?, is used to define changes of variability over time.
Snyder (2002) claims that the advantage of the methods is that they can be applied to
both fast and slow moving demand. However, they have the disadvantages of
underestimating the variability of leadtime demand due to ignoring the effects of
estimation error, and being based on the Normal distribution, which may not be
appropriate when demand sizes are small.
2.3.2.2 Willemain et al. (2004)
Bootstrapping (Efron, 1979) produces pseudodata by sampling with replacement
from the observations. Willemain et al. (2004) develop a modified bootstrap to
forecast the distribution of the sum of intermittent demands over a fixed lead time.
The modification allows for autocorrelation, frequent repeated values, and relatively
short series which are ignored in conventional bootstrapping.
The underlying model of intermittent demand incidence that they assume is a two
state, first order Markov process.
As for the demand sizes, two models have been discussed. The first model only
assumes that demand sizes are stationary and they can be obtained by the method of
sampling from the nonzero values that have appeared in the past. The problem here,
as also mentioned by Willemain et al. (2004), is that no different values would
appear in the future. As a result, they suggest an ad hoc method to deal with demand
sizes called ?jittering? based on no model for demand sizes.
The method is based on generating a sequence of zero and nonzero values over the L
periods of the lead time. The state transition probabilities are estimated from the
historical demand series using started counts. Then values are assigned to nonzero
forecasts. Here, instead of only choosing nonzero values that have appeared in the
past, which results in having no different values in the future, they jitter the selected
value from the past, i.e. add some random variation to it.
Although some methods for intermittent demand forecasting are based on the
M.Mohammadipour, 2009, Chapter 2 27
assumptions that demands in each time period are independent and normally
distributed, with neither of them necessarily being valid for intermittent demand
(Willemain et al., 2004), Willemain et al.?s bootstrap method requires neither
assumption. It only assumes that demand is stationary.
The main advantage of bootstrapping is that it does not rely on any distribution while
all other methods do. However, the model of Willemain et al. (2004) for demand
occurrence is not general and also the method of producing variable demand sizes is
ad hoc.
They compare the accuracy of the developed bootstrap against exponential
smoothing and Croston?s method by applying them to over 28,000 items provided by
nine industrial companies. The results suggest that, although Croston?s method
provides more accurate estimates of the mean level of demand at the moments when
demand occur, it does not outperform exponential smoothing when forecasting the
lead time distribution. However, the bootstrap provides an improvement on
exponential smoothing, especially for short lead times.
Gardner and Koehler (2005) argue that Willemain et al. (2004) did not use the
correct lead time demand distribution for exponential smoothing and Croston's
method in their comparison. They also argue that Willemain et al. (2004) should
have considered the modifications to the Croston's method that have shown
improvements.
2.3.2.3 Porras and Dekker (2007)
Porras and Dekker (2007) propose another procedure to specify the lead time
demand (LTD) distribution. Known as the Empirical Method, it differs from
Willemain?s in that it constructs a histogram of demand over lead time without
sampling. Therefore, it has the benefit of capturing the autocorrelation of LTD. It is
also easier than Willemain?s bootstrap to implement. However, when lead time is
long and the length of demand history is short, which is often the case for
intermittent demand data, there are few blocks of LTD to select from.
Another problem with this method is that, particularly for short lead times (say ?=
M.Mohammadipour, 2009, Chapter 2 28
1), it is not possible to attain high percentiles in the demand distribution. Even if a
high percentile can be reached, it might be an outlier and not representative of the
population. Willemain et al. (2004) tackle this issue by introducing jittering, as
explained in the previous section.
An empirical study is conducted to compare the performance of the new method
from an inventory control perspective to other methods including: Willemain?s
bootstrap, Normal distribution, and Poisson distribution. For the normalbased
model, it is assumed that demand per period follows a Normal distribution. The
average and standard deviation of the observed demand are then used to estimate the
parameters of the Normal lead time demand. When the lead time demand is assumed
to have a Poisson distribution, the parameter is estimated from the demand data for
the different items.
The results show that although demand of spare parts does not generally follow a
Normal distribution, the Normal model performs well, producing the highest savings
in inventory costs. It has also been found that the Empirical Method outperforms the
Willemain method.
2.3.3 Causal Models
As pointed out by Hua et al. (2007), demand of spare parts can be attributed both to
the status of the equipment and the spare parts, and to the maintenance policy. Hua et
al. (2007) establish the impact of the maintenance policy through finding the
relationship between explanatory variables and the nonzero demand of spare parts.
They develop a method called the Integrated Forecasting Method (IFM) that first
forecasts the occurrence of nonzero demand and then estimates the lead time
demand. The first is done using an autocorrelation function to choose either a
Markov process or explanatory variables, while the second is done by sampling from
the nonzero values observed in the past and summing them over the lead time. They
compare the results of the Markov Bootstrapping (MB) method (same as
Willemain?s bootstrapping method except that the jittering technique is not used)
with IFM. The performance of SES, Croston?s method, MB and IFM is also
compared based on the mean absolute percentage error of lead time demand
M.Mohammadipour, 2009, Chapter 2 29
(MAPELTD). The results confirm that Croston?s method provides more accurate
estimates of mean LTD than the other methods.
Ghobbar and Friend (2002) study the source of lumpiness of demand for aircraft
maintenance repair parts, in order to reduce the occurrence of part shortages. They
study the effect of five environmental factors on lumpiness. These factors are: the
primary maintenance process (PMP) (including hardtime and conditional
monitoring), the aircraft utilization rate (AUR), the component?s overhaul life
(COL), the squared coefficient of variation of demand (CV2) and the average inter
demand interval (ADI). The first three factors are independent variables and the last
two are dependent variables in the general linear model (GLM). The results show
that the demand variability increases when the level of aircraft utilization and flying
hours increases. It shows that AUR can be a major source of lumpiness because it
increases the CV2 and decreases the ADI for the observed demand.
In another study, they compare the results of 13 forecasting methods including
Croston?s method, SES, and also those used by aviation companies (Ghobbar and
Friend, 2003). Four environmental factors considered in this study are: the seasonal
period length, primary maintenance process (PMP), squared coefficient of variation
(CV2) and the average interdemand interval (ADI). CV2 and ADI are taken as
covariate factors while the other two are considered as categorical factors. The
variation attributable to each factor and their interactions is studied through using
analysis of variance (ANOVA).
Ghobbar and Friend (2003) also establish a predictive errorforecasting model
(PEFM) to compare different forecasting methods based on their factor levels to
evaluate which method is the best in any situation. The model is based on a GLM
that predicts a response variable using its relationship with factor variables.
The results of studies on causal models show an interesting line to pursue. These
models are especially useful when a short demand history is available and time series
methods cannot be used (Boylan and Syntetos, 2008). However, these models have
not yet been well developed in the literature. As a future line of study, the effect of
incorporating these models and INARMA models on the accuracy of forecasts can be
investigated (see chapter 10).
M.Mohammadipour, 2009, Chapter 2 30
2.3.4 Conclusions on IDF Methods
Three classes of methods for intermittent demand forecasting have been reviewed in
this section, namely Croston?s method and its variants, bootstrapping methods, and
causal methods.
Empirical studies have proved the ability of Croston?s method and variants
(especially SBA) to produce reasonable forecasts (Eaves and Kingsman, 2004;
Syntetos and Boylan, 2005). However, properties of these methods (such as bias)
have been derived based on assumptions of independence of demand, independence
of demand sizes and interarrival times. As mentioned, some of these assumptions
are not valid for realworld data. Another problem with Croston?s method is that it
has not yet been shown to be optimal for a specific demand model. Modelling
intermittent demand by INARMA models makes it possible to take into account the
correlation between demands. Another advantage is that optimal methods are known
to exist for these models.
Bootstrap methods have the advantage of having no distributional and independence
assumptions. The main disadvantage of these methods is that they are rather
complex. These methods along with causal methods have not been well developed in
the literature and more comparative studies with the best benchmark methods (as
criticized by Gardner and Koehler (2005)) need to be done to prove their
performance.
2.4 Assessing Forecast Accuracy
The nature of intermittent demand data makes some of the conventional accuracy
measures inappropriate. For example, when one or more of the observed demands is
zero, the mean absolute percentage error (MAPE) is undefined and, therefore, cannot
be used.
In fact, none of the relativetotheseries accuracy measures can be used because zero
observations may yield ?division by zero? problems. This includes MAPE, Median
Absolute Percentage Error (MdAPE), Root Mean Square Percentage Error (RMSPE),
M.Mohammadipour, 2009, Chapter 2 31
and Root Mean Square Percentage Error (RMSPE).
Although the symmetric MAPE introduced by Makridakis and Hibon (1995) tackles
this issue by dividing the absolute error by the average of the actual observation and
the forecast:
sMAPE =?
?????
(??+??)/2
?
?=1
/?? 100
Equation 221
it is known that it suffers from asymmetry problems in its treatment of negative and
positive errors (Goodwin and Lawton, 1999). Also, if the actual value ?? is zero,
sMAPE would always be 200%, which is meaningless (Syntetos, 2001).
As discussed by Syntetos and Boylan (2005), all relativetoabase accuracy
measures that relate the forecast error to a benchmark, usually na?ve 1 method,
should also be discarded because the error would often be zero.
The mean absolute scaled error (MASE) proposed by Hyndman and Koehler (2006)
does not have the problems seen with previous measures. This measure is obtained
by scaling the absolute error based on the insample MAE from a benchmark forecast
method. Assuming the benchmark method is the na?ve method, the MASE is defined
as:
MASE =
1
?
? ?
?????
1
??1
? ??????1?
?
?=2
?
?
?=1
Equation 222
A
1MASE ?
means that the forecasting method is on average better than the na?ve
forecasts, and a
1MASE ?
indicates that the method is on average worse than the
na?ve method. As explained by Hyndman (2006), the insample MAE is always
available and more reliably nonzero than any outofsample measures. The MASE
would be infinite or undefined only if all historical observations are equal. However,
using insample MAE has a disadvantage of making MASE vulnerable to outliers in
the historical time series (Kolassa and Sch?tz, 2007). As a result, the MASE of two
time series with the same forecasts and identical true demands during the forecast
M.Mohammadipour, 2009, Chapter 2 32
horizon will be different if the two series differed in their historical demands. This
makes it a more complicated metric to interpret.
Syntetos and Boylan (2005) categorize the accuracy measures for the purpose of
comparing methods in an intermittent demand context into two categories: absolute
accuracy measures and accuracy measures relative to another method. Each of these
categories is reviewed as follows.
2.4.1 Absolute Accuracy Measures
These measures are calculated as a function of the forecast errors alone. Examples
include the mean square error (MSE) and the mean absolute error (MAE).
Theoretically, these measures can be computed for intermittent demand. However,
when averaged across many time series, they do not take into account the scale
differences between them (Syntetos and Boylan, 2005). Mean error (ME) can be
considered as an exception to the above rule because it takes into account the sign of
the error and is less susceptible to scale effects. ME is given by:
ME =?
?????
?
?
?=1
Equation 223
If ME is divided by the average demand per unit time period, the scale dependencies
are eliminated.
2.4.2 Accuracy Measures Relative to another Method
These measures are calculated as a ratio to other forecasting methods. Examples
include percent better (PB) and the relative geometric root mean square error
(RGRMSE).
The percentage better measure counts and reports the percentage of time that a given
method has a smaller forecasting error than another method.
M.Mohammadipour, 2009, Chapter 2 33
The RGRMSE for methods A and B in a time series is:
RGRMSE =
?? (?????,?)
2?
?=1 ?
1
2?
?? (?????,?)2
?
?=1 ?
1
2?
Equation 224
The theoretical properties of RGRMSE are discussed by Fildes (1992). He assumes
that the squared errors of a particular series have the form of:
??,?+?
2 =??,?+?
2 .??+?
Equation 225
where (??,?+?=??+????,?+?), ??+? are assumed positive and can be thought of as
errors due to the particular time period affecting all methods equally, while ??,?+?
are the method?s (M) specific errors. He argues that the model of Equation 225
represents the case where data (and therefore errors) are contaminated by occasional
outliers. He then shows that the geometric (rather than arithmetic) RMSE is
independent of ??+?.
It can be seen from Equation 224 that GRMSE is identical to GMAE because the
square and the square root cancel each other (Hyndman, 2006):
GRMSE =
?? (?????,?)
2?
?=1 ?
1
2
?? (?????,?)2
?
?=1 ?
1
2
=
? ?????,?
?
?=1
? ?????,?
?
?=1
= GMAE
The only issue with calculating RGRMSE by Equation 224 is that if ?? and ??,? are
identical for a specific time period ?, the measure will be undefined. Despite this
problem, RGRMSE has been suggested as an appropriate measure for intermittent
demand data (see for example Syntetos and Boylan, 2005), because in most of the
cases the forecasts (??,?) are not integer (although averaging methods might produce
integer forecasts) and therefore, the denominator is not zero. When all of the
observations are zero, this shows that there has been no demand for the item in a long
time and in reality no forecasts are made for such items. Therefore, such series can
be excluded from the study.
M.Mohammadipour, 2009, Chapter 2 34
2.4.3 Conclusions on Accuracy Measures
When comparing the performance of different forecasting methods, a range of
accuracy measures should be used. This is because different measures are designed
to assess different aspects. For example, MSE puts heavier penalties on higher errors
while MAE is designed to lessen the effect of outliers. It has been found through
forecasting competition studies such as the Mcompetition (Makridakis et al., 1982)
and the M3competition (Makridakis and Hibon, 2000) that the performance of
different methods changes considerably depending on the accuracy measure being
used. As a result, such studies have used several accuracy measures. As discussed
previously, not all accuracy measures can be used for intermittent demand data.
MASE has been suggested for intermittent demand studies (Hyndman, 2006).
Because it is based on insample MAE of the na?ve method, there is no ?division by
zero? problem unless all of the observations are equal. However, MASE, as well as
MAE, can sometimes be misleading when comparing forecasting methods.
The following example illustrates how the Zero Forecasting method (ZF) can show
superiority to exponential smoothing when only MASE is concerned. The idea of
using ZF as a benchmark comes from Teunter and Duncan (2009). They only used
MAE and MSE, but we also include ME and MASE in our comparison. The data, the
same as that presented by Hyndman (2006), are given in Table 22.
Table ?22 Example data (Hyndman, 2006)
Insample Outofsample
Actual ?? 0 2 0 1 0 1 0 0 0 0 2 0 6 3 0 0 0 0 0 7 0 0 0 0 0 0 0 3 1 0 0 1 0 1 0 0
Na?ve forecast ?? 0 2 0 1 0 1 0 0 0 0 2 0 6 3 0 0 0 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
We have compared Exponential Smoothing (ES) with ZF. The results of ME, MASE,
MAE, and MSE (average over both insample and outofsample errors) are given in
Table 23.
Table ?23 Comparing ES with ZF based on different accuracy measures
ME MASE MAE MSE
Exponential Smoothing
?=?.?
0.1722 0.7114 1.1754 3.3458
Zero Forecast 0.8 0.4842 0.8 3.3143
M.Mohammadipour, 2009, Chapter 2 35
The results of Table 23 suggest that ZF is the best method when MASE, MAE and
MSE are used. Teunter and Duncan (2009) suggest that therefore these accuracy
measures should not be used in this regard and inventory implication metrics should
be used instead. However, we believe that these measures should be interpreted
based on what they are designed to measure and, as a result, different measures
should be used to compare forecasting methods. MASE and MAE are designed to
measure the absolute errors while ME is designed to measure bias. Because
intermittent data have many zero observations, when ZF is used on these
observations, the error would be zero. Therefore, ZF produces better results on
MASE and MAE than ES does. But looking at ME reveals that the forecasts are
much more biased than forecasts based on ES.
Table 24 summarizes the absolute accuracy measures which will be used by this
study.
Table ?24 Accuracy measures for simulation and empirical studies
Theory Simulation Empirical
MSE ? ? ?
ME ? ? ?
MASE ? ?
RGRMSE ?
PB ?
MSE is a widely used measure in the forecasting literature, is mathematically easy to
handle, can be linked to a quadratic loss function and therefore it will be used for
theoretical comparisons. MSE is a sensible measure for evaluating an individual time
series and, although it is scale dependent and cannot be used for assessing a method?s
accuracy across multiple series, it can be used in simulation where all the series are
theoretically generated. Keeping in mind the scaledependency and sensitivity to
outliers issues, we will also use MSE for empirical analysis.
ME can also be used for theoretical comparisons and, as discussed earlier, it does not
have the scaledependency problem. Therefore it will be used for both simulation and
empirical analysis.
Based on the above mentioned properties, MASE will be used for simulation and
empirical analysis. It should be mentioned that as both MAE and MASE are used to
M.Mohammadipour, 2009, Chapter 2 36
measure the same factor (absolute errors), we do not see any benefit in using both
and only MASE will be calculated.
In addition to the above measures, two relativetoanothermethod measures will be
used for empirical analysis: PB and RGRMSE. The PB is based on comparing the
MASEs of two methods and determines how often a method is better than another,
but not by how much. RGRMSE is used to calculate the magnitude of improvement
(in terms of MSE) of one method over another. Both of these measures have been
used in an intermittent demand context (Syntetos and Boylan, 2005; Teunter and
Duncan, 2009). They are insensitive to outliers and have been used in other studies,
allowing comparative analyses to be undertaken.
2.5 Comparative Studies
A number of studies have been undertaken to compare methods for forecasting
intermittent demand (Willemain et al., 1994; Johnston and Boylan, 1996; Eaves and
Kingsman, 2004; Lev?n and Segerstedt, 2004; Willemain et al., 2004; Syntetos and
Boylan, 2005). Three of the main comparison studies will be discussed here.
Eaves and Kingsman (2004) compare the performance of exponential smoothing
(ES), Croston?s method, SyntetosBoylan Approximation (SBA), a moving average
(MA12) and a simple average method. Mean absolute deviation (MAD), root mean
square error (RMSE) and MAPE have been used to compare the accuracy of
forecasts obtained by each method, although it has been concluded that these
measures are not ideal for slowmoving demands. It is also suggested to use stock
holdings as a measure instead of the abovementioned conventional accuracy
measures. The results show the superiority of SBA with regards to stockholdings.
The study by Willemain et al. (2004) compares three methods of ES, Croston?s, and
Willemain?s bootstrap based on the uniformity of observed leadtime demand (LTD)
percentiles. It is found that Croston?s method provides a more accurate estimate of
mean demand than ES, but the same is not true for forecasting the distribution of
LTD. The results also show that bootstrapping is the most accurate method,
especially for short lead times.
M.Mohammadipour, 2009, Chapter 2 37
Another study by Syntetos and Boylan (2005) compares the accuracy of forecasts
obtained by simple moving average (SMA13), SES, Croston?s method, and SBA.
ME, scaled mean error, RGRMSE, Percentage Better (PB), and Percentage Best
(PBt) have been used and the results suggest that using different accuracy measures
leads to different conclusions, agreeing with the findings of Eaves and Kingsman
(2004). However, the SBA seems to be the most accurate method for faster
intermittent demand (p values close to 1).
It is also argued by Teunter and Duncan (2009) that the conflicting results of
comparative studies in the literature are due to use of inappropriate measures. Instead
of comparing ?per period forecast error? (measures such as MAD and MSE), they
propose analyzing the effect of forecasting methods on inventory control parameters
and also comparing the resulting average inventory and service levels. Boylan and
Syntetos (2006) also highlight the distinction between these two approaches which
they refer to as forecastaccuracy metrics and accuracyimplication metrics,
respectively.
As discussed previously, because this research only focuses on forecasting and not
on inventory control, the forecastaccuracy metrics suggested in section 2.4 will be
used. The proposed method based on INARMA modeling of intermittent demand
will be compared with Croston?s method, SBA and SBJ. The reason for considering
Croston?s is its simplicity, popularity (for example, it is used in Forecast Pro), and its
superiority over SES and SMA (Syntetos and Boylan, 2005). SBA has also been
proved by a number of studies discussed above to perform reasonably well and
therefore is included for comparison in this research. SBJ is included because it is
based on Poisson demand arrivals. The reason for ignoring bootstrapping is that we
have restricted our model to Poisson marginal distribution which makes it an
inappropriate comparison considering the fact that there is no distributional
assumption for bootstrapping.
2.6 INARMA Models
As discussed by Willemain et al. (1994), stochastic models of intermittent demand
have assumed that the successive intervals between demands, successive demand
M.Mohammadipour, 2009, Chapter 2 38
sizes, and intervals and sizes are mutually independent. However, their sample data
contradict this assumption (with some autocorrelations as high as 0.53 and 0.39).
This emphasizes the necessity to use models that take into account autocorrelation.
Shenstone and Hyndman (2005) also suggest using integer autoregressive moving
average (INARMA) models with Poisson marginal distribution or other time series
models for counts for intermittent demand forecasting.
This research focuses on using a class of models called INARMA models to model
intermittent demand. The forecasting method will then be the minimum mean square
error (MMSE) of the model. We start from a stochastic model for intermittent
demand and then build a method for forecasting which is optimal for the underlying
model. It has also the advantage of enabling us to directly find the mean and variance
of leadtime demand. The conditional mean of leadtime demand will be compared
with the leadtime forecasts of the benchmark methods in chapters 8 and 9.
Table 25 summarizes the methods for intermittent demand forecasting based on
their dependence, distributional, and stationarity assumptions. Although this research
only focuses on INARMA models with Poisson marginal distribution, other discrete
distributions such as negative binomial can be considered as a future line of study.
Table ?25 The categorization of methods of intermittent demand forecasting based on their assumptions
Models Dependence structure Distribution
Croston?s method independent
Normal sizes of demand
Geometric interarrival times
SBA independent
Normal sizes of demand
Geometric interarrival times
SBJ independent
Geometric sizes of demand
Negative Exponential or Erlang
interarrival times
Bootstrapping:
Willemain et al.
(2004)
Markov demand incidence

Porras and
Dekker (2007)

INARMA modelling autocorrelated demand Poisson
2.7 Conclusions
The literature on forecasting intermittent demand has been reviewed in this chapter.
Different definitions of intermittent demand have been reviewed. In this study, a
M.Mohammadipour, 2009, Chapter 2 39
series is called intermittent if it contains nonnegative integer values with some
values being zero (Shenstone and Hyndman, 2005). As explained, intermittence and
lumpiness are two different concepts and should be distinguished. This study focuses
on INARMA models with a Poisson marginal distribution. Although other marginal
distributions could be used to allow for more demand size variability, in general
these models are not designed for very lumpy demand. The two classification
schemes for intermittent demand by Eaves and Kingsman (2004) and Syntetos et al.
(2005) have been reviewed. As explained, the INARMA approach is universal and
can be used for both intermittent and smooth demand but we will use the Syntetos et
al. classification to find the best benchmark method for comparison.
Different intermittent demand forecasting methods have been reviewed. These
methods are:
? Croston?s method and its variants
? Bootstrapping
? Causal methods
Although Croston's method is the most widely used approach for intermittent
demand forecasting, it has been shown that it is biased (Syntetos and Boylan, 2001).
Some modifications have been suggested to reduce its bias (e.g. SBA and SBJ) while
some add even more bias (Modified Croston). The optimal demand model
underlying Croston's method has not yet been established, although some studies
have focused on finding such a model (Shenstone and Hyndman, 2005; Snyder and
Beamont, 2007).
Different bootstrapping methods have been established in the literature to estimate
the distribution of lead time demand. The parametric bootstrap by Snyder (2002) has
the disadvantage of being based on the Normal distribution. The method of
Willemain et al. (2004), however, is not based on any distributional assumptions.
Porras and Dekker (2007) develop another procedure for identification of the lead
time demand.
The studies on using causal models for intermittent demand forecasting have also
been reviewed. Both bootstrapping methods and causal models seems promising but
the literature on these methods is not yet welldeveloped. As will be discussed in
chapter 10, incorporating causal factors into INARMA models can be pursued as a
M.Mohammadipour, 2009, Chapter 2 40
future line of study. The performance of INARMA models in forecasting the lead
time distribution can also be compared to that of bootstrapping methods.
The fact that intermittent data include zero values makes some of the conventional
accuracy measures inappropriate. Different accuracy measures that can be used for
intermittent data are reviewed. Three absolute accuracy measures will be used in this
research: ME, MSE, and MASE. For the empirical study, two relativetoanother
method accuracy measures will be added to the abovementioned measures: PB and
RGRMSE.
Using the INARMA procedure for intermittent demand forecasting has the advantage
of utilizing a modelbased forecasting method which takes into account the
correlation between demands. Unlike bootstrapping, a marginal distribution has to be
assumed. The Poisson distribution has been selected because it has been used in the
IDF literature and also due to its interesting properties in INARMA models, to be
reviewed in chapter 3.
As discussed in section 2.5, the results of INARMA method will be compared to
Croston?s method, SyntetosBoylan Approximation (SBA) and ShaleBoylan
Johnston (SBJ) method.
The next chapter introduces INARMA models and reviews their properties in detail.
M.Mohammadipour, 2009, Chapter 3 41
Chapter 3 INTEGER AUTOREGRESSIVE
MOVING AVERAGE MODELS
3.1 Introduction
Continuousvalued autoregressive integrated moving average (ARIMA) models, also
known as BoxJenkins models, developed by Box and Jenkins (1970) are used to
model stationary processes under the assumption of Gaussianity, i.e. all the joint
distributions of the time series are multivariate normal (Brockwell and Davis, 1996).
However, this assumption is not valid for modelling count data, especially for low
frequency count data that cannot be suitably approximated by continuous models.
Therefore, a number of models using different approaches have been proposed for
integervalued time series in the literature (e.g. Smith, 1979; Zeger, 1988; Zeger and
Qaqish, 1988; Harvey and Fernandes, 1989).
M.Mohammadipour, 2009, Chapter 3 42
These models are divided into two categories: observationdriven and parameter
driven. Observationdriven models specify a direct link between current and past
observations, while parameterdriven models rely on a latent process connecting the
observations (Jung and Tremayne, 2006a). As an example of parameterdriven
models for time series of counts, Zeger and Qaqish (1988) proposed Poisson
regression to model trend and seasonality explicitly and an unobserved stationary
process (latent process) to model the autocorrelation.
One of the early observationdriven models for count data is the model suggested by
Jacobs and Lewis (1978a; 1978b; 1983) called DARMA (discrete autoregressive
moving average) models which will be discussed in section 3.2.
Another class of observationdriven models has been developed by McKenzie (1985)
and later generalized by AlOsh and Alzaid (1987) as integervalued autoregressive
(INAR) models for modelling series with correlated counting data. Examples of this
kind of time series include the number of patients in a hospital at a specific point of
time, the number of people in a queue waiting to receive a service at a certain
moment (Silva and Oliveira, 2004), and the number of accidents in a firm each three
months (McKenzie, 2003).
Table 31 briefly describes the most frequently used count models and summarizes
their strengths and weaknesses (McKenzie, 2003; Rengifo, 2005).
This thesis is focused on integer autoregressive moving average (INARMA) models.
As explained in chapter 2, the need for a modelbased method for intermittent
demand forecasting has motivated this research. The similarities that these models
have with the conventional ARMA models are also an advantage. Because DARMA
models are also based on ARMA models, we introduce them in section 3.2. The
main reasons for excluding these models from our study are explained in this
section.
This chapter mainly focuses on introducing the INARMA models and reviewing
their statistical properties. Identification of these models, estimation of their
parameters and INARMA forecasting will be discussed in subsequent chapters.
This chapter is structured as follows. In section 3.2, DARMA models are introduced.
M.Mohammadipour, 2009, Chapter 3 43
INARMA models are reviewed in section 3.3. Summary of literature review and the
conclusions are provided in sections 3.4 and 3.5.
Table ?31 Review of count models in time series
Model Description Strengths and weaknesses
Markov chains First, the transition probabilities between all
the possible values that the count variable
can take are defined. Then, the appropriate
order of the time series is determined
(e.g. see Raftery 1985, and Pegram 1980).
This method can only be reasonably
used when the possible values that
the observations can take are very
limited. It is generally
overparametrized and too limited in
correlation structure.
Discrete
Autoregressive
Moving Average
(DARMA) models
These models are structurally based of
ARMA models and are probabilistic mixtures
of discrete i.i.d. random variables with a
marginal distribution (see section 3.2).
When the serial correlation is high,
the data will be characterized by a
series of runs of a single value.
Therefore, they are rarely used. The
main application is in the hydrological
literature.
Integer
Autoregressive
Moving Average
(INARMA) models
These models are a generalization of the
linear ARMA models for count data (see
section 3.3).
The models have the same serial
correlation structure as ARMA
models. The marginal distribution of
the model is same as the distribution
of the innovations if the latter is
Poisson.
Regression models
(or generalized linear
models)
These are regression models for the special
case where the dependent variable is a non
negative integer with a correction for
autocorrelation (Zeger, 1988; Br?nn?s and
Johansson, 1994).
These models are generally easy to
construct and have the
ovserdispersion property. An explicit
joint density function cannot be
obtained which restricts the class of
possible predictors.
The hidden Markov
models
These models are extension of the basic
Markov chains models, in which various
regimes characterizing the possible values
of the mean are identified. It is then
assumed that the transition from one regime
to another is ruled by a Markov chain
(MacDonald and Zucchini 1997).
There is no accepted way of
determining the appropriate order for
the Markov chain. There can be too
many parameters to estimate,
especially when the number of
regimes is large.
State Space models These models specify the conditional
distribution of the series to depend on
stochastic parameters that evolve according
to a specified distribution. The parameters of
such distributions are determined by some
regressors (Harvey and Fernandes, 1989;
Durbin and Koopman, 2000).
These models are very general and
can be used in a wide range of
applications. The behaviour of
different components of the series
can be modelled separately and then
put together.
3.2 DARMA Models
This class of models was first introduced by Jacobs and Lewis (1978a) for a
stationary sequence of dependent discrete random variables. Because we focus on
INARMA models in this thesis, only a brief review of the first order discrete
autoregressive model is provided in this section. The applications of DARMA
models will then be discussed.
M.Mohammadipour, 2009, Chapter 3 44
3.2.1 DAR(1) Model
The first order discrete autoregressive form, DAR(1), is given by:
??=?????1 + (1???)??
Equation ?31
where {??} are i.i.d. binary random variables with ????= 1?=? and {??} are i.i.d.
discrete random variables with the probability mass function (hereafter called a
?distribution? as by the authors) ?1 (Jacobs and Lewis, 1978a). Therefore, in this
model, the current observation is either the previous observation, with probability ?,
or another, independent, sample from a specific distribution. This approach is similar
to the BoxJenkins approach except that a probabilistic mixture replaces the linear
combination in the continuousvalued model and, as the authors mentioned, a
realization of the process will generally contain many runs of a constant value. This
can be a significant disadvantage of these models. It is obvious that the larger the
value of ?, the longer the runs.
It can be seen that, in the model of Equation 31, ???1 contains all information about
the past. Because there is randomization between ???1 and ??, if ?? is chosen, the
memory of the process before time ? is gone forever.
The autocorrelation function (ACF) of {??} is given by:
??=?
? for ?= 0,1,?
Equation ?32
The conditional mean of ?? given ???1 is linear in ???1 and the conditional variance is
quadratic in ???1. It also follows from Equation 31 that {??} is a Markov chain since:
????+1 =??1,?,???=????+1 =????
Equation 33
The transition matrix for this Markov chain is given by:
1 ????=??=?(?) ?= 0,1,?
M.Mohammadipour, 2009, Chapter 3 45
????+1 =???=??=?
?1????(?) for ???
?+?1????(?) for ?=?
?
Equation 34
It can be seen from Equation 31 that when ?= 0, the model is reduced to a
stationary sequence of i.i.d. random variables with distribution ?.
The mixed DARMA(p,q+1) model is built by adding the two autoregressive and
moving average components as follows:
??=???????+ (1???)?????1
Equation 35
??=???????+ (1???)??
Equation 36
where {??} and {??} are as before. {??} are i.i.d. random variables defined on the set
?= {1,2,?,?} in such a way that ????=??=??. {??} are i.i.d. random variables
so that ????=??=?? for ?= {0,1,?,?} and {??} are i.i.d. binary random
variables with ????= 1?=?.
3.2.2 Applications of DARMA Models
Applications of DARMA models can be found mainly in the hydrological literature.
This is not surprising given the structure of the model, which represents dependence
as runs (McKenzie, 2003). These applications are briefly discussed below.
In climatic analysis, these models were introduced by Buishand (1978). He proposes
a binary discrete autoregressive moving average (BDARMA) process to model daily
rainfall sequences and finds that this model is promising in tropical and monsoonal
areas.
Chang et al. (1984) use the same type of models (BDARMA) in their study of daily
precipitation by transforming the daily level of precipitation into a discrete variable
based on its magnitude. They conclude that the statistical properties of the daily
rainfall process can be preserved by DARMA models.
M.Mohammadipour, 2009, Chapter 3 46
Salas et al. (2001) use DARMA models to simulate the return period and risk of
extreme droughts. A series of wet and dry years is obtained from a continuousvalued
hydrologic series, such as annual stream flows, and a method is presented for relating
the autocorrelation functions of these two series. The analysis of 23 series of annual
flows reveals that this relationship is applicable and reliable.
Ksenija (2006) employs a DARMA(1,1) model to describe the wetdry day
sequences in Split, on the middle Adriatic coast of Croatia. The results are compared
to the DAR(1) model and reveal that, although both models underestimate dry spell
runs, the DARMA(1,1) model provides a better fit to the empirical distribution both
for short (one day) and long runs (more than 10 days). But, for short wet spells, the
DAR(1) model estimates are closer to the observed frequencies of short spells in the
months studied.
As previously mentioned, the main disadvantage of the DARMA models, which
makes their application areas very limited, is that a realization of the process will
generally contain many runs of a constant value. This is especially true when the
serial correlation is high. Therefore we exclude these models and concentrate on
another class of models for count data, called integer autoregressive moving average
models, which overcome these problems.
3.3 INARMA Models
The integervalued autoregressive models are equivalent to autoregressive models for
Gaussian time series. By ?equivalent? we mean that they share some similar
properties, which will be discussed later in relevant subsections.
First, we introduce the firstorder integer autoregressive, INAR(1), model. It has
been shown that this model belongs to a more general class of models known as non
Gaussian conditional linear AR(1), CLAR(1), models (Grunwald et al., 2000). Other
integer autoregressive, moving average and mixed models are then described.
Although many papers discuss the statistical aspects of INARMA models, fewer
studies have been done regarding their practical application (Jung and Tremayne,
M.Mohammadipour, 2009, Chapter 3 47
2003). Some applications of these models are reviewed at the end of this section.
3.3.1 INAR(1) Model
Before describing the INAR(1) model, we first introduce the meaning of the
binomial thinning operation defined by Sueutel and van Harn (1979). Suppose Y is a
nonnegative integervalued random variable. Then, for any ??[0,1], the thinning
operation ??? is defined by:
???=? ??
?
?=1
Equation 37
where {??} is a sequence of i.i.d. Bernoulli random variables, independent of ?, and
with a constant probability that the variable will take the value of unity:
????= 1?= 1?????= 0?=?
Equation 38
From the above definition, some of the properties of the thinning operation can be
obtained as follows:
(1) 0??= 0
(2) 1??=?
(3) ??(???) (??)??
(4) ??????=??(?)
(5) ??????2 =?2???2?+??1????(?)
(6) var?????=?2var???+??1????(?)
where stands for equal in distribution.
Now, the integervalued first order autoregressive, INAR(1), model is defined by the
Equation 39. A discrete time stochastic process, ????, is called an INAR(1) process
if it satisfies the equation:
??=?????1 +??
Equation 39
d
=
d
=
M.Mohammadipour, 2009, Chapter 3 48
where ??[0,1] and {??} is a sequence of i.i.d. nonnegative integervalued random
variables, independent of ?? with mean ?? and finite variance ??
2. ?? and ???1 are
assumed to be stochastically independent for all points in time. The process obtained
by the Equation 39 is stationary and it resembles the Gaussian AR(1) process except
that it is nonlinear due to the thinning operation replacing the scalar multiplication in
continuous models. It should be noted that, in Equation 39, subsequent thinning
operations are performed independently of each other.
Equation 39 shows that, based on the definition of the thinning operation, unlike the
DAR(1) model, the memory of an INAR(1) model decays exponentially (AlOsh and
Alzaid, 1987).
The two independence limitations we have assumed so far ? independence of {??} in
the thinning operation, and independence of ?? and ???1? have been relaxed in a
study by Br?nn?s and Hellstr?m (2001).
It is worth mentioning that the probability ? is assumed to be constant here. Alzaid
and AlOsh (1993) develop a model in which this probability of retaining an element
is not constant. Also, Zheng et al. (2007) develop a random coefficient model where
?? are i.i.d. random variables that can take values in the interval [0,1).
A realization of ?? in an INAR(1) model of Equation 39 has two components: (i) the
survivors of elements of the process at time (??1), ???1, each with probability of
survival ? and (ii) the innovation term, ??, which represents new entrants to the
system in the interval (??1,?].
The mean and variance of the process {??} are:
?????=?
??(?0) +??? ?
?
??1
?=0
Equation 310
var????=?
2?var(?0) +?1???? ?
2??1
?
?=1
???????+??
2? ?2(??1)
?
?=1
Equation 311
It is shown by AlOsh and Alzaid (1987) that the autocorrelation function (ACF) of
M.Mohammadipour, 2009, Chapter 3 49
this process is given by ??=?
? for ?= 1,2,?. This is identical to the ACF of a
linear Gaussian AR(1) process. The only difference is that the ACF of the INAR(1)
model is always positive.
The derivation of the first and second order unconditional moments of the INAR(1)
process is straightforward:
?????=??/(1??)
Equation 312
var????= (???+??
2)/(1??2)
Equation 313
It can be seen that both the first order (regression function) and the second order
conditional moments are linear in ???1.
???????1?=????1 +??
Equation 314
var??????1?=?(1??)???1 +??
2
Equation 315
Another similarity between the model of Equation 39 and the Gaussian AR(1) is that
the distribution of the innovation term (??) plays the same role in determining the
distribution of ?? that the normal distribution of the shock term plays in the AR(1)
model. In fact, AlOsh and Alzaid (1987) argue that the distribution of ?? is uniquely
determined by the distribution of ??.
In Equation 39, {??} can have any nonnegative discrete distribution. A natural
first choice of interest for these variables is Poisson. AlOsh and Alzaid (1987)
show that if ??~??(?), the marginal distribution of the process ?? is also Poisson
??~??(?/(1??)). In this case the model is called PoINAR(1) (Jung and
Tremayne, 2006b) (or PAR(1) as suggested, for example, by Freeland and McCabe
(2004b)). Hence, the role of the Poisson distribution in the INAR(1) process is
analogous to the role of the Gaussian distribution in the AR(1) process.
The INAR(1) process is a member of a class of models introduced by Grunwald et al.
(2000). They suggest that nearly all the nonGaussian AR(1) models are in fact a part
of a class of conditional linear AR(1) models, CLAR(1).
M.Mohammadipour, 2009, Chapter 3 50
If ????,?= 0,1,? is a timehomogeneous firstorder Markov process on a sample
space ???, then it is said to have CLAR(1) structure if it satisfies:
?????1?=????1 +?
where ?????1?=?(?????1) and ? and ? are real numbers. Grunwald et al. (2000)
show that stochastic properties of CLAR(1) models are similar to those of the
Gaussian AR(1) model.
3.3.2 INAR(2) Model
To take into account higher order dependence in the data, higher order INAR models
are developed. In this section, the second order model, INAR(2), will be reviewed
briefly.
A discrete time stochastic process, {??} is called an INAR(2) process if it satisfies the
equation:
??=?1????1 +?2????2 +??
Equation 316
with all the previously mentioned definitions, except for the thinning mechanism.
There are two approaches regarding the binomial thinning mechanism.
The first approach is proposed by Alzaid and AlOsh (1990) and the other by Du and
Li (1991). The corresponding processes will henceforth be denoted by INAR(2)AA
and INAR(2)DL, respectively, as in Jung and Tremayne (2006b).
In the INAR(2)AA process, the random variables ?1????2 and ?2????2 which are
elements of ???1 and ??, are connected in a powerful way. In fact, ??1?? and ??2??
and ???2 are dependent although they appear in different times. The vector (?1?
??,?2???) given ??=?? is multinomial with parameters (?1,?2,??). To understand
this structure, consider the simulated process: at time ?, ?? is observed and from the
previous time period we have ???1 and we have formed ?1????1, and ?2????1.
Now, we form ??=?1??? and ??=?2???. Then we have ??+1 =??+???1 +
??+1 and ?? is available to obtain ??+2.
M.Mohammadipour, 2009, Chapter 3 51
This dependence results in two important consequences: (i) the process maintains its
physical interpretation in terms of the counts evolving as a birth and death process,
and (ii) a moving average structure is included into the process in such a way that the
ACF of the INAR(2)AA process mimics that of a Gaussian ARMA(2,1) process
(Alzaid and AlOsh, 1990).
The stationarity condition for the INAR(2)AA process is ?1 +?2 < 1 and,
assuming
??~??(?), the marginal distribution of the process ?? is ??~??(?/(1?
?1??2)). However, the conditional mean function for this model is not linear,
which shows that this process is not a member of the CLAR class.
The ACF of the INAR(2)AA process is given by:
??=?1???1 +?2???2 for ??2
Equation 317
where the starting values are ?0 = 1 and ?1 =?1 (Jung and Tremayne, 2006b).
The other specification of the INAR(2) model is that of Du and Li (1991). The
concept of their model is closer to the Gaussian higher order AR models in which ??
is obtained by a direct multiplication of the constants ?1 and ?2 to ???1 and ???2,
independent of all previous stochastic structures. It means that at time ? we have
already observed ???1 and ???2 and the thinning operations are applied independently
of the previous period which result in: ??= (?1????1) + (?2????2) +??. Du and
Li (1991) show that the unconditional mean of the process is again ??/(1??1?
?2) and the stationarity condition remains the same. However, they show that the
correlation properties of their model are identical to those of the Gaussian AR(2)
model which is one of the main differences that distinguishes INAR(2)DL from
INAR(2)AA. An additional difference is that the conditional mean of the ?? in
INAR(2)DL is given by:
?(?????1,???2,??=?1???1 +?2???2 +??
Equation 318
which obviously is linear, while that of the INAR(2)AA process is nonlinear.
Therefore, the INAR(2)DL model has a CLAR(2) structure. It should also be noted
that, even with Poisson innovations, the marginal distribution of ?? is not Poisson,
M.Mohammadipour, 2009, Chapter 3 52
which is in contrast with the INAR(2)AA model.
Because the specification by Du and Li (1991) is similar to the conventional AR(2)
model and we do not use the physical interpretation of the INAR models (as a birth
and death process which is maintained in the specification by Alzaid and AlOsh
(1990)), we use the Du and Li approach in this study.
3.3.3 INAR(p) Model
Realizations of some counting process {??} might be attributed not only to its
immediate predecessors {???1} and {???2} as in INAR(2), but also to previous
realizations of the process, {????}?=3
?
. The pth order integervalued autoregressive,
INAR(p), process is defined by Alzaid and AlOsh (1990) as follows:
??=?1????1 +?2????2 +?+???????+??
Equation 319
with all the previously mentioned definitions. {??} are nonnegative constants such
that the process remains stationary and ?1,?,???1 ?[0,1] and ???(0,1]. The
stationarity condition for the INAR(p) process is that the roots of the equation
????1?
??1??????1????= 0 lie inside the unit circle (Alzaid and AlOsh,
1990).
Like the INAR(2), there are two approaches concerning the binomial thinning
mechanisms. The first model is that of Alzaid and AlOsh (1990) which will be
denoted by INAR(p)AA and the other is the model of Du and Li (1991), known as
INAR(p)DL. The idea behind these approaches is the same as that explained for
INAR(2). The INAR(p)AA process shares the same correlation properties with the
Gaussian ARMA(p,p1) process (Alzaid and AlOsh, 1990), while the INAR(p)DL
mimics the AR(p) process (Du and Li, 1991).
The unconditional first moment of the INAR(p)AA process is given by:
?????=? ???????
?
?=1
+??
Equation 320
M.Mohammadipour, 2009, Chapter 3 53
The conditional moments of ?? are given by:
???????1,?,?????=? ?????
?
?=1
+??
var??????1,?,?????=? ??(1???)????
?
?=1
+??
2
Assuming ??~??(?), the marginal distribution of the process ?? would be
??~??(?/(1? ??
?
?=1 )).
The autocovariance at lag ? of the INAR(p)AA process satisfies the equation:
??=? ?????
?
?=1
+ ? ????,??
?
?=?+1
+??(0)??
2
Equation 321
where ????,???cov?????+?,?????????? and ???0?= 1 if ?= 0 and zero
otherwise. The autocovariance of the INAR(p)AA has the same form of that of the
Gaussian ARMA(p,p1).
As for the INAR(2)DL, when the approach of Du and Li (1991) is taken, the
marginal distribution of ?? is not the same as the distribution of innovations. The
autocovariance function of the INAR(p)DL satisfies:
??=?1???1 +?2???2 +?+??????
Equation 322
Therefore, the ACF of this process is found from equations of the form:
??=?1???1 +?2???2 +?+??????
Equation 323
which implies that the correlation structures of INAR(p) and AR(p) processes are the
same. This makes these models similar to the standard AR(p) models not only in
form, but also in stationarity conditions and correlation structure.
M.Mohammadipour, 2009, Chapter 3 54
3.3.4 INMA(1) Model
Having introduced integer autoregressive models for count series, AlOsh and Alzaid
(1988) then developed a class of models for integervalued moving average (INMA)
processes. In INMA models, a stationary sequence of random variables {??} is
formed from a sequence {??} of i.i.d. random variables which are nonnegative and
also integervalued. The first order model, which we are going to describe in this
section, is the case in which adjacent members of the sequence are correlated. A
process {??} is called an INMA(1) process if it satisfies the equation:
??=?????1 +??
Equation 324
where ??[0,1] and {??} are as before and the thinning operation is defined via:
???=? ??
Z
?=1
Equation 325
where {??} is a sequence of i.i.d. Bernoulli random variables, independent of ? and
satisfying:
????= 1?= 1??(??= 0) =?
Equation 326
The INMA(1) model defined by Equation 324 is similar to the Gaussian MA(1)
process except that scalar multiplication is replaced by the thinning operation. Jung
and Tremayne (2006a) present a physical interpretation of this model as follows. If
we consider ?? as the number of particles in a welldefined space at time point ?, it
can be assumed that this number is made of two components: (i) particles entering
during (??1,?], and (ii) survivors of those who entered the space during (??2,??
1]. Therefore, the thinning at time ?, is applied to only immigrants at time ??1, not
all particles in space, as in an INAR(1) process. Examples of this process include the
number of patients staying in a hospital or the number of customers in a department
store (AlOsh and Alzaid, 1988).
It can be inferred from the Equation 324 that each element stays in the system no
longer than two periods. This is in contrast to the INAR(1) process in which there is
M.Mohammadipour, 2009, Chapter 3 55
no limit on the survival of elements in the system.
The unconditional first and second moments of the INMA(1) process are:
?????= (1 +?)??
Equation 327
var????=??1?????+ (1 +?
2)??
2
Equation 328
It is shown by AlOsh and Alzaid (1988) that the autocorrelation function (ACF) of
this process is given by:
??
INMA (1)
=?
???
2
??1?????+ (1 +?2)??
2 for ?= 1
0 for ?> 1
?
Equation 329
which is analogous (but not identical) to that of the Gaussian MA(1) process, where
??=????1 +?? and the ACF is given by:
??
MA (1)
=?
1 for ?= 0
?
1 +?2
for ?= ?1
0 for ??> 1
?
Equation 330
Another property of the INMA(1) process which is similar to MA(1) is that if ?= 0,
the sequence {??} becomes a sequence of i.i.d. random variables with the distribution
of ??. Also, if ?= 1 the process will have the highest ?1 which again agrees with the
MA(1) process.
As for the INAR(1) process, a natural candidate for the marginal distribution of an
INMA(1) process is the Poisson distribution. It is shown by AlOsh and Alzaid
(1988) that assuming ??~??(?), the marginal distribution of the process ?? would be
??~??(?(1 +?)). This process is referred to as a PoINMA(1) process (Jung and
Tremayne, 2006a).
M.Mohammadipour, 2009, Chapter 3 56
3.3.5 INMA(2) Model
The secondorder moving average process is an extension of the INMA(1) process
introduced in the previous section. The model is given by:
??=?1????1 +?2????2 +??
Equation 331
where both parameters ?1 and ?2 lie in the interval [0,1]. The individual thinning
operations ??????? for ?= 1,2 follow the Equation 325 and it is assumed that they
perform independently of each other. Two approaches arise regarding the thinning
mechanisms, as in higher order autoregressive models.
One approach is proposed by AlOsh and Alzaid (1988), assuming dependence
between the thinnings of terms ?1??? and ?2??? (similar to that of INAR(2)AA).
The unconditional expected value and variance of such a process, henceforth called
INMA(2)AA, are given by:
?(??) =??(1 +?1 +?2)
Equation 332
var????=??? ???1????
2
?=1
+??
2? ??
2
2
?=0
Equation 333
with ?0 = 1. The ACF of the INMA(2)AA process is:
??=?
? [??(?????+?)??
2??
?=0 +????+???
2]
??? ??(1???)
2
?=1 +??
2? ??
22
?=0
for ?= 1,2
0 for ?> 2
?
Equation 334
It can be seen that the cutoff property of the INMA(2)AA process is the same as
that of the Gaussian MA(2) process.
The other approach concerning the thinning operation in an INMA(2) process is
introduced by McKenzie (1988). In an INMA(2)MK process, it is assumed that the
individual thinning operations ??????? for ?= 1,2 are performed independently not
only from each other, but also from corresponding operations at previous times in
M.Mohammadipour, 2009, Chapter 3 57
Equation 331. The unconditional moments of this process with Poisson innovations
are the same as INMA(2)AA, ?????= var????=?(1 +?1 +?2), and the ACF,
which is again the same as that of the Gaussian MA(2) process, is given by:
??=?
? ????+?
2??
?=0
1 +?1 +?2
for ?= 1,2
0 for ?> 2
?
Equation 335
Unlike INAR(2)DL, for an INMA(2)MK process, if ??~??(?), then
??~??(?(1 +?1 +?2)) which is the same as INMA(2)AA. In this PhD thesis, we
adopt the approach by McKenzie (1988) because his model is more similar to the
classic MA(q) model (Br?nn?s and Hall, 2001).
3.3.6 INMA(q) Model
The ?th order integer moving average model, introduced by AlOsh and Alzaid
(1988) and McKenzie (1988) is defined by:
??=?1????1 +?2????2 +?+???????+??
Equation 336
where {??} is defined as before and the parameters ?1,?,???1 ?[0,1] and ???(0,1].
Using the properties of the thinning operation, it is shown by Br?nn?s and Hall
(2001) that:
?(??) =??(1 +? ??
?
?=1
)
Equation 337
var????=??
2 +? [??
2??
2 +????(1???)]
?
?=1
Equation 338
As for an INMA(2) process, there are two approaches based on the thinning
mechanisms. The ACF of the INMA(q)AA process is given by:
M.Mohammadipour, 2009, Chapter 3 58
??=?
? [??(?????+?)??
???
?=0 +????+???
2]
??? ??(1???)
?
?=1 +??
2? ??
2?
?=0
for ?= 1,?,?
0 for ?>?
?
Equation 339
On the other hand, the ACF of the INMA(q)MK process is:
??=?
? ????+?
???
?=0
? ??
?
?=0
for ?= 1,?,?
0 for ?>?
?
Equation 340
It can be seen that the autocorrelation function of an INMA(q) process is analogous
to than of the classical MA(q) process. The difference is that all autocorrelations are
positive (Br?nn?s and Hall, 2001).
3.3.7 INARMA(1,1) Model
Having introduced INAR and INMA processes, Alzaid and AlOsh (1990) suggested
that these two processes can be mixed in a manner similar to that of the standard
ARMA processes to provide the mixed integer autoregressive moving average class
of models. There are two approaches regarding the modelling of this kind of
processes.
The first approach was introduced by McKenzie (1988) for INARMA processes with
Poisson marginal distributions. He suggests the mixed process should be constructed
by coupling the two AR and MA processes and a common innovation process.
According to this viewpoint, the AR component of the INARMA(1,1) process is
given by:
??=?????1 +??
Equation 341
and the MA component is:
??=???1 +????
Equation 342
M.Mohammadipour, 2009, Chapter 3 59
where all the thinning operations are independent, ?,??[0,1] and {??} is a
sequence of i.i.d. Poisson variables.
The second approach suggested by Neal and Rao (2007) is what we follow because it
is similar to the Gaussian ARMA process. A discrete time stochastic process, {??}, is
called an INARMA(1,1) process if it satisfies the equation:
??=?????1 +??+?????1
Equation ?343
where ?,??[0,1] and {??} is a sequence of i.i.d. nonnegative integervalued
random variables, independent of ?? with mean ?? and finite variance ??
2. Here, the
two thinning operations are independent of each other and also of the corresponding
operations at previous times, and are defined as follows:
???=? ??
?
?=1
Equation 344
???=? ??
?
?=1
Equation 345
To ensure the stationarity and invertibility of the above INARMA(1,1) process given
by Equation 343, the two conditions of ?< 1 and ?< 1 must hold.
The unconditional first and second moments of this process are:
?(??) =?
1 +?
1??
???
Equation 346
var(??) =
1
1??2
???+??+???2???+ (1 +?
2 + 2??)??
2?
Equation 347
The autocorrelation function (ACF) of this process is given by (see Appendix 3.A for
the proof):
M.Mohammadipour, 2009, Chapter 3 60
??
INARMA ?1,1?=
?
??2 +?2?+?????2???+ (?+??
2 +?2?+?)??
2
??+??+???2???+ (1 +?2 + 2??)??
2 for ?= 1
????1 for ?> 1
?
Equation 348
It can be seen that the ACF of an INARMA(1,1) dies exponentially, which is
analogous to the ACF of the Gaussian ARMA(1,1) which is as follows (for ??= 0
and ??
2 = 1):
??
ARMA (1,1)
=?
?+?+?2?+??2
1 +?2 + 2??
for ?= 1
????1 for ?> 1
?
Equation 349
3.3.8 INARMA(p,q) Model
The INARMA(p,q) process is given by the following difference equation:
??=? ???????
?
?=1
+??+? ???????
?
?=1
Equation 350
where ??,???[0,1] and {??} is as before and thinning operations are performed
independently of each other and also of the corresponding operations at previous
times.
The stationarity conditions of this process are the same as those of an INAR(p)
process: to ensure that the above process is stationary, it is required that
(?1,?2,?,??) are such that the roots of the porder polynomial ?
???1?
??1???
???1????= 0 lie inside the unit circle.
Neal and Rao (2007) discuss the invertibility conditions for an INARMA(p,q)
process for the moving average parameters (?1,?2,?,??). They assume that these
conditions are the same as the those of an MA(q) process. However, they have not
provided any proof in this regard and left it as an open question to investigate if this
condition is sufficient for an INARMA(p,q) process to be invertible.
M.Mohammadipour, 2009, Chapter 3 61
3.3.8.1 First and Second Unconditional Moments
As mentioned before, the stochastic properties, including the autocorrelation
function, of the general INARMA(p,q) process have not been found in the literature.
Therefore, to answer the first research question ?How can the appropriate integer
autoregressive moving average (INARMA) model be identified for a time series of
counts??, here we investigate these properties.
Obtaining the first unconditional moment of the INARMA process of Equation 350
is straightforward. It is given by:
?(??) =?
1 +? ??
?
?=1
1?? ??
?
?=1
???
Equation 351
However, the derivation of the second unconditional moment is more challenging.
We have found the unconditional variance of the INARMA process of Equation 350
as follows (see Appendix 3.B for the proof):
var????=
??
1?? ??
2?
?=1
?
1 +? ??
?
?=1
1?? ??
?
?=1
? ??1???
?
?=1
+? ??1???
?
?=1
?
+
??
2
1?? ??
2?
?=1
?1 +? ??
2
?
?=1
+ 2 ? ???
min (?,?)
?=1
+
2? ? ???+???
???
?=1
??1
?=1 + 2? ? ???????
???
?=?+1
min (?,?)
?=1
1?? ??
2?
?=1
Equation 352
where ??
?? is the crosscovariance function derived in Appendix 3.C and ?? is the
autocovariance at lag ? which can be expressed in terms of ?0 (or var????) from the
equations obtained in the next section.
It can be seen that if ?= 0, for an INAR(p) process, the unconditional variance
would be:
var????=
1
1?? ??
2?
?=1
??
? ??1???
?
?=1
1?? ??
?
?=1
???+??
2 + 2? ? ???+???
???
?=1
??1
?=1
?
Equation 353
M.Mohammadipour, 2009, Chapter 3 62
In the above equation, ?? can be expressed in terms of ?0 based on the YuleWalker
equations of Equation 412. For example, for an INAR(2) process we have:
?1 =?1?0 +?2?1
and as a result:
?1 =
?1
1??2
?0
The unconditional variance of an INAR(2) process can then be found from Equation
353 to be:
var????=
??1??1
2 +?2?2?2
2??1?2 +?1
2?2 +?2
3???+ (1??1?2?2 +?1?2 +?2
2)??
2
1??1??1
2 +?1
3?2?2 + 2?2
3??2
4 +?1?2 +?1?2
2??1?2
3 +?1
2?2
It can be seen from Equation 353 that the unconditional mean and variance of a
PoINAR(p) process are not equal when the distribution of the innovations is Poisson.
This has been also confirmed in the literature (Bu and McCabe, 2008).
Also, when ?= 0, for an INMA(q) process, the variance would be:
var????=?? ??1???
?
?=1
???+?1 +? ??
2
?
?=1
???
2
Equation 354
which agrees with the result found by Br?nn?s and Hall (2001) (Equation 338).
Equation 352 is the first new result found in this PhD study.
3.3.8.2 Autocorrelation Function (ACF)
The next step is finding the autocorrelation function of an INARMA(p,q) process. In
order to do so, first, we need to find the covariance of this process. The covariance of
INARMA(p,q) at lag ? is:
??= cov(??,????)
According to the relation between ? and ? and ?, there are four cases:
M.Mohammadipour, 2009, Chapter 3 63
? ???, ???
? ???, ?>?
? ?>?, ???
? ?>?, ?>?
Each of these cases will be considered in sequence.
1. If ???, ??? (? can be only equal to either ? or ? if ???)
Figure ?31 The covariance at lag , , when
qkpk ?? ,
??= cov???,?????
= cov???1????1 +?+???????+??+?1????1 +?+????????,?????
= cov???1????1?,?????+?+ cov??????????,?????
+cov??????????,?????+ cov????+1??????1?,?????+?
+ cov?????1?????+1?,?????+ cov??????????,?????
??=?1???1 +?+???1?1 +??var??????+??+1?1 +?+??????
+???0
??+??+1?1
??+?+???1????1???
?? +??????
??
where ??
?? is the crosscovariance between ? and ? at lag ? (see Appendix 3.C).
Then, the above equation can be written as:
k
k?
? ?????
??1
?=1 ? ?????
?
?=?+1
??var(????)
? ? 0
? ? 0
? ???????
2?
?=?+1
????
2
?
?
M.Mohammadipour, 2009, Chapter 3 64
??= ? ?????
??1
?=1
+??var??????+ ? ????
?
?=?+1
+???Z
2 + ? ??????
??
?
?=?+1
Note that if 2???, there will be a ?? in ? ????
?
?=?+1 which has to be considered. This
is because when ?= 2?, we have ?2??2???=?2???.
2. If ???, ?>?
Figure ?32 The covariance at lag at lag , , when
qk pk ?? ,
??= cov???,?????
= cov???1????1 +?+???????+??+?1????1 +?+????????,?????
=?1???1 +?2???2 +?+???1?1 +??var??????+??+1?1 +??+2?2 +?+??????
Therefore, for the second case, the autocovariance at lag ? can be obtained from:
??= ? ?????
??1
?=1
+??var??????+ ? ????
?
?=?+1
Again, if 2???, there will be a ?? in ? ????
?
?=?+1 which has to be considered.
k
k?
? ?????
??1
?=1 ? ?????
?
?=?+1
? ? 0
??var(????)
? ? 0
?
?
M.Mohammadipour, 2009, Chapter 3 65
3. If ?>?, ???
Figure ?33 The covariance at lag , , when
qk pk ?? ,
??= cov???,?????
= cov???1????1 +?+???????+??+?1????1 +?+????????,?????
=?1???1 +?2???2 +?+??????+???0
??+??+1?1
??+?+???1????1???
?? +??????
??
Therefore, the autocovariance at lag ? for the third case can be obtained from:
??=? ?????
?
?=1
+???Z
2 + ? ??????
??
?
?=?+1
4. If ?>?, ?>?
Figure ?34 The covariance at lag , , when
qkpk ?? ,
k
k?
k
k?
? ? 0
?
0 ? ?
? ?????
?
?=1
?
0 ? ?
? ?????
?
?=1
? ? 0
? ???????
2?
?=?+1
????
2
?
?
M.Mohammadipour, 2009, Chapter 3 66
??= cov???,?????
= cov???1????1 +?+???????+??+?1????1 +?+????????,?????
=?1???1 +?2???2 +?+??????=? ?????
?
?=1
Finally, the autocovariance at lag ? for the fourth case is given by:
??=? ?????
?
?=1
Therefore, we can write the autocorrelation function of an INARMA(p,q) process as
follows:
??
=
?
?
?
?
?
?
?
?
?
?
?
?
? ?????
??1
?=1 +?????+? ?????
2??1
?=?+1 +? ?????
?
?=2?+1 +????
2 +? ?????
???
?=?+1
?1??2?????
for 2???
? ?????
??1
?=1 +?????+? ?????
?
?=?+1 +????
2 +? ?????
???
?=?+1
???
for 2?>?
????,???
?
?
?
? ?????
??1
?=1 +?????+? ?????
2??1
?=?+1 +? ?????
?
?=2?+1
?1??2?????
for 2???
? ?????
??1
?=1 +?????+? ?????
?
?=?+1
???
for 2?>?
? ???,?>?
?
Equation 355
where ??? is the second unconditional moment of the process given by the Equation
352 and ??
?? is the crosscovariance given by the Equation 3.C1 (see Appendix
3.C). The other two cases of ?>?,??? and ?>?,?>? are special cases of the
above expressions.
The above equation can be simply written as:
??=?
?1???1 +?2???2 +?+??????+???0
??+??+1?1
??+?+??????
??
?0
???
?1???1 +?2???2 +?+?????? ?>?
?
Equation 356
Equation 356 is another new result of this research. Identification of the order of an
INARMA(p,q) process requires both the autocorrelation function (ACF) and the
partial autocorrelation function (PACF) of the process. The structure of the PACF of
an INARMA(p,q) process will be discussed in chapter 4.
M.Mohammadipour, 2009, Chapter 3 67
In order to test the Equation 356, we check if it results in the correct ACF for
INAR(p) and INMA(q) processes. When ?= 0, i.e. for an INAR(p) process, it can
be seen that the ACF based on the Equation 356 will be:
??=?1???1 +?2???2 +?+??????
which agrees with the result found by Du and Li (1991).
When ?= 0, i.e. for an INMA(q) process, it can be seen that the ACF based on
Equation 356 will be:
??=?
????
2 +? ?????
???
?=?+1
?0
???
0 ?>?
?
where ???=?? ??(1???)
?
?=1 ???+?1 +? ??
2?
?=1 ???
2. For an INMA(q) process
with Poisson marginal distribution (??=??
2 =?), the variance is given by:
???=?1 +? ??
?
?=1
??
It can be seen from Equation 3.C1 that for an INMA(q) process ??
??=??? for
0????, so we have:
??=?
??+? ?????
?
?=?+1
1 +? ??
?
?=1
???
0 ?>?
?
which agrees with the result in the literature (Br?nn?s and Hall, 2001).
3.3.9 Applications of INARMA Models
Applications of INAR processes in the medical sciences can be found in, for
example, Franke and Seligmann (1993) and Cardinal et al. (1999); and applications
to economics in, for example, B?ckenholt (1999), Berglund and Br?nn?s (2001),
Br?nn?s and Hellstr?m (2001), Rudholm (2001) and Freeland and McCabe (2004b).
M.Mohammadipour, 2009, Chapter 3 68
Br?nn?s (1995) studies the consequences and required adaptations when explanatory
variables are included in the INAR(1) model. He obtains new conditional least
squares (CLS) and generalized method of moments estimators for the model
containing explanatory variables and applies the INAR(1) model with Poisson
marginal distribution to the number of Swedish mechanical paper and pulp mills
during 19211981.
INAR(1) models also have applications in inventory control. Aggoun et al. (1997)
use an INAR(1) inventory model for perishable items, i.e. each item in the stock
perishes in a given time period with an unknown probability. The sequence of these
probabilities is assumed to be a homogeneous Markov chain and the paper finds the
conditional probability distribution of this sequence and estimates the transition
probabilities of the Markov chain. However, the model is not applied to realworld
data.
Cardinal et al. (1999) represent infectious disease incidence time series by INAR
models. They state that realvalued time series models have been used in the analysis
of infectious disease surveillance data, but argue that these models are not suitable in
some cases such as the analysis of a rare disease. Meningococcal infection is
considered as a rare disease in their study and the integervalued INAR(5) model is
fitted to the data set.
B?ckenholt (1999) introduces the application of INAR models in investigating
regularity and predictability of purchase behaviour over time. He uses a PoINAR(1)
model for the analysis of longitudinal purchase data because there is a notion that
purchase behaviour of nondurable goods is welldescribed by a Poisson process. The
population of consumers is then divided to an unknown number of mutually
exclusive and exhaustive segments, and within each a PoINAR(1) process is used to
model the counts. The mixed PoINAR(1) model is finally applied to a powder
detergent purchase data set of about 5000 households.
Another straightforward application of INAR models can be found in a paper by
Berglund and Br?nn?s (2001) in which they study the entry and exit of plants in
Swedish municipalities as an INAR(1) process. In their model, they incorporate the
variables affecting survival and entry and employ generalized method of moments
M.Mohammadipour, 2009, Chapter 3 69
for estimation.
Br?nn?s and Hellstr?m (2001) apply the INAR(1) model to the number of Swedish
mechanical paper and pulp mills. They consider the number of firms in a region at a
certain time to be equal to the number of firms surviving from the previous time
period plus the number of new firms. They assume that the survival of a firm
depends on the survival of other firms (dependent exits) and there is dependence
between the survival of a firm and entry of a new firm (dependent entry and exit).
The results for the set of data show that the correlation between exits is not
significant, but that of entry and exit is significant. In another Swedish application,
Rudholm (2001) analyses the factors affecting entry into the Swedish
pharmaceuticals market using an INAR(1) model.
Br?nn?s et al. (2002) find another application of INAR models in forecasting hotel
guest nights which conventionally is based on economic demand models, pure time
series analytical models, or on a mixture of them. They suggest that the daily number
of guest nights for a specific hotel follows an INAR(1) process, and then proceed by
crosssectional and temporal aggregation of the model. The former means
aggregation over more than two hotels which yields an INAR(1) model, and the
latter means aggregation over time that results in an INARMA(1,1) model, later
simplified as an INMA(1) model.
Karlis (2002) introduces an INAR(1) model with a general mixed Poisson
distribution for the innovation term which allows for overdispersion. The model is
then applied to the number of forest fires in Greece in a twomonth period (daily
observations).
Blundell et al. (2002) apply a Linear Feedback Model (LFM), which is derived from
the INAR process, to the panel data of Hall et al. (1986). In their model, the
technological output of a firm is a function of the corresponding R&D investment in
current and previous periods, some unknown technology parameters, and the firm
specific propensity to patent. As another application, Gourieroux and Jasiak (2004)
use an INAR(1) model to update premiums in car insurance and compare it to the
standard negative binomial approach.
Freeland and McCabe (2004b) use a PoINAR model for a monthly count data set of
M.Mohammadipour, 2009, Chapter 3 70
claimants for wage loss benefit. They propose a method for producing coherent
forecasts based on the conditional median rather than the conventional conditional
mean. It is also argued that when the counts are low, the median should be
accompanied by estimates of the probabilities associated with the point masses of the
?stepahead conditional distribution. A method for calculating confidence intervals
for these probabilities is also presented.
Quddus (2008) uses a PoINAR(1) model for analysis of traffic accidents in Great
Britain. The results of his study show that an ARIMA model performs better than
INAR(1) for geographically and temporally aggregated time series. However, as
expected, the reverse is true for disaggregated low count time series (see section
3.3.10 for aggregation in INARMA models).
It can be seen that INARMA models have been generally used for time series of
counts (counts of objects, events, or individuals). Although this application area is
based on the direct physical interpretation of the INARMA models, as suggested in
the literature (McKenzie, 2003), this should not restrict these models to only such
applications. This research is an attempt to use INARMA models beyond their direct
interpretation to model intermittent demand.
3.3.10 Aggregation in INARMA Models
Time series aggregation is a widely discussed subject for continuousvalued time
series. It goes back over 50 years (Quenouille, 1958) and since then many papers
have considered different aspects of aggregation in continuousvalued time series
(see for example: Amemiya and Wu (1972), Brewer (1973), Harvey and Pierse
(1984), Nijman and Palm (1990), Drost and Nijman (1993), Marcellino (1999), Teles
and Wei (2002), and Man (2004)).
Three types of aggregation have been identified in the literature which can be
classified as:
a. temporal aggregation
b. crosssectional aggregation
c. over a forecast horizon aggregation
M.Mohammadipour, 2009, Chapter 3 71
Temporal aggregation, also called flow scheme, refers to aggregation in which a low
frequency time series (e.g. annual) is achieved from a high frequency time series
(e.g. quarterly or monthly). It means that the low frequency variable is the sum of k
consecutive periods of the high frequency variable. For example, the annual
observations are the sum of the monthly observations every twelve periods. For
Gaussian models, it has been proved that the aggregation of an AR(p) process
produces an ARMA(p,q) process where ???.
Jung and Tremayne (2003) provide a modified statistic based on the meanvariance
equality property of the Poisson distribution. The modified statistic is given by:
??=??
? ????1???(????)
?
?=2
? (????)2
?
?=1
Equation 43
which is asymptotically equivalent to the Sstatistic under the null hypothesis. Here,
again the null hypothesis is rejected if ??>??.
4.2.3 Portmanteautype Tests
Jung and Tremayne (2003) use two portmanteautype tests originally designed by
Venkataraman (1982) and Mills and Seneta (1989) to measure the goodnessoffit in
branching processes as tests for independence in a time series of counts.
M.Mohammadipour, 2009, Chapter 4 89
Under the null hypothesis (?0) of no serial dependence or i.i.d. random variables, the
modified version of the statistic presented by Venkataraman (1982) is:
???????=
? ??+1
2?
?=1 ?? (????)
2?
?=1 ?
2
? ??????2(?????1??)2
?
?=?+2
Equation 44
where ??=? ??????(??????)
?
?=?+1 /? (????)
2?
?=1 and ??1 is an arbitrary
integer (in their MonteCarlo study, Jung and Tremayne (2003) use ?= 1, 5, 10).
Under the null hypothesis of i.i.d. Poisson variables {??}, ????(?)
?
??2(?) as
???.
The second test is an adapted version of the test presented by Mills and Seneta
(1989) with the statistic:
????????=
? ??+1
2?
?=1 ?? (????)
2?
?=1 ?
2
? ??????2(?????1??)2
?
?=?+2
Equation 45
It can be easily seen that this is exactly the same as ????(?) except that ??+1 is
replaced by ??+1. Here, ?? is the kth order sample partial autocorrelation and ? is as
defined before. Similarly, ?????(?)
?
??2(?) as ???.
It should be noted that the firstorder lag sample correlations are ignored in both
statistics. These two tests can be used to distinguish between INAR(1) and INMA(1)
structures.
Two portmanteau tests used in the ARMA literature to find if the data has any serial
dependence are the BoxPierce and the LjungBox tests. The latter is an enhancement
of the former to improve the performance of the test for small sample sizes (Ljung
and Box, 1978). The LjungBox statistic is given by:
??=???+ 2??
??
2
???
?
?=1
Equation 46
where ? is the number of observations and ?? is the sample autocorrelation at lag ?.
A large value of ?? indicates that the model is inadequate.
M.Mohammadipour, 2009, Chapter 4 90
4.3 Identification based on ACF and PACF
The sample autocorrelation function (SACF) and sample partial autocorrelation
function (SPACF) have been widely used in the literature for identification of the
autoregressive and moving average order of the INARMA models (Latour, 1998;
Jung and Tremayne, 2006a; Zheng et al., 2006; Zhu and Joe, 2006; Bu and McCabe,
2008). In this section, we review these functions for different INARMA models.
4.3.1 Autocorrelation Function (ACF)
The Autocorrelation Function (ACF) is defined as a plot of the autocorrelations at lag
? versus the lag ?. In this section, we investigate the autocorrelation function of an
INARMA(p,q) process. First, we recall that the ACF of an INAR(p) process of:
??=?1????1 +?2????2 +?+???????+??
is determined by Du and Li (1991) as:
??=?1???1 +?2???2 +?+??????
Equation 47
It can be seen that the correlation structures of INAR(p) and AR(p) processes are the
same. For an INMA(q) process of:
??=??+?1????1 +?2????2 +?+???????
it is shown by Br?nn?s and Hall (2001) that the ACF for an INMA(q) process with
Poisson marginal distribution is given by:
??=?
? ????+?
???
?=0
? ??
?
?=0
for ?= 1,?,?
0 for ?>?
?
Equation 48
Again, it can be seen that this is analogous to the ACF of an MA(q) process.
In chapter 3, we showed that the autocorrelation function of an INARMA(p,q)
process is as follows:
M.Mohammadipour, 2009, Chapter 4 91
??=?
?1???1 +?2???2 +?+??????+???0
??+??+1?1
??+?+??????
??
?0
???
?1???1 +?2???2 +?+?????? ?>?
?
Equation 49
When ???+ 1, the autocorrelation is:
??=?1???1 +?2???2 +?+?????? for ???+ 1
Equation 410
Therefore, the ACF of an INARMA(p,q) process is analogous to that of an
ARMA(p,q) process and it can be used in identifying the integervalued time series.
4.3.2 Partial Correlation Function (PACF)
The correlation between two variables can be used as a measure of interdependence.
However, when a variable is correlated with a second variable, this may be due to the
fact that they both are correlated with another variable(s). Therefore, it may be of
interest to examine the correlations between variables when other variables are held
constant. These are called partial correlations (Hamilton, 1994).
PACF is a device to identify the autoregressive order of a stationary time series. It
has been shown that, although an AR(p) process has an infinite ACF, it can be
described in terms of p nonzero functions of the autocorrelations (Box et al., 1994).
In this section, we examine the partial autocorrelation function of INARMA
processes. The section is organized as follows. First the PACF of an INAR(p)
process is studied. The PACF of INMA(q) and INARMA(p,q) processes are then
discussed.
4.3.2.1 PACF of an INAR(p) Model
In this section, we examine the partial autocorrelation function of INAR processes.
The INAR(p) process is defined by the recursion:
M.Mohammadipour, 2009, Chapter 4 92
??=?1????1 +?2????2 +?+???????+??
Here we assume that ?? has a Poisson distribution with parameter ?. Multiplying the
above equation by ???? produces:
??????=????(?1????1) +????(?2????2) +?+????(???????) +??????
Equation 411
but we know that:
?(??) =?1?(???1) +?2?(???2) +?+???(????) +?(??)
?=???1 +?2 +?+???+?
So, if we take the expected value of Equation 411 and subtract ?(??)?(????) from
it, we have:
??=?1???1 +?2???2 +?+??????+ cov(????,??) ??0
Equation 412
considering the fact that ????????????????=???(????????) (Silva and Oliveira,
2004). The last term in the RHS is zero because ???? can only involve innovation
terms up to time ??? and therefore is uncorrelated with ??. Dividing the Equation
412 by ?0 yields:
??=?1???1 +?2???2 +?+?????? ??0
Equation 413
which is analogous to the difference equation for Gaussian AR processes. The Yule
Walker equations are:
?1 = ?1 + ?2?1 + ? + ?????1
?2 = ?1?1 + ?2 + ? + ?????2
? ? ? ? ?
?? = ?1???1 + ?2???2 + ? + ??
Equation 414
The autocorrelation function of an INAR(p) process can be described in terms of ?
M.Mohammadipour, 2009, Chapter 4 93
nonzero functions of the autocorrelations.
In an integer autoregressive process of order ?:
??=??1???1 +??2???2 +?+??(??1)????+1 +??????? for ?= 1,2,?,?
Equation 415
where ??? is the ?th coefficient in an integer autoregressive representation of order ?.
Therefore ??? is the last coefficient and ???= 0 for ?>?.
The Equation 415 leads to the YuleWalker equations which may be written as:
?
1 ?1 ?2 ? ???1
?1 1 ?1 ? ???2
? ? ? ? ?
???1 ???2 ???3 ? 1
??
??1
??2
?
???
?=?
?1
?2
?
??
?
Equation 416
As defined by Hamilton (1994), the ?th population partial autocorrelation (denoted
by ???) is the last coefficient in a linear (for Gaussian ARMA processes) projection
of ? on its ? most recent values. In the case of integer autoregressive models, it can
be stated as:
??=??1????1 +??2????2 +?+????????
Equation 417
which results in the same sets of difference equations based on autocorrelations. The
justification for using partial autocorrelations is that, if data really were generated by
an INAR(p) process, only the ? most recent values of ? would be useful for
forecasting and the coefficients on ?'s more than ? periods in the past are equal to
zero, which means that:
???= 0 for ?=?+ 1,?+ 2,?
We can express the INAR(p) process as follows:
??=????1????1??2????2??????????=???? ???????
?
?=1
It can be seen that the series in the RHS of the above equation is finite. Therefore, the
M.Mohammadipour, 2009, Chapter 4 94
PACF of an INAR (p) process is finite.
Although an INAR(p) process differs from the AR(p) process due to the thinning
operations, these two processes share some properties including the ACF and PACF
structure. Therefore, using the sample partial autocorrelation function (SPACF) may
help us in identifying the autoregressive order of an INAR time series.
4.3.2.2 PACF of an INMA(q) Model
This section focuses on finding the partial autocorrelation function of an INMA
process. The INMA(q) process has the following form:
??=??+?1????1 +?2????2 +?+???????
We again assume that ?? has a Poisson distribution with parameter ?. Using the same
argument as for an INAR(p) process, it can be seen that the autocovariance function
of an INMA(q) process is:
??=?[(??+?1????1 +?2????2 +?+???????).
(????+?1??????1 +?2??????2 +?+?????????)]
so
??=?
(??+?1??+1 +?2??+2 +?+??????)??
2 ?= 1,2,?,?
0 ?>?
?
Equation 418
Thus, the autocorrelation function (ACF) is given by:
??=?
??+?1??+1 +?2??+2 +?+??????
1 +?1 +?+??
?= 1,2,?,?
0 ?>?
?
Equation 419
It can be seen that the autocorrelation function of an INMA(q) process is zero,
beyond the order ? of the process.
As discussed earlier in section 4.3.2.1, if the data were generated by an INAR(p)
M.Mohammadipour, 2009, Chapter 4 95
process, only the ? most recent values of ? would be useful for forecasting and
therefore the PACF cuts off after lag ?. However, if the data were generated by an
INMA(q) process with ??1, then the partial autocorrelation ??? asymptotically
approaches zero instead of cutting off abruptly.
The PACF of the process (???) can be obtained by solving the set of equations given
by Equation 415 using the ACF of the process. For example, AlOsh and Alzaid
(1988) find the first and second partial autocorrelations of an INMA(1) process as:
?11 =?1
?22 =
?2??1
2
1??1
2 =
??2
1 + 2?
which can be seen is the same as that of a MA(1) process.
4.3.2.3 PACF of an INARMA(p,q) Model
The partial autocorrelation function of an INARMA(p,q) process is examined in this
section. This process satisfies the difference equation:
??=? ???????
?
?=1 +??+? ???????
?
?=1
An INARMA(p,q) process can be written in form of (see Appendix 4.A):
??=???? ? ???????
??
?=1
?
?=1
Equation 420
where ??=?
?? ????
?
?=1 + 1 0 ?
?.
The values of ?? can be found by repeated multiplications of the parameters of
INARMA(p,q) process.
It can be seen that the series in the RHS of the Equation 420 is infinite. Therefore,
the PACF of an INARMA(p,q) process, similar to an INMA(q) process, is infinite.
M.Mohammadipour, 2009, Chapter 4 96
As shown in section 4.3.1, the structure of the autocorrelation function of the
INARMA(p,q) process is analogous to that of an ARMA process. The structure of
the partial autocorrelation function of this process is also similar to that of an ARMA
process, i.e. the PACF of a mixed integer autoregressive moving average process is
infinite and it has the same shape as the PACF of a pure integer moving average
process.
4.4 Residual Analysis
The residuals of the INARMA models can provide a check of model adequacy (Jung
and Tremayne, 2006b). After identification of the appropriate model using ACF and
PACF and estimation of the parameters of the identified model, the residuals should
be examined to check for any serial dependence. Any dependence in the residuals
would suggest that a different model should be used. The ACF and PACF of the
residuals should be plotted for this reason.
Freeland and McCabe (2004a) define two sets of residuals: one for the arrivals
component and another for the continuation process of a PoINAR(p) process. The
residuals for the continuation component are given by:
?1?=?????????????? for ?=?+ 1,?,?
Equation 421
The residuals for the arrivals component are:
?2?=????
Equation 422
However, as they mention, these definitions are not practical because ??????? and
?? cannot be observed and should be replaced with their conditional expected values.
Adding the new components then results in the usual definition of residuals for a
PoINAR(p) process:
??=???? ?????
?
?=1
??
Equation 423
M.Mohammadipour, 2009, Chapter 4 97
Bu and McCabe (2008) suggest that in order to check the adequacy of the selected
model, checking the traditional residual of the Equation 423 is not enough and the
components residuals (?1? and ?2?) should also be tested to examine the suitability of
each component in the model. The expected value of ??????? and ?? are provided
in terms of the conditional probabilities. This resolves the impracticality issue
mentioned in Freeland and McCabe (2004a).
The residuals of an INMA(q) process are given by (Br?nn?s and Hall, 2001):
??=?????? ?????
?
?=1
Equation 424
Based on Equation 423 and Equation 424, the residuals of an INARMA(p,q) model
can be obtained from:
??=???? ?????
?
?=1
???? ?????
?
?=1
Equation 425
It will be explained in section 4.5 that the identification methods used in this PhD
thesis are not based on ACF and PACF. However, as an example, here we show what
the ACF and PACF of one INAR(1) series look like. The series is selected from
those of the 16,000 series data set of chapter 9. The time series plot for all 72 periods
is provided in Figure 41. The sample ACF and sample PACF of the above series are
presented in Figure 42. The sample PACFs suggest that an INAR(1) model is
appropriate.
The parameters of the identified INAR(1) model are then estimated using the Yule
Walker (YW) estimation method (see section 5.3.1) to be ?= 0.5581 and ?=
0.0552.
The next step is to check the model?s adequacy using the residual analysis. The
residuals of the INAR(1) model are defined as:
??=?????????1????
M.Mohammadipour, 2009, Chapter 4 98
Figure ?41 Time series plot of one demand series among 16,000 series
Figure ?42 Correlograms of the selected series among 16,000 series
where ??? and ??? are the YuleWalker estimates of parameters in the INAR(1)
model.
If any dependence structure exists in the residuals, a different model specification
example would be considered. In order to check if such dependence exists in our
example, the SACFs and SPACFs of the residuals of the estimated INAR(1) model
are depicted in Figure 43. The figure suggests that there is no obvious dependence
structure left in the residuals.
M.Mohammadipour, 2009, Chapter 4 99
Figure ?43 Correlograms of the residuals of the INAR(1) model
4.5 Identification based on Penalty Functions
It has been argued that distinguishing between autoregressive moving average
models based on the BoxJenkins procedure is difficult (Chatfield and Prothero,
1973; Newbold and Granger, 1974). This is because the ACF and PACF plots cannot
easily identify mixed ARMA models. Moreover, the identification of ARMA models
usually involves subjective judgment. The same is true for identification of
INARMA models using ACF and PACF.
The KullbackLeibler information (Kullback and Leibler, 1951) is used to measure
the difference between two probability density functions ?(?) and ?(?):
???,??=?????log?
?(?)
?(??)
???
Equation 426
In the above equation, ???,?? denotes the information lost when ? is used to
approximate ?. ? is considered to be fixed and ? varies over ?.
Akaike (1973) introduces the Akaike information criterion (AIC) as an approximately
unbiased estimate of KullbackLeibler information. The AIC is given by:
M.Mohammadipour, 2009, Chapter 4 100
AIC =??2?log?maximum likelihood?+ 2?
Equation 427
where ? is the number of estimable parameters in the approximating model. Ozaki
(1977) shows that the AIC of the ARMA(p,q) model is given by:
AIC??log??
2 + 2?
Equation 428
where ??
2 is the residual variance and ?=?+?+ 1. When the sample size is small
(?/?< 40), the above expression is biased and the following bias correction is
necessary (Hurvich and Tsai, 1989; Hurvich and Tsai, 1995):
AICC ??log??
2 + 2?+ 2?(?+ 1)/(????1)
Equation 429
AIC has also been used in the INARMA literature (see for example: B?ckenholt,
1999; Brandt et al., 2000; Zhu and Joe, 2006). However, the complexity of the
likelihood function of these models has led to some limitations in the use of AIC.
The likelihood function of an INAR(p) process with Poisson innovations is given by
(Bu, 2006):
???1,?,??,??= ? ?(?????1,?,????)
?
?=?+1
Equation 430
where the conditional probability function ???????1,?,????? is:
???????1,?,?????= ? ?
???1
?1
??1
?1?1??1?
???1??1
min????1 ,???
?1=0
? ?
???2
?2
??2
?1?1??2?
???2??2
min????2 ,????1?
?2=0
?
? ?
????
??
???
??(1???)
???????
?????????1+?+???1?
??????1 +?+???1??!
min [????,?????1+?+???1?]
??=0
Equation 431
M.Mohammadipour, 2009, Chapter 4 101
It can be seen that the logarithm of Equation 431 cannot be simplified as in ARMA
models. It should also be mentioned that the likelihood function of the
INARMA(p,q) process is not established yet.
4.6 The Identification Procedure
For the simulation experiment, we need an automated method for identification that
can be used for thousands of replications. Because the likelihood function of an
INARMA(p,q) process is not yet found and even the likelihood function of an
INAR(p) process is very complicated, the penalty functions for these models are not
easy to find. It has been shown in chapter 3 that an INAR process is analogous to an
AR process in autocorrelation structure and also forecasting. Therefore, it has been
suggested that the standard programmes for AR processes, which are mainly based
on AIC or BIC, could also be used for INAR processes (Latour, 1998). The same is
true for an INMA process regarding the ACF structure and the conditional expected
value.
As mentioned in the previous section, the use of AIC in the INARMA literature is
limited to those processes for which likelihood functions have been derived. This
excludes the INMA and mixed models. Based on the above argument by Latour
(1998) we test the performance of AIC of Equation 428 (or where applicable, AICC
of Equation 429) for INARMA models.
As discussed in section 4.2, Jung and Tremayne (2003) suggest that in analysing the
time series of counts, any serial dependence should first be detected. If no such
dependence is found, the complicated INARMA methods can be replaced with easier
methods for independent data. Based on the above argument, we use two
identification procedures in this thesis: a twostage and a onestage method.
In the twostage method, the first stage distinguishes between the INARMA(0,0) and
the other INARMA models. The LjungBox statistic of Equation 46 is used for this
reason. This is because it is a standard test used for conventional ARMA models.
Therefore, it is included in most software packages (including MATLAB which is
used in this thesis) and, based on the argument by Latour (1998), it can be used for
M.Mohammadipour, 2009, Chapter 4 102
INARMA models as well. It will be shown in chapter 8 that the rejection percentages
under the null hypothesis of i.i.d. Poisson are comparable to the results of the tests
suggested by Jung and Tremayne (2003).
The AIC of Equation 428 is then used for identification among the other INARMA
models. This is again based on the argument of Latour (1998) to use the standard
programmes for ARMA models for INARMA models. It should also be mentioned
that the AIC of ARMA models has been used in the INARMA literature (e.g.
Br?nn?s and Quoreshi, 2004). The reliability of this identification procedure will be
tested in the simulation chapter. To our knowledge, this has not been done in the
literature before. The impact of misidentification on the forecast accuracy will also
be checked. The onestage method only uses the AIC of Equation 428 (or AICC of
Equation 429) to identify the most appropriate model.
The performance of these two methods will be compared. This will be done in terms
of the percentage of time that the correct model is identified and also the accuracy of
forecasts based on each method.
4.7 Conclusions
In this chapter, the methods of identification of the autoregressive and moving
average orders of an INARMA process have been reviewed. It has been shown that
the autocorrelation and partial autocorrelation functions of an INAR(p) process have
the same structure as those of an AR(p) process. The same is true for INMA(q) and
INARMA(p,q) processes. Therefore, the estimates of the functions (SACF and
SPACF) can identify the moving average and autoregressive orders, respectively.
The residuals of the estimated INARMA process then need to be checked for any
remaining correlations.
Two identification procedures will be used in this thesis. A twostage method is
based on first using the LjungBox statistic to identify any correlation in the data
series. The next step involves using the AIC of ARMA models to select from the
other possible INARMA models. The performance of this procedure in terms of the
percentage of time that the model is identified correctly and also the effect of
M.Mohammadipour, 2009, Chapter 4 103
misidentification on forecast accuracy will be tested in chapter 8. To our knowledge,
this has not been done in INARMA literature. The two stage method will then be
compared to a onestage identification method based on using the AIC to select
among the INARMA models including INARMA(0,0).
M.Mohammadipour, 2009, Chapter 5 104
Chapter 5 ESTIMATION IN INARMA MODELS
5.1 Introduction
Having identified the appropriate INARMA model, we then need to estimate the
parameters of the selected model. Different estimation methods have been used in the
literature to estimate the parameters of INAR(p) and INMA(q) models, namely:
? YuleWalker (YW)
? Conditional least squares (CLS)
? Maximum likelihood (ML)
? Generalized method of moments (GMM)
Each of these methods is briefly reviewed. Table 51 lists the main studies on
estimation of parameters of INARMA models.
M.Mohammadipour, 2009, Chapter 5 105
Table ?51 Research papers on estimation of parameters of INARMA models
Model YW CLS ML GMM
INAR(1) AlOsh and
Alzaid, 1987
AlOsh and
Alzaid, 1987
AlOsh and
Alzaid, 1987
Br?nn?s, 1994
Br?nn?s and
Hellstr?m, 2001
INAR(p) Du and Li, 1991
Jung and
Tremayne, 2006b
for INAR(2)
Du and Li, 1991
Bu et al., 2008
INMA(1) Br?nn?s and Hall,
2001
Br?nn?s and Hall,
2001
Br?nn?s and Hall,
2001
INMA(q) Br?nn?s and Hall,
2001 for INMA(2)
Br?nn?s and Hall,
2001
Br?nn?s and Hall,
2001
The YW method is based on using the YuleWalker equations of:
?1 = ?1 + ?2?1 + ? + ?????1
?2 = ?1?1 + ?2 + ? + ?????2
? ? ? ? ?
?? = ?1???1 + ?2???2 + ? + ??
and replacing the theoretical autocorrelations ?? by the sample autocorrelations, ??
(Box et al., 1994):
??=
? ??????(??????)
?
?=?+1
? ??????2
?
?=1
Equation 51
Lawrence et al. (1978) develop the CLS estimation procedure for stochastic
processes based on the minimization of a sum of squared deviations about
conditional expectation.
Maximum likelihood estimation method, as the name suggests, finds the parameters
that maximize the likelihood of the sample data. The likelihood of the sample data is
the probability of obtaining that particular set of data, given the specific probability
distribution.
In the generalized method of moments, a set of population moment conditions is first
derived based on the assumptions of the model. Then the GMM estimates of the
parameters are obtained such that these moment conditions are satisfied for the
M.Mohammadipour, 2009, Chapter 5 106
sample data.
This chapter is organized as follows. The estimate of the parameter of a Poisson
INARMA(0,0) process is provided in section 5.2. The YW, CLS, CML, and GMM
estimates for the parameters of INAR(1) and INAR(p) processes are reviewed in
sections 5.3 and 5.4 (GMM only for INAR(1) process). The corresponding
estimation methods for INMA(1) and INMA(q) processes are reviewed in sections
5.5 and 5.6. The YW and CLS estimates of the parameters of an INARMA(1,1)
process are derived in section 5.7. Finding the ACF of an INARMA(p,q) model in
chapter 3 enables us to find the YW estimates of these models. As an example, the
YW estimates of an INARMA(2,2) model are derived in section 5.8. The conclusions
are given in section 5.9.
As will be discussed in chapter 7, four INARMA models are selected for simulation
and empirical analysis to compete against the benchmark methods. These models
are: INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1). The estimation
methods used in this thesis for these models are CLS and YW for the last three and
also CML for INAR(1). Therefore, these estimates are specifically given in this
section. It will be discussed in section 5.2 that the CLS and ML estimation methods
result in the same estimate for a Poisson INARMA(0,0) process.
5.2 Estimation in an INARMA(0,0) Model
The INARMA(0,0) process with Poisson marginal distribution is simply an i.i.d.
Poisson process of:
??=??
Equation 52
where ?? are i.i.d. Poisson random variables. The conditional expected value of ??
given ???1 is therefore given by:
???????1?=?
Equation 53
where ? is the only parameter to be estimated. The conditional least squares estimate
M.Mohammadipour, 2009, Chapter 5 107
of ? is obtained by minimizing the function:
?????=? [??????????1?]
2
?
?=1
=? (????)
2
?
?=1
Equation 54
with respect to ? for a sample of {?1,?2,?,??}.
?=
? ??
?
?=1
?
Equation 55
The likelihood function of a sample of ? observations of an INARMA(0,0) process
can be written as:
????=?
??????
????!
?
?=1
Equation 56
The ML estimator of ? can be obtained by maximizing the log of the likelihood
function in Equation 56. It can be seen that this results in the same estimator as that
of CLS (Equation 55).
5.3 Estimation in an INAR(1) Model
5.3.1 YW for INAR(1)
The YuleWalker estimator for ? in an INAR(1) model was found by AlOsh and
Alzaid (1987) to be as follows:
?=
? ??????(??+1??)
??1
?=0
? ??????2
?
?=0
Equation 57
where ? is the sample mean. Since ?? is assumed to have a Poisson distribution with
parameter ?, the estimator for ? is:
M.Mohammadipour, 2009, Chapter 5 108
???=
? ??
?
?=1
?
Equation 58
where ??=???????1.
Jung and Tremayne (2006b) propose the same estimator for ?, but a slightly different
estimator for ?, which is indicated by ???. They argue that, as the first order moment
of the INAR(1) model is ?????=
?
1??
, ? can be estimated from:
???= (1??)
? ??
?
?=1
?
Equation 59
In this thesis we use the Equation 57 and Equation 59 to obtain the YW estimates
of an INAR(1) process. This is because the Equation 59 is based on observed data
and not estimates of the innovations as in Equation 58.
5.3.2 CLS for INAR(1)
It can be easily seen that in the INAR(1) model, ?? given ???1 is still a random
variable due to the definition of the thinning operation. The conditional mean of ??
given ???1, which is the best onestepahead predictor (Br?nn?s and Hall, 2001), is:
???????1?=????1 +?=?(?,???1)
Equation 510
where ?= (?,?)? is the vector of parameters to be estimated. AlOsh and Alzaid
(1987) employ a procedure developed by Klimko and Nelson (1978) and derive the
estimators for ? and ? as follows:
?=
? ?????1
?
?=1 ?(? ??
?
?=1 ? ???1
?
?=1 )/?
? ???1
2?
?=1 ?(? ???1
?
?=1 )
2/?
Equation 511
and
M.Mohammadipour, 2009, Chapter 5 109
?=?? ??
?
?=1
??? ???1
?
?=1
?/?
Equation 512
It can be easily verified that ?, ??/??, ??/??, ?2?/???? satisfy the regularity
conditions proposed by Klimko and Nelson (1978). It follows that the CLS
estimators are strongly consistent and asymptotically normally distributed as:
??????0?~?(?,??1??)
where ?0 = (?0,?0)? denotes the true values of the parameters and:
??=??
??(?0,???1)
???
.
??(?0,???1)
???
? ?,?= 1,2
??=????
2(?0)
??(?0,???1)
???
.
??(?0,???1)
???
? ?,?= 1,2
with ????
0?= ????(?
0,???1).
Freeland and McCabe (2005) show that the distributions of the CLS and YW
estimators of a PoINAR(1) process are asymptotically equivalent.
5.3.3 CML for INAR(1)
The Conditional Maximum Likelihood (CML) and Maximum Likelihood (ML)
estimators for the PoINAR(1) process are provided by AlOsh and Alzaid (1987).
The likelihood function of a sample of ? observations from an INAR(1) process can
be written as:
???,??=?(?1)? ?(?????1)
?
?=2
Equation 513
where ?(?????1) is given by:
M.Mohammadipour, 2009, Chapter 5 110
???????1?= ? ?
???1
?
???(1??)???1??
????????
??????!
min (???1 ,??)
?=0
Equation 514
Because the marginal distribution of the PoINAR(1) process is Poisson with mean
?/(1??), the unconditional likelihood function is:
???,??=
???/(1??)[
?
1??]
?1
??1?!
? ? ? ?
???1
?
???(1??)???1??
????????
??????!
min (???1 ,??)
?=0
?
?
?=2
Equation 515
In order to find the conditional maximum likelihood estimation (CML), ?1 is
assumed to be given and the conditional likelihood function is reduced to:
???,??=? ? ? ?
???1
?
???(1??)???1??
????????
??????!
min (???1 ,??)
?=0
?
?
?=2
Equation 516
Then, the unconditional and conditional maximum likelihood estimators can be
derived by maximizing the logarithm of the likelihood functions of Equation 515
and Equation 516, respectively.
AlOsh and Alzaid (1987) used the procedure of Sprott (1983) to eliminate one of the
parameters in the derivatives of the loglikelihood function.
?log[???,??]
??
=? ?(?)
?
?=2
????1?= 0
Equation 517
?log[???,??]
??
=?
????????1????(?)
?(1??)
?
?=2
= 0
Equation 518
where ????=?????1?/?????. The Equation 518 results in:
?=
? ??
?
?=2 ??? ???1
?
?=2
??1
Equation 519
M.Mohammadipour, 2009, Chapter 5 111
The Equation 519 can then be used in Equation 517 to find ?. The ML estimators
of ? and ? have the following asymptotic distribution:
???
???
???
?~?(?,??1)
where the matrix ? is the Fisher information (Bu, 2006).
AlOsh and Alzaid (1987) compare the performance of YW, CLS and CML
estimates in terms of bias and MSE in a simulation study. Their results suggest that,
in general, CML is worth the extra effort because it has the least bias and MSE of all.
However, for small sample sizes (??75) and small autoregressive parameter
??= 0.1?, because the sample contains many zero values, CML is not as good as
YW in terms of bias and MSE. It is worth mentioning that their study only compares
the accuracy of estimates and not their impact on forecast accuracy, which is done in
this PhD thesis (see section 8.4).
5.3.4 Conditional GMM for INAR(1)
Br?nn?s (1994) uses the conditional GMM estimation method of Hansen (1982) to
estimate the parameters of a PoINAR(1) process. It is called a conditional GMM
since the moment restrictions used are conditional. The GMM estimator is based on
minimization of the function:
?=?(?)???1?(?)
Equation 520
where ?= (?,?)? is the vector of the unknown parameters to be estimated and ?(?)
is the vector of moment restrictions. When ? is the asymptotic covariance of ?(?),
the GMM estimator is efficient. ? is first minimized based on using an identity
matrix ? for ?. Then the estimates ? are used to from ?.
The moment restrictions used are:
1
?
? ??
?
?=2
= 0
M.Mohammadipour, 2009, Chapter 5 112
1
?
? ?(???1??)
?
?=2
= 0
where ?? is the onestep ahead prediction error, ??=????( ?????1).
Although the above moments are unconditional, they are equal to the conditional
ones (Br?nn?s, 1994). It can be seen that when ? =?2, the GMM and the CLS are
the same.
Br?nn?s (1994) then compares the performance of CLS, ML, and GMM. The results
show that for large values of ?, the GMM estimates have smaller MSE than CLS. In
general, for large values of ?, ?GMM is close to ?ML in terms of MSE. But for small
values of ?, ?ML outperforms ?GMM . Moreover, ?ML always has smaller MSE than
?GMM , although when ? increases this difference decreases. The results do not show
any conclusive advantage by using GMM over ML in terms of bias.
To conclude, we do not see any benefit in using GMM for an INAR(1) process,
considering the fact that it does not outperform the maximum likelihood method,
which is used in this research, in terms of MSE.
5.4 Estimation in an INAR(p) Model
5.4.1 YW for INAR(p)
Du and Li (1991) generalize the YuleWalker estimation method to estimate the
parameters of an INAR(p) process. The YuleWalker equations are:
??=?
Equation 521
where ?= [?????]???, ?= (?1,?2 ,?,??)?, and ?= (?1,?2 ,?,??)?. Replacing the
theoretical autocorrelations ?? by the sample autocorrelations ?? results in the YW
estimate of ?.
Then, ? can be estimated from the expected value of the INAR(p) process:
M.Mohammadipour, 2009, Chapter 5 113
?= (1??1?????)?
Equation 522
These estimators are strongly consistent and asymptotically normally distributed
(Silva and Silva, 2006). Also, for a large number of observations, YW estimators are
very close to the CLS estimators (Du and Li, 1991).
5.4.2 CLS for INAR(p)
The CLS estimators of an INAR(p) process are also derived by Du and Li (1991).
The conditional expected value of the process is:
???????1,?,?2,?1?=?1???1 +?+??????+?
Equation 523
The least squares criterion to be minimized is then:
?????= ? [??????????1,?,?2,?1?]
2
?
?=?+1
Equation 524
where ?= (?1,?2,?,??,?)
is the vector of parameters to be estimated. ? can be
found by setting the partial derivatives to zero.
??????
???
= 0 (?= 1,2,?,?)
??????
??
= 0
Du and Li (1991) suggest that, for large samples, the CLS estimators for INAR(p)
process are very close to YuleWalker estimators. They also are strongly consistent
and asymptotically normal.
5.4.3 CML for INAR(p)
Bu et al. (2008) study the maximum likelihood estimators of a general INAR(p)
M.Mohammadipour, 2009, Chapter 5 114
process. The conditional likelihood function of an INAR(p) process is:
???1,?2,?,??,??= ? ?(?????1,?,????)
?
?=?+1
Equation 525
where the conditional probabilities are:
???????1,?,?????= ? ?
???1
?1
??1
?1?1??1?
???1??1
min????1 ,???
?1=0
?
? ?
???2
?2
??2
?1?1??2?
???2??2
min????2 ,????1?
?2=0
???
? ?
????
??
???
??(1???)
???????
?????????1+?+???1?
??????1 +?+???1??!
min [????,?????1+?+???1?]
??=0
Equation 526
The asymptotic distribution of the maximum likelihood estimator is Normal:
???????~?(?,??1)
where ?= (?1,?,??,?)? is the parameter vector and the matrix ? is the Fisher
information.
Bu et al. (2008) investigate the asymptotic benefit of ML over CLS for a PoINAR(2)
process using the asymptotic relative efficiency (ARE) ratio between the two
estimators, i.e. the ratio of their asymptotic variances. Their results show that for
persistent processes (high values of ?1, ?2 or both), the ML estimates are
asymptotically more efficient than the CLS estimates. For low values of ?1 and ?2 and
also for high values of ?2 and low values of ?1, the benefit of ML over CLS is slight.
They also compare the performance of ML and CLS in a simulation study for
?= 100 and ?= 500. The results suggest that there is a gain in terms of MSE in
using ML for larger samples. For smaller samples, when ?1 and ?2 are small, CLS
has lower MSE than ML.
M.Mohammadipour, 2009, Chapter 5 115
5.5 Estimation in an INMA(1) Model
5.5.1 YW for INMA(1)
The YuleWalker estimator for ? in a PoINMA(1) process (??=??+?????1) is as
follows:
?=
?1
1??1
Equation 527
where ?1 is the lag one sample autocorrelation given by the Equation 51. Then, ?
can be estimated from the expected value of the process:
?=
1
1 +?
? ??
?
?=1
?
Equation 528
Note that ? can also be estimated form the unconditional variance of the INMA(1)
(Equation 328), but the estimator based on the unconditional mean has smaller
variance (Br?nn?s and Hall, 2001).
5.5.2 CLS for INMA(1)
The conditional expected value of ?? given ???1 for an INMA(1) process is given by:
???????1?=????1 +?
Equation 529
The prediction error is:
??=???????1??
Equation 530
The CLS estimates of ? and ? can then obtained by minimizing the following
function:
?????=? [???(????1 +?)]
2
?
?=1
Equation 531
M.Mohammadipour, 2009, Chapter 5 116
with respect to ?, where ?= (?,?)? is the parameter vector to be estimated. The
CLS estimates for ? and ? are:
?=
? ?????1
?
?=1 ?(? ??
?
?=1 ? ???1
?
?=1 )/?
? ???1
2?
?=1 ?(? ???1
?
?=1 )
2/?
Equation 532
?=?? ??
?
?=1
??? ???1
?
?=1
?/?
Equation 533
5.6 Estimation in an INMA(q) Model
5.6.1 YW for INMA(q)
The autocorrelation function of an INMA(q) process of ??=??+?1????1 +?+
??????q is given by the Equation 340. The ACF can be used to find the YW
estimates of (?1,?,??).
Once these parameters have been estimated, ? can be estimated from the expected
value of the process:
?=
1
1 +? ??
?
?=1
? ??
?
?=1
?
Equation 534
When the order of the INMA model increases, the equations to be solved become
more complex. This is shown in Table 52.
Table ?52 The relationship between the order of the model and the type of YW equation
Model Equation
INMA(1) Linear
INMA(2) quadratic (2)
INMA(3) quartic (4)
INMA(4) sextic (6)
M.Mohammadipour, 2009, Chapter 5 117
The results of Table 52 can be found by direct expansion of expressions in the Yule
Walker equations. As can be seen from the above table, for INMA processes with
order higher than two, it becomes more complex to find the estimators and although
such equations can be solved numerically, it is computationally expensive to find the
YuleWalker estimators for such processes.
5.6.2 CLS for INMA(q)
The conditional first moment of the INMA(q) process is:
???????1?= ?+? ?????j
?
?=1
Equation 535
Hence, the forecast error is:
??=?????? ??????
?
?=1
Equation 536
The least squares criterion is then:
?????= ? ??????? ??????
?
?=1
?
2
?
?=?+1
Equation 537
The corresponding parameter vector ?= (?1,?,??,?)? for the minimum ????? can
be obtained.
5.6.3 GMM based on Probability Generation Functions for INMA(q)
In this estimation method, the probability generation functions (pgf) ?(?) and
??(?1,?2) are evaluated at any ?. Based on the law of large numbers:
M.Mohammadipour, 2009, Chapter 5 118
? (1???)
???
?=1
?
?
??(??)
? (1???)
??(1???)
?????
?=?+1
???
?
???(??,??)
where
?
? denotes convergence in probability. Therefore, the moment conditions are
formed as:
?1?=
? (1???)
???
?=1
?
??(??)
?2?,??=
? (1???)
??(1???)
?????
?=?+1
???
???(??,??)
The GMM criterion to be minimized is then:
?=?? ?1?
where ? is the vector of moment restrictions and ? is the covariance matrix of ?.
Similar to section ?5.3.4, first it is assumed that ? is equal to the identity matrix. This
results in a consistent and asymptotically normal estimator. However, if ? is known
or a consistent estimator of ? is used, the GMM estimators would be more efficient
than if ? is equal to ?.
For a PoINMA(1) process, GMM estimators provided by Br?nn?s and Hall (2001)
are as follows:
?=
?ln????ln?????1?ln????2
?ln????ln?????1?ln????2(1??1)
Equation 538
?=?
ln(?)
?1(1 +?)
Equation 539
where ? and ? are the sample moments corresponding to ?(?1) and ?1(?1,?2),
respectively.
Br?nn?s and Hall (2001) compare the performance of YW, CLS and GMM for an
M.Mohammadipour, 2009, Chapter 5 119
INMA(2) process. The results for the case that ?1 =?2 = 0.9 (note that the process
is not invertible) suggest that GMM has the smallest bias and MSE among all
estimators except for small sample sizes where CLS is the best. For all ?2 values
except ?2 = 0.9, CLS is better than GMM in terms of MSE.
Although they mention that for other values of ?1 and ?2 GMM has smaller bias and
MSE in most cases, they did not present any results for further comparisons.
Br?nn?s and Hall (2001) explain that the performance of the GMM estimator
depends highly on the selection of ?values. Finally, because of the better overall
performance and its simplicity, they advocate the use of CLS instead of GMM.
5.7 Estimation in an INARMA(1,1) Model
In this section, the YW and CLS estimators for the parameters of an INARMA(1,1)
process, derived in this PhD research, are presented.
5.7.1 YW for INARMA(1,1)
The ACF of an INARMA(1,1) process of ??=?????1 +??+?????1 is given by
Equation 348. When the distribution of the innovations {??} is Poisson, the ACF is:
??=?
?+?+??+?2 + 2?2?
1 +?+?+ 3??
for ?= 1
????1 for ?> 1
?
Equation 540
Hence, ? and ? can be estimated from:
?=
?2
?1
=
? ??????(???2??)
?
?=3
? ??????(???1??)
?
?=2
Equation 541
?=
?1 +??(???1)
?1?1 + 3???1???2?2
Equation 542
M.Mohammadipour, 2009, Chapter 5 120
Then, ? can be estimated using the expected value of an INARMA(1,1) process:
?=
1??
1 +?
? ??
?
?=1
?
Equation 543
5.7.2 CLS for INARMA(1,1)
The conditional expected value of an INARMA(1,1) process is:
?(?????1) =????1 +?+????1
Equation 544
The conditional least squares criterion is therefore:
?????=? [????????1 +?+????1?]
2
?
?=1
Equation 545
with ?= (?,?,?)? is the parameter vector to be estimated. The estimators for ?, ?,
and ? can then be obtained by minimizing the above function with respect to ?.
?=
?
?2??????1????1
2 ?????????1????1
2 ????????1?????1?
2 +?????1????1??????1
??2??????1????1???1 +????????1????1???1
?
?2????1
2 ????1
2 ???????1?2????1
2 ??????1
2 ?????1?2 + 2?????1????1????1???1??2?????1???1?2
Equation 546
?=
???????1???????1???1????????1 +?????1????1
?????1
2 ??????1?2
Equation 547
?=
?????????1??????1
?
Equation 548
where all the summations are from 1 to ? (see Appendix 5.A for the proof).
M.Mohammadipour, 2009, Chapter 5 121
5.8 YW Estimators of an INARMA(2,2) Model
The YW estimators of an INARMA(2,2) model can be obtained from the ACF of an
INARMA(p,q) model derived in chapter 3. The INARMA(2,2) model has the form:
??=?1????1 +?2????2 +??+?1????1 +?2????2
Equation 549
From the Equation 352, the variance of the above process, when the innovations are
Poisson distributed, the variance of an INARMA(2,2) process can be found as (see
Appendix 5.B):
var????=
?
(1??1
2??2
2)(1??2)?2?1
2?2
?
??1??2??
1 +?1 +?2
1??1??2
??1??1
2 +?2??2
2?+ 1 +?1 +?2 + 2?1?1
+ 2?2?2?+ 2?1?2?1 + 2?1
2?2 + 2?1?1?2?
Equation 550
The autocorrelation function of an INARMA(2,2) process can be found from the
Equation 355 as:
??=
?
?
?
?
?
?1?0 +?2?1 +?1?+?2(?1 +?1)?
?0
??? ?= 1
?1?1 +?2?0 +?2?
?0
??? ?= 2
?1???1 +?2???2 ??? ?> 2
?
Equation 551
Based on the Equation 551, the autocorrelation of lags one to four can be used to
estimate ?1, ?2, ?1, and ?2.
?1 =?1 +?2?1 +
?1?+?2(?1 +?1)?
?0
Equation 552
?2 =?1?1 +?2 +
?2?
?0
Equation 553
?3 =?1?2 +?2?1
Equation 554
M.Mohammadipour, 2009, Chapter 5 122
?4 =?1?3 +?2?2
Equation 555
The last two equations can be used to find ?1 and ?2.
?1 =
?1?4??2?3
?1?3??2
2
Equation 556
?2 =
?3??1?2
?1
Equation 557
where ?? is the sample autocorrelation at lag ? given by the Equation 51. Then, ?1
and ?2 can be found from Equation 552 and Equation 553.
Finally, the expected value of the process can be used to estimate ?.
?=?
1??1??2
1 +?1 +?2
??
Equation 558
where ?=
? ??
?
?=1
?
.
5.9 Conclusions
Different methods for estimating the parameters of INARMA models provided in the
literature have been reviewed in this chapter. This includes YW, CLS, CML, and
GMM. Not all these methods have been developed for all INARMA models. For
example, the maximum likelihood function and therefore the CML estimators have
been developed only for INAR(p) models.
The performance of these estimators has been compared in some studies (AlOsh and
Alzaid, 1987; Br?nn?s, 1994; Br?nn?s, 1995; Br?nn?s and Hall, 2001; Bu et al.,
2008). The results generally suggest that the CML is worth the extra effort especially
for high values of ? (and ?1 for an INAR(2) process) and reasonably large samples.
For lower values of ? and smaller samples, the CLS has lower MSE than ML.
M.Mohammadipour, 2009, Chapter 5 123
The YW and CLS estimates for an INARMA(1,1) process are provided for the first
time. The YW results are based on the ACF of an INARMA(p,q) model of Equation
355. These results along with YW, CLS, CML for INAR(1) process and YW and
CLS for INMA(1) process will be used in simulation and empirical analyses of this
PhD research (chapters 8 and 9).
Finding the ACF of an INARMA(p,q) process in chapter 3 has enabled us to derive
the YW estimators for these processes. As a further example, these estimators are
obtained for an INARMA(2,2) process.
We have decided not to follow the GMM method. This is because, for an INAR(1)
process, it does not outperform the maximum likelihood estimator, which is covered
by this study, in terms of MSE. For the INMA(1) process, no comparison has been
done in the literature and the one that compared CLS and GMM for INMA(2)
(Br?nn?s and Hall, 2001) does not provide results for all parameter sets.
It is worth mentioning that all of the abovementioned studies only compared the bias
and MSE of the estimates and not their impact on forecast accuracy. This will be
done in our simulation experiment (see section 8.4).
M.Mohammadipour, 2009, Chapter 6 124
Chapter 6 FORECASTING IN INARMA
MODELS
6.1 Introduction
Having discussed some stochastic properties of INARMA models, and identification
and estimation of the parameters of these models, we now investigate how these
models can be used in forecasting future values of an observed time series. This
section is organized as follows. The minimum mean square error (MMSE) forecasts
for INAR(p), INMA(q) and INARMA(p,q) processes are reviewed in section 6.2.
The lead time aggregation and forecasting of INARMA processes is then discussed
in section 6.3. Results on lead time aggregation and forecasting for INAR(1),
INMA(1), and INARMA(1,1) processes are presented. The conclusions are given in
section 6.4.
M.Mohammadipour, 2009, Chapter 6 125
6.2 MMSE Forecasts
The most common forecasting procedure discussed in the time series literature is
using the conditional expectation (Freeland and McCabe, 2004b). The main
advantage of this method, apart from being simple, is that it produces forecasts with
minimum mean square error (MMSE).
Freeland and McCabe (2004b) argue that this method does not produce coherent
forecasts for INARMA models. Coherency means that forecasts should comply with
time series restrictions, in this case being integers. Freeland and McCabe (2004b)
suggest the median of the distribution and use the ?stepahead conditional
distribution to produce coherent forecasts for the PoINAR(1) model. Bu and McCabe
(2008) present a procedure to produce hstepahead distribution forecasts for the
PoINAR(p) process using the transition probability function of the process. Jung and
Tremayne (2006b) introduce a Monte Carlo procedure to estimate the ?stepahead
forecast distribution for INAR(1) and INAR(2) processes.
This PhD research tries to apply INARMA models for intermittent demand
forecasting. We are especially interested in comparing the accuracy of forecasts
produced by INARMA methods to nonoptimal smoothingbased methods (the last
research question, p.7). For this reason, we compare the point forecasts of all
methods using the accuracy measures suggested in section 2.4.3 (including MSE).
We focus on the conditional expectation since it provides the MMSE forecasts for
INARMA methods. It will be discussed in chapter 10 that forecasting the whole
distribution can be considered as a further research avenue.
6.2.1 MMSE Forecasts for an INAR(p) Model
Minimum mean square error (MMSE) forecasts are used to find ??+?, ?= 1,2,?,?
of the process ?? based on the observed series of {?1,?,??}. The MMSE forecast of
the process is given by:
??+?=????+???,?,?1?
Equation ?61
M.Mohammadipour, 2009, Chapter 6 126
As indicated by its title, this method yields forecasts with minimum MSE. For an
INAR(p) model of Equation 319, we have:
??+?=?1??+??1 +?2??+??2 +?+????+???+?
Equation 62
where the ? values on the RHS of Equation 62 may be either actual or forecast
values (Du and Li, 1991; Jung and Tremayne, 2006b). This is shown in Figure 61
for the case where ???.
Figure ?61 hstepahead forecast for an INAR(p) model when
ph ?
This is called using a single model for all horizons. For example, for an INAR(2)
process, the hstep ahead forecast is given by:
??+?=?1??+??1 +?2??+??2 +?
Equation 63
This implies that for large ?, the forecasts converge to the unconditional mean of the
INAR(2) process that is:
??+??
?
1??1??2
Some authors suggested that using different models for different horizons can
improve forecast accuracy (Cox, 1961; Tiao and Xu, 1993; Kang, 2003). For an
AR(p) model, this is:
??+?=?1,???+?2,????1 +?+??(?),????????+1 +??,?
Equation 64
??+?
??+??1 ??+1 ? ?? ???1 ??+??? ?
??1 nonobserved terms ???+ 1 observed terms
M.Mohammadipour, 2009, Chapter 6 127
It can be seen from Equation 64 that even the order of the AR model depends on the
forecast horizon ?.
6.2.2 MMSE Forecasts for an INMA(q) Model
As studied by Br?nn?s and Hall (2001), for an INMA(q) process of Equation 336,
the MMSE onestep ahead forecast can be obtained from:
??+1 =?1??+?2???1 +?+??????+1 +?
Equation 65
The forecast error variance is:
var(??) =??1 +? ??(1???)
?
?=1
?
Note that when ? is a random variable, var?????=?2var???+??1????(?),
but when ? is given as in the above case, var?????=??1????(?).
The hstep ahead forecast when ??? is given by:
??+?=????+?+????+???+?(1 +?1 +?+???1)
Equation 66
This is shown in Figure 62. In the above equation, the ? values on the RHS can be
estimated from the previous estimated ?s and observed ?s based on Equation 336.
Figure ?62 hstepahead forecast for an INMA(q) model when
qh ?
??+?
?(??+??1) =? ?(??+1) =? ? ?? ???1 ??+??? ?
??1 nonobserved terms ???+ 1 observed terms
M.Mohammadipour, 2009, Chapter 6 128
The forecast error variance for ??? is:
var(??) =??1 +? ??
??1
?=1
+? ??(1???)
?
?=?
?
When ?>?, the ?step ahead forecast becomes:
??+?=??1 +? ??
?
?=1
?
Equation 67
with the forecast error variance of var(??) =??1 +? ??
?
?=1 ?.
6.2.3 MMSE Forecasts for an INARMA(p,q) Model
The above results can be generalized for an INARMA(p,q) process. The MMSE one
stepahead forecast is then:
??+1 =?1??+?+??????+1 +?+?1??+?+??????+1
Equation 68
The hstep ahead forecast when ??? will be:
??+?=?1??+??1 +?+????+???+?+????+?+????+???+?(?1 +?+???1)
Equation 69
where ? values on the RHS of Equation 69 may be either actual or forecast values.
When ?>?, the hstep ahead forecast becomes:
??+?=?1??+??1 +?+????+???+?? ??
?
?=0
Equation 610
where again ? values on the RHS of the above equation may be either actual or
forecast values and ?0 = 1.
M.Mohammadipour, 2009, Chapter 6 129
6.3 Forecasting over Lead Time
In this section, forecasting over a lead time is discussed. Lead time forecasting has
applications in many areas, particularly in an inventory management context, where
forecasts are needed over the period that it takes from placing an order to receiving it
from the supplier.
It can be easily seen that for an INARMA(0,0) process, the lead time aggregated
process is:
? ??+?
?+1
?=1
=??+1 +??+2 +?+??+?+1 =? ??+?
?+1
?=1
Therefore, the conditional expected value and variance of the above equation are:
??? ??+?
?+1
?=1
???= var?? ??+?
?+1
?=1
???= (?+ 1)?
which is expected as the aggregated process is the sum of (?+ 1) independent Poisson
random variables which is in fact a Poisson variable with parameter (?+ 1)?.
This section is organized as follows. First, the results of overlead time aggregation
and forecasting of INAR(1) and INMA(1) processes are presented. These results,
along with similar results for INAR(2) and INARMA(1,2) processes in Appendices
6.A and 6.B, will then help us to find the overlead time aggregation of the
INARMA(p,q) process. The corresponding results for an INARMA(1,1) process are
also provided, which will be used in chapters 8 and 9.
6.3.1 Lead Time Forecasting for an INAR(1) Model
For the INAR(1) process of ??=?????1 +??, the cumulative ? over lead time ? is
given by:
? ??+?
?+1
?=1
=??+1 +??+2 +?+??+?+1
M.Mohammadipour, 2009, Chapter 6 130
=?????+??+1?+??
2???+????+1 +??+2?
+?+???+1???+?
????+1 +?
??1???+2 +?+??+?+1?
Equation 611
Because ???+????(?+?)??, the above equation can be written as:
? ??+?
?+1
?=1
=? ? ??
1 ???
??
1
?=1
?+1
?=1
+? ? ??
2 ???+??
??
2
?=1
?+1
?=1
Equation 612
where ??
1 is the number of ?? terms in each of {??+?}?=1
?+1 in Equation 611, ??
1 is the
corresponding coefficient for each ??, ??
2 is the number of ??+?? terms in each of
{??+?}?=1
?+1 in Equation 611, and ??
2 is the corresponding coefficient for each ??+??.
All of these terms are explained below.
It can be seen that because the process is an integer autoregressive of order one, each
of {??+?}?=1
?+1 yields only one ?? in Equation 611; therefore, ??
1 = 1. The
corresponding coefficient for ?? in each of {??+?}?=1
?+1 (say ??+2) is obtained from ?
thinned the coefficient of ?? in the previous term (in this case ??+1). As a result,
??
1 =??. These coefficients are shown in Table 61.
Table ?61 Coefficients of in each of for an INAR(1) model
?= 1,?= 1 ?11
1 =?
?= 2,?= 1 ?12
1 =?2
? ?
?=?+ 1,?= 1
?1(?+1)
1 =??+1
It can be seen from Equation 611 that due to the repeated substitution of ??+?, the
number of ??+?? increases in each of {??+?}?=1
?+1 . This number, shown by ??
2, can be
obtained from ???1
2 + 1. This means that each of {??+?}?=1
?+1 (say ??+2) has one more ?
compared to the previous one (which is ??+1 in this case). The corresponding
coefficient for each ??+??, shown by ??
2 , is ? thinned the corresponding coefficient
in the previous term (???????1?
2 ). ?+?? is the subscript of innovation terms in
tY 11
?
??
l
jjtY }{
M.Mohammadipour, 2009, Chapter 6 131
each of {??+?}?=1
?+1 and from Equation 611 it can be easily seen that ?? is given by
Equation 613. All of these terms are shown in Table 62.
??=?
??(??1) for 1??????1
2
? for ???1
2 ??(??1)
? ??
?=
?
?
?
?
?
?
?
?
?
?
?
?
????(???)
? ?= 1,?,????
?
? ?
?1??(??1)
? ?=???2
? + 1,?,???2
? +???1
?
??+(??1) ?=???1
? + 1
?????(??1)
?
????(???)
? ?= 1,?,????
?
? ?
?1??(??1)
? ?=???2
? + 1,?,???2
? +???1
?
??>??(??1)
?
??
?+1 =?? ????
?+1?
?=1 ?+ (?+ 1)
??
?+1 =
?
?
?
?
?
????(???)
?+1 ?= 1,?,????
?+1
? ?
?1??(??1)
?+1 ?=???2
?+1 + 1,?,???2
?+1 +???1
?+1
??,?,?1 , 1 ?=???1
?+1 + 1,?,???1
?+1 +??
?+1
?
??=
?
?
?
?
?
{??(???)} ?= 1,?,????
?+1
? ?
{??(??1)} ?=? ????
?+1?
?=2 + 1,?, (? ????
?+1?
?=2 ) +???1
?+1
???,?,??1,? ?=? ????
?+1?
?=1 + 1,?,??
?+1
?
Now, in order to find the forecast over lead time, we need to calculate the expected
value of the aggregated process given the pprevious observations.
??? ??+?
?+1
?=1
????+1,?,???1 ,???=?? ? ??
1
??
1
?=1
?+1
?=1
???+?? ? ??
2
??
2
?=1
?+1
?=1
????1 +?
+?? ? ??
?
??
?
?=1
?+1
?=1
?????+1 +?? ? ??
?+1
??
?+1
?=1
?+1
?=1
??
Equation 627
M.Mohammadipour, 2009, Chapter 6 136
6.4 Conclusions
Forecasting with an INARMA process is discussed in this chapter. The minimum
mean square error (MMSE) forecasts for INAR(p), INMA(q) and INARMA(p,q)
processes are reviewed. This includes both onestep and hstep ahead forecasts.
These forecasts are based on the conditional expected value of the process and, as
argued by McCabe and Martin (2005), these are not coherent forecasts. This means
that the results are not necessarily integers. However, using the conventional
forecasting method of conditional expectations is the most widely used approach in
the literature even for count series.
It is shown in this chapter that the aggregation of an INARMA(p,q) process with
Poisson innovations (with mean ?) over a lead time ? results in an INARMA(p,q)
process with the same autoregressive and moving average parameters and the
innovation parameter of (?+ 1)?. The lead time aggregation and forecasting for the
INARMA(p,q) process is obtained. In order to understand the implications of the
results, some examples including a range of autoregressive and moving average
processes are provided.
It will be discussed in chapter 7 that four INARMA models will be used in
simulation and empirical analysis of this thesis. These models are INARMA(0,0),
INAR(1), INMA(1), and INARMA(1,1). Therefore, the lead time forecast for the last
three processes are presented in this chapter. The lead time forecast of an
INARMA(0,0) can simply be obtained from ??+ 1??.
M.Mohammadipour, 2009, Chapter 7 137
Chapter 7 SIMULATION DESIGN
7.1 Introduction
This chapter addresses a simulation experiment based on theoretically generated
data. A modelbased simulation shows the evolution through time of a stochastic
process, represented by a mathematical model through multiple realizations of the
process. In this research, simulation is used for various reasons including:
? to assess the effects of the approximations made for the mathematical model
? to test the performance of identification methods
? to measure the accuracy of estimates of the model?s parameters
? to assess the sensitivity of forecast accuracy to control parameters such as the
number of observations and the sparsity of data (based on the INARMA
parameters)
M.Mohammadipour, 2009, Chapter 7 138
? to compare the forecasts of the mathematical model with other benchmark
methods.
The chapter is organized as follows. The reasons for conducting simulation are
discussed in section 7.2. The simulation design is defined in section 7.3, including
the range of INARMA models to be used in the simulation, the control parameters
and the performance metrics. Verification of the simulation is discussed in section
7.4 and, finally, section 7.5 provides the conclusions.
7.2 Rationale for Simulation
In chapter 4, two identification methods were discussed, namely the twostage and
the onestage methods. Simulation enables us to find the percentage of theoretically
generated INARMA time series that can be identified correctly by each of these
methods. A further application of the simulation model is to investigate the effect of
identifying an incorrect model for a specific series, or misidentification, on the
accuracy of forecasts.
The next step in the INARMA methodology is estimating the parameters of the
identified model. As explained in detail in chapter 5, Conditional Least Squares
(CLS) and YuleWalker (YW) are the two estimation methods used (CML will also
be used for the INAR(1) process). The role of simulation is to compare the results of
these methods in terms of: (i) how close are the estimates to the real parameters,
which are known when theoretically generated data are being used, and (ii) which
estimation method results in better forecasts.
The simulation model will be based on the assumption that the distribution of the
innovations is Poisson. Although other distributions have been proposed in the
literature including compound Poisson (McKenzie, 2003), negative Binomial
(McKenzie, 1985; AlOsh and Alzaid, 1987; Br?nn?s and Hall, 2001) and the
Geometric (McKenzie, 1986; Alzaid and AlOsh, 1988), this research only focuses
on the Poisson. The sensitivity of the results to the distributional assumption can be
analyzed but this will not be covered in this thesis. Other marginal distributions are
beyond this research?s scope. The Poisson distribution is probably the most
commonly used distribution in modelling counting processes (Alzaid and AlOsh,
M.Mohammadipour, 2009, Chapter 7 139
1990). It is the only distribution among the class of discrete selfdecomposable1
distributions which has a finite mean (Silva and Oliveira, 2004). Another property of
interest is that in the INAR(1) and INMA(q) processes, the Poisson distribution plays
a role similar to that of the Gaussian distribution in the AR(1) process. However,
Jung and Tremayne (2006b) argued that only an INAR(2)AA process with Poisson
innovations results in a process with Poisson marginal distribution and the same is
not true for an INAR(2)DL process. Another advantage of the Poisson over other
distributions is that it has only one parameter to estimate.
One of the main concerns in forecasting intermittent series is the length of available
data history. This is because in practice we may be limited by short length of history.
For example, the 3,000 series that we use in empirical analysis (see chapter 9) only
has 24 periods of monthly data. Simulation enables us to check the sensitivity of the
identification, estimation, and forecasting results to the length of the series.
Once the forecasting results have been established, simulation can be used to
compare these results with benchmark methods, Croston, SBA, and SBJ methods
(see chapter 2 for detailed discussion on benchmark methods). This includes onestep
ahead, ?step ahead, and lead time forecasts.
In a nutshell, simulation is conducted to analyze the sensitivity of results to: the
sparsity of data, the length of history, the parameters? ranges, the estimation methods,
and the effect of misidentification. It also enables us to compare the INARMA
forecasts with those of benchmark methods using different accuracy measures.
7.3 Simulation Design
7.3.1 The Range of Series
Different integer autoregressive moving average processes will be used to test the
1 A distribution with probability generating function (p.g.f) ? is called discrete selfdecomposable if:
????=??1??+?????(?) ??1 ??(0,1)
where ?? is a p.g.f. The above equation can also be written in the form of:
?=????+??
where ???? and ?? are independent and ?
? is distributed as ? (Sueutel, F. W. and K. van Harn
(1979). Discrete analogues of selfdecomposability and stability. Annals of Probability 7(5): 893
899.).
M.Mohammadipour, 2009, Chapter 7 140
mathematical findings. We consider an INAR process, an INMA process and a
mixed INARMA process. In order to test the performance of the benchmark
methods, the special case of INARMA(0,0) (or simply an i.i.d. Poisson process) is
also used. Therefore, the following four processes are assumed for this study:
INARMA(0,0), INARMA(1,0), INARMA(0,1), INARMA(1,1).
An extension to this study would be to examine higher order INARMA processes.
However, as shown in a later chapter, the simpler models ((0,0) and (1,0)) perform
very well on empirical data. Using the above models also has the benefit of having
few parameters to be estimated.
7.3.2 Producing INARMA(p,q) Series
Since it has been assumed that the innovations are Poisson distributed (??~???(?)),
we first need to generate i.i.d. Poisson random numbers.
The simulation code is written in MATLAB 6.1. Hence, we use the poissrnd function
from MATLAB?s statistics toolbox. The performance of this function is tested by the
Poisson dispersion and the score tests (see section 4.2.2) and the results confirm the
accuracy of the function.
Next, by assuming the values of autoregressive parameters {??}?=1
?
and moving
average parameters {??}?=1
?
, the autoregressive and moving average components are
generated using a Binomial random number generator. This is because, based on the
properties of binomial thinning discussed in chapter 3, ??? given ? has a binomial
distribution with parameters (?,?). Therefore, for example in an INAR(1) model
(??=?????1 +??), ?????1 is obtained from generating a random Binomial
number with parameters (???1,?). The Binomial numbers are generated using the
binornd function from the MATLAB?s statistics toolbox as a sum of Bernoulli
random variables. The performance of this function is also tested by the score test
and a builtin goodnessoffit test (based on chisquare). The results, again, support
the use of this function.
Then, the INARMA series is generated from the model:
M.Mohammadipour, 2009, Chapter 7 141
??=? ???????
?
?=1
+??+? ???????
?
?=1
Equation 71
In order to obtain a stationary series, the series is initialized with the expected value
of each process (AlOsh and Alzaid, 1987; Br?nn?s, 1994). The expected value of
the above process is given by:
?????=
??1 +? ??
?
?=1 ?
1?? ??
?
?=1
Equation 72
7.3.3 Control Parameters
The control parameters of the simulation are: the mean of the Poisson innovations
(?), autoregressive and moving average parameters ({??}?=1
? , {??}?=1
? ), the length of
the series (?), the forecast horizon (?), the length of the lead time (?), and the
benchmark methods? parameters. In this section, the ranges of these control
parameters are reviewed.
7.3.3.1 INARMA Parameters
From the definition of the thinning operation it is obvious that the autoregressive and
moving average parameters represent the chance of surviving for elements of the
process at time ??1 (????s and ????s, respectively). Therefore, these parameters are
probabilities and can only take values in the range [0,1].
Other restrictions have to be applied on the autoregressive and moving average
parameters in order to assure the stationarity and invertibility of the process. Table
71 reviews the range of values that these parameters can take for the INARMA
processes selected in section 7.3.1 (see section 3.3.8 for stationarity and invertibility
conditions of an INARMA(p,q) process).
M.Mohammadipour, 2009, Chapter 7 142
Table ?71 Range of autoregressive and moving average parameters
INARMA(p,q) models
Range of autoregressive
parameters
Range of moving average
parameters
INARMA(0,0)  
INARMA(1,0)
0??< 1
stationarity condition: ??1

INARMA(0,1) 
0??< 1
invertibility condition: ??1
INARMA(1,1)
0??< 1
stationarity condition: ??1
0??< 1
invertibility condition:
??1
The range of parameters for some simulation studies reported in the literature are
reviewed in Table 72.
Table ?72 Range of INARMA parameters studied in the literature
Study
Number of observations
?
AR or MA parameter
? or ?
Innovation
parameter
?
Number of
replications
AlOsh and Alzaid (1987) ?= 50, 75, 100, 200 ?= 0.1, 0.2, 0.9 ?= 1 ?0.5?, 3 200
Br?nn?s and Hall (2001) ?= 10 ?10?, 100 ?100?, 500 ??= 0.1, 0.5, 0.9 ?= 5 1000
Br?nn?s and Hellstr?m
(2001)
?= 50 , 100 , 200 ?= 0.5, 0.7, 0.9 ?= 5, 10 1000
Silva and Oliveira (2004) ?= 64, 128, 512, 1024 ?= 0.1, 0.5, 0.9 ?= 1, 3 200
Silva et al. (2005) ?= 25, 50, 100 ?= 0.1, 0.3, 0.7, 0.9 ?= 1, 3
Bu et al. (2008) ?= 100, 500 ??= 0.1, 0.3, 0.5, 0.7 ?= 1 1000
Based on the constraints of Table 71, and taking into account previous experiments
(Table 72), the parameter space for the four selected INARMA models used in this
thesis is shown in Table 73.
If the discrete variates are large numbers, they can be approximated by continuous
variates. It is when they are relatively small integers that using integer autoregressive
moving average models becomes justifiable (McKenzie, 2003). Therefore, the
innovation term (?) has to be defined to assure the observations are small integers.
As can be seen from Table 73, we assume a range of ?= [0.5,5] for most models.
For INARMA(0,0) we consider two other values of ?= 0.3 and ?= 20. This is to
test the CrostonSBA categorization for highly intermittent and barely intermittent
series.
M.Mohammadipour, 2009, Chapter 7 143
Table ?73 Parameter space for the selected INARMA models
INARMA(p,q)
models
Parameters
INARMA(0,0) ?= 0.3, 0.5, 0.7, 1, 3, 5, 20
INARMA(1,0)
?= 0.1,?= 0.5
?= 0.5,?= 0.5
?= 0.9,?= 0.5
?= 0.1,?= 1
?= 0.5,?= 1
?= 0.9,?= 1
?= 0.1,?= 3
?= 0.5,?= 3
?= 0.9,?= 3
?= 0.1,?= 5
?= 0.5,?= 5
?= 0.9,?= 5
INARMA(0,1)
?= 0.1,?= 0.5
?= 0.5,?= 0.5
?= 0.9,?= 0.5
?= 0.1,?= 1
?= 0.5,?= 1
?= 0.9,?= 1
?= 0.1,?= 3
?= 0.5,?= 3
?= 0.9,?= 3
?= 0.1,?= 5
?= 0.5,?= 5
?= 0.9,?= 5
INARMA(1,1)
?= 0.1,?= 0.1,?= 0.5
?= 0.1,?= 0.9,?= 0.5
?= 0.5,?= 0.5,?= 0.5
?= 0.9,?= 0.1,?= 0.5
?= 0.1,?= 0.1,?= 1
?= 0.1,?= 0.9,?= 1
?= 0.5,?= 0.5,?= 1
?= 0.9,?= 0.1,?= 1
?= 0.1,?= 0.1,?= 5
?= 0.1,?= 0.9,?= 5
?= 0.5,?= 0.5,?= 5
?= 0.9,?= 0.1,?= 5
7.3.3.2 Length of Series
Different lengths of series are considered in order to test the sensitivity of results
(identification, estimation, and forecasts accuracy) to the length of history. Because
in real cases, we are often restricted by the short lengths of history (as will be seen in
empirical analysis of this thesis) we use ?= 24, 36, 48, 96. Only for investigating
the accuracy of estimates in terms of bias and MSE (section 8.3), ?= 500 is also
added to the above cases.
The first half of the observations is assigned for identification and estimation, and is
referred to as the estimation period. This also includes the benchmark methods of
Croston, SBA and SBJ. The second half is left for forecasting and is called the
performance period.
7.3.3.3 Forecast Horizon and Lead Time
Threestep and sixstep ahead forecasts are calculated in addition to onestep ahead
forecasts. The lead times considered are also three and six periods.
The number of replications is set to 1000. However, for the INARMA(0,0) model
with very small mean (??1) more replications are used to reduce the sampling
M.Mohammadipour, 2009, Chapter 7 144
error. Therefore, the number of replications for ?= 0.3, 0.5 is 30,000 and for
?= 0.7, 1 is 10,000.
7.3.3.4 Benchmark Methods??Parameters
As discussed in chapter 2, three methods of forecasting intermittent demand are
selected to compete against the INARMA method. These methods are: Croston
(Croston, 1972), SBA (Syntetos and Boylan, 2005) and SBJ (Shale et al., 2006).
All of these methods are based on separate smoothing of demand sizes and on the
interval between positive demands using a common smoothing parameter for size
and interval. Therefore a smoothing constant needs to be selected. It has been
suggested in the literature (e.g. Brown, 1959; Croston, 1972) that, especially when
the length of history is short, it is best to use fixed values of the smoothing
parameter.
We choose two arbitrary values for smoothing parameter: ?= 0.2 and ?= 0.5. The
first value is selected because in intermittent demand context low smoothing constant
values are suggested (Syntetos and Boylan, 2005). However, as can be seen from
Table 73, some generated series have high autocorrelation; therefore ?= 0.5 is also
used.
For initialization of the methods, the first interdemand interval is used as the first
smoothed interdemand interval. For the first smoothed size, the average of the first
two positive demands is used. If fewer than two positive demands is observed in the
estimation period, the estimation period for that particular replication is extended
until two nonzero demands are observed.
7.3.4 Identification Procedure
As argued in chapter 4, the sample autocorrelation function (SACF) and partial
autocorrelation function (SPACF) of INARMA models have the same structure as
those of ARMA models and therefore can be used in identifying the moving average
M.Mohammadipour, 2009, Chapter 7 145
and autoregressive orders of the model. However, as argued in section 4.6, for
simulation purposes automated methods such as penalty functions should be used.
Jung and Tremayne (2003) argue that the first step in analysing time series of counts
is to investigate if the data exhibit any serial dependence. If such dependence does
not exist, standard methods for independent data should be used. Based on this
argument, two identification procedures were suggested in chapter 4, namely, two
stage and onestage methods.
In the twostage identification method, a LjungBox test of Equation 46 is first used
to test if data has serial dependence. The reasons for the selection of this test were
discussed in section 4.6. The second step involves using the AIC of Equation 428
(or where applicable, AICC of Equation 429) to select the appropriate model among
the three possible INARMA models (see section 4.6 for discussion on the application
of AIC of ARMA models for INARMA series).
In the onestage identification method, the first step of the previous method is
ignored. This means that the AIC is used to select among all possible INARMA
models (INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1)).
The results of these two methods will be compared in terms of the percentage of
series for which the model is identified correctly. This can be done in simulation
because the correct model from which the series is produced is known. Another
aspect that can be tested is the accuracy of forecasts obtained from each
identification method (the accuracy measures are reviewed in section 7.3.7).
No identification method can guarantee that the correct model is identified at all
times. In such cases, the effect of misidentification on the accuracy of forecasts is of
interest. This will also be tested in the next chapter.
7.3.5 Estimation of Parameters
As discussed in chapter 5, YuleWalker (YW) and conditional least squares (CLS)
methods are used for estimation of parameters of INAR(1), INMA(1), and
INARMA(1,1) processes. Because the conditional maximum likelihood (CML)
M.Mohammadipour, 2009, Chapter 7 146
estimation has been established only for INAR(p) processes, we can use it only for
an INAR(1) process. For an INARMA(0,0) process, the three estimation methods
result in the same estimator. All of these estimators are given in chapter 5.
The performance of these estimators has been tested in the literature (AlOsh and
Alzaid, 1987; Br?nn?s, 1994; Bu, 2006). However, this has been done for sample
sizes greater than 50. Because we also use smaller numbers of observations, we
compare the performance of these estimators. Since the true values of the parameters
are known in simulation, we compare the bias and the MSE of the estimates. The
impact of estimates on forecast accuracy is also an important issue that has not been
looked at before and is covered in this thesis.
7.3.6 Forecasting Method
This thesis focuses on comparing the accuracy of forecasts produced by INARMA
and benchmark methods. The accuracy measures include MSE and MASE (see
section 7.3.7). We use the conditional expected value which yields minimum mean
square error (MMSE) forecasts. It has been argued in the literature that this method is
not coherent in that it does not produce integervalued forecasts (Freeland and
McCabe, 2004b). Other methods such as conditional median, Markov Chains, and
bootstrapping have been suggested to tackle this problem (Cardinal et al., 1999;
Freeland and McCabe, 2004b; Jung and Tremayne, 2006b; Bu and McCabe, 2008).
However, none of these methods produces MMSE forecasts. Also, those methods
that produce the distribution forecast instead of point forecasts are not used for our
comparison. Such methods are definitely useful for competing against bootstrap
methods for intermittent demand forecasting such as Willemain?s bootstrap
(Willemain et al., 2004) and can be considered as a future line of study.
The ?step ahead forecasts and lead time forecasts for INARMA models are
discussed in chapter 6 in detail. The Croston, SBA and SBJ forecasts are given in
chapter 2. For these methods, the ?step ahead forecasts are the same as the onestep
ahead forecasts and the lead time forecast is simply the onestep ahead forecast
multiplied by the length of lead time.
M.Mohammadipour, 2009, Chapter 7 147
Finally, two cases regarding the forecast timing are considered: all points in time or
focusing on those periods immediately after a positive demand is occurred (issue
points). This is because Croston?s method is designed to outperform the SES for issue
points and it is of interest to test the performance of the INARMA method for issue
points.
7.3.7 Performance Metrics
In this section, the performance measures to be used in the simulation are reviewed.
In the identification stage, where we want to examine the capability of the two
identification procedures, the percentage of correctly identified models is calculated.
The accuracy of the forecasts produced by each identification method is also
compared.
In order to compare the estimation methods (YW, CLS and CML only for INAR(1)),
the bias (using Mean Error) and Mean Square Error (MSE) of parameters? estimates
are calculated. The performance of the estimates is also compared in terms of their
impact on forecast accuracy.
Finally, selecting the appropriate forecasting accuracy measure is an important issue
for intermittent processes. As discussed in section 2.4, the fact that intermittent
demand series include zeros, makes some of the conventional measures
inappropriate. The following accuracy measures are used in this thesis: Mean Error
(ME), Mean Square Error (MSE), Mean Absolute Scaled Error (MASE) for
simulation, along with Percentage Better (PB) of MASE and Relative Geometric
RootMeanSquare Error (RGRMSE) for empirical analysis (see section 2.4.3 for
more details).
7.4 Verification
Verification is the process to make sure that no programming error has been made
(Kleijnen and Groenendaal, 1992). This can be done by calculating some intermediate
results manually and comparing them with the results obtained by the program. This
M.Mohammadipour, 2009, Chapter 7 148
is called tracing (Kleijnen and Groenendaal, 1992). Eyeballing or reading through the
code and looking for bugs is another way of verification (Kleijnen and Groenendaal,
1992). The following steps have been done in order to verify the simulation model:
? The MATLAB code has been read through to make sure that the correct logic
and functions have been used.
? The intermediate and also the final results have been compared for a limited
number of replications (e.g. 20 replications) with MS Excel.
? The average and standard deviation of the generated INARMA series is
calculated and compared to the theoretical mean and standard deviation of the
process to test the generated data.
The selection of parameters was made to make sure that both highlyintermittent and
lessintermittent data are considered. Interarrival times are also obtained for each
time series.
7.5 Conclusions
In this chapter, a simulation experiment was developed to assess the accuracy of
approximations made for the mathematical analysis, to measure the accuracy of
estimates, to assess the sensitivity of forecast accuracy to control parameters, and to
compare the INARMA forecasts with those of benchmark methods.
Four integer autoregressive moving average models have been selected for the
purpose of simulation (models with ?,??1). The marginal distribution is assumed
to be Poisson. The control parameters used are: autoregressive and moving average
parameters, innovation parameter, forecast horizon, the length of lead time, and the
smoothing parameter for the benchmark methods.
As previously discussed in section 2.4.3, different accuracy measures are needed to
assess the accuracy of estimates and forecasts. The accuracy of estimates is measured
using ME and MSE. Demand being intermittent makes some forecast accuracy
measures not applicable. We have selected ME, MSE, and MASE for simulation. The
PB of MASE and RGRMSE will be added to the above measures for empirical
analysis.
M.Mohammadipour, 2009, Chapter 8 149
Chapter 8 SIMULATION RESULTS
8.1 Introduction
The simulation results are presented in this chapter. As discussed in chapter 7, the
main objective of simulation is to test whether using an INARMA model results in
better forecasts compared to benchmark methods of Croston, SyntetosBoylan
Approximation (SBA) and ShaleBoylanJohnston (SBJ). This is discussed in section
8.6. The simulation experiment also enables us to test the applicability of the
CrostonSBA categorization (Syntetos et al., 2005) when demand is an INARMA
process.
As discussed in section 7.3.1, four processes are simulated: INARMA(0,0),
INAR(1), INMA(1), and INARMA(1,1). Based on the arguments in section 2.4, the
ME, MSE and MASE of the forecasts are compared to those of Croston, SBA and
M.Mohammadipour, 2009, Chapter 8 150
SBJ methods. A range of INARMA parameters and different lengths of history are
used (see section 7.3.3).
The estimation methods used in this study are YW and CLS for INAR(1), INMA(1)
and INARMA(1,1) processes and CML for INAR(1) (see Chapter 5 for detailed
discussion). As another objective of simulation, the accuracy of parameters? estimates
needs to be tested. The performance of the estimators can be tested not only by
comparing the accuracy of the estimates, but also by comparing their impact on the
forecast accuracy. The former has been undertaken by comparing the ME and MSE of
the parameters? estimates (see section 8.3 and Appendix 8.A). The latter, the results
of which are presented in section 8.4, has been accomplished by comparing the ME,
MSE and MASE of forecasts obtained using each estimation method.
The chapter is structured as follows. Details of the simulation design are reviewed in
section 8.2. Sections 8.3 and 8.4 compare the accuracy of different estimates of the
parameters of INARMA processes. The CrostonSBA categorization (Syntetos et al.,
2005) for data produced by INARMA models is validated in section 8.5. The
INARMA forecasts are then compared to the benchmark methods in section 8.6. It is
first assumed that the order of the INARMA model is known. The results for the case
where the order needs to be identified are presented in section 8.6.2. The leadtime
forecasts are compared in section 8.6.3 and the conclusions are provided in section
8.7.
8.2 Details of Simulation
As mentioned in chapter 7, the number of replications is set to 1000. However, in
order to reduce the sampling error for the case of INARMA(0,0) process with small
parameters, (?= 0.3, 0.5) and (?= 0.7, 1), 30,000 and 10,000 replications are used,
respectively.
It has been suggested in the literature that, especially with short length of history, it is
best to use fixed values of smoothing parameters (Brown, 1959; Croston, 1972).
Because with intermittent demand, data history is short in most cases, we use two
arbitrary values for the smoothing parameter for Croston, SBA and SBJ (?= 0.2 and
M.Mohammadipour, 2009, Chapter 8 151
?= 0.5).
As summarized in chapter 7, the initialization for Croston, SBA and SBJ is based on
using the first interdemand interval as the first smoothed interdemand interval and
the average of the first two nonzero observations as the first smoothed size. The
observations are divided into two categories: estimation period and performance
period. Initialization and estimation of parameters are conducted in the estimation
period and the estimates? accuracy and forecasting accuracy are assessed in the
performance period. If at least two nonzero demands are observed in the estimation
period, the first half of the observations is assigned for the estimation period and the
other half for the performance period. However, if fewer than two nonzero demands
are observed in the estimation period, this period will be extended until the second
nonzero demand is observed.
In order to obtain a stationary series, we initialize the INARMA methods with the
expected value of each model. As discussed in chapter 7, the forecasting accuracy is
obtained for both cases of all points in time and issue points (i.e. after a positive
demand is observed). Finally, if there is no nonzero observation in the performance
period, the error measures for issue points are excluded (only for the corresponding
replication). If the insample MAE is zero, the MASE for that replication is
excluded.
8.3 Accuracy of INARMA Parameter Estimates
As previously discussed in chapter 5, two methods (YW and CLS) have been used to
estimate the parameters of all four INARMA processes. In this section, the accuracy
of these parameter estimates is evaluated using MSE. Out of the four INARMA
processes of this study, only three are included for comparison of estimation
methods: INAR(1), INMA(1) and INARMA(1,1). The YW, CLS and CML estimates
for INARMA(0,0) are the same (see section 5.2).
As previously mentioned, CML is also used in addition to YW and CLS in order to
estimate the parameters of an INAR(1) process. The reason for excluding CML for
other processes is that the maximum likelihood functions for INMA(1) and
M.Mohammadipour, 2009, Chapter 8 152
INARMA(1,1) processes have not been developed in the literature (see Chapter 5).
The parameters may fall out of the region [0,1]. In order to tackle this issue, the
parameters are set equal to their closest boundary value in each case (Br?nn?s and
Hall, 2001).
The accuracy of YW, CLS and CML estimates of the parameters of an INAR(1)
process for the case of ?= 24 are compared in Table ?81. For high values of ? and
also when the mean of the process, ?/(1??), is high, the CML becomes
computationally expensive.
Table 81 MSE of YW, CLS and CML estimates for INAR(1) series when
Parameters
? ?
YW CLS CML YW CLS CML
?= 0.1,?= 0.5 0.0203 0.0238 0.0337 0.0391 0.0394 0.0447
?= 0.5,?= 0.5 0.0772 0.0762 0.0734 0.1381 0.1337 0.1043
?= 0.9,?= 0.5 0.1436 0.1056 0.0042 3.9057 2.8683 0.0891
?= 0.1,?= 1 0.0196 0.0226 0.0364 0.0877 0.0910 0.1104
?= 0.5,?= 1 0.0723 0.0704 0.0631 0.4136 0.3950 0.3125
?= 0.9,?= 1 0.1429 0.1067 0.0028 14.7487 11.0440 0.2526
?= 0.1,?= 3 0.0188 0.0215 0.0419 0.4225 0.4509 0.6832
?= 0.5,?= 3 0.0716 0.0684 0.0658 2.9762 2.8352 2.6277
?= 0.9,?= 3 0.1462 0.1124 0.0024 134.0715 102.9696 2.0292
?= 0.1,?= 5 0.0197 0.0227 0.0390 0.9650 1.0462 1.6005
?= 0.5,?= 5 0.0710 0.0686 0.0606 7.6814 7.4755 6.4135
The results confirm that, as suggested by AlOsh and Alzaid (1987), the MSE of
estimates produced by CML is generally less than that of YW and CLS (with the
exception of the cases where ?= 0.1). However, it will be seen in a later section that
the results of CML in terms of its effect on forecast accuracy are not very far from
those by YW and CLS. This is also true for those cases in Table ?81 that the MSE of
CML is much less than that of the other methods (e.g. ?= 0.9 and ?= 3).
The results of comparing the MSE of YW and CLS estimates of the parameters of
INAR(1), INMA(1) and INARMA(1,1) processes are shown in Table ?82, Table ?83,
and Table ?84, respectively.
AlOsh and Alzaid (1987) suggest that the accuracy of YW and CLS estimates for
parameters of an INAR(1) process are close. The results of Table ?82 confirm this
24?n
M.Mohammadipour, 2009, Chapter 8 153
when the number of observations is high. However, for fewer observations, the
difference is high when the autoregressive parameter is high.
Table 82 Accuracy of YW and CLS estimates for INAR(1) series
Parameters
MSE(???)/MSE(????) MSE(???)/MSE(????)
?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96
?=
500
?= 0.1,?= 0.5 0.8402 0.8978 0.9351 0.9700 1.0000 0.9868 0.9918 0.9890 1.0000 1.0000
?= 0.5,?= 0.5 1.0291 1.0291 1.0199 1.0248 1.0000 1.0408 1.0289 1.0194 1.0182 1.0000
?= 0.9,?= 0.5 1.3036 1.3487 1.3063 1.2404 1.1250 1.3089 1.3355 1.2935 1.2440 1.0874
?= 0.1,?= 1 0.8522 0.9222 0.9388 0.9691 1.0000 0.9655 0.9728 0.9762 0.9891 1.0000
?= 0.5,?= 1 1.0255 1.0227 1.0278 1.0286 1.0000 1.0213 1.0348 1.0227 1.0230 1.0000
?= 0.9,?= 1 1.3502 1.2990 1.3029 1.2294 1.1429 1.3550 1.3101 1.3004 1.2291 1.0957
?= 0.1,?= 3 0.8610 0.9102 0.9379 0.9655 1.0000 0.9228 0.9422 0.9644 0.9806 0.9946
?= 0.5,?= 3 1.0319 1.0349 1.0224 1.0148 1.0000 1.0317 1.0336 1.0307 1.0185 1.0024
?= 0.9,?= 3 1.3285 1.3063 1.3038 1.2526 1.1429 1.3273 1.3012 1.3010 1.2485 1.0954
?= 0.1,?= 5 0.8649 0.9118 0.9416 0.9670 1.0000 0.9094 0.9370 0.9565 0.9776 0.9959
?= 0.5,?= 5 1.0290 1.0396 1.0224 1.0216 1.0000 1.0355 1.0400 1.0240 1.0204 1.0068
?= 0.9,?= 5 1.3288 1.2945 1.3123 1.2376 1.1429 1.3236 1.2904 1.3110 1.2391 1.0960
The results show that for an INAR(1) process, when the number of observations is
small, for high values of ?, CLS produces much better estimates for both ? and ? in
terms of MSE (up to 35 percent improvement in MSE). On the other hand, for small
values of ?, YW results in better estimates (up to 16 percent improvement in MSE).
The results of section 8.4 show that this is also true for the accuracy of forecasts
produced by these estimates.
The MSE of ? for both YW and CLS estimates increases with an increase in ? but
this is not necessarily the case for the MSE of ? (see Appendix 8.A). This confirms
the argument by AlOsh and Alzaid (1987).
The results of Table ?83 show that for an INMA(1) series, for a small number of
observations, CLS has smaller MSE than YW except for the case of ?= 0.9. When
the number of observations increases, for high values of ?, the MSE of YW
estimates decreases with a greater pace compared to CLS. However, as will be
discussed in section 8.4, it does not have a great effect on the accuracy of forecasts
produced by each method. The MSE of ? for both YW and CLS estimates increases
with an increase in ? but the same is not necessarily true for the MSE of ? (see
Appendix 8.A).
M.Mohammadipour, 2009, Chapter 8 154
Table 83 Accuracy of YW and CLS estimates for INMA(1) series
Parameters
MSE(???)/MSE(????) MSE(???)/MSE(????)
?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500
?= 0.1,?= 0.5 2.0558 1.9491 2.0409 1.9897 1.3929 0.9838 1.0084 1.0265 1.0510 1.0500
?= 0.5,?= 0.5 1.1761 1.2907 1.1692 1.1307 0.5213 0.9008 0.9189 0.9032 0.8563 0.6098
?= 0.9,?= 0.5 0.8054 0.6843 0.6217 0.5340 0.2336 0.9462 0.8521 0.8112 0.7927 0.5000
?= 0.1,?= 1 1.9449 2.0335 2.0395 1.7292 1.3793 0.9952 1.0249 1.0518 1.0909 1.0962
?= 0.5,?= 1 1.1879 1.1665 1.2214 1.1247 0.5354 0.8940 0.9045 0.8977 0.8772 0.5968
?= 0.9,?= 1 0.7304 0.6812 0.5904 0.4636 0.2263 0.8650 0.8296 0.8080 0.7352 0.4012
?= 0.1,?= 3 2.2829 2.1860 2.1367 1.8646 1.4615 1.0799 1.1157 1.1649 1.2067 1.1815
?= 0.5,?= 3 1.1216 1.1679 1.1264 1.0872 0.4928 0.8906 0.9078 0.9011 0.8400 0.5050
?= 0.9,?= 3 0.6900 0.6048 0.5664 0.3868 0.1909 0.8005 0.7554 0.7359 0.6084 0.3300
?= 0.1,?= 5 2.4159 2.4242 2.2715 1.8780 1.4074 1.1174 1.2171 1.2425 1.2495 1.1940
?= 0.5,?= 5 1.0589 1.0776 1.1294 1.0726 0.4634 0.8654 0.8406 0.8873 0.8217 0.4519
?= 0.9,?= 5 0.6060 0.5175 0.4474 0.3523 0.1727 0.7006 0.6274 0.5856 0.5120 0.2597
The results of Table ?84 show that, for INARMA(1,1) series, CLS produces better
estimates especially when the number of observations is small and the autoregressive
parameter is high. This is also true for the accuracy of forecasts produced by CLS
compared to those by YW (as shown later).
To conclude, for INAR(1), INMA(1), and INARMA(1,1) processes, the
autoregressive and moving average parameters and the number of observations
determine which estimation method produces more accurate estimates. For an
INAR(1) process, CLS outperforms YW for high values of ?. The same is generally
true for an INMA(1) process with low values of ? and small number of observations.
Finally, for an INARMA(1,1) process, CLS generally produces better estimates than
YW with a few exceptions.
8.4 Forecasting Accuracy of INARMA Estimation Methods
As previously discussed in chapter 5, two methods (CLS and YW) have been used to
estimate the parameters of all four INARMA processes. In this section, the accuracy of
these estimates in terms of their effect on forecast accuracy is evaluated. The forecast
accuracy is measured by ME, MSE, and MASE (see section 2.4 for detailed
discussion). We focus on MSE in this section. MSE is specially selected because of its
theoretical tractability. Also due to the fact that data is theoretically generated, the scale
dependency problem is not an issue when we average across multiple series.
M.Mohammadipour, 2009, Chapter 8 155
Table 84 Accuracy of YW and CLS estimates for INARMA(1,1) series
Parameters
MSE(???)/MSE(????) MSE(???)/MSE(????) MSE(???)/MSE(????)
?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500
?= 0.1,?= 0.1,?= 0.5 4.8614 5.4738 6.0749 7.0578 3.4853 1.7669 1.9854 1.9831 1.9153 1.6269 2.0720 2.6889 3.1239 3.9265 3.7576
?= 0.1,?= 0.9,?= 0.5 0.8613 0.7344 0.6736 0.6839 0.9770 0.6373 0.5349 0.4602 0.3182 0.1052 1.0420 0.9860 0.9155 0.7370 0.3245
?= 0.5,?= 0.5,?= 0.5 1.3778 1.5195 1.5427 1.4870 1.1607 1.2698 1.2589 1.3484 1.3281 0.5957 1.0321 0.9228 0.8333 0.6433 0.2514
?= 0.9,?= 0.1,?= 0.5 2.1067 2.0306 1.8965 1.5271 1.2222 26.3171 27.4400 28.9136 20.4444 3.8571 1.2033 1.1088 1.0270 0.9594 0.8696
?= 0.1,?= 0.1,?= 1 4.9796 6.3395 6.5819 5.9756 3.6176 2.0247 1.8398 1.7630 1.7983 1.5588 2.7496 3.5359 3.9433 4.7348 4.6538
?= 0.1,?= 0.9,?= 1 0.8243 0.5925 0.6154 0.6291 0.8710 0.5975 0.4733 0.4273 0.2969 0.1013 1.0431 0.9933 0.9032 0.7263 0.2915
?= 0.5,?= 0.5,?= 1 1.6988 1.7842 1.7343 1.6031 1.2157 1.0136 1.0992 1.1177 1.0436 0.5140 1.0337 0.9391 0.8374 0.6111 0.2446
?= 0.9,?= 0.1,?= 1 2.3724 2.2172 2.0000 1.7658 1.3750 43.0375 35.2133 31.3239 19.9841 2.8772 1.1029 1.0416 0.9534 0.9135 0.8853
?= 0.1,?= 0.1,?= 5 5.2880 6.5479 7.3636 8.0300 3.3701 2.5138 2.3421 2.1606 1.9722 1.6857 4.4277 5.6560 6.6359 8.6472 5.6930
?= 0.1,?= 0.9,?= 5 0.5406 0.4332 0.3695 0.3471 0.4227 0.4187 0.3415 0.2803 0.1872 0.0582 1.1328 1.0459 0.9843 0.7719 0.2663
?= 0.5,?= 0.5,?= 5 2.2217 2.3401 2.5780 2.1421 1.0517 0.8303 0.8293 0.8479 0.7495 0.4093 0.9961 0.9877 0.9281 0.7039 0.2679
?= 0.9,?= 0.1,?= 5 2.5276 2.3845 2.2099 1.8218 1.5000 45.5789 37.9867 31.5135 17.3108 2.4211 1.0838 0.9427 0.8756 0.8296 0.8635
M.Mohammadipour, 2009, Chapter 8 156
The effect of CLS and YW estimates on onestep ahead forecasts is presented in
Table 85, Table 86, and Table 87. The results for threestep ahead and sixstep
ahead forecasts are presented in Table 88 to Table ?813.
For an INAR(1) process, the results of YW and CLS are compared to those of CML
for ?= 24. For longer length of history, the CML results become computationaly
expensive. The results for an INAR(1) process show that when the history is short
and data is highly autocorrelated (?= 0.9), CLS produces more accurate forecasts
(up to 11 percent improvement in MSE) than YW. For ??0.5, YW produces better
forecasts, but the magnitude of improvement is small (up to a maximum of 3 percent
improvement in MSE). The results also confirm that when the number of
observations increases, the two methods yield very similar forecast errors (AlOsh
and Alzaid, 1987; Bu, 2006).
Table 85 Onestep ahead forecast error comparison (YW, CLS and CML) for INAR(1) series
Parameters
MSEYW /MSECLS MSEYW /MSECML
?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24
?= 0.1,?= 0.5 0.9752 0.9841 0.9936 0.9978 1.0000 0.9727
?= 0.5,?= 0.5 0.9847 0.9954 0.9974 0.9999 1.0001 0.9799
?= 0.9,?= 0.5 1.1087 1.0957 1.0560 1.0248 1.0010 1.2084
?= 0.1,?= 1 0.9774 0.9926 0.9956 0.9985 0.9999 0.9751
?= 0.5,?= 1 0.9871 1.0006 0.9984 0.9995 1.0001 0.9899
?= 0.9,?= 1 1.1268 1.0836 1.0637 1.0249 1.0012 1.2329
?= 0.1,?= 3 0.9842 0.9935 0.9962 0.9987 0.9999 0.9694
?= 0.5,?= 3 0.9877 1.0009 1.0019 1.0000 1.0001 0.9850
?= 0.9,?= 3 1.0993 1.0807 1.0568 1.0228 1.0009 1.2605
?= 0.1,?= 5 0.9849 0.9922 0.9960 0.9987 0.9999 0.9687
?= 0.5,?= 5 0.9925 1.0000 1.0004 1.0003 1.0001 1.0019
?= 0.9,?= 5 1.1133 1.0866 1.0647 1.0252 1.0011 
As noted previously, it is computationally expensive to calculate CML when the mean
of the process is high. Therefore, no result is presented for the last case in Table ?85.
The results also show that, except for the case where the autoregressive parameter is
high, YW forecasts have smaller MSE than CML forecasts for ?= 24. The above
discussion about YW and CLS suggests that for such cases CLS is better than YW,
but the results show that CML is still better than CLS for these cases.
The results for an INMA(1) process show that the forecast accuracy of YW and CLS
estimates are generally close. When the history is short, CLS produces better
M.Mohammadipour, 2009, Chapter 8 157
forecasts for lower values of ? (up to 1.4 percent improvement in MSE). As shown in
Table ?86, for high values of ?, YW outperforms CLS (up to 3 percent improvement
in MSE).
Table 86 Onestep ahead forecast error comparison (YW and CLS) for INMA(1) series
Parameters
MSEYW /MSECLS
?= 24 ?= 36 ?= 48 ?= 96 ?= 500
?= 0.1,?= 0.5 1.0000 1.0003 1.0008 1.0004 1.0000
?= 0.5,?= 0.5 1.0065 1.0080 1.0054 1.0034 1.0008
?= 0.9,?= 0.5 1.0137 1.0156 1.0111 1.0069 1.0013
?= 0.1,?= 1 0.9998 1.0009 1.0005 1.0004 1.0001
?= 0.5,?= 1 1.0058 1.0040 1.0045 1.0037 1.0031
?= 0.9,?= 1 1.0084 1.0062 1.0075 1.0019 1.0003
?= 0.1,?= 3 0.9985 0.9999 1.0003 1.0004 1.0000
?= 0.5,?= 3 1.0020 1.0032 1.0006 1.0019 0.9998
?= 0.9,?= 3 0.9930 0.9888 0.9915 0.9914 0.9944
?= 0.1,?= 5 0.9954 0.9986 0.9992 0.9999 0.9999
?= 0.5,?= 5 0.9972 0.9957 0.9964 0.9990 0.9989
?= 0.9,?= 5 0.9888 0.9850 0.9857 0.9835 0.9885
For an INARMA(1,1) process the results show that CLS always produces better
forecasts than YW. As shown in Table ?87, when ??0.5, CLS outperforms YW by
up to 20 percent. The difference is much greater when ?= 0.9 (up to 90 percent
improvement in MSE). However, with an increase in the number of observations, the
two methods become closer, especially for ?= 0.9.
Table 87 Onestep ahead forecast error comparison (YW and CLS) for INARMA(1,1) series
Parameters
MSEYW /MSECLS
?= 24 ?= 36 ?= 48 ?= 96 ?= 500
?= 0.1,?= 0.1,?= 0.5 1.0850 1.1076 1.1175 1.0991 1.0259
?= 0.1,?= 0.9,?= 0.5 1.0273 1.0556 1.0551 1.0499 1.0185
?= 0.5,?= 0.5,?= 0.5 1.0633 1.0842 1.0805 1.0467 1.0087
?= 0.9,?= 0.1,?= 0.5 1.4006 1.2431 1.1654 1.0475 1.0032
?= 0.1,?= 0.1,?= 1 1.0992 1.1077 1.1015 1.0874 1.0252
?= 0.1,?= 0.9,?= 1 1.0527 1.0624 1.0700 1.0568 1.0188
?= 0.5,?= 0.5,?= 1 1.1293 1.1172 1.1031 1.0565 1.0104
?= 0.9,?= 0.1,?= 1 1.4797 1.2853 1.1771 1.0561 1.0031
?= 0.1,?= 0.1,?= 5 1.1110 1.1142 1.1175 1.1074 1.0246
?= 0.1,?= 0.9,?= 5 1.1254 1.1285 1.1397 1.1273 1.0693
?= 0.5,?= 0.5,?= 5 1.1939 1.1694 1.1603 1.0789 1.0232
?= 0.9,?= 0.1,?= 5 1.9098 1.4115 1.2442 1.0676 1.0042
Based on the above results for onestep ahead forecasts, for INMA(1), YW and CLS
M.Mohammadipour, 2009, Chapter 8 158
are close. For INAR(1), when the history is short and the autoregressive parameter is
high, CLS is considerably better than YW. But when ??0.5, the difference is much
smaller. For INARMA(1,1) and especially for short history, CLS estimates produce
better results than YW.
Although the above results are based on MSE, using MASE produces similar results
(see Appendix 8.B). For INAR(1), CLS produces more accurate forecasts (up to 9
percent improvement in MASE) when the history is short and the autoregressive
parameter is high (?= 0.9). For ??0.5, YW produces better forecasts, but the
magnitude of improvement is small (up to a maximum of 3 percent improvement in
MASE). For an INMA(1) process, the forecasting accuracy of YW and CLS
forecasts using MASE are very close. For an INARMA(1,1) process, CLS produces
better forecasts than YW in most of the cases (up to 30 percent improvement in
MASE). Finally, the results confirm that when the number of observations increases,
the two methods become very close in terms of MASE.
The results for threestep and sixstep ahead forecasts for INAR(1) process are
shown in Table ?88 and Table ?89. Although the threestep ahead results follow the
same pattern as onestep ahead forecasts, both threestep and sixstep ahead forecasts
produced by YW and CLS estimates are very close.
The results of YW and CLS threestep and sixstep ahead forecasts for INMA(1)
process are presented in Table ?810 and Table ?811. It can be seen that the two
estimation methods result in very close forecasts.
Table 88 Threestep ahead forecast error comparison (YW and CLS) for INAR(1) series
Parameters
MSEYW /MSECLS
?= 24 ?= 36 ?= 48 ?= 96
?= 0.1,?= 0.5 0.9988 0.9992 0.9997 0.9998
?= 0.5,?= 0.5 0.9906 0.9993 1.0009 1.0013
?= 0.9,?= 0.5 1.0081 1.0189 1.0229 1.0382
?= 0.1,?= 1 0.9989 0.9996 0.9996 1.0000
?= 0.5,?= 1 0.9965 0.9995 1.0001 1.0012
?= 0.9,?= 1 1.0181 1.0210 1.0170 1.0367
?= 0.1,?= 3 0.9982 0.9994 0.9998 0.9999
?= 0.5,?= 3 0.9933 0.9995 1.0003 1.0014
?= 0.9,?= 3 1.0070 1.0135 1.0245 1.0362
?= 0.1,?= 5 0.9990 0.9998 0.9999 0.9999
?= 0.5,?= 5 0.9938 0.9997 1.0005 1.0014
?= 0.9,?= 5 1.0033 1.0157 1.0118 1.0352
M.Mohammadipour, 2009, Chapter 8 159
Table 89 Sixstep ahead forecast error comparison (YW and CLS) for INAR(1) series
Parameters
MSEYW /MSECLS
?= 24 ?= 36 ?= 48 ?= 96
?= 0.1,?= 0.5 0.9986 0.9995 0.9998 1.0000
?= 0.5,?= 0.5 0.9848 0.9938 0.9979 0.9993
?= 0.9,?= 0.5 0.9701 0.9786 0.9936 1.0139
?= 0.1,?= 1 0.9989 0.9997 0.9997 0.9999
?= 0.5,?= 1 0.9833 0.9936 0.9969 0.9994
?= 0.9,?= 1 0.9509 0.9723 0.9897 1.0138
?= 0.1,?= 3 0.9987 0.9995 0.9998 0.9999
?= 0.5,?= 3 0.9907 0.9918 0.9980 0.9994
?= 0.9,?= 3 0.9576 0.9679 0.9880 1.0175
?= 0.1,?= 5 0.9984 0.9996 0.9996 0.9999
?= 0.5,?= 5 0.9883 0.9952 0.9960 0.9992
?= 0.9,?= 5 0.9566 0.9738 0.9854 1.0101
Table 810 Threestep ahead forecast error comparison (YW and CLS) for INMA(1) series
Parameters
MSEYW /MSECLS
?= 24 ?= 36 ?= 48 ?= 96
?= 0.1,?= 0.5 0.9940 0.9957 0.9980 0.9988
?= 0.5,?= 0.5 0.9895 0.9909 0.9939 0.9967
?= 0.9,?= 0.5 0.9811 0.9886 0.9918 0.9959
?= 0.1,?= 1 0.9948 0.9964 0.9988 0.9984
?= 0.5,?= 1 0.9934 0.9916 0.9929 0.9968
?= 0.9,?= 1 0.9774 0.9851 0.9896 0.9954
?= 0.1,?= 3 0.9947 0.9961 0.9968 0.9981
?= 0.5,?= 3 0.9878 0.9940 0.9953 0.9963
?= 0.9,?= 3 0.9951 0.9917 0.9932 0.9953
?= 0.1,?= 5 0.9927 0.9957 0.9976 0.9985
?= 0.5,?= 5 0.9940 0.9935 0.9948 0.9988
?= 0.9,?= 5 0.9970 0.9998 0.9970 0.9978
Table 811 Sixstep ahead forecast error comparison (YW and CLS) for INMA(1) series
Parameters
MSEYW /MSECLS
?= 24 ?= 36 ?= 48 ?= 96
?= 0.1,?= 0.5 0.9927 0.9955 0.9970 0.9984
?= 0.5,?= 0.5 0.9846 0.9946 0.9925 0.9973
?= 0.9,?= 0.5 0.9798 0.9953 0.9942 0.9960
?= 0.1,?= 1 0.9921 0.9970 0.9975 0.9989
?= 0.5,?= 1 0.9907 0.9908 0.9939 0.9955
?= 0.9,?= 1 0.9821 0.9844 0.9909 0.9945
?= 0.1,?= 3 0.9929 0.9931 0.9968 0.9989
?= 0.5,?= 3 0.9917 0.9911 0.9942 0.9963
?= 0.9,?= 3 0.9919 0.9936 0.9953 0.9934
?= 0.1,?= 5 0.9935 0.9954 0.9969 0.9980
?= 0.5,?= 5 0.9919 0.9910 0.9967 0.9984
?= 0.9,?= 5 1.0013 0.9958 0.9935 0.9980
M.Mohammadipour, 2009, Chapter 8 160
For an INARMA(1,1) process, as can be seen from Table ?812 and Table 813, CLS
does not always produce better forecasts than YW. For high number of observations,
the results are close.
Table 812 Threestep ahead forecast error comparison (YW and CLS) for INARMA(1,1) series
Parameters
MSEYW /MSECLS
?= 24 ?= 36 ?= 48 ?= 96
?= 0.1,?= 0.1,?= 0.5 1.0301 1.0092 1.0051 1.0020
?= 0.1,?= 0.9,?= 0.5 0.9986 0.9928 0.9977 0.9973
?= 0.5,?= 0.5,?= 0.5 0.9752 0.9976 0.9965 1.0009
?= 0.9,?= 0.1,?= 0.5 0.8939 0.9184 0.9467 1.0027
?= 0.1,?= 0.1,?= 1 1.0217 1.0082 1.0100 1.0022
?= 0.1,?= 0.9,?= 1 0.9825 0.9949 0.9958 0.9968
?= 0.5,?= 0.5,?= 1 0.9531 1.0014 1.0013 1.0004
?= 0.9,?= 0.1,?= 1 0.9139 1.0007 0.9974 1.0353
?= 0.1,?= 0.1,?= 5 1.0147 1.0143 1.0072 1.0003
?= 0.1,?= 0.9,?= 5 0.9696 0.9956 0.9892 0.9921
?= 0.5,?= 0.5,?= 5 0.9635 0.9725 0.9923 0.9919
?= 0.9,?= 0.1,?= 5 1.1665 1.0307 1.0383 1.0639
Table 813 Sixstep ahead forecast error comparison (YW and CLS) for INARMA(1,1) series
Parameters
MSEYW /MSECLS
?= 24 ?= 36 ?= 48 ?= 96
?= 0.1,?= 0.1,?= 0.5 1.0094 1.0088 1.0050 1.0006
?= 0.1,?= 0.9,?= 0.5 0.9936 0.9933 0.9977 0.9975
?= 0.5,?= 0.5,?= 0.5 0.9720 0.9965 0.9975 0.9976
?= 0.9,?= 0.1,?= 0.5 0.8577 0.8722 0.9262 0.9797
?= 0.1,?= 0.1,?= 1 1.0157 1.0127 1.0047 1.0010
?= 0.1,?= 0.9,?= 1 0.9735 0.9956 0.9964 0.9963
?= 0.5,?= 0.5,?= 1 0.9659 0.9947 0.9939 0.9959
?= 0.9,?= 0.1,?= 1 0.9251 0.9431 0.9648 0.9977
?= 0.1,?= 0.1,?= 5 1.0072 1.0193 1.0027 1.0017
?= 0.1,?= 0.9,?= 5 0.9688 0.9872 0.9930 0.9919
?= 0.5,?= 0.5,?= 5 0.9167 0.9718 0.9769 0.9907
?= 0.9,?= 0.1,?= 5 1.1674 1.0402 1.0173 1.0190
For an INAR(1) process, the results of this section show that CLS produces better
onestep ahead forecasts than YW for high autoregressive parameters. For lower
autoregressive parameters, YW slightly outperform CLS. For an INMA(1) process,
generally for high values of ?, YW is slightly better than CLS in terms of MSE of
onestep ahead forecasts; but the opposite is true for lower values of ?. For an
INARMA(1,1) process, CLS always produces better onestep ahead forecast than
M.Mohammadipour, 2009, Chapter 8 161
YW using MSE.
However, the results of threestep and sixstep ahead forecasts show that although
the YW and CLS results are generally very close for all of the three INARMA
processes, YW results are slightly better in many cases.
Therefore, based on the superior performance of CLS for onestep ahead forecasts, in
the following sections where we compare the INARMA forecasts with those of
benchmark methods, we use the CLS to estimate the parameters for onestep ahead
INARMA forecasts. For threestep and sixstep ahead forecasts, on the other hand,
we use YW to estimate the parameters of the INARMA process.
8.5 CrostonSBA Categorization
Syntetos et al. (2005) compare Croston and SBA, based on MSE, to establish the
areas that each method should be used over the other. The squared coefficient of
variation (??2) of demand size and the average interdemand interval (?) are used
to identify the areas.
The coefficient of variation is defined by ?=?/?1
?, where ?? is the value of the
mean measured from some arbitrary origin. It is estimated using the formula:
?=
?
1
?
? (????1
?)2??=1 ?
1/2
1
?
? ??
?
?=1
Equation 81
where ? is the sample size. The results show that for the smoothing parameter
? = 0.2, when p > 1.31, SBA is superior to Croston?s method in terms of MSE. For
p?1.31, if CV2 > 0.47 then MSECroston > MSESBA , but if CV
2 ?0.47 then
Croston?s method performs better in terms of MSE (Syntetos et al., 2005). This is
shown in Figure 81. The cutoff values are slightly different for different smoothing
parameters (Syntetos et al., 2005).
The CrostonSBA categorization is based on the assumption that demand occurs as
an i.i.d. Bernoulli process. Therefore it is worth testing if it also holds for an i.i.d.
M.Mohammadipour, 2009, Chapter 8 162
Poisson process. An INARMA(0,0) process produces such data series.
Figure ?81 Cutoff values for Croston and SBA when (Syntetos et al., 2005)
The simulation results show that the squared coefficient of variation of demand size
(??2) is always less than the cutoff value determined by Syntetos et al. (2005)
(0.47). This is due to the Poisson assumption of demand. The pvalues, however,
vary below and beyond the cutoff value (1.31). Therefore, the demand series
produced by an INARMA(0,0) process could belong to either region 3 or 4 in Figure
81.
The results of simulation confirm the CrostonSBA categorization for both cases of
?= 0.2, 0.5. Therefore, for ??1.31 (or ??3), MSECroston < MSESBA and for
?> 1.31 (or ?< 3), MSESBA < MSECroston when either all points in time or issue
points are considered.
Although the CrostonSBA categorization is based on MSE, the results show that it
generally holds for MASE as well. This was expected due to the similarities between
the two error measures. However, there are some exceptions. For the case of ?= 3
where Croston?s method should outperform SBA, MASESBA < MASECroston for both
cases of ?= 0.2, 0.5. Because the corresponding ?value and ??2 of size for these
cases are ?= 1.0538 and ??2 = 0.2622, these exceptions can be attributed to the
nonlinear boundaries between region 3 and others in Figure 81 (Syntetos, 2001;
Kostenko and Hyndman, 2006). This is an interesting finding because MASE is a
relatively new measure and has been suggested for intermittent demand studies
20.??
Syntetos &
Boylan
2
Syntetos &
Boylan
1
Syntetos &
Boylan
4
Croston
3
?= 0.47
?= 1.31
M.Mohammadipour, 2009, Chapter 8 163
(Hyndman, 2006).
The results show that when the number of observations increases, the advantage of
SBA over Croston decreases. This is a new finding and this issue has not been
discussed previously in the literature. It is shown is Table 814.
Table 814 The advantage of SBA over Croston for and all points in time
Parameters
MSESBA ?MSECroston MASESBA ?MASECroston
?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500
?= 0.3 0.0186 0.0085 0.0051 0.0025 0.0025 0.0507 0.0431 0.0392 0.0355 0.0342
?= 0.5 0.0168 0.0084 0.0061 0.0051 0.0052 0.0245 0.0197 0.0176 0.0175 0.0174
?= 0.7
0.0173 0.0100 0.0082 0.0079 0.0077 0.0116 0.0076 0.0067 0.0069 0.0063
?= 1 0.0173 0.0118 0.0111 0.0107 0.0102 0.0152 0.0131 0.0134 0.0130 0.0131
Table 815 The advantage of SBA over Croston for and all points in time
Parameters
MSESBA ?MSECroston MASESBA ?MASECroston
?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24 ?= 36 ?= 48 ?= 96 ?= 500
?= 0.3 0.0407 0.0266 0.0229 0.0209 0.0215 0.1285 0.1168 0.1102 0.1043 0.1000
?= 0.5 0.0495 0.0430 0.0403 0.0407 0.0390 0.0733 0.0673 0.0629 0.0624 0.0612
?= 0.7 0.0623 0.0595 0.0547 0.0562 0.0558 0.0486 0.0473 0.0438 0.0461 0.0439
?= 1 0.0780 0.0743 0.0761 0.0715 0.0696 0.0442 0.0427 0.0441 0.0415 0.0416
It can be seen from Table ?814 that the advantage of SBA over Croston decreases
when ? increases.
The MSE of one step ahead forecast for a stationary mean model is:
MSE = var?Estimates?+ Bias2 + var(Actual Demand)
Equation 82
Syntetos (2001) assumes an infinite history for SES estimates:
??= ? ?(1??)
?????
?
?=0
Equation 83
and, therefore, Brown?s expression for the variance of estimates is independent of the
length of demand history, ?:
20.??
50.??
M.Mohammadipour, 2009, Chapter 8 164
var(??) =
?
2??
var(??)
Equation 84
However, when this assumption is relaxed, the finite representation of SES becomes
(Graves, 1999):
??=????1 +??1??????2 +?+??1???
??1????+1 +?1???
?????
Equation 85
Therefore, the variance of the estimates produced by SES with a finite history is:
var????=
?+ 2(1??)2?+1
2??
var????
Equation 86
The variances of the exponentially smoothed size of demand and interdemand
interval with finite observations are then:
var???
? =
?+ 2(1??)2?+1
2??
var????
Equation 87
var???
? =
?+ 2(1??)2?+1
2??
var????
Equation 88
where var????=?
2 and var????=?(??1), since the interdemand interval
follows the geometric distribution. The variance of the estimates produced by
Croston?s method is:
var???
? = var?
??
?
??
?
Equation 89
The variance of the ratio of two independent variables is given by (Stuart and Ord,
1994):
var?
?
?
?=?
????
????
?
2
?
var(?)
[????]2
+
var(?)
[????]2
?
Equation 810
M.Mohammadipour, 2009, Chapter 8 165
The variance of the estimates produced by Croston?s method with finite sample is
therefore:
var???
? =
?+ 2(1??)2?+1
2??
?
(??1)2?2
?4
+
?2
?2
?
Equation 811
It can be seen that for ??(0,1], as ? increases, the above coefficient decreases until
it reaches a limit of [?/(2??)]. For high values of ?, the limit is approached very
quickly.
The difference between MSE of SBA and Croston?s method is:
MSESBA ?MSECroston
=?var?EstimatesSBA?+ BiasSBA
2 ??[var?EstimatesCroston ?+ BiasCroston
2 ]
Equation 812
But when ? increases, the bias of both Croston and SBA decreases and is close to
zero (this has been confirmed by simulation results). So the difference is
approximately:
MSESBA ?MSECroston ???1?
?
2
?
2
?1?var?EstimatesCroston ?
Equation 813
From Equation 811 we have:
MSESBA ?MSECroston ???1?
?
2
?
2
?1??
?+ 2(1??)2?+1
2??
??
(??1)2?2
?4
+
?2
?2
?
Equation 814
As the results of Table 814 show, when ? = 0.2, the above coefficient decreases
when ? increases; therefore the difference between MSE of Croston and SBA also
decreases. However, because the above coefficient reaches a limit of ?
?2
4
????
?
2??
?,
it can be seen from Table 814 that the advantage of SBA over Croston does not
change perceptibly when the number of observations is high.
For ?= 0.5, the results of Table ?815 confirm that, as expected, the difference
between MSE of Croston and SBA changes little with changes in ?.
M.Mohammadipour, 2009, Chapter 8 166
Although the CrostonSBA categorization is for i.i.d. demand, we have also tested it
when demand is an INAR(1), INMA(1) or an INARMA(1,1) process. The results
confirm that the CrostonSBA categorization generally holds for all of the above
mentioned processes. There is only one exception when there is an autoregressive
component (either INAR(1) or INARMA(1,1)). For the INMA(1) case, there are two
exceptions to the CrostonSBA categorization. The results are presented in Appendix
8.C.
Therefore, because the CrostonSBA categorization generally holds when the data is
produced by any of the four INARMA processes, the best benchmark can be used to
compete with INARMA forecasting methods.
8.6 INARMA vs Benchmark Methods
This research has suggested using INARMA models to forecast intermittent demand.
In order to answer the last research question of ?Do INARMA models provide more
accurate forecasts for intermittent demand than nonoptimal smoothingbased
methods??, the performance of INARMA forecasts based on ME, MSE and MASE
has been compared to that of benchmark methods. As previously mentioned, the
benchmarks are Croston (Croston, 1972), SBA (Syntetos and Boylan, 2005) and SBJ
(Shale et al., 2006) methods.
The first steps in the INARMA methodology are identification and estimation. These
steps make INARMA more complicated than the benchmarks and result in two types
of errors: error of identification and error of estimation. In order to investigate the
effect of the identification error we first assume that the order of the model is known.
The results are studied in section 8.6.1. Then we relax this assumption and examine
the results for unknown model orders in section 8.6.2. Finally, the leadtime forecasts
are compared in section 8.6.3.
8.6.1 INARMA with Known Order
In this section, we first compare the onestep ahead forecasts produced by each
M.Mohammadipour, 2009, Chapter 8 167
method (INARMA, Croston, SBA, and SBJ). The threestep and sixstep ahead
forecasts are then compared.
The results show that INARMA almost always produces the lowest MSE for all four
processes (INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1)) when all points
in time are considered. This is expected because the INARMA onestep ahead
forecasts are MMSE forecasts and therefore when the demand follows an INARMA
process, INARMA forecasts should outperform the benchmarks in terms of MSE.
The results also confirm that when only issue points are considered, the INARMA
forecasts are biased (see Appendix 8.D). This is expected because the least squares
criterion and therefore the CLS estimates are developed for the case where
parameters are updated each period regardless of the demand being positive or zero.
As parameters need to be estimated for INARMA models, with an increase in the
number of observations, the forecasts? accuracy increases.
The degree of improvement over benchmarks using the MSE measure, when all
points in time are considered, is shown in Table 816 to Table ?821. It should be
noted that in some tables ? is used in two different ways: when it is below the
benchmark methods, it is the smoothing parameter for that specific method and when
it is in the first column of the table, it is the autoregressive parameter of the
INARMA process.
Table 816 Onestep ahead for INARMA(0,0) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.8691 0.9114 0.9150 0.9473 0.9711 0.9731 0.9740 0.9894 0.9904 0.9794 0.9872 0.9875
?= 0.5 0.9502 0.9773 0.9789 0.9747 0.9895 0.9904 0.9762 0.9870 0.9876 0.9675 0.9769 0.9772
?= 0.7
0.9694 0.9900 0.9917 0.9732 0.9857 0.9862 0.9706 0.9812 0.9814 0.9587 0.9688 0.9690
?= 1 0.9749 0.9892 0.9894 0.9688 0.9789 0.9788 0.9575 0.9670 0.9668 0.9478 0.9572 0.9569
?= 3 0.9596 0.9525 0.9484 0.9424 0.9369 0.9331 0.9323 0.9257 0.9218 0.9208 0.9136 0.9096
?= 5 0.9498 0.9255 0.9178 0.9367 0.9112 0.9034 0.9269 0.9048 0.8974 0.9126 0.8889 0.8814
?= 20 0.9529 0.8212 0.7940 0.9315 0.8023 0.7756 0.9281 0.7990 0.7721 0.9124 0.7851 0.7586
The results show that the improvement increases when more observations are
available for higher values of ? (??0.7). With more observations, the accuracy of
parameters? estimates and therefore forecasts of INARMA and benchmark methods
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Chapter 8 168
become more accurate. However, the results suggest that the accuracy of INARMA
forecasts improves at a faster rate than the benchmarks.
The simulation results also show that for INARMA(0,0) and INMA(1) processes, the
improvement over benchmarks is not considerable. However, with the presence of an
autoregressive component, as in the INAR(1) and INARMA(1,1) cases, the
improvement is considerable for the cases in which the autoregressive parameter is
high (?= 0.9).
As can be seen from Table ?816, when ?= 20, the improvement of MSE of
INARMA over SBA and SBJ is very high. This is because these methods are
designed for highly intermittent demand, but when ?= 20, the demand is barely
intermittent and the methods do not reduce to SES. In this case, Croston?s method is
equivalent to SES.
Table 817 Onestep ahead for INMA(1) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9421 0.9733 0.9755 0.9634 0.9816 0.9829 0.9715 0.9833 0.9837 0.9583 0.9704 0.9709
?= 0.5,?= 0.5 0.9124 0.9470 0.9490 0.9395 0.9645 0.9660 0.9400 0.9607 0.9620 0.9282 0.9483 0.9495
?= 0.9,?= 0.5 0.9124 0.9543 0.9569 0.9227 0.9503 0.9521 0.9273 0.9503 0.9515 0.9118 0.9352 0.9367
?= 0.1,?= 1 0.9739 0.9871 0.9871 0.9629 0.9752 0.9751 0.9566 0.9673 0.9671 0.9426 0.9532 0.9530
?= 0.5,?= 1 0.9872 1.0027 1.0024 0.9685 0.9793 0.9785 0.9550 0.9674 0.9669 0.9425 0.9545 0.9539
?= 0.9,?= 1 1.0070 1.0146 1.0129 0.9849 0.9945 0.9931 0.9750 0.9839 0.9825 0.9628 0.9706 0.9690
?= 0.1,?= 3 0.9719 0.9611 0.9563 0.9587 0.9490 0.9442 0.9500 0.9390 0.9341 0.9334 0.9249 0.9204
?= 0.5,?= 3 1.0174 0.9948 0.9870 1.0035 0.9805 0.9728 0.9954 0.9730 0.9655 0.9763 0.9536 0.9461
?= 0.9,?= 3 1.0720 1.0282 1.0166 1.0657 1.0262 1.0150 1.0405 1.0045 0.9937 1.0300 0.9917 0.9809
?= 0.1,?= 5 0.9788 0.9474 0.9383 0.9499 0.9252 0.9169 0.9445 0.9168 0.9082 0.9308 0.9028 0.8944
?= 0.5,?= 5 1.0318 0.9787 0.9648 1.0241 0.9679 0.9537 1.0019 0.9461 0.9322 0.9866 0.9376 0.9244
?= 0.9,?= 5 1.0969 1.0211 1.0025 1.0777 1.0051 0.9867 1.0635 0.9917 0.9736 1.0345 0.9622 0.9445
When data is produced by an INARMA(0,0) or an INMA(1) process, the results of
INARMA forecasts are only compared to those of Croston, SBA and SBJ with
smoothing parameter 0.2. But for INAR(1) and INARMA(1,1) processes where an
autoregressive component is present, the benchmark methods with smoothing
parameter 0.5 are also included in comparisons (Table ?819 and Table ?821). This is
because when the autoregressive parameter is high, the benchmark methods with
higher smoothing parameter produce better forecasts than those with smoothing
parameter 0.2.
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Chapter 8 169
Table 818 Onestep ahead with smoothing parameter 0.2 for INAR(1)
series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9752 1.0033 1.0026 0.9868 1.0061 1.0074 0.9747 0.9877 0.9884 0.9609 0.9726 0.9733
?= 0.5,?= 0.5 0.7962 0.8190 0.8212 0.7715 0.7929 0.7941 0.7688 0.7852 0.7858 0.7408 0.7582 0.7590
?= 0.9,?= 0.5 0.5848 0.5134 0.4988 0.5333 0.4666 0.4536 0.5251 0.4684 0.4559 0.4917 0.4396 0.4280
?= 0.1,?= 1 0.9971 1.0153 1.0158 0.9746 0.9860 0.9858 0.9636 0.9725 0.9721 0.9433 0.9532 0.9529
?= 0.5,?= 1 0.8845 0.8890 0.8870 0.8280 0.8308 0.8288 0.8122 0.8145 0.8124 0.7924 0.7962 0.7943
?= 0.9,?= 1 0.5825 0.4598 0.4379 0.5478 0.4285 0.4078 0.5275 0.4196 0.3997 0.5068 0.4054 0.3865
?= 0.1,?= 3 0.9922 0.9794 0.9742 0.9621 0.9539 0.9494 0.9538 0.9433 0.9385 0.9317 0.9222 0.9176
?= 0.5,?= 3 0.9342 0.8928 0.8815 0.8807 0.8408 0.8301 0.8586 0.8180 0.8073 0.8356 0.7960 0.7857
?= 0.9,?= 3 0.5985 0.3502 0.3175 0.5379 0.3104 0.2816 0.5215 0.3001 0.2723 0.4984 0.2848 0.2584
?= 0.1,?= 5 0.9982 0.9644 0.9550 0.9675 0.9353 0.9261 0.9515 0.9224 0.9135 0.9325 0.9031 0.8943
?= 0.5,?= 5 0.9170 0.8396 0.8217 0.8934 0.8194 0.8019 0.8586 0.7846 0.7677 0.8383 0.7676 0.7511
?= 0.9,?= 5 0.5881 0.2457 0.2165 0.5374 0.2341 0.2067 0.5208 0.2322 0.2052 0.5022 0.2190 0.1934
Table 819 Onestep ahead with smoothing parameter 0.5 for INAR(1)
series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.8996 0.9844 1.0063 0.8776 0.9584 0.9960 0.8577 0.9340 0.9680 0.8445 0.9192 0.9505
?= 0.5,?= 0.5 0.7901 0.8582 0.7930 0.7459 0.8201 0.7663 0.7356 0.8014 0.7470 0.7034 0.7688 0.7252
?= 0.9,?= 0.5 0.9191 0.3860 0.2257 0.8746 0.3646 0.2094 0.8416 0.3604 0.2104 0.8066 0.3418 0.1975
?= 0.1,?= 1 0.8912 0.9544 0.9629 0.8619 0.9153 0.9243 0.8548 0.9036 0.9074 0.8318 0.8836 0.8915
?= 0.5,?= 1 0.9069 0.8851 0.7594 0.8500 0.8284 0.7064 0.8316 0.8128 0.6949 0.8058 0.7894 0.6785
?= 0.9,?= 1 0.9332 0.2532 0.1451 0.8807 0.2362 0.1349 0.8583 0.2296 0.1316 0.8248 0.2256 0.1294
?= 0.1,?= 3 0.8630 0.8141 0.7700 0.8310 0.7906 0.7558 0.8293 0.7850 0.7434 0.8105 0.7676 0.7264
?= 0.5,?= 3 0.9728 0.7309 0.5562 0.9190 0.6883 0.5230 0.8957 0.6672 0.5067 0.8703 0.6505 0.4946
?= 0.9,?= 3 0.9337 0.1106 0.0630 0.8746 0.1003 0.0569 0.8514 0.0972 0.0552 0.8186 0.0922 0.0523
?= 0.1,?= 5 0.8603 0.7393 0.6599 0.8312 0.7100 0.6363 0.8165 0.6998 0.6270 0.8010 0.6854 0.6142
?= 0.5,?= 5 0.9588 0.5936 0.4247 0.9311 0.5748 0.4116 0.8952 0.5533 0.3949 0.8720 0.5369 0.3847
?= 0.9,?= 5 0.9364 0.0642 0.0362 0.8728 0.0620 0.0350 0.8586 0.0612 0.0346 0.8171 0.0578 0.0326
The degree of improvement over benchmarks, using the MASE measure, is shown in
Appendix 8.E. The results show that for INARMA(0,0) and INMA(1) processes, the
improvement over benchmarks, in terms of MASE, is not considerable. But for
INAR(1) and INARMA(1,1) processes, the improvement is high for the cases in
which the autoregressive parameter is high. This confirms the results using MSE.
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Chapter 8 170
Table 820 Onestep ahead with smoothing parameter 0.2 for INARMA(1,1)
series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.9881 1.0188 1.0202 0.9798 1.0007 1.0021 0.9716 0.9882 0.9892 0.9516 0.9667 0.9674
?= 0.1,?= 0.9,?= 0.5 0.8263 0.8614 0.8633 0.8117 0.8382 0.8400 0.8272 0.8483 0.8495 0.8113 0.8311 0.8322
?= 0.5,?= 0.5,?= 0.5 0.7641 0.7846 0.7846 0.7271 0.7439 0.7443 0.7000 0.7128 0.7130 0.6491 0.6638 0.6641
?= 0.9,?= 0.1,?= 0.5 0.5621 0.4725 0.4577 0.5314 0.4674 0.4542 0.4982 0.4397 0.4276 0.4740 0.4208 0.4091
?= 0.1,?= 0.1,?= 1 1.0022 1.0176 1.0176 0.9872 1.0013 1.0012 0.9657 0.9782 0.9780 0.9410 0.9530 0.9528
?= 0.1,?= 0.9,?= 1 0.9076 0.9143 0.9127 0.8657 0.8765 0.8754 0.8613 0.8670 0.8655 0.8523 0.8595 0.8581
?= 0.5,?= 0.5,?= 1 0.8216 0.8156 0.8119 0.7701 0.7676 0.7645 0.7533 0.7488 0.7456 0.7082 0.7073 0.7045
?= 0.9,?= 0.1,?= 1 0.5736 0.4402 0.4180 0.5206 0.4057 0.3860 0.4975 0.3911 0.3723 0.4610 0.3665 0.3492
?= 0.1,?= 0.1,?= 5 1.0117 0.9720 0.9613 0.9802 0.9453 0.9352 0.9653 0.9306 0.9206 0.9442 0.9096 0.8998
?= 0.1,?= 0.9,?= 5 0.9149 0.8425 0.8257 0.8857 0.8149 0.7987 0.8560 0.7915 0.7762 0.8482 0.7853 0.7701
?= 0.5,?= 0.5,?= 5 0.8271 0.7310 0.7106 0.7831 0.6908 0.6713 0.7541 0.6713 0.6531 0.7249 0.6439 0.6262
?= 0.9,?= 0.1,?= 5 0.5341 0.2424 0.2140 0.5230 0.2214 0.1951 0.4948 0.2153 0.1901 0.8477 0.8073 0.0154
Table 821 Onestep ahead with smoothing parameter 0.5 for INARMA(1,1) series
(known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.9038 1.0019 1.0255 0.8647 0.9572 0.9917 0.8475 0.9364 0.9709 0.8258 0.9106 0.9474
?= 0.1,?= 0.9,?= 0.5 0.7641 0.8583 0.8595 0.7157 0.8114 0.8250 0.7357 0.8203 0.8208 0.7211 0.8043 0.8037
?= 0.5,?= 0.5,?= 0.5 0.7960 0.8529 0.7388 0.7567 0.7983 0.6949 0.7268 0.7625 0.6618 0.6713 0.7108 0.6191
?= 0.9,?= 0.1,?= 0.5 0.9343 0.3702 0.2060 0.8703 0.3592 0.2061 0.8319 0.3426 0.1959 0.7895 0.3213 0.1843
?= 0.1,?= 0.1,?= 1 0.8982 0.9543 0.9513 0.8780 0.9369 0.9357 0.8621 0.9150 0.9112 0.8355 0.8885 0.8884
?= 0.1,?= 0.9,?= 1 0.8859 0.8900 0.7964 0.8257 0.8452 0.7746 0.8359 0.8396 0.7565 0.8209 0.8294 0.7517
?= 0.5,?= 0.5,?= 1 0.9256 0.8391 0.6537 0.8635 0.7895 0.6188 0.8433 0.7673 0.6014 0.7877 0.7255 0.5714
?= 0.9,?= 0.1,?= 1 0.9236 0.2359 0.1346 0.8665 0.2240 0.1270 0.8322 0.2161 0.1228 0.7864 0.2054 0.1164
?= 0.1,?= 0.1,?= 5 0.8940 0.7391 0.6392 0.8696 0.7208 0.6244 0.8625 0.7127 0.6135 0.8371 0.6939 0.6003
?= 0.1,?= 0.9,?= 5 0.9156 0.5979 0.4408 0.8931 0.5789 0.4277 0.8616 0.5651 0.4186 0.8536 0.5581 0.4132
?= 0.5,?= 0.5,?= 5 0.9496 0.4851 0.3221 0.8921 0.4558 0.3031 0.8605 0.4491 0.2998 0.8230 0.4266 0.2851
?= 0.9,?= 0.1,?= 5 0.8881 0.0630 0.0357 0.8565 0.0576 0.0325 0.8252 0.0568 0.0320 0.9035 0.6340 0.5079
The results of comparing INARMA with benchmark methods for threestep ahead
forecasts are given in Table ?822 to Table ?827 (See Appendix 8.F for the sixstep
ahead results). The results of comparing the hstep ahead forecasts (?= 3, 6) of
INARMA with benchmarks for INARMA(0,0) are very close to the results of one
step ahead forecasts.
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Chapter 8 171
Table 822 Threestep ahead for INARMA(0,0) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.8459 0.8930 0.8980 0.9447 0.9692 0.9714 0.9751 0.9886 0.9898 0.9826 0.9901 0.9904
?= 0.5 0.9458 0.9761 0.9785 0.9756 0.9910 0.9919 0.9735 0.9846 0.9851 0.9665 0.9757 0.9759
?= 0.7
0.9693 0.9909 0.9924 0.9751 0.9868 0.9871 0.9720 0.9824 0.9827 0.9586 0.9686 0.9687
?= 1 0.9758 0.9901 0.9903 0.9686 0.9791 0.9789 0.9644 0.9724 0.9720 0.9475 0.9564 0.9561
?= 3 0.9578 0.9475 0.9432 0.9411 0.9346 0.9307 0.9305 0.9222 0.9182 0.9196 0.9137 0.9100
?= 5 0.9522 0.9320 0.9249 0.9389 0.9149 0.9073 0.9263 0.9029 0.8954 0.9118 0.8908 0.8836
?= 20 0.9549 0.8234 0.7960 0.9331 0.8011 0.7740 0.9265 0.7968 0.7699 0.9152 0.7911 0.7649
Table 823 Threestep ahead for INMA(1) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9359 0.9747 0.9782 0.9738 0.9916 0.9928 0.9660 0.9812 0.9820 0.9593 0.9711 0.9716
?= 0.5,?= 0.5 0.9514 0.9890 0.9922 0.9541 0.9782 0.9798 0.9503 0.9713 0.9726 0.9280 0.9474 0.9486
?= 0.9,?= 0.5 0.9481 0.9968 1.0009 0.9423 0.9752 0.9776 0.9297 0.9595 0.9615 0.9142 0.9367 0.9380
?= 0.1,?= 1 0.9727 0.9880 0.9882 0.9664 0.9784 0.9783 0.9530 0.9666 0.9667 0.9357 0.9470 0.9468
?= 0.5,?= 1 0.9614 0.9816 0.9820 0.9430 0.9580 0.9579 0.9274 0.9431 0.9431 0.9066 0.9214 0.9213
?= 0.9,?= 1 0.9597 0.9746 0.9741 0.9269 0.9385 0.9377 0.9113 0.9312 0.9312 0.8884 0.9023 0.9017
?= 0.1,?= 3 0.9542 0.9490 0.9449 0.9369 0.9290 0.9246 0.9246 0.9208 0.9169 0.9097 0.9042 0.9001
?= 0.5,?= 3 0.9426 0.9304 0.9246 0.9146 0.9060 0.9007 0.8970 0.8871 0.8818 0.8797 0.8728 0.8677
?= 0.9,?= 3 0.9444 0.9307 0.9238 0.9052 0.8900 0.8830 0.8915 0.8789 0.8723 0.8626 0.8496 0.8431
?= 0.1,?= 5 0.9535 0.9341 0.9265 0.9339 0.9077 0.8996 0.9143 0.8929 0.8853 0.9010 0.8786 0.8710
?= 0.5,?= 5 0.9401 0.9164 0.9067 0.9199 0.8872 0.8769 0.8997 0.8650 0.8546 0.8741 0.8418 0.8319
?= 0.9,?= 5 0.9493 0.9035 0.8903 0.9075 0.8631 0.8503 0.8929 0.8528 0.8406 0.8618 0.8220 0.8101
Table 824 Threestep ahead with smoothing parameter 0.2 for INAR(1)
series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9779 1.0054 1.0076 0.9766 0.9924 0.9934 0.9688 0.9841 0.9850 0.9538 0.9672 0.9678
?= 0.5,?= 0.5 1.0023 1.0387 1.0413 0.9757 1.0063 1.0083 0.9403 0.9708 0.9728 0.8924 0.9209 0.9228
?= 0.9,?= 0.5 0.9722 0.9494 0.9365 0.9039 0.8583 0.8446 0.8577 0.8218 0.8098 0.7963 0.7682 0.7574
?= 0.1,?= 1 0.9789 0.9993 1.0002 0.9623 0.9752 0.9752 0.9497 0.9607 0.9605 0.9347 0.9469 0.9469
?= 0.5,?= 1 1.0005 1.0231 1.0231 0.9571 0.9759 0.9757 0.9256 0.9445 0.9444 0.8840 0.9012 0.9009
?= 0.9,?= 1 0.9861 0.8725 0.8463 0.9031 0.8023 0.7786 0.8624 0.7758 0.7537 0.8057 0.7239 0.7029
?= 0.1,?= 3 0.9586 0.9553 0.9512 0.9445 0.9363 0.9318 0.9253 0.9201 0.9161 0.9050 0.8995 0.8955
?= 0.5,?= 3 0.9964 0.9755 0.9670 0.9330 0.9170 0.9095 0.9118 0.8958 0.8884 0.8730 0.8639 0.8573
?= 0.9,?= 3 0.9680 0.6638 0.6176 0.8894 0.6050 0.5624 0.8873 0.6099 0.5665 0.8018 0.5559 0.5167
?= 0.1,?= 5 0.9576 0.9332 0.9251 0.9296 0.9014 0.8931 0.9172 0.8933 0.8853 0.9035 0.8787 0.8707
?= 0.5,?= 5 1.0170 0.9605 0.9454 0.9463 0.8926 0.8779 0.9114 0.8690 0.8558 0.8659 0.8225 0.8096
?= 0.9,?= 5 0.9575 0.5535 0.5009 0.8854 0.8709 0.8602 0.8585 0.4990 0.4524 0.7998 0.4644 0.4210
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Chapter 8 172
For an INMA(1) process, the performance of INARMA compared to benchmark
methods is improved for hstep ahead forecasts compared to onestep ahead
forecasts. This could be because the benchmark methods use the same forecast as
onestep ahead forecast, but the INMA(1) method updates the forecasts.
Table 825 Threestep ahead with smoothing parameter 0.5 for INAR(1)
series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.8885 0.9715 0.9774 0.8680 0.9432 0.9476 0.8537 0.9305 0.9356 0.8285 0.9070 0.9135
?= 0.5,?= 0.5 0.9040 1.0122 1.0132 0.8470 0.9572 0.9595 0.8166 0.9260 0.9295 0.7684 0.8704 0.8741
?= 0.9,?= 0.5 1.0808 0.7588 0.5810 1.0428 0.6862 0.5234 0.9866 0.6757 0.5192 0.9210 0.6331 0.4861
?= 0.1,?= 1 0.8477 0.9174 0.9078 0.8337 0.8935 0.8821 0.8181 0.8761 0.8642 0.8076 0.8658 0.8543
?= 0.5,?= 1 0.8654 0.9204 0.8876 0.8261 0.8750 0.8432 0.8023 0.8504 0.8206 0.7673 0.8105 0.7808
?= 0.9,?= 1 1.1085 0.5490 0.3835 1.0376 0.5089 0.3552 1.0002 0.4991 0.3480 0.9346 0.4595 0.3190
?= 0.1,?= 3 0.8006 0.7790 0.7181 0.7972 0.7646 0.7029 0.7724 0.7476 0.6896 0.7575 0.7331 0.6762
?= 0.5,?= 3 0.8384 0.7401 0.6411 0.7946 0.7033 0.6100 0.7727 0.6852 0.5947 0.7383 0.6629 0.5761
?= 0.9,?= 3 1.0951 0.2602 0.1617 1.0240 0.2353 0.1460 1.0154 0.2317 0.1432 0.9261 0.2125 0.1313
?= 0.1,?= 5 0.7956 0.7032 0.6145 0.7698 0.6751 0.5888 0.7597 0.6681 0.5823 0.7511 0.6579 0.5729
?= 0.5,?= 5 0.8714 0.6602 0.5333 0.8028 0.6011 0.4821 0.7715 0.5927 0.4782 0.7308 0.5574 0.4492
?= 0.9,?= 5 1.1015 0.1672 0.0996 1.0322 0.7275 0.5586 1.0000 0.1529 0.0914 0.9226 0.1424 0.0852
Table 826 Threestep ahead with smoothing parameter 0.2 for INARMA(1,1)
series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.9621 0.9993 0.9992 0.9806 0.9994 1.0006 0.9730 0.9897 0.9907 0.9526 0.9673 0.9680
?= 0.1,?= 0.9,?= 0.5 0.9515 0.9996 1.0031 0.9404 0.9728 0.9753 0.9278 0.9577 0.9597 0.8986 0.9229 0.9244
?= 0.5,?= 0.5,?= 0.5 0.9945 1.0227 1.0239 0.9531 0.9855 0.9875 0.9317 0.9631 0.9651 0.8928 0.9228 0.9247
?= 0.9,?= 0.1,?= 0.5 1.3727 1.2345 1.2092 1.5485 1.4763 1.4539 1.5901 1.5111 1.4876 1.7403 1.6794 1.6553
?= 0.1,?= 0.1,?= 1 0.9927 1.0168 1.0178 0.9670 0.9810 0.9810 0.9561 0.9699 0.9698 0.9220 0.9359 0.9360
?= 0.1,?= 0.9,?= 1 0.9532 0.9718 0.9715 0.9177 0.9347 0.9344 0.9064 0.9227 0.9224 0.8833 0.8989 0.8985
?= 0.5,?= 0.5,?= 1 0.9933 1.0106 1.0097 0.9427 0.9644 0.9641 0.9201 0.9347 0.9336 0.8862 0.9034 0.9027
?= 0.9,?= 0.1,?= 1 1.3867 1.2511 1.2147 1.5431 1.3756 1.3331 1.5887 1.4125 1.3700 1.7368 1.5550 1.5095
?= 0.1,?= 0.1,?= 5 0.9832 0.9584 0.9495 0.9375 0.9126 0.9041 0.9181 0.8978 0.8898 0.8928 0.8704 0.8625
?= 0.1,?= 0.9,?= 5 0.9710 0.9245 0.9105 0.9130 0.8713 0.8584 0.8894 0.8487 0.8365 0.8556 0.8185 0.8068
?= 0.5,?= 0.5,?= 5 1.0173 0.9483 0.9303 0.9598 0.9053 0.8889 0.9245 0.8736 0.8579 0.8803 0.8285 0.8133
?= 0.9,?= 0.1,?= 5 1.7356 0.9574 0.8627 1.6699 0.9654 0.8737 1.7683 1.0071 0.9128 1.7151 0.9770 0.8836
For INAR(1) and INARMA(1,1) processes, the performance of INARMA over the
benchmark methods is improved compared to the onestep ahead case when the
autoregressive parameter is low. But, as discussed in chapter 6, when the
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Chapter 8 173
autoregressive parameter is high, the fact that the forecasts converge to the mean of
the process results in poor forecasts compared to the onestep ahead case. As
explained in chapter 6, some authors suggest using different models for different
horizons in order to improve forecast accuracy (Cox, 1961; Tiao and Xu, 1993;
Kang, 2003).
Table 827 Threestep ahead with smoothing parameter 0.5 for INARMA(1,1)
series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.8645 0.9711 1.0200 0.8670 0.9500 0.9861 0.8402 0.9282 0.9735 0.8223 0.9071 0.9479
?= 0.1,?= 0.9,?= 0.5 0.8371 0.9597 1.0175 0.7945 0.9047 0.9622 0.7757 0.8869 0.9431 0.7477 0.8483 0.9002
?= 0.5,?= 0.5,?= 0.5 0.8855 0.9781 0.9853 0.8377 0.9372 0.9549 0.8185 0.9158 0.9341 0.7771 0.8723 0.8951
?= 0.9,?= 0.1,?= 0.5 1.5506 0.9825 0.6679 1.8084 1.2198 0.8406 1.8603 1.2357 0.8578 2.0440 1.3820 0.9572
?= 0.1,?= 0.1,?= 1 0.8588 0.9342 0.9719 0.8309 0.8943 0.9200 0.8203 0.8794 0.9068 0.7889 0.8504 0.8796
?= 0.1,?= 0.9,?= 1 0.7944 0.8515 0.8782 0.7608 0.8154 0.8461 0.7513 0.8033 0.8338 0.7334 0.7820 0.8109
?= 0.5,?= 0.5,?= 1 0.8670 0.9014 0.8896 0.8186 0.8594 0.8564 0.7895 0.8212 0.8123 0.7642 0.7988 0.7923
?= 0.9,?= 0.1,?= 1 1.5546 0.7857 0.5145 1.7618 0.8418 0.5498 1.8280 0.8853 0.5789 2.0404 0.9911 0.6449
?= 0.1,?= 0.1,?= 5 0.8070 0.7083 0.6504 0.7714 0.6736 0.6199 0.7510 0.6620 0.6146 0.7323 0.6432 0.5955
?= 0.1,?= 0.9,?= 5 0.7803 0.6129 0.5297 0.7334 0.5807 0.5030 0.7137 0.5707 0.4969 0.6856 0.5474 0.4774
?= 0.5,?= 0.5,?= 5 0.8725 0.6197 0.4987 0.8124 0.5889 0.4761 0.7889 0.5699 0.4608 0.7503 0.5401 0.4354
?= 0.9,?= 0.1,?= 5 1.9454 0.2820 0.1644 1.9418 0.2908 0.1712 2.0688 0.3114 0.1832 1.9978 0.2951 0.1731
8.6.2 INARMA with Unknown Order
It was assumed in section 8.6.1 that the order of the INARMA process is known.
However, in reality this is not the case and the autoregressive and moving average
orders of the model need to be identified. As discussed in chapter 4, when simulating
a high number of replications, automated approaches such as AIC and BIC should be
used for identification rather than sample autocorrelation (SACF) and sample partial
autocorrelation functions (SPACF).
In chapter 4, two identification procedures were suggested. A twostage
identification procedure is based on using the LjungBox test to distinguish between
INARMA(0,0) and other processes and then using the AIC (or AICC for small
sample sizes) to select among the other possible INARMA processes. On the other
hand, the onestage method only uses the AIC to select among all possible INARMA
models including INARMA(0,0).
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Chapter 8 174
For simplicity, we first assume that data can be produced by either an INARMA(0,0)
or an INAR(1) process. The results of identification among these two processes are
summarized in section 8.6.2.1. As an alternative approach to identification, we
suggest that the model with the highest order in the set of models (in this case
INAR(1)) can be used for forecasting. The results are presented in section 8.6.2.2.
Then we assume that data can be produced by any of the four processes. The results
of identification based using twostage and onestage identification procedures are
presented in section 8.6.2.3. The mostgeneralmodel approach is also tested and the
results are analyzed in section 8.6.2.4. The results of treating all models as INAR(1)
are compared to the benchmark methods in section 8.6.2.5.
8.6.2.1 Identification among Two Processes
In this section, it is assumed that data is either INARMA(0,0) or INAR(1). As
suggested by Jung and Tremayne (2003), in order to distinguish between the
INARMA(0,0) (or an i.i.d. Poisson process) and INAR(1), we test if the data show a
significant serial dependence or not.
This is done using a portmanteau test called the LjungBox test explained in section
4.2.3. The test is based on a ??statistic given by:
??=?(?+ 2)? ??????1??
2
?
?=1
Equation 815
where ? is the sample size, ? is the number of autocorrelation lags included in the
statistic (we assume ?= 10), ?? is the sample autocorrelation at lag ?. The ?
?
statistic can be used when a univariate model is fitted to a time series. It can be used
as a lackoffit test for a departure from randomness. Under the null hypothesis that
the model fit is adequate, the test statistic is asymptotically chisquare distributed.
Results are presented for a significance level of 0.05.
The results in terms of percentage of series for which the model is correctly
identified are summarized in Table 828 and Table 829 for both LjungBox and AIC
identification procedures. The results of Table 828 show identification with the
M.Mohammadipour, 2009, Chapter 8 175
LjungBox test provides better results than with the AIC for INARMA(0,0) series.
Comparing the results of Table 828 to the results by Jung and Tremayne (2003)
shows that the ?? statistic produces similar results to those suggested in their study.
The simulation results show that for those cases where an INARMA(0,0) is
misidentified as an INAR(1) process, the estimated autoregressive parameter is close
to zero.
Table 828 The percentage of correct identification for INARMA(0,0) series
Parameters
LjungBox AIC
?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=??
?= 0.3 94.42 95.53 97.10 95.00 91.00 89.60 87.70 81.50
?= 0.5 93.35 94.50 94.60 95.60 92.20 88.30 83.70 78.70
?= 0.7
93.16 94.11 96.00 95.60 91.80 87.80 83.10 74.80
?= 1 93.64 94.25 94.60 95.30 91.10 83.20 80.40 71.40
?= 3 95.00 92.50 94.20 95.20 83.10 79.60 74.70 69.70
?= 5 92.80 93.70 93.80 94.60 84.20 77.20 74.40 68.80
?= 20 93.20 94.20 93.60 94.80 83.70 75.90 72.40 68.50
It can be seen from Table ?829 that, for an INAR(1) process, when the autoregressive
parameter is small (?= 0.1), the model is often misidentified as INARMA(0,0). The
AIC is always better than the LjungBox method. For high values of ?, the correct
model in identified by the AIC in most cases. Both identification methods perform
better when more observations are available. For high values of ? (?= 0.9) and ?
(?= 96) the two identification methods are close.
Table 829 The percentage of correct identification for INAR(1) series
Parameters
LjungBox AIC
?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=??
?= 0.1,?= 0.5 5.90 6.00 5.10 5.40 13.70 19.90 28.40 47.20
?= 0.5,?= 0.5 15.40 23.90 34.30 71.60 49.00 73.60 86.00 98.60
?= 0.9,?= 0.5 34.20 65.50 84.80 100.00 90.00 98.50 99.70 100.00
?= 0.1,?= 1 6.50 7.20 6.90 6.30 14.90 26.10 35.40 52.20
?= 0.5,?= 1 16.30 22.60 35.40 73.90 57.70 80.70 90.10 99.70
?= 0.9,?= 1 35.10 63.30 86.50 99.80 89.80 98.30 100.00 100.00
?= 0.1,?= 3 9.00 5.90 7.20 6.30 22.30 33.90 39.10 57.00
?= 0.5,?= 3 14.50 23.90 34.40 71.30 62.70 83.40 92.50 99.60
?= 0.9,?= 3 34.50 66.00 83.50 99.90 91.10 98.70 99.80 100.00
?= 0.1,?= 5 7.40 6.50 4.80 7.70 22.80 37.10 40.10 59.40
?= 0.5,?= 5 16.80 24.40 35.30 72.50 67.00 86.60 93.20 99.70
?= 0.9,?= 5 33.90 64.40 83.60 99.70 91.10 98.50 99.70 100.00
M.Mohammadipour, 2009, Chapter 8 176
Because the INARMA(0,0) process is correctly identified in most of the cases, the
forecast accuracy of the INARMA(0,0) with identification is close to that of the case
when the order is known. The same is true for an INAR(1) process with high values
of ? and ?. As a result, the performance of these two processes compared to
benchmarks is similar to what was discussed in section 8.6.1.
The accuracy results of INAR(1) forecasts using ME, MSE and MASE for the two
identification methods are shown in Table ?830. Similar results for the case that the
order of the model is known are provided in Table ?831 for comparison.
The results of Table ?830 show that the AIC produces better forecasts than the Ljung
Box method in most of the cases, except for the case where ?= 0.1 and ? and ? are
small.
The results also show that when ?= 0.1, although the percentage of correct
identification is small, INARMA with identification produces more accurate
forecasts compared to the case where the order is known. This means that, in this
case, using an INARMA(0,0) forecast based on the average of the previous
observations produces better results than estimating ? and ? and forecasting using an
INAR(1) model.
However, when the autoregressive parameter is high (?= 0.9) and the number of
observations is small (?= 24), although the percentage of correct identification is
considerable, the difference between INARMA without identification and with
identification for the LjungBox method is huge. This is expected because here a
time series with high autocorrelation is wrongly identified as a series with no
autocorrelation. Therefore, instead of putting a high weight on the last observation,
the forecast is based on the INARMA(0,0) model which uses the simple average of
all previous observations with equal weights. However, as the length of history
increases, the percentage of correct identification also increases and the forecast
accuracy of the two cases become very close. This is also true when comparing the
LjungBox and AIC identification methods. For high values of ?, the latter is
considerably better than the former due to the higher percentage of correct
identification and the fact that misidentification of an INAR(1) process with high
autoregressive parameter as an INARMA(0,0) has a huge penalty.
M.Mohammadipour, 2009, Chapter 8 177
Table 830 Accuracy of INAR(1) forecasts for LjungBox and AIC identification procedures
Parameters
?=?? ?=?? ?=?? ?=??
LjungBox AIC LjungBox AIC LjungBox AIC LjungBox AIC
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0105 0.6244 1.0977 0.0006 0.6343 1.1239 0.0064 0.6235 1.0925 0.0104 0.6038 1.0563 0.0056 0.5928 1.0145 0.0084 0.6196 1.0273 0.0024 0.5691 0.9758 0.0038 0.5660 0.9690
?= 0.5,?= 0.5 0.0047 1.0475 1.4131 0.0144 1.0372 1.3421 0.0072 1.0201 1.2497 0.0102 0.9106 1.1900 0.0016 0.9858 1.2016 0.0023 0.8703 1.1342 0.0062 0.8633 1.1078 0.0039 0.7970 1.0655
?= 0.9,?= 0.5 0.0547 2.7057 1.9921 0.0082 1.4192 1.4380 0.0188 2.0866 1.6381 0.0122 1.1614 1.3010 0.0193 1.5043 1.3871 0.0223 1.0431 1.1720 0.0058 1.0078 1.1428 0.0091 1.0118 1.1512
?= 0.1,?= 1 0.0022 1.2865 0.9486 0.0063 1.3055 0.8974 0.0019 1.1974 0.8462 0.0011 1.2412 0.8653 0.0053 1.1850 0.8292 0.0045 1.1771 0.8355 0.0106 1.1682 0.8102 0.0064 1.1372 0.7995
?= 0.5,?= 1 0.0108 2.1835 1.1859 0.0187 1.9727 1.1119 0.0284 2.1543 1.1541 0.0017 1.8151 1.0594 0.0131 1.9543 1.0778 0.0031 1.7068 1.0146 0.0040 1.6895 0.9885 0.0046 1.5869 0.9646
?= 0.9,?= 1 0.0246 5.3777 1.7945 0.0147 2.8294 1.3379 0.0015 4.2493 1.5135 0.0247 2.3516 1.1871 0.0190 2.8263 1.2406 0.0047 2.1486 1.1096 0.0084 2.0473 1.0789 0.0035 2.0222 1.0807
?= 0.1,?= 3 0.0226 3.9401 0.8599 0.0091 3.8786 0.8505 0.0248 3.6785 0.8039 0.0103 3.8087 0.8416 0.0014 3.6089 0.8094 0.0014 3.6078 0.8150 0.0065 3.4860 0.7868 0.0025 3.4883 0.7823
?= 0.5,?= 3 0.0856 6.5598 1.1244 0.0293 5.9107 1.0177 0.0051 6.2115 1.0573 0.0050 5.4093 0.9960 0.0089 5.9927 1.0129 0.0133 5.1793 0.9548 0.0333 5.2292 0.9589 0.0359 4.6773 0.9000
?= 0.9,?= 3 0.1048 15.6396 1.6548 0.0026 8.1910 1.2077 0.0602 11.6692 1.3805 0.0020 6.9766 1.1231 0.0335 9.4088 1.2402 0.0010 6.5496 1.0791 0.0076 6.1685 1.0349 0.0080 6.0951 1.0283
?= 0.1,?= 5 0.0021 6.6522 0.8473 0.0089 6.4835 0.8336 0.0023 6.1105 0.8153 0.0198 6.2402 0.8273 0.0329 5.8728 0.7829 0.0073 6.0014 0.7918 0.0209 5.7109 0.7718 0.0015 5.8188 0.7695
?= 0.5,?= 5 0.0828 10.9849 1.1122 0.0084 9.6383 1.0246 0.0380 10.2881 1.0358 0.0419 8.7787 0.9678 0.0139 9.9559 1.0261 0.0012 8.4644 0.9364 0.0196 8.4499 0.9316 0.0009 7.9078 0.8969
?= 0.9,?= 5 0.0928 26.7365 1.6375 0.0277 13.8802 1.2078 0.0658 21.3689 1.4308 0.0124 11.6831 1.1051 0.0208 15.1822 1.2121 0.0341 10.8485 1.0563 0.0020 10.2432 1.0281 0.0286 10.1068 1.0197
M.Mohammadipour, 2009, Chapter 8 178
Table 831 Accuracy of INAR(1) forecasts when the order is known
Parameters
?=?? ?=?? ?=?? ?=??
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0026 0.6614 1.1418 0.0089 0.6286 1.0555 0.0052 0.6047 1.0237 0.0045 0.5822 0.9822
?= 0.5,?= 0.5 0.0026 0.9466 1.3290 0.0089 0.8743 1.1807 0.0044 0.8532 1.1570 0.0054 0.7834 1.0534
?= 0.9,?= 0.5 0.0017 1.2082 1.3901 0.0042 1.1489 1.2894 0.0043 1.0811 1.2176 0.0033 1.0164 1.1486
?= 0.1,?= 1 0.0149 1.2880 0.9376 0.0059 1.2072 0.8496 0.0052 1.1967 0.8382 0.0029 1.1663 0.8057
?= 0.5,?= 1 0.0195 1.8846 1.0837 0.0018 1.7244 1.0202 0.0045 1.6745 1.0059 0.0026 1.5905 0.9618
?= 0.9,?= 1 0.0121 2.5216 1.2401 0.0054 2.2655 1.1594 0.0146 2.1925 1.1262 0.0105 2.0319 1.0686
?= 0.1,?= 3 0.0100 3.9224 0.8544 0.0176 3.6703 0.8228 0.0010 3.6641 0.8136 0.0085 3.4701 0.7846
?= 0.5,?= 3 0.0308 5.6926 1.0126 0.0035 5.1557 0.9491 0.0078 4.9742 0.9331 0.0143 4.7749 0.9160
?= 0.9,?= 3 0.0906 7.5862 1.1509 0.0298 6.7494 1.1026 0.0243 6.4376 1.0658 0.0093 6.0230 1.0205
?= 0.1,?= 5 0.0371 6.6926 0.8565 0.0209 6.1869 0.8143 0.0282 6.0095 0.7973 0.0139 5.7529 0.7742
?= 0.5,?= 5 0.0123 9.3467 0.9840 0.0082 8.6581 0.9653 0.0116 8.3583 0.9392 0.0034 7.8699 0.9011
?= 0.9,?= 5 0.0013 11.9986 1.1483 0.0126 11.2051 1.0914 0.0467 10.8985 1.0720 0.0185 10.1102 1.0272
8.6.2.2 AllINAR(1)
In this section, it is again assumed that data is produced by either an INARMA(0,0)
or an INAR(1) process. Instead of identification, we assume that the most general
model, INAR(1) in this case, is used for estimation and forecasting. It is expected
that if the data is in fact an INARMA(0,0) process, the estimated autoregressive
parameter should be close to zero and the results confirm this. In general, the
forecasting accuracy deteriorates slightly in this case compared to the case of using
LjungBox for identification among two possible models. The results for all points in
time are shown in Table ?832.
This shows that, instead of adding an extra step to the INARMA forecasting
procedure for identification, treating everything as an INAR(1) process produces
close results to those with identification and it has the advantage of being simple.
The degree of deterioration caused by skipping identification is on average 2 percent
for both MSE and MASE.
M.Mohammadipour, 2009, Chapter 8 179
Table 832 Accuracy of forecasts with identification and allINAR(1) for INARMA(0,0) series
Parameters
?=?? ?=?? ?=?? ?=??
LjungBox AllINAR(1) LjungBox AllINAR(1) LjungBox AllINAR(1) LjungBox AllINAR(1)
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.3 0.0000 0.3509 1.1781 0.0040 0.3610 1.1919 0.0005 0.3309 1.1476 0.0008 0.3383 1.1552 0.0032 0.3161 1.0892 0.0066 0.3295 1.0741 0.0039 0.3075 1.0115 0.0025 0.3131 1.0161
?= 0.5 0.0021 0.5796 1.0779 0.0054 0.5967 1.0959 0.0012 0.5512 1.0077 0.0019 0.5613 1.0133 0.0009 0.5368 0.9725 0.0013 0.5561 0.9737 0.0039 0.5183 0.9365 0.0017 0.5191 0.9412
?= 0.7
0.0003 0.8039 0.9807 0.0062 0.8331 0.9786 0.0039 0.7708 0.9005 0.0017 0.7855 0.9152 0.0066 0.7572 0.8817 0.0016 0.7510 0.8937 0.0048 0.7241 0.8537 0.0012 0.7374 0.8466
?= 1 0.0012 1.1651 0.8716 0.0028 1.1930 0.8947 0.0002 1.0979 0.8120 0.0011 1.1178 0.8275 0.0000 1.0774 0.8025 0.0068 1.0679 0.7956 0.0115 1.0490 0.7596 0.0023 1.0481 0.7666
?= 3 0.0210 3.4439 0.8065 0.0273 3.5655 0.8128 0.0154 3.3509 0.7703 0.0024 3.3711 0.7799 0.0087 3.2019 0.7589 0.0083 3.2774 0.7651 0.0054 3.1061 0.7422 0.0086 3.0890 0.7322
?= 5 0.0099 5.8831 0.8153 0.0158 5.8972 0.8071 0.0067 5.5456 0.7624 0.0150 5.4933 0.7567 0.0001 5.4289 0.7581 0.0010 5.4727 0.7642 0.0123 5.2099 0.7375 0.0003 5.2125 0.7325
?= 20 0.0248 22.8486 0.7730 0.0053 23.8074 0.8088 0.0543 22.0121 0.7581 0.0683 22.0701 0.7631 0.0183 21.2026 0.7420 0.0101 22.2400 0.7589 0.0079 20.5646 0.7248 0.0335 20.8631 0.7307
M.Mohammadipour, 2009, Chapter 8 180
8.6.2.3 Identification among Four Processes
In this section it is assumed that data can be produced by one of the four processes
that we focus on in this study: INARMA(0,0), INAR(1), INMA(1), or
INARMA(1,1). As previously mentioned, two methods of identification are used.
The twostage method is based on first using the LjungBox ??statistic of Equation
815 to distinguish between the INARMA(0,0) or random Poisson process from the
other possible INARMA models. Then, the other three models (INAR(1), INMA(1),
and INARMA(1,1)) are distinguished using the Akaike information criterion based
on the expression for ARMA models:
AIC??log??
2 + 2?
Equation 816
As explained in section 4.5, when the sample size is small (?/?< 40), the
following bias correction is necessary:
AICC ??log??
2 + 2?+ 2?(?+ 1)/(????1)
Equation 817
On the other hand, the onestage method only uses the AIC to select the appropriate
model. As discussed in chapter 4, although the above equations have been developed
for ARMA models with a Normal distribution, as the likelihood function for INMA
processes has not been established in the literature and AIC is a method of
identification that can be automated, we use these equations to test the performance
of AIC for INARMA processes.
The percentage of correct identification for each of the four INARMA processes for
both twostage and onestage methods is shown in Table ?833, Table ?834, Table
?835, and Table ?836.
The results of Table ?833 confirm the results of section 8.6.2.1 in that the twostage
method identifies the INARMA(0,0) processes more frequently than the onestage
method.
M.Mohammadipour, 2009, Chapter 8 181
Table 833 The percentage of correct identification for INARMA(0,0) series
Parameters
Twostage identification Onestage identification
?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=??
?= 0.3 94.62 95.87 96.30 94.90 89.40 89.00 88.60 80.80
?= 0.5 93.29 94.45 94.40 95.10 91.90 88.10 84.30 73.30
?= 0.7
92.81 94.07 95.30 95.90 88.80 82.80 78.80 69.00
?= 1 93.54 94.32 94.10 95.10 90.40 79.60 76.00 67.20
?= 3 93.40 92.40 94.40 94.70 82.70 72.50 65.40 57.00
?= 5 92.90 93.40 92.80 93.50 78.50 70.90 61.70 55.00
?= 20 92.70 93.40 94.10 95.30 76.20 66.00 61.00 51.40
For an INAR(1) case, the results of Table ?834 confirm that when the autoregressive
parameter is low, the process is misidentified in most cases for both identification
methods. However, the onestage method produces better results. On the other hand,
with a high autoregressive parameter, the performance of both methods improves.
The results also show that when more observations are available, the percentage of
correct identification increases for both methods. The onestage method outperforms
the twostage method in most of the cases, especially for small samples.
Table 834 The percentage of correct identification for INAR(1) series
Parameters
Twostage identification Onestage identification
?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=??
?= 0.1,?= 0.5 2.30 2.50 2.60 2.90 9.60 15.80 21.20 26.40
?= 0.5,?= 0.5 9.50 19.80 25.80 52.70 38.40 55.60 67.70 75.80
?= 0.9,?= 0.5 30.60 55.00 61.50 73.80 73.30 71.70 71.70 72.50
?= 0.1,?= 1 1.70 1.50 1.80 3.00 13.20 19.20 22.20 30.00
?= 0.5,?= 1 10.90 18.70 26.60 49.70 46.90 59.20 65.90 72.40
?= 0.9,?= 1 28.60 52.40 62.50 71.50 74.70 71.80 72.20 69.30
?= 0.1,?= 3 1.90 2.50 1.70 2.90 16.30 22.10 22.40 28.00
?= 0.5,?= 3 11.20 16.10 24.30 50.20 47.40 57.90 60.70 68.00
?= 0.9,?= 3 24.40 46.30 63.90 75.30 75.50 74.90 72.30 75.50
?= 0.1,?= 5 2.70 2.40 2.70 2.50 17.50 22.70 22.50 26.10
?= 0.5,?= 5 12.10 17.60 26.30 44.70 50.20 59.10 61.50 68.30
?= 0.9,?= 5 27.30 49.40 63.00 73.80 76.20 75.40 73.30 73.90
As can be seen from Table ?835, an INMA(1) process is misidentified in most of the
cases regardless of the size of the moving average parameter. However, the results
show that it does not affect the forecasting accuracy to a great extent. This can be
seen by comparing the results of Table ?838 and Table ?841. The onestage
identification method again outperforms the twostage method.
M.Mohammadipour, 2009, Chapter 8 182
Table 835 The percentage of correct identification for INMA(1) series
Parameters
Twostage identification Onestage identification
?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=??
?= 0.1,?= 0.5 3.30 2.70 3.30 3.10 4.10 6.70 8.10 12.00
?= 0.5,?= 0.5 4.30 4.00 4.60 7.60 13.40 19.80 23.00 23.20
?= 0.9,?= 0.5 7.90 10.60 11.60 32.50 21.70 33.70 37.80 45.10
?= 0.1,?= 1 4.40 3.90 3.30 1.50 5.60 6.80 8.10 11.50
?= 0.5,?= 1 3.20 3.90 3.30 4.10 9.80 14.60 17.20 16.80
?= 0.9,?= 1 7.80 7.60 6.70 14.90 19.80 23.50 24.70 21.50
?= 0.1,?= 3 3.80 3.40 2.50 1.20 5.40 6.70 9.00 14.00
?= 0.5,?= 3 2.00 2.70 2.50 2.10 10.90 14.60 15.10 11.50
?= 0.9,?= 3 4.60 4.00 4.30 7.10 18.30 16.70 14.40 11.00
?= 0.1,?= 5 4.80 2.30 2.50 2.00 5.70 7.40 12.10 12.10
?= 0.5,?= 5 3.90 3.20 2.20 1.80 10.60 14.20 11.20 9.60
?= 0.9,?= 5 3.80 4.30 3.20 5.00 17.30 15.80 13.40 7.50
The results of Table ?836 suggest that, as expected, when the autoregressive
parameter is high, the correct model is identified more often than the case with low
autoregressive parameter. The identification performance improves when the length
of history increases. The onestage identification method produces better results than
the twostage method in most of the cases.
Table 836 The percentage of correct identification for INARMA(1,1) series
Parameters
Twostage identification Onestage identification
?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=?? ?=??
?= 0.1,?= 0.1,?= 0.5 0.60 0.40 0.90 1.70 1.30 2.40 4.40 13.20
?= 0.1,?= 0.9,?= 0.5 1.70 2.70 6.90 14.00 5.80 11.20 10.80 12.00
?= 0.5,?= 0.5,?= 0.5 3.10 6.20 10.80 14.50 7.80 12.10 13.90 16.90
?= 0.9,?= 0.1,?= 0.5 6.20 15.50 19.60 29.50 14.20 23.70 26.10 25.60
?= 0.1,?= 0.1,?= 1 0.10 0.80 1.30 1.70 1.70 6.90 11.80 20.10
?= 0.1,?= 0.9,?= 1 2.30 4.20 7.10 11.20 7.90 14.40 15.30 12.80
?= 0.5,?= 0.5,?= 1 3.00 8.60 11.80 15.20 11.50 18.80 20.10 14.70
?= 0.9,?= 0.1,?= 1 7.30 20.00 29.70 31.30 24.20 31.60 34.70 34.40
?= 0.1,?= 0.1,?= 5 0.50 1.70 2.20 3.70 8.80 18.70 23.70 33.10
?= 0.1,?= 0.9,?= 5 7.00 13.20 17.20 23.80 29.50 40.40 42.30 27.20
?= 0.5,?= 0.5,?= 5 12.20 18.80 32.50 37.90 35.60 46.30 50.40 43.40
?= 0.9,?= 0.1,?= 5 14.20 31.40 42.10 47.80 37.10 49.50 48.70 47.80
The accuracy of INAR(1), INMA(1) and INARMA(1,1) forecasts using ME, MSE
and MASE for two identification methods are presented in Table ?837, Table ?838,
and Table ?839. Similar results for the cases that the order of the model is known are
M.Mohammadipour, 2009, Chapter 8 183
provided in Table ?840, Table ?841, and Table ?842 for comparison reasons.
The results show that for processes with an AR component, when the autoregressive
parameter is high, misidentification has a great effect on the accuracy of forecasts.
However, because the onestage identification method identifies the correct model
more frequently than the twostage method, the forecasts are closer to the case of
known order. But when the autoregressive parameter is small, the effect of
misidentification on forecasting accuracy is also small. For INARMA processes
without an AR component, the effect of misidentification on forecasting accuracy is
small, regardless of the size of the MA parameter. When the number of observations
increases, forecasts with identification and without identification have similar
accuracy.
M.Mohammadipour, 2009, Chapter 8 184
Table 837 Accuracy of INAR(1) forecasts for onestage and twostage identification procedures
Parameters
?=?? ?=?? ?=?? ?=??
Twostage
identification
Onestage
identification
Twostage
identification
Onestage
identification
Twostage
identification
Onestage
identification
Twostage
identification
Onestage
identification
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0013 0.6357 1.1459 0.0005 0.6347 1.1208 0.0073 0.6327 1.1044 0.0115 0.6260 1.1063 0.0008 0.6026 1.0373 0.0056 0.6005 1.0295 0.0019 0.5765 0.9804 0.0010 0.5826 0.9799
?= 0.5,?= 0.5 0.0034 1.1192 1.4704 0.0381 1.0614 1.3967 0.0278 1.0414 1.2999 0.0050 0.9311 1.2226 0.0179 1.0165 1.2303 0.0064 0.8655 1.1425 0.0200 0.8786 1.1276 0.0055 0.8014 1.0694
?= 0.9,?= 0.5 0.0283 2.6093 1.9312 0.0279 1.4243 1.4776 0.0005 2.0101 1.6350 0.0034 1.1610 1.2703 0.0090 1.6472 1.4275 0.0103 1.0725 1.2157 0.0025 1.0257 1.1445 0.0120 0.9981 1.1395
?= 0.1,?= 1 0.0188 1.3041 0.9162 0.0063 1.3144 0.9415 0.008 1.2309 0.8614 0.0024 1.2606 0.8758 0.0029 1.2061 0.8395 0.0071 1.2258 0.8523 0.0005 1.1699 0.8088 0.0044 1.1590 0.8104
?= 0.5,?= 1 0.0409 2.3062 1.2365 0.0196 2.0526 1.1509 0.0131 2.0748 1.1167 0.0117 1.8415 1.0550 0.0019 1.9812 1.0876 0.0093 1.7273 1.0262 0.0109 1.7521 1.0034 0.0039 1.6017 0.9651
?= 0.9,?= 1 0.018 5.2825 1.7681 0.0212 2.7024 1.2864 0.0258 4.2144 1.4907 0.0037 2.2686 1.1378 0.0222 3.1025 1.2932 0.0043 2.1714 1.1203 0.0087 2.0297 1.0845 0.0012 2.0015 1.0693
?= 0.1,?= 3 0.0055 3.7732 0.8183 0.0310 3.8137 0.8345 0.0096 3.7235 0.8334 0.0042 3.7900 0.8337 0.0036 3.5162 0.7995 0.0041 3.6297 0.8083 0.0098 3.5198 0.7902 0.0184 3.4845 0.7845
?= 0.5,?= 3 0.0356 6.6692 1.0898 0.0256 6.0453 1.0472 0.0322 6.4381 1.0708 0.0170 5.5157 0.9955 0.0377 6.0613 1.0365 0.0103 5.1944 0.9556 0.0263 5.1238 0.9517 0.0027 4.7714 0.9155
?= 0.9,?= 3 0.0466 17.2237 1.6973 0.0198 8.0490 1.2229 0.0386 12.4798 1.4291 0.0090 6.7598 1.1070 0.0380 8.8346 1.1905 0.0301 6.4818 1.0660 0.0107 6.1671 1.0259 0.0099 6.0402 1.0160
?= 0.1,?= 5 0.0215 6.3926 0.8466 0.0440 6.7467 0.8632 0.0314 6.1703 0.8199 0.0022 6.1377 0.8190 0.0108 6.0691 0.8034 0.0241 6.1470 0.8087 0.0073 5.7416 0.7720 0.0252 5.9063 0.7726
?= 0.5,?= 5 0.0479 10.7311 1.0682 0.0116 9.9783 1.0548 0.1183 10.4885 1.0480 0.0001 9.0103 0.9843 0.0082 9.7144 1.0139 0.0479 8.4696 0.9420 0.0108 8.5805 0.9368 0.0078 7.9000 0.9092
?= 0.9,?= 5 0.0995 26.0358 1.6186 0.1313 13.5067 1.1973 0.0334 21.0114 1.4054 0.0554 11.3171 1.0809 0.0524 15.4657 1.1925 0.0363 10.8754 1.0706 0.0140 10.2025 1.0191 0.0036 10.0731 1.0171
M.Mohammadipour, 2009, Chapter 8 185
Table 838 Accuracy of INMA(1) forecasts for onestage and twostage identification procedures
Parameters
?=?? ?=?? ?=?? ?=??
Twostage
identification
Onestage
identification
Twostage
identification
Onestage
identification
Twostage
identification
Onestage
identification
Twostage
identification
Onestage
identification
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0263 0.6370 1.1790 0.0006 0.6474 1.1537 0.0081 0.6080 1.0762 0.0042 0.6022 1.0458 0.0030 0.5913 1.0025 0.0005 0.5732 0.9729 0.0064 0.5719 0.9804 0.0088 0.6604 0.7515
?= 0.5,?= 0.5 0.0166 0.8479 1.3582 0.0210 0.8392 1.2970 0.0125 0.8181 1.2071 0.0001 0.7935 1.1635 0.0022 0.7871 1.1224 0.0013 0.7284 1.0496 0.0082 0.7597 1.0724 0.0401 0.9122 0.9183
?= 0.9,?= 0.5 0.0246 1.0729 1.4336 0.0101 1.9689 1.1383 0.0207 1.0169 1.3105 0.0269 0.9813 1.2952 0.0240 1.0123 1.2547 0.0304 0.8882 1.1203 0.0250 0.9228 1.1422 0.1589 1.0747 1.0178
?= 0.1,?= 1 0.0003 1.2975 0.9130 0.0104 1.2574 0.8783 0.0070 1.2071 0.8535 0.0014 1.2389 0.8692 0.0017 1.1671 0.8266 0.0103 1.1444 0.8014 0.0036 1.1466 0.8073 0.0335 1.2159 0.8080
?= 0.5,?= 1 0.0098 1.7287 1.0880 0.0241 1.7225 1.0816 0.0153 1.6183 1.0208 0.0065 1.5827 0.9817 0.0051 1.6029 1.0149 0.0020 1.4386 0.9239 0.0077 1.5257 0.9653 0.0177 1.5554 0.9547
?= 0.9,?= 1 0.0081 2.1437 1.2136 0.0131 1.9568 1.1574 0.0027 2.0207 1.1294 0.0036 1.8299 1.0827 0.0221 2.0155 1.1022 0.0179 1.6855 0.9954 0.0025 1.7263 1.0134 0.0595 1.7732 1.0167
?= 0.1,?= 3 0.0143 3.8219 0.8451 0.0170 3.9592 0.8653 0.0095 3.6174 0.8141 0.0217 3.7610 0.8249 0.0132 3.5934 0.8019 0.0052 3.4847 0.7859 0.0158 3.3791 0.7756 0.0006 3.5011 0.8056
?= 0.5,?= 3 0.0029 5.1085 0.9854 0.0143 5.2555 0.9867 0.0086 4.8587 0.9522 0.0237 4.8094 0.9361 0.0259 4.7425 0.9197 0.0040 4.2445 0.8681 0.0024 4.5513 0.8899 0.0040 4.2567 0.8749
?= 0.9,?= 3 0.0104 6.3963 1.1064 0.0283 6.0747 1.0769 0.0162 5.9595 1.0526 0.0190 5.4653 1.0106 0.0191 5.7408 1.0273 0.0034 4.8673 0.9248 0.0221 5.0070 0.9420 0.0028 4.8710 0.9265
?= 0.1,?= 5 0.0349 6.4620 0.8532 0.0388 6.5789 0.8436 0.0228 6.1804 0.8130 0.0241 6.1703 0.8007 0.0035 5.9428 0.7934 0.0168 5.7806 0.7811 0.0003 5.7474 0.7794 0.0160 5.7818 0.7843
?= 0.5,?= 5 0.0000 8.3916 0.9442 0.0841 8.5133 0.9644 0.0077 8.0429 0.9225 0.0184 7.8372 0.9231 0.0038 8.0922 0.9232 0.0028 7.1503 0.8623 0.0291 7.5896 0.8869 0.0027 7.1503 0.8628
?= 0.9,?= 5 0.0320 10.5829 1.1034 0.0552 9.9051 1.0576 0.0166 10.2863 1.0590 0.0005 8.9502 0.9893 0.0596 9.3927 0.9941 0.0135 7.9759 0.9193 0.0112 8.3874 0.9427 0.0135 7.9764 0.9194
M.Mohammadipour, 2009, Chapter 8 186
Table 839 Accuracy of INARMA(1,1) forecasts for onestage and twostage identification procedures
Parameters
?=?? ?=?? ?=?? ?=??
Twostage
identification
Onestage
identification
Twostage
identification
Onestage
identification
Twostage
identification
Onestage
identification
Twostage
identification
Onestage
identification
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.1,?= 0.5 0.0011 0.7193 1.1489 0.0064 0.7234 1.2251 0.0071 0.6856 1.0945 0.0163 0.6807 1.1061 0.0040 0.6758 1.0704 0.0071 0.6736 1.0554 0.0059 0.6447 1.0204 0.0001 0.6425 1.0050
?= 0.1,?= 0.9,?= 0.5 0.0368 1.3416 1.5159 0.0243 1.2303 1.4021 0.0201 1.2114 1.2766 0.0094 1.0992 1.2406 0.0173 1.1478 1.2195 0.0162 1.0410 1.1899 0.0159 1.0410 1.1174 0.0138 1.0175 1.1117
?= 0.5,?= 0.5,?= 0.5 0.0536 1.9040 1.5660 0.0198 1.6235 1.4007 0.0186 1.6852 1.3458 0.0318 1.4292 1.2942 0.0301 1.5869 1.3092 0.0198 1.3254 1.1971 0.0064 1.1989 1.0976 0.0049 1.1844 1.1020
?= 0.9,?= 0.1,?= 0.5 0.1342 2.8645 2.0436 0.0480 1.5286 1.4458 0.0377 2.1973 1.6213 0.0289 1.3400 1.3159 0.0277 1.6027 1.3899 0.0310 1.1833 1.2270 0.0105 1.1303 1.1566 0.0057 1.1247 1.1435
?= 0.1,?= 0.1,?= 1 0.0004 1.4375 0.9427 0.0167 1.4091 0.9407 0.0130 1.3668 0.8973 0.0079 1.3705 0.9013 0.0038 1.3390 0.8764 0.0020 1.3314 0.8871 0.0064 1.2803 0.8575 0.0080 1.2693 0.8538
?= 0.1,?= 0.9,?= 1 0.0137 2.5594 1.2132 0.0161 2.3860 1.1767 0.0010 2.4152 1.1409 0.0137 2.1568 1.0963 0.0109 2.3587 1.0905 0.0001 1.9950 1.0265 0.0028 2.0246 1.0093 0.0032 1.9137 0.9745
?= 0.5,?= 0.5,?= 1 0.0028 3.7251 1.3345 0.0434 3.0511 1.1857 0.0317 3.4130 1.2229 0.0026 2.7933 1.1045 0.0225 3.0826 1.1397 0.0027 2.5683 1.0629 0.0053 2.3995 1.0005 0.0111 2.2977 0.9876
?= 0.9,?= 0.1,?= 1 0.0911 6.0831 1.7798 0.0259 3.0317 1.2791 0.0230 4.8164 1.5115 0.0462 2.4860 1.1595 0.0248 3.4115 1.2711 0.0062 2.3640 1.0960 0.0036 2.2186 1.0714 0.0052 2.1915 1.0725
?= 0.1,?= 0.1,?= 5 0.0126 7.1217 0.8862 0.0116 7.3120 0.8927 0.0082 6.8366 0.8465 0.0300 6.7345 0.8344 0.0131 6.6983 0.8327 0.0088 6.6701 0.8331 0.0071 6.4563 0.8119 0.0136 6.3122 0.8035
?= 0.1,?= 0.9,?= 5 0.0523 12.8778 1.1181 0.0211 11.8098 1.0700 0.0125 11.8990 1.0497 0.0111 10.4597 0.9831 0.0224 11.3640 1.0295 0.0163 9.9553 0.9482 0.0102 9.9313 0.9337 0.0177 9.2879 0.9064
?= 0.5,?= 0.5,?= 5 0.0698 18.7993 1.2511 0.0942 14.7010 1.0942 0.0161 16.7775 1.1609 0.0720 12.9606 1.0033 0.0238 15.2377 1.0874 0.0601 12.3897 0.9876 0.0213 11.8909 0.9624 0.0353 11.4452 0.9463
?= 0.9,?= 0.1,?= 5 0.0827 29.4367 1.6336 0.0109 15.2016 1.2041 0.0115 23.6024 1.4174 0.0590 12.3696 1.0847 0.0741 16.4158 1.2023 0.0188 11.9975 1.0709 0.0215 11.0569 1.0183 0.0287 11.1985 1.0193
M.Mohammadipour, 2009, Chapter 8 187
Table 840 Accuracy of INAR(1) forecasts when the order in known
Parameters
?=?? ?=?? ?=?? ?=??
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0026 0.6614 1.1418 0.0089 0.6286 1.0555 0.0052 0.6047 1.0237 0.0045 0.5822 0.9822
?= 0.5,?= 0.5 0.0026 0.9466 1.3290 0.0089 0.8743 1.1807 0.0044 0.8532 1.1570 0.0054 0.7834 1.0534
?= 0.9,?= 0.5 0.0017 1.2082 1.3901 0.0042 1.1489 1.2894 0.0043 1.0811 1.2176 0.0033 1.0164 1.1486
?= 0.1,?= 1 0.0149 1.2880 0.9376 0.0059 1.2072 0.8496 0.0052 1.1967 0.8382 0.0029 1.1663 0.8057
?= 0.5,?= 1 0.0195 1.8846 1.0837 0.0018 1.7244 1.0202 0.0045 1.6745 1.0059 0.0026 1.5905 0.9618
?= 0.9,?= 1 0.0121 2.5216 1.2401 0.0054 2.2655 1.1594 0.0146 2.1925 1.1262 0.0105 2.0319 1.0686
?= 0.1,?= 3 0.0100 3.9224 0.8544 0.0176 3.6703 0.8228 0.0010 3.6641 0.8136 0.0085 3.4701 0.7846
?= 0.5,?= 3 0.0308 5.6926 1.0126 0.0035 5.1557 0.9491 0.0078 4.9742 0.9331 0.0143 4.7749 0.9160
?= 0.9,?= 3 0.0906 7.5862 1.1509 0.0298 6.7494 1.1026 0.0243 6.4376 1.0658 0.0093 6.0230 1.0205
?= 0.1,?= 5 0.0371 6.6926 0.8565 0.0209 6.1869 0.8143 0.0282 6.0095 0.7973 0.0139 5.7529 0.7742
?= 0.5,?= 5 0.0123 9.3467 0.9840 0.0082 8.6581 0.9653 0.0116 8.3583 0.9392 0.0034 7.8699 0.9011
?= 0.9,?= 5 0.0013 11.9986 1.1483 0.0126 11.2051 1.0914 0.0467 10.8985 1.0720 0.0185 10.1102 1.0272
Table 841 Accuracy of INMA(1) forecasts when the order in known
Parameters
?=?? ?=?? ?=?? ?=??
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0031 0.6295 1.1297 0.0027 0.6035 1.0473 0.0114 0.5993 1.0493 0.0007 0.5702 0.9736
?= 0.5,?= 0.5 0.0395 0.8793 1.3407 0.0279 0.8552 1.2527 0.0229 0.7997 1.1494 0.0031 0.7885 1.0948
?= 0.9,?= 0.5 0.0767 1.1019 1.5472 0.0671 1.0609 1.3632 0.0628 1.0229 1.2606 0.0445 0.9878 1.1797
?= 0.1,?= 1 0.0234 1.2748 0.9163 0.0000 1.2038 0.8547 0.0082 1.1724 0.8365 0.0048 1.1313 0.7957
?= 0.5,?= 1 0.0347 1.7554 1.0666 0.0440 1.6455 1.0247 0.0310 1.6074 1.0005 0.0204 1.5302 0.9648
?= 0.9,?= 1 0.1249 2.2869 1.2485 0.1025 2.1650 1.1687 0.1148 2.0762 1.1223 0.1116 1.9944 1.0920
?= 0.1,?= 3 0.0452 3.9039 0.8547 0.0402 3.6622 0.8158 0.0150 3.5353 0.7960 0.0062 3.4237 0.7767
?= 0.5,?= 3 0.1260 5.2415 0.9971 0.0781 4.9565 0.9450 0.0929 4.8942 0.9424 0.0608 4.6821 0.9036
?= 0.9,?= 3 0.2970 6.7422 1.1268 0.3209 6.5993 1.1023 0.2796 6.2599 1.0587 0.2718 6.0678 1.0364
?= 0.1,?= 5 0.0838 6.4549 0.8431 0.0132 6.2194 0.8182 0.0226 5.9292 0.7937 0.0111 5.7446 0.7806
?= 0.5,?= 5 0.1407 8.7278 0.9676 0.1984 8.3020 0.9417 0.1738 8.0993 0.9245 0.1269 7.8404 0.9063
?= 0.9,?= 5 0.4387 11.6802 1.1864 0.4081 10.9042 1.0713 0.4804 10.7677 1.0895 0.4479 10.1863 1.0242
Table 842 Accuracy of INARMA(1,1) forecasts when the order in known
Parameters
?=?? ?=?? ?=?? ?=??
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.1,?= 0.5 0.0007 0.7482 1.2163 0.0066 0.7138 1.1186 0.0019 0.6777 1.0787 0.0047 0.6468 1.0076
?= 0.1,?= 0.9,?= 0.5 0.0401 1.2112 1.4173 0.0336 1.1340 1.2974 0.0267 1.0734 1.1958 0.0163 1.0429 1.1248
?= 0.5,?= 0.5,?= 0.5 0.0485 1.5791 1.4864 0.0209 1.4070 1.2666 0.0445 1.3825 1.1924 0.0261 1.2261 1.1137
?= 0.9,?= 0.1,?= 0.5 0.0920 1.4003 1.4228 0.0532 1.2843 1.3142 0.0487 1.2226 1.2608 0.0324 1.1110 1.1489
?= 0.1,?= 0.1,?= 1 0.0041 1.4908 0.9689 0.0108 1.4146 0.9145 0.0129 1.3358 0.8925 0.0119 1.2833 0.8575
?= 0.1,?= 0.9,?= 1 0.0192 2.3265 1.1462 0.0185 2.2569 1.0930 0.0368 2.2224 1.0713 0.0149 2.0900 1.0210
?= 0.5,?= 0.5,?= 1 0.0296 3.1322 1.1914 0.0250 2.7101 1.1033 0.0409 2.6748 1.0798 0.0220 2.4363 1.0245
?= 0.9,?= 0.1,?= 1 0.0665 2.8324 1.2617 0.0494 2.5623 1.1748 0.0640 2.4301 1.1219 0.0413 2.2588 1.0786
?= 0.1,?= 0.1,?= 5 0.0279 7.4165 0.8824 0.0070 6.9501 0.8502 0.0011 6.7534 0.8307 0.0006 6.4253 0.8079
?= 0.1,?= 0.9,?= 5 0.0436 11.2147 1.0506 0.0580 10.5085 0.9941 0.0521 10.1101 0.9589 0.0606 9.7366 0.9338
?= 0.5,?= 0.5,?= 5 0.1082 14.0754 1.0425 0.1239 13.0227 1.0240 0.1074 12.7060 0.9952 0.1065 11.8259 0.9556
?= 0.9,?= 0.1,?= 5 0.1214 13.8720 1.1795 0.1895 12.4636 1.0932 0.1569 12.0389 1.0813 0.0906 11.2782 1.0311
M.Mohammadipour, 2009, Chapter 8 188
8.6.2.4 AllINARMA(1,1)
In this section the method of section 8.6.2.2 is extended to include all four processes.
Therefore data can be produced by either an INARMA(0,0), INAR(1), INMA(1) or
an INARMA(1,1) process. Then, for estimation of parameters and forecasting, an
INARMA(1,1) process is used. We expect that when data is in fact INARMA(0,0)
the estimated autoregressive and moving average parameters (?,?) will be close to
zero, and for INAR(1) and INMA(1) data, the estimated ? or ? will be close to zero,
respectively.
The results for all points in time are shown in Table ?843, Table ?844, and Table ?845.
The results show that, for INARMA(0,0), identification produces better forecasts
than the allINARMA(1,1) approach. When the number of observations increases,
the results of two approaches become close. For ?= 96, the degree of improvement
by using identification rather than allINARMA(1,1) is on average 2.3 percent.
For INAR(1) and INMA(1) processes, when the number of observations is small, the
allINARMA(1,1) approach produces better results in many cases. But when the
number of observations increases, the results of identification improve and the two
methods produce close results.
Based on the results of Table ?843, Table ?844, and Table ?845, using the most general
model could be a good substitute for identification especially when less data is
available as is often the case for intermittent demand data. Although it has not been
looked at in the literature, the results suggest that treating the data as the general
INARMA process and eliminating the complexity of identification, can be considered
as a potentially useful approach.
The forecast accuracy of allINARMA(1,1) is compared to those of allINAR(1) in
Appendix 8.G. It is expected that for INARMA(0,0) and INAR(1) series the latter
outperforms the former and the results confirm this for most of the cases. However,
the results show that, even for INMA(1) and INARMA(1,1) series, and even for high
moving average parameters, allINAR(1) method produces more accurate forecasts
(in terms of MSE and MASE) than allINARMA(1,1) method in most of the cases.
The difference increases for longer history.
M.Mohammadipour, 2009, Chapter 8 189
Table 843 Accuracy of forecasts with identification and allINARMA(1,1) for INARMA(0,0) series
Parameters
?=?? ?=?? ?=?? ?=??
Twostage
identification
AllINARMA(1,1)
Twostage
identification
AllINARMA(1,1)
Twostage
identification
AllINARMA(1,1)
Twostage
identification
AllINARMA(1,1)
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.3 0.0020 0.3496 1.1877 0.0032 0.3692 1.2046 0.0019 0.3307 1.1525 0.0048 0.3470 1.1612 0.0003 0.3190 1.0924 0.0064 0.3374 1.1003 0.0028 0.3146 1.0229 0.0032 0.3182 1.0161
?= 0.5 0.0010 0.5789 1.0799 0.0088 0.6195 1.1038 0.0002 0.5493 1.0109 0.0072 0.5774 1.0240 0.0050 0.5209 0.9639 0.0114 0.5404 0.9630 0.0054 0.5099 0.9213 0.0011 0.5416 0.9403
?= 0.7
0.0008 0.8074 0.9693 0.0137 0.8690 0.9951 0.0011 0.7706 0.9111 0.0057 0.8129 0.9260 0.0023 0.7429 0.8764 0.0193 0.7647 0.8797 0.0022 0.7186 0.8378 0.0010 0.7351 0.8524
?= 1 0.0001 1.1538 0.8665 0.0203 1.2508 0.8980 0.0011 1.1076 0.8157 0.0111 1.1551 0.8358 0.0152 1.0705 0.7820 0.0017 1.1163 0.8011 0.0061 1.0289 0.7508 0.0045 1.0540 0.7640
?= 3 0.0042 3.4186 0.8175 0.0324 3.5960 0.8346 0.0064 3.3024 0.7818 0.0082 3.3893 0.7871 0.0162 3.2304 0.7654 0.0151 3.3233 0.7691 0.0025 3.0952 0.7386 0.0081 3.1778 0.7455
?= 5 0.0161 5.8873 0.8155 0.0507 6.0384 0.8235 0.0107 5.5208 0.7711 0.0144 5.7687 0.7783 0.0042 5.4208 0.7613 0.0135 5.5495 0.7689 0.0153 5.2324 0.7339 0.0056 5.1751 0.7306
?= 20 0.0509 22.9286 0.7872 0.0146 24.9564 0.8225 0.0421 21.9667 0.7702 0.0514 22.8435 0.7773 0.0156 21.5521 0.7400 0.0269 22.0828 0.7551 0.0252 20.5910 0.7322 0.0400 21.1039 0.7334
Table 844 Accuracy of forecasts with identification and allINARMA(1,1) for INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Onestage
identification
AllINARMA(1,1)
Onestage
identification
AllINARMA(1,1)
Onestage
identification
AllINARMA(1,1)
Onestage
identification
AllINARMA(1,1)
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0005 0.6347 1.1208 0.0008 0.6699 1.1275 0.0115 0.6260 1.1063 0.0049 0.6447 1.0710 0.0056 0.6005 1.0295 0.0111 0.6389 1.0340 0.0010 0.5826 0.9799 0.0010 0.5857 0.9845
?= 0.5,?= 0.5 0.0381 1.0614 1.3967 0.0130 1.0056 1.3278 0.0050 0.9311 1.2226 0.0039 0.9436 1.2602 0.0064 0.8655 1.1425 0.0041 0.9131 1.1864 0.0055 0.8014 1.0694 0.0033 0.8186 1.0727
?= 0.9,?= 0.5 0.0279 1.4243 1.4776 0.0347 1.2681 1.4405 0.0034 1.1610 1.2703 0.0344 1.1681 1.3181 0.0103 1.0725 1.2157 0.0066 1.0919 1.2150 0.0120 0.9981 1.1395 0.0206 1.0235 1.1707
?= 0.1,?= 1 0.0063 1.3144 0.9415 0.0152 1.3755 0.9455 0.0024 1.2606 0.8758 0.0028 1.2778 0.8819 0.0071 1.2258 0.8523 0.0066 1.1947 0.8427 0.0044 1.1590 0.8104 0.0045 1.1875 0.8235
?= 0.5,?= 1 0.0196 2.0526 1.1509 0.0092 1.9908 1.1188 0.0117 1.8415 1.0550 0.0050 1.8491 1.0640 0.0093 1.7273 1.0262 0.0046 1.7735 1.0251 0.0039 1.6017 0.9651 0.0052 1.6469 0.9944
?= 0.9,?= 1 0.0212 2.7024 1.2864 0.0544 2.4325 1.2304 0.0037 2.2686 1.1378 0.0378 2.3301 1.1748 0.0043 2.1714 1.1203 0.0484 2.1853 1.1423 0.0012 2.0015 1.0693 0.0312 2.0343 1.0747
?= 0.1,?= 3 0.0310 3.8137 0.8345 0.0275 4.1292 0.8818 0.0042 3.7900 0.8337 0.0177 3.7457 0.8236 0.0041 3.6297 0.8083 0.0144 3.6442 0.8115 0.0184 3.4845 0.7845 0.0222 3.4929 0.7828
?= 0.5,?= 3 0.0256 6.0453 1.0472 0.0237 5.8020 1.0279 0.0170 5.5157 0.9955 0.0291 5.4831 0.9848 0.0103 5.1944 0.9556 0.0027 5.2166 0.9710 0.0027 4.7714 0.9155 0.0268 4.9748 0.9319
?= 0.9,?= 3 0.0198 8.0490 1.2229 0.1097 7.3168 1.1422 0.0090 6.7598 1.1070 0.1077 6.9143 1.1046 0.0301 6.4818 1.0660 0.0627 6.5204 1.0742 0.0099 6.0402 1.0160 0.0612 6.1440 1.0290
?= 0.1,?= 5 0.0440 6.7467 0.8632 0.0124 6.5975 0.8522 0.0022 6.1377 0.8190 0.0016 6.3370 0.8206 0.0241 6.1470 0.8087 0.0078 6.2455 0.8197 0.0252 5.9063 0.7726 0.0063 5.8121 0.7805
?= 0.5,?= 5 0.0116 9.9783 1.0548 0.0798 9.5203 1.0114 0.0001 9.0103 0.9843 0.0050 8.8544 0.9603 0.0479 8.4696 0.9420 0.0325 8.4735 0.9460 0.0078 7.9000 0.9092 0.0003 7.9638 0.9095
?= 0.9,?= 5 0.1313 13.5067 1.1973 0.0773 12.2386 1.1385 0.0554 11.3171 1.0809 0.1196 11.3739 1.0820 0.0363 10.8754 1.0706 0.1021 10.9289 1.0679 0.0036 10.0731 1.0171 0.0956 10.1452 1.0232
M.Mohammadipour, 2009, Chapter 8 190
Table 845 Accuracy of forecasts with identification and allINARMA(1,1) for INMA(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Onestage
identification
AllINARMA(1,1)
Onestage
identification
AllINARMA(1,1)
Onestage
identification
AllINARMA(1,1)
Onestage
identification
AllINARMA(1,1)
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0006 0.6474 1.1537 0.0056 0.6500 1.1831 0.0042 0.6022 1.0458 0.0038 0.6155 1.0589 0.0005 0.5732 0.9729 0.0071 0.6070 1.0025 0.0088 0.6604 0.7515 0.0063 0.5733 0.9809
?= 0.5,?= 0.5 0.0210 0.8392 1.2970 0.0205 0.8681 1.3428 0.0001 0.7935 1.1635 0.0151 0.8047 1.1628 0.0013 0.7284 1.0496 0.0011 0.7815 1.1069 0.0401 0.9122 0.9183 0.0038 0.7423 1.0637
?= 0.9,?= 0.5 0.0101 1.9689 1.1383 0.0430 1.0089 1.3941 0.0269 0.9813 1.2952 0.0352 0.9383 1.2649 0.0304 0.8882 1.1203 0.0122 0.9436 1.1917 0.1589 1.0747 1.0178 0.0250 0.9208 1.1448
?= 0.1,?= 1 0.0104 1.2574 0.8783 0.0202 1.3447 0.9356 0.0014 1.2389 0.8692 0.0154 1.2458 0.8487 0.0103 1.1444 0.8014 0.0017 1.1956 0.8443 0.0335 1.2159 0.8080 0.0016 1.1525 0.8117
?= 0.5,?= 1 0.0241 1.7225 1.0816 0.0060 1.6743 1.0374 0.0065 1.5827 0.9817 0.0128 1.5905 0.9873 0.0020 1.4386 0.9239 0.0030 1.5879 0.9894 0.0177 1.5554 0.9547 0.0077 1.4733 0.9413
?= 0.9,?= 1 0.0131 1.9568 1.1574 0.0145 1.9996 1.1350 0.0036 1.8299 1.0827 0.0180 1.9324 1.0960 0.0179 1.6855 0.9954 0.0165 1.8242 1.0545 0.0595 1.7732 1.0167 0.0184 1.7825 1.0406
?= 0.1,?= 3 0.0170 3.9592 0.8653 0.0153 3.8953 0.8642 0.0217 3.7610 0.8249 0.0022 3.7206 0.8211 0.0052 3.4847 0.7859 0.0041 3.6216 0.7983 0.0006 3.5011 0.8056 0.0215 3.5003 0.7873
?= 0.5,?= 3 0.0143 5.2555 0.9867 0.0306 4.8945 0.9557 0.0237 4.8094 0.9361 0.0137 4.7351 0.9278 0.0040 4.2445 0.8681 0.0013 4.6193 0.9143 0.0040 4.2567 0.8749 0.0061 4.3870 0.8783
?= 0.9,?= 3 0.0283 6.0747 1.0769 0.0763 5.8039 1.0545 0.0190 5.4653 1.0106 0.0192 5.5915 1.0116 0.0034 4.8673 0.9248 0.0118 5.2638 0.9726 0.0028 4.8710 0.9265 0.0439 5.1879 0.9612
?= 0.1,?= 5 0.0388 6.5789 0.8436 0.0108 6.5867 0.8570 0.0241 6.1703 0.8007 0.0108 6.2327 0.8241 0.0168 5.7806 0.7811 0.0045 6.0357 0.7906 0.0160 5.7818 0.7843 0.0103 5.7236 0.7712
?= 0.5,?= 5 0.0841 8.5133 0.9644 0.0335 8.3565 0.9718 0.0184 7.8372 0.9231 0.0231 7.7461 0.9228 0.0028 7.1503 0.8623 0.0136 7.6924 0.8954 0.0027 7.1503 0.8628 0.0160 7.2509 0.8698
?= 0.9,?= 5 0.0552 9.9051 1.0576 0.0927 9.8826 1.0531 0.0005 8.9502 0.9893 0.0643 8.8871 0.9798 0.0135 7.9759 0.9193 0.0683 8.9059 0.9673 0.0135 7.9764 0.9194 0.0524 8.4545 0.9443
M.Mohammadipour, 2009, Chapter 8 191
8.6.2.5 AllINAR(1) vs Benchmark Methods
Based on the argument in the previous section, the degree of improvement by
treating all INARMA series as an INAR(1) model over the benchmark methods is
investigated in this section. The MSE results for INARMA(0,0), INMA(1), and
INARMA(1,1) series are shown in Table 846 to Table ?849. The results for INAR(1)
series are the same as the results of known order (Table ?818 and Table ?819). The
results are for the case that all points in time are considered.
As previously mentioned in section 8.6.1, there was a slight improvement over the
benchmark methods when demand is INARMA(0,0) or INMA(1). This was for the
case that the order of the INARMA model was known. Considering the fact that the
identification errors result in deterioration of forecasting accuracy for INARMA
models, we except benchmark methods to outperform INARMA especially for more
sparse demand. The results of Table 846 and Table ?847 confirm this.
Table 846 for INARMA(0,0) series (unknown order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.8998 0.9435 0.9473 0.9746 0.9991 1.0012 1.0070 1.0230 1.0239 0.9918 0.9997 1.0000
?= 0.5 0.9838 1.0119 1.0136 0.9998 1.0150 1.0159 1.0045 1.0157 1.0163 0.9756 0.9850 0.9854
?= 0.7
1.0048 1.0262 1.0280 0.9922 1.0049 1.0054 0.9845 0.9952 0.9955 0.9663 0.9764 0.9767
?= 1 1.0031 1.0179 1.0181 0.9785 0.9887 0.9885 0.9440 0.9533 0.9531 0.9624 0.9719 0.9716
?= 3 1.0170 1.0095 1.0051 0.9479 0.9424 0.9385 0.9496 0.9428 0.9388 0.9198 0.9126 0.9087
?= 5 0.9647 0.9399 0.9321 0.9383 0.9128 0.9050 0.9593 0.9364 0.9288 0.9290 0.9048 0.8972
?= 20 0.9677 0.8339 0.8063 0.9323 0.8030 0.7762 0.9686 0.8339 0.8058 0.9276 0.7982 0.7712
Table 847 for INMA(1) series (unknown order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 1.0350 1.0693 1.0717 0.9922 1.0109 1.0122 0.9695 0.9813 0.9818 0.9558 0.9678 0.9683
?= 0.5,?= 0.5 0.8597 0.8923 0.8941 0.8599 0.8828 0.8842 0.8933 0.9129 0.9141 0.8192 0.8369 0.8380
?= 0.9,?= 0.5 0.7818 0.8177 0.8200 0.7585 0.7812 0.7826 0.7387 0.7571 0.7580 0.7201 0.7387 0.7398
?= 0.1,?= 1 1.0022 1.0157 1.0157 0.9694 0.9819 0.9818 0.9695 0.9804 0.9801 0.9598 0.9705 0.9703
?= 0.5,?= 1 0.9325 0.9471 0.9468 0.9167 0.9269 0.9261 0.8776 0.8891 0.8885 0.8698 0.8809 0.8803
?= 0.9,?= 1 0.8373 0.8436 0.8421 0.7809 0.7884 0.7873 0.7796 0.7868 0.7856 0.7468 0.7528 0.7515
?= 0.1,?= 3 0.9677 0.9569 0.9521 0.9527 0.9430 0.9383 0.9385 0.9276 0.9228 0.9381 0.9295 0.9250
?= 0.5,?= 3 0.9425 0.9215 0.9143 0.9388 0.9173 0.9101 0.9063 0.8860 0.8791 0.8884 0.8678 0.8610
?= 0.9,?= 3 0.8611 0.8259 0.8166 0.8248 0.7942 0.7855 0.8116 0.7835 0.7752 0.8002 0.7705 0.7621
?= 0.1,?= 5 0.9625 0.9316 0.9227 0.9444 0.9198 0.9116 0.9639 0.9356 0.9269 0.9229 0.8952 0.8868
?= 0.5,?= 5 0.9521 0.9031 0.8903 0.9356 0.8843 0.8713 0.9162 0.8651 0.8525 0.8655 0.8224 0.8108
?= 0.9,?= 5 0.8754 0.8149 0.8001 0.8617 0.8037 0.7889 0.8093 0.7546 0.7409 0.7908 0.7355 0.7219
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Chapter 8 192
The results for INARMA(1,1) series again show that with the presence of a high
autoregressive parameter, INARMA has a considerably smaller MSE than the
benchmark methods. The results of MASE also confirm this (see Appendix 8.H).
Table 848 with smoothing parameter 0.2 for INARMA(1,1) series
(unknown order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.9452 0.9745 0.9759 0.9301 0.9500 0.9513 0.9733 0.9899 0.9910 0.9297 0.9444 0.9451
?= 0.1,?= 0.9,?= 0.5 0.7664 0.7989 0.8007 0.7325 0.7564 0.7581 0.7616 0.7811 0.7821 0.7025 0.7196 0.7206
?= 0.5,?= 0.5,?= 0.5 0.6855 0.7039 0.7039 0.6828 0.6985 0.6990 0.6243 0.6358 0.6359 0.6193 0.6332 0.6336
?= 0.9,?= 0.1,?= 0.5 0.5274 0.4433 0.4294 0.5164 0.4542 0.4414 0.4968 0.4385 0.4264 0.4740 0.4208 0.4091
?= 0.1,?= 0.1,?= 1 0.9771 0.9922 0.9922 0.9486 0.9621 0.9620 0.9571 0.9695 0.9693 0.9168 0.9284 0.9282
?= 0.1,?= 0.9,?= 1 0.8802 0.8867 0.8851 0.7635 0.7730 0.7721 0.7569 0.7619 0.7605 0.7493 0.7556 0.7544
?= 0.5,?= 0.5,?= 1 0.7720 0.7663 0.7629 0.7394 0.7370 0.7340 0.6899 0.6858 0.6828 0.6710 0.6701 0.6675
?= 0.9,?= 0.1,?= 1 0.5481 0.4206 0.3994 0.5058 0.3942 0.3750 0.4906 0.3856 0.3671 0.4525 0.3597 0.3428
?= 0.1,?= 0.1,?= 5 0.9920 0.9531 0.9426 0.9574 0.9233 0.9135 0.9436 0.9097 0.8999 0.9182 0.8846 0.8750
?= 0.1,?= 0.9,?= 5 0.9232 0.8501 0.8332 0.8544 0.7861 0.7705 0.8255 0.7633 0.7485 0.7985 0.7393 0.7249
?= 0.5,?= 0.5,?= 5 0.8379 0.7406 0.7199 0.7689 0.6783 0.6591 0.7251 0.6455 0.6280 0.7068 0.6278 0.6106
?= 0.9,?= 0.1,?= 5 0.5482 0.2488 0.2197 0.5407 0.2289 0.2017 0.4987 0.2170 0.1916 0.4535 0.2013 0.1779
Table 849 with smoothing parameter 0.5 for INARMA(1,1) series
(unknown order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.8646 0.9584 0.9809 0.8208 0.9087 0.9414 0.8490 0.9381 0.9726 0.8068 0.8896 0.9256
?= 0.1,?= 0.9,?= 0.5 0.7087 0.7961 0.7972 0.6459 0.7323 0.7446 0.6773 0.7553 0.7557 0.6244 0.6964 0.6958
?= 0.5,?= 0.5,?= 0.5 0.7141 0.7652 0.6628 0.7106 0.7497 0.6526 0.6482 0.6801 0.5903 0.6404 0.6781 0.5906
?= 0.9,?= 0.1,?= 0.5 0.8766 0.3474 0.1933 0.8457 0.3490 0.2002 0.8296 0.3417 0.1954 0.7896 0.3213 0.1843
?= 0.1,?= 0.1,?= 1 0.8757 0.9304 0.9275 0.8436 0.9003 0.8991 0.8544 0.9068 0.9031 0.8140 0.8656 0.8655
?= 0.1,?= 0.9,?= 1 0.8592 0.8631 0.7723 0.7282 0.7454 0.6831 0.7346 0.7378 0.6648 0.7217 0.7292 0.6608
?= 0.5,?= 0.5,?= 1 0.8696 0.7883 0.6142 0.8291 0.7581 0.5942 0.7723 0.7027 0.5508 0.7463 0.6873 0.5413
?= 0.9,?= 0.1,?= 1 0.8825 0.2253 0.1286 0.8418 0.2177 0.1234 0.8206 0.2131 0.1211 0.7720 0.2016 0.1143
?= 0.1,?= 0.1,?= 5 0.8766 0.7247 0.6268 0.8494 0.7041 0.6099 0.8431 0.6967 0.5997 0.8140 0.6748 0.5838
?= 0.1,?= 0.9,?= 5 0.9239 0.6033 0.4448 0.8616 0.5584 0.4126 0.8308 0.5450 0.4037 0.8036 0.5254 0.3890
?= 0.5,?= 0.5,?= 5 0.9620 0.4915 0.3263 0.8759 0.4476 0.2977 0.8274 0.4318 0.2883 0.8025 0.4159 0.2780
?= 0.9,?= 0.1,?= 5 0.9115 0.0647 0.0366 0.8855 0.0596 0.0336 0.8317 0.0572 0.0323 0.7628 0.0531 0.0300
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Chapter 8 193
8.6.3 Lead Time Forecasts
In chapter 6, the lead time forecasts for the INARMA models were presented. For
INARMA(0,0) and INMA(1) processes, the lead time forecast is simply given by:
??? ??+?
?+1
?=1
???= (?+ 1)?
Equation 818
??? ??+?
?+1
?=1
???=??+ 1?(1 +?)?
Equation 819
For INAR(1) and INARMA(1,1) processes, the lead time forecasts are:
??? ??+?
?+1
?=1
???=
?(1???+1)
1??
??+
?
1??
???+ 1??? ??
?+1
?=1
?
Equation 820
??? ??+?
?+1
?=1
???=
?(1???+1)
1??
??+
?(1 +?)
1??
???+ 1??? ??
?+1
?=1
?
Equation 821
The results of sections 8.6.2.4 and 8.6.2.5 show that the accuracy of forecasts of an
allINAR(1) method are generally better than those of an allINARMA(1,1) method
even for INARMA(1,1) series. Therefore, in this section we use an allINAR(1)
method and compare the lead time forecasts of this method with those of benchmarks.
The results of comparing the MSE of INARMA with that of benchmark methods for
INARMA(0,0), INMA(1), INAR(1) and INARMA(1,1) series are presented in Table
?850 to Table ?861. This includes both cases of ?= 3 and ?= 6. The results using
MASE are presented in Appendix 8.I.
For INARMA(0,0) series, the results of Table ?850 show that the allINAR(1) lead
time forecasts are better than the best benchmark in most of the cases (with an
exception of ?= 0.7,?= 24). The same is true for INMA(1) series, with some
exceptions for ?= 24.
M.Mohammadipour, 2009, Chapter 8 194
Table 850
BenchmarkINARMA MSEMSE /
of leadtime forecasts for INARMA(0,0) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.7373 0.8307 0.8409 0.8726 0.9393 0.9457 0.9482 0.9880 0.9912 0.9448 0.9681 0.9695
?= 0.5 0.9128 0.9879 0.9946 0.9573 0.9968 0.9992 0.9467 0.9804 0.9822 0.9097 0.9357 0.9367
?= 0.7
0.9583 1.0130 1.0166 0.9541 0.9862 0.9872 0.9332 0.9606 0.9611 0.8914 0.9142 0.9143
?= 1 0.9589 0.9878 0.9876 0.9361 0.9625 0.9620 0.9040 0.9265 0.9257 0.8601 0.8835 0.8830
?= 3 0.9238 0.9077 0.8984 0.8642 0.8432 0.8338 0.8292 0.8141 0.8055 0.8021 0.7884 0.7803
?= 5 0.9120 0.8519 0.8342 0.8657 0.8127 0.7967 0.8270 0.7755 0.7598 0.7897 0.7418 0.7272
?= 20 0.9025 0.6417 0.5990 0.8521 0.6085 0.5675 0.8197 0.5886 0.5494 0.7793 0.5561 0.5189
Table 851 of leadtime forecasts for INMA(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9226 1.0030 1.0104 0.9503 0.9928 0.9956 0.9429 0.9749 0.9765 0.9075 0.9372 0.9386
?= 0.5,?= 0.5 0.9668 1.0400 1.0463 0.9292 0.9766 0.9800 0.9077 0.9468 0.9493 0.8554 0.8898 0.8918
?= 0.9,?= 0.5 0.9745 1.0630 1.0708 0.9331 0.9877 0.9916 0.9030 0.9444 0.9469 0.8419 0.8803 0.8826
?= 0.1,?= 1 0.9935 1.0339 1.0348 0.9341 0.9628 0.9627 0.9005 0.9262 0.9258 0.8557 0.8814 0.8812
?= 0.5,?= 1 1.0054 1.0366 1.0366 0.9199 0.9444 0.9438 0.8925 0.9156 0.9148 0.8340 0.8579 0.8575
?= 0.9,?= 1 1.0143 1.0469 1.0465 0.9382 0.9631 0.9621 0.9013 0.9218 0.9203 0.8360 0.8568 0.8557
?= 0.1,?= 3 0.9407 0.9237 0.9139 0.8797 0.8710 0.8626 0.8487 0.8312 0.8223 0.8107 0.7981 0.7900
?= 0.5,?= 3 0.9729 0.9451 0.9334 0.9029 0.8737 0.8627 0.8729 0.8508 0.8408 0.8190 0.7970 0.7873
?= 0.9,?= 3 1.0083 0.9679 0.9541 0.9234 0.8801 0.8666 0.8841 0.8521 0.8401 0.8274 0.7961 0.7846
?= 0.1,?= 5 0.9367 0.8801 0.8631 0.8710 0.8150 0.7988 0.8364 0.7885 0.7732 0.8075 0.7592 0.7443
?= 0.5,?= 5 0.9695 0.9048 0.8853 0.8995 0.8289 0.8103 0.8683 0.8063 0.7887 0.8225 0.7612 0.7442
?= 0.9,?= 5 0.9940 0.9124 0.8899 0.9071 0.8238 0.8029 0.8743 0.7985 0.7784 0.8297 0.7582 0.7392
Table 852 of leadtime forecasts with smoothing parameter 0.2 for
INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9441 1.0142 1.0203 0.9422 0.9906 0.9942 0.9398 0.9719 0.9736 0.9012 0.9277 0.9289
?= 0.5,?= 0.5 0.9037 0.9525 0.9560 0.8704 0.9106 0.9131 0.8268 0.8611 0.8631 0.7714 0.8063 0.8085
?= 0.9,?= 0.5 0.8049 0.6984 0.6780 0.7157 0.6624 0.6473 0.6897 0.6391 0.6243 0.6219 0.5734 0.5605
?= 0.1,?= 1 0.9950 1.0303 1.0307 0.9124 0.9404 0.9403 0.9040 0.9314 0.9313 0.8533 0.8793 0.8792
?= 0.5,?= 1 0.9647 0.9843 0.9824 0.8941 0.9126 0.9110 0.8683 0.8820 0.8799 0.8117 0.8272 0.8254
?= 0.9,?= 1 0.7938 0.6568 0.6283 0.7219 0.6001 0.5749 0.6752 0.5758 0.5528 0.6244 0.5242 0.5024
?= 0.1,?= 3 0.9404 0.9214 0.9117 0.8876 0.8741 0.8649 0.8494 0.8370 0.8286 0.8052 0.7906 0.7825
?= 0.5,?= 3 0.9868 0.9416 0.9267 0.9043 0.8639 0.8504 0.8770 0.8385 0.8255 0.8203 0.7864 0.7745
?= 0.9,?= 3 0.7825 0.4666 0.4243 0.7136 0.4197 0.3827 0.6830 0.4084 0.3724 0.6364 0.3850 0.3508
?= 0.1,?= 5 0.9398 0.8872 0.8696 0.8577 0.8118 0.7964 0.8487 0.7989 0.7833 0.8011 0.7536 0.7389
?= 0.5,?= 5 0.9849 0.8914 0.8674 0.9053 0.8171 0.7949 0.8794 0.8054 0.7844 0.8246 0.7455 0.7253
?= 0.9,?= 5 0.7978 0.3639 0.3223 0.7284 0.3349 0.2968 0.6803 0.3201 0.2843 0.6294 0.3002 0.2666
)( 3?l
BenchmarkINARMA MSEMSE / )( 3?l
BenchmarkINARMA MSEMSE / )( 3?l
M.Mohammadipour, 2009, Chapter 8 195
Table 853
BenchmarkINARMA MSEMSE /
of leadtime forecasts with smoothing parameter 0.5 for
INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.8202 0.9889 1.0012 0.7103 0.8761 0.8946 0.7140 0.8540 0.8617 0.6728 0.8053 0.8140
?= 0.5,?= 0.5 0.8152 0.9551 0.9479 0.7505 0.8866 0.8825 0.7104 0.8328 0.8258 0.6566 0.7783 0.7749
?= 0.9,?= 0.5 1.0550 0.5086 0.3537 0.9908 0.5117 0.3575 0.9417 0.4827 0.3368 0.8686 0.4493 0.3156
?= 0.1,?= 1 0.7599 0.8681 0.8403 0.6779 0.7799 0.7567 0.6688 0.7718 0.7501 0.6283 0.7240 0.7033
?= 0.5,?= 1 0.8607 0.8891 0.8204 0.7999 0.8315 0.7695 0.7669 0.7907 0.7273 0.7220 0.7482 0.6894
?= 0.9,?= 1 1.0446 0.3595 0.2321 0.9967 0.3383 0.2190 0.9427 0.3302 0.2135 0.8687 0.2953 0.1905
?= 0.1,?= 3 0.6633 0.6147 0.5303 0.6263 0.5800 0.4968 0.6038 0.5614 0.4833 0.5732 0.5315 0.4577
?= 0.5,?= 3 0.8636 0.6624 0.5257 0.7882 0.6057 0.4822 0.7691 0.5909 0.4702 0.7130 0.5554 0.4434
?= 0.9,?= 3 1.0440 0.1492 0.0887 0.9923 0.1413 0.0843 0.9500 0.1361 0.0812 0.8741 0.1263 0.0751
?= 0.1,?= 5 0.6527 0.5102 0.4046 0.5950 0.4720 0.3774 0.5958 0.4671 0.3722 0.5596 0.4401 0.3514
?= 0.5,?= 5 0.8592 0.5293 0.3882 0.7813 0.4819 0.3539 0.7587 0.4735 0.3476 0.7183 0.4396 0.3219
?= 0.9,?= 5 1.0677 0.0967 0.0564 0.9947 0.0895 0.0522 0.9433 0.0866 0.0505 0.8708 0.0809 0.0472
The results of Table ?853 for INAR(1) series reveal that, for high autoregressive
parameters, the improvement of INARMA over the best benchmark is narrow for short
length of history (in case of ?= 24, INARMA is even worse). However with more
observations, the improvement also increases. For small autoregressive parameters, the
results of Table ?852 show that INARMA always outperforms the benchmark methods.
Again, the improvement increases with an increase in the length of history.
Table 854 of leadtime forecasts with smoothing parameter 0.2 for
INARMA(1,1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.9369 1.0169 1.0242 0.9479 0.9921 0.9951 0.9330 0.9650 0.9668 0.8867 0.9178 0.9195
?= 0.1,?= 0.9,?= 0.5 1.0164 1.0937 1.1002 0.9385 0.9915 0.9953 0.8892 0.9311 0.9338 0.8316 0.8679 0.8700
?= 0.5,?= 0.5,?= 0.5 0.9728 1.0101 1.0118 0.8611 0.8924 0.8939 0.8120 0.8471 0.8490 0.7593 0.7894 0.7908
?= 0.9,?= 0.1,?= 0.5 0.7710 0.6826 0.6646 0.7032 0.6297 0.6135 0.6573 0.6016 0.5875 0.6034 0.5641 0.5519
?= 0.1,?= 0.1,?= 1 0.9975 1.0335 1.0342 0.9151 0.9411 0.9409 0.9053 0.9308 0.9305 0.8528 0.8780 0.8779
?= 0.1,?= 0.9,?= 1 1.0291 1.0487 1.0470 0.9294 0.9539 0.9530 0.8891 0.9065 0.9048 0.8321 0.8514 0.8502
?= 0.5,?= 0.5,?= 1 0.9776 0.9953 0.9927 0.8819 0.8938 0.8909 0.8278 0.8396 0.8372 0.7817 0.7921 0.7897
?= 0.9,?= 0.1,?= 1 0.7905 0.6207 0.5913 0.7148 0.5925 0.5671 0.6650 0.5543 0.5310 0.6202 0.5197 0.4979
?= 0.1,?= 0.1,?= 5 0.9627 0.9084 0.8908 0.8969 0.8410 0.8240 0.8528 0.7949 0.7788 0.8188 0.7691 0.7541
?= 0.1,?= 0.9,?= 5 0.9897 0.9024 0.8794 0.9067 0.8267 0.8062 0.8628 0.7857 0.7659 0.8240 0.7549 0.7362
?= 0.5,?= 0.5,?= 5 0.9816 0.8746 0.8481 0.8829 0.7884 0.7650 0.8442 0.7589 0.7369 0.7974 0.7113 0.6900
?= 0.9,?= 0.1,?= 5 0.8126 0.3508 0.3100 0.7160 0.3352 0.2973 0.6811 0.3100 0.2747 0.6252 0.2905 0.2576
For INARMA(1,1) series, the results of Table ?854 reveal that for small autoregressive
parameters, INARMA outperforms the benchmark methods in most cases (except for
)( 3?l
BenchmarkINARMA MSEMSE / )( 3?l
M.Mohammadipour, 2009, Chapter 8 196
sparse data and short history). The improvement increases with an increase in the
length of history. For high autoregressive parameters, the improvement of INARMA
over the best benchmark is narrow for short length of history (in case of ?= 24,
INARMA is even worse). However, with more observations, the improvement also
increases.
Table 855 of leadtime forecasts with smoothing parameter 0.5 for
INARMA(1,1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.7926 0.9844 1.0046 0.7387 0.8943 0.9066 0.7080 0.8508 0.8602 0.6609 0.8020 0.8151
?= 0.1,?= 0.9,?= 0.5 0.8284 1.0145 1.0283 0.7348 0.8943 0.9020 0.7039 0.8473 0.8523 0.6502 0.7844 0.7891
?= 0.5,?= 0.5,?= 0.5 0.8944 1.0074 0.9783 0.7886 0.8929 0.8719 0.7309 0.8355 0.8170 0.6878 0.7789 0.7584
?= 0.9,?= 0.1,?= 0.5 1.0603 0.5206 0.3644 0.9695 0.4808 0.3352 0.9265 0.4691 0.3282 0.8538 0.4430 0.3099
?= 0.1,?= 0.1,?= 1 0.7832 0.8878 0.8586 0.6954 0.7909 0.7656 0.6774 0.7742 0.7505 0.6433 0.7359 0.7132
?= 0.1,?= 0.9,?= 1 0.8336 0.8953 0.8471 0.7351 0.7975 0.7544 0.7160 0.7657 0.7200 0.6662 0.7168 0.6756
?= 0.5,?= 0.5,?= 1 0.9017 0.9086 0.8233 0.8102 0.8075 0.7288 0.7721 0.7688 0.6955 0.7151 0.7143 0.6467
?= 0.9,?= 0.1,?= 1 1.0454 0.3365 0.2171 0.9817 0.3287 0.2121 0.9322 0.3129 0.2021 0.8667 0.2929 0.1888
?= 0.1,?= 0.1,?= 5 0.6847 0.5329 0.4230 0.6441 0.4943 0.3906 0.6123 0.4713 0.3743 0.5900 0.4578 0.3637
?= 0.1,?= 0.9,?= 5 0.7799 0.5191 0.3907 0.7282 0.4872 0.3678 0.6804 0.4559 0.3444 0.6462 0.4364 0.3297
?= 0.5,?= 0.5,?= 5 0.8976 0.5027 0.3589 0.8053 0.4581 0.3287 0.7748 0.4442 0.3189 0.7296 0.4119 0.2948
?= 0.9,?= 0.1,?= 5 1.0675 0.0921 0.0538 1.0085 0.0898 0.0523 0.9440 0.0829 0.0483 0.8732 0.0781 0.0455
When ?= 6, the results of Table ?856 and Table ?857 show that the improvement by
using INARMA (an allINAR(1) method) over the benchmarks is generally greater
than the case of ?= 3 for INARMA(0,0) and INMA(1) series. This is also true for
INAR(1) and INARMA(1,1) series with small autoregressive parameters.
Table 856 of leadtime forecasts for INARMA(0,0) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.7796 0.9163 0.9315 0.7748 0.8717 0.8815 0.9112 0.9772 0.9824 0.9101 0.9488 0.9509
?= 0.5 0.8894 1.0054 1.0161 0.8839 0.9565 0.9615 0.8798 0.9363 0.9394 0.8343 0.8782 0.8801
?= 0.7
0.9194 1.0260 1.0341 0.9137 0.9761 0.9785 0.8802 0.9208 0.9212 0.7996 0.8416 0.8424
?= 1 0.9390 1.0161 1.0193 0.8674 0.9197 0.9202 0.8225 0.8631 0.8623 0.7647 0.7959 0.7945
?= 3 0.8688 0.8472 0.8330 0.7845 0.7530 0.7397 0.7503 0.7332 0.7213 0.6768 0.6616 0.6507
?= 5 0.8601 0.7770 0.7533 0.7800 0.6999 0.6783 0.7253 0.6539 0.6336 0.6543 0.5956 0.5778
?= 20 0.8767 0.5377 0.4903 0.7646 0.4681 0.4264 0.7239 0.4344 0.3956 0.6602 0.3997 0.3638
BenchmarkINARMA MSEMSE / )( 3?l
BenchmarkINARMA MSEMSE / )( 6?l
M.Mohammadipour, 2009, Chapter 8 197
Table 857 of leadtime forecasts for INMA(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9555 1.0866 1.0992 0.9010 0.9718 0.9768 0.8826 0.9284 0.9306 0.8254 0.8694 0.8715
?= 0.5,?= 0.5 0.9758 1.1192 1.1334 0.9130 0.9811 0.9859 0.8869 0.9499 0.9540 0.7785 0.8321 0.8355
?= 0.9,?= 0.5 0.9807 1.1343 1.1492 0.9248 1.0144 1.0212 0.8610 0.9261 0.9304 0.7866 0.8389 0.8420
?= 0.1,?= 1 0.9389 0.9944 0.9952 0.8628 0.9081 0.9082 0.8151 0.8529 0.8524 0.7649 0.7994 0.7987
?= 0.5,?= 1 1.0045 1.0576 1.0578 0.8718 0.9157 0.9156 0.8293 0.8652 0.8644 0.7372 0.7696 0.7690
?= 0.9,?= 1 1.0814 1.1128 1.1094 0.8983 0.9348 0.9332 0.8351 0.8787 0.8780 0.7429 0.7737 0.7725
?= 0.1,?= 3 0.9124 0.8748 0.8592 0.8079 0.7739 0.7602 0.7394 0.7149 0.7030 0.6741 0.6553 0.6447
?= 0.5,?= 3 0.9662 0.9205 0.9032 0.8329 0.8094 0.7957 0.7654 0.7379 0.7251 0.7011 0.6748 0.6630
?= 0.9,?= 3 1.0009 0.9633 0.9456 0.8596 0.8297 0.8139 0.7917 0.7518 0.7364 0.7021 0.6679 0.6544
?= 0.1,?= 5 0.9127 0.8509 0.8281 0.7848 0.7108 0.6897 0.7241 0.6617 0.6424 0.6698 0.6068 0.5887
?= 0.5,?= 5 0.9715 0.8770 0.8489 0.8084 0.7152 0.6921 0.7664 0.6978 0.6771 0.6977 0.6281 0.6085
?= 0.9,?= 5 0.9922 0.8905 0.8600 0.8249 0.7123 0.6872 0.7697 0.6726 0.6487 0.7043 0.6223 0.6010
Table 858 of leadtime forecasts with smoothing parameter 0.2 for INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.8625 0.9847 0.9967 0.9076 0.9769 0.9817 0.8948 0.9484 0.9515 0.8221 0.8695 0.8720
?= 0.5,?= 0.5 1.0555 1.1742 1.1849 0.9302 0.9968 1.0014 0.8466 0.9094 0.9138 0.7524 0.8044 0.8079
?= 0.9,?= 0.5 1.0075 0.9778 0.9611 0.8818 0.8539 0.8397 0.8450 0.8217 0.8080 0.7595 0.7278 0.7149
?= 0.1,?= 1 0.9882 1.0425 1.0431 0.8784 0.9175 0.9170 0.8314 0.8690 0.8684 0.7560 0.7902 0.7896
?= 0.5,?= 1 1.0573 1.0997 1.0987 0.9195 0.9563 0.9552 0.8406 0.8812 0.8809 0.7540 0.7849 0.7841
?= 0.9,?= 1 1.0224 0.8642 0.8311 0.8859 0.7750 0.7469 0.8428 0.7480 0.7224 0.7515 0.6479 0.6238
?= 0.1,?= 3 0.9045 0.8885 0.8752 0.8091 0.7854 0.7727 0.7458 0.7196 0.7073 0.6899 0.6730 0.6625
?= 0.5,?= 3 1.0213 0.9597 0.9405 0.8731 0.8446 0.8301 0.8273 0.8022 0.7887 0.7472 0.7167 0.7039
?= 0.9,?= 3 1.0113 0.6310 0.5780 0.8916 0.5759 0.5289 0.8439 0.5512 0.5061 0.7476 0.4875 0.4483
?= 0.1,?= 5 0.9003 0.8337 0.8110 0.7897 0.7115 0.6909 0.7370 0.6712 0.6516 0.6750 0.6152 0.5977
?= 0.5,?= 5 1.0195 0.9360 0.9091 0.8660 0.7777 0.7532 0.8137 0.7304 0.7077 0.7336 0.6596 0.6391
?= 0.9,?= 5 1.0231 0.5077 0.4525 0.8918 0.4638 0.4151 0.8473 0.4276 0.3828 0.7593 0.3885 0.3471
Table 859 of leadtime forecasts with smoothing parameter 0.5 for INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.7025 0.9582 0.9857 0.6247 0.8199 0.8331 0.5745 0.7630 0.7799 0.5215 0.6905 0.7043
?= 0.5,?= 0.5 0.8524 1.1231 1.1434 0.7197 0.9131 0.9156 0.6479 0.8299 0.8336 0.5690 0.7232 0.7255
?= 0.9,?= 0.5 1.1450 0.7467 0.5466 1.0414 0.6744 0.4948 1.0005 0.6378 0.4654 0.8948 0.5661 0.4149
?= 0.1,?= 1 0.6911 0.8284 0.7860 0.5767 0.6950 0.6643 0.5338 0.6439 0.6137 0.4794 0.5823 0.5562
?= 0.5,?= 1 0.8340 0.9221 0.8532 0.7122 0.7868 0.7257 0.6494 0.7205 0.6662 0.5808 0.6405 0.5915
?= 0.9,?= 1 1.1542 0.4903 0.3295 1.0352 0.4408 0.2939 0.9963 0.4401 0.2945 0.8878 0.3718 0.2478
?= 0.1,?= 3 0.5518 0.5061 0.4185 0.4869 0.4383 0.3619 0.4381 0.3939 0.3249 0.4151 0.3808 0.3157
?= 0.5,?= 3 0.7784 0.5975 0.4667 0.6520 0.5253 0.4146 0.6223 0.4970 0.3921 0.5562 0.4413 0.3479
?= 0.9,?= 3 1.1314 0.2106 0.1279 1.0516 0.1967 0.1188 0.9973 0.1881 0.1133 0.8818 0.1691 0.1023
?= 0.1,?= 5 0.5334 0.4028 0.3086 0.4696 0.3467 0.2660 0.4351 0.3195 0.2432 0.4036 0.2995 0.2290
?= 0.5,?= 5 0.7723 0.4940 0.3607 0.6419 0.4011 0.2909 0.6079 0.3820 0.2778 0.5447 0.3420 0.2489
?= 0.9,?= 5 1.1593 0.1391 0.0817 1.0545 0.1297 0.0765 0.9927 0.1220 0.0720 0.8994 0.1079 0.0635
BenchmarkINARMA MSEMSE / )( 6?l
BenchmarkINARMA MSEMSE / )( 6?l
BenchmarkINARMA MSEMSE / )( 6?l
M.Mohammadipour, 2009, Chapter 8 198
Table 860 of leadtime forecasts with smoothing parameter 0.2 for
INARMA(1,1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.9060 1.0221 1.0332 0.9114 0.9817 0.9867 0.8773 0.9416 0.9460 0.8151 0.8622 0.8649
?= 0.1,?= 0.9,?= 0.5 1.1265 1.2635 1.2754 0.9392 1.0143 1.0196 0.8486 0.9209 0.9263 0.7638 0.8196 0.8232
?= 0.5,?= 0.5,?= 0.5 1.1516 1.2325 1.2381 0.9878 1.0498 1.0535 0.8733 0.9306 0.9341 0.7564 0.8034 0.8062
?= 0.9,?= 0.1,?= 0.5 1.0244 0.9352 0.9148 0.9012 0.8209 0.8031 0.8309 0.7946 0.7805 0.7339 0.7067 0.6946
?= 0.1,?= 0.1,?= 1 0.9968 1.0472 1.0475 0.8069 0.7319 0.7102 0.8184 0.8626 0.8631 0.7493 0.7859 0.7857
?= 0.1,?= 0.9,?= 1 1.0785 1.1185 1.1166 0.9066 0.9557 0.9558 0.8358 0.8781 0.8777 0.7402 0.7684 0.7670
?= 0.5,?= 0.5,?= 1 1.1171 1.1825 1.1829 0.9376 0.9673 0.9650 0.8410 0.8700 0.8683 0.7467 0.7730 0.7715
?= 0.9,?= 0.1,?= 1 1.0111 0.8583 0.8239 0.9020 0.7835 0.7542 0.8249 0.7473 0.7226 0.7269 0.6397 0.6174
?= 0.1,?= 0.1,?= 5 0.9241 0.8565 0.8332 0.8016 0.7424 0.7218 0.7275 0.6645 0.6456 0.6887 0.6289 0.6111
?= 0.1,?= 0.9,?= 5 0.9912 0.8566 0.8262 0.8560 0.7602 0.7341 0.7844 0.7027 0.6792 0.7024 0.6251 0.6044
?= 0.5,?= 0.5,?= 5 1.0538 0.9423 0.9107 0.9001 0.7968 0.7694 0.8238 0.7342 0.7098 0.7444 0.6602 0.6380
?= 0.9,?= 0.1,?= 5 1.0067 0.4899 0.4369 0.8862 0.4498 0.4017 0.8352 0.4233 0.3787 0.7291 0.3757 0.3360
Table 861 of leadtime forecasts with smoothing parameter 0.5 for
INARMA(1,1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.7042 0.9695 1.0084 0.6267 0.8313 0.8481 0.5555 0.7524 0.7762 0.5135 0.6838 0.7022
?= 0.1,?= 0.9,?= 0.5 0.8479 1.1332 1.1494 0.6814 0.8807 0.8870 0.5838 0.7769 0.7926 0.5222 0.6861 0.6938
?= 0.5,?= 0.5,?= 0.5 0.9535 1.1744 1.1651 0.8018 0.9757 0.9598 0.6933 0.8499 0.8383 0.6112 0.7413 0.7295
?= 0.9,?= 0.1,?= 0.5 1.1333 0.7081 0.5236 1.0656 0.6303 0.4630 0.9797 0.6255 0.4586 0.8789 0.5565 0.4072
?= 0.1,?= 0.1,?= 1 0.6951 0.8151 0.7736 0.4846 0.3506 0.2659 0.5338 0.6524 0.6276 0.4804 0.5845 0.5592
?= 0.1,?= 0.9,?= 1 0.7753 0.8666 0.8034 0.6309 0.7326 0.6886 0.5860 0.6681 0.6238 0.5191 0.5816 0.5393
?= 0.5,?= 0.5,?= 1 0.8907 0.9756 0.8926 0.7496 0.7941 0.7204 0.6635 0.7086 0.6475 0.5940 0.6314 0.5747
?= 0.9,?= 0.1,?= 1 1.1266 0.4673 0.3099 1.0550 0.4426 0.2935 0.9801 0.4372 0.2912 0.8728 0.3766 0.2513
?= 0.1,?= 0.1,?= 5 0.5502 0.4146 0.3174 0.4870 0.3610 0.2743 0.4375 0.3241 0.2474 0.4218 0.3097 0.2358
?= 0.1,?= 0.9,?= 5 0.6697 0.4177 0.3047 0.5673 0.3623 0.2632 0.5251 0.3368 0.2448 0.4660 0.3026 0.2219
?= 0.5,?= 0.5,?= 5 0.8193 0.4787 0.3404 0.7079 0.4066 0.2885 0.6455 0.3788 0.2698 0.5771 0.3386 0.2420
?= 0.9,?= 0.1,?= 5 1.1254 0.1355 0.0798 1.0412 0.1238 0.0729 0.9907 0.1195 0.0705 0.8700 0.1045 0.0615
In general, the lead time forecasts produced by an allINAR(1) method beat the
benchmark methods except for the cases when the data is sparse and the sample is
small. This can be attributed to the fact that there is less positive data available for
estimation of parameters for an INAR(1) process. The improvement increases with
an increase in the length of history. When the lead time increases, the improvement
of INARMA over benchmarks generally increases except for the case where the
autoregressive parameter is high.
BenchmarkINARMA MSEMSE / )( 6?l
BenchmarkINARMA MSEMSE / )( 6?l
M.Mohammadipour, 2009, Chapter 8 199
8.7 Conclusions
In this chapter, the results of the simulation experiment have been presented. The
performance of YW and CLS estimation methods (and CML for INAR(1) process) in
terms of the accuracy of estimates and also their impact on forecast accuracy has
been examined. The results show that when the length of history is short, CLS
produces better forecasts than YW especially in the presence of a high autoregressive
parameter. For cases where the autoregressive parameter is low, and also for the
INMA(1) process, the two estimation methods are close. Also when the number of
observations increases, the two estimation methods will produce close forecasts (in
terms of MSE and MASE).
The CrostonSBA categorization (Syntetos et al., 2005) has been tested and validated
for an i.i.d. Poisson process (an INARMA(0,0) process). Although the categorization
was originally developed using MSE of forecasts, the results of simulation show that
it also holds when MASE of forecasts are considered. The simulation results show
that the CrostonSBA categorization also holds for INAR(1), INMA(1) and
INARMA(1,1) processes.
It has been found that when the number of observation increases, the advantage of
SBA over Croston in terms of MSE decreases until it reaches a limit.
Four INARMA processes have been used in this study: INARMA(0,0), INAR(1),
INMA(1) and INARMA(1,1). The identification is therefore limited to selecting the
best process among them. As discussed in chapter 4, two identification procedures
are used. A twostage identification procedure first uses the LjungBox statistic to
distinguish between INARMA(0,0) and other processes. The AIC is then used to
select among the other INARMA models. A onestage identification procedure only
uses AIC to select among all INARMA models including INARMA(0,0). The results
show that the twostage method provides better results for the INARMA(0,0) model
(in terms of the percentage of series for which the correct model is identified).
However, for other models, the onestage method produces better results. In terms of
the accuracy of forecasts, for an INARMA(0,0) process, the twostage method
produces better forecasts using MSE and MASE in most of the cases. For other
processes, when the autoregressive parameter is high, the onestage method produces
M.Mohammadipour, 2009, Chapter 8 200
much better forecasts. This is also true for high moving average parameters but the
difference is smaller. When more observations are available, the two methods
produce similar forecasts. The results also suggest that, as expected, misidentification
has a high effect on forecast accuracy when the autoregressive parameter is high. But
for MA processes and for AR processes with low autoregressive parameters, the
effect of misidentification is not high.
As a potential substitute to identification, the most general INARMA model can be
used. For example, if data is in fact an INAR(1) process, the estimated MA
parameter should be close to zero. The results show that, in the presence of an AR
component, using the most general INARMA model results in more accurate
forecasts (in terms of MSE and MASE) than those of identification, especially for
short history and high AR parameter. When the number of observations increases,
the results of two methods will be close.
We have also tested using an INAR(1) method to forecast all four INARMA models
and have compared the results to the case of using an INARMA(1,1) method for all
four models. The results show that the allINAR(1) method generally produces better
forecasts than the allINARMA(1,1) method even for MA series.
The INARMA forecasts are compared to those of Croston, SBA and SBJ. For
INARMA(0,0) and INMA(1) processes, the improvement by using INARMA over
benchmark methods is small. But when data is produced by INAR(1) or
INARMA(1,1) and the autoregressive parameter is high, INARMA produces much
more accurate onestep ahead forecasts than the benchmark methods. The degree of
improvement generally increases when more observations are available.
The results for threestep and sixstep ahead forecasts show that, for INMA(1), and
INAR(1) and INARMA(1,1) processes with small autoregressive parameters, the
forecast accuracy of INARMA over benchmarks is improved compared to the one
step ahead forecasts. However, the same is not true for INARMA processes with
high autoregressive parameters.
Finally, the lead time forecasts of an allINAR(1) method have been compared to
those of the benchmark methods. The results show that the allINAR(1) method
generally beats the benchmark methods and the improvement increases when more
M.Mohammadipour, 2009, Chapter 8 201
observations are available. The only exception is when both autoregressive parameter
and lead time are high and the number of observations is small. In that case, the best
benchmark method outperforms the allINAR(1) method. Even for such cases, when
the number of observations increases, INARMA starts to produce more accurate lead
time forecasts than benchmarks.
M.Mohammadipour, 2009, Chapter 9 202
Chapter 9 EMPIRICAL ANALYSIS
9.1 Introduction
As discussed in chapter 3, INARMA models have been developed for forecasting
count data. The application areas of these models have been mainly for counts of
events or individuals such as the number of patients in a hospital?s emergency unit
each hour.
Intermittent series, as a series of nonnegative integer values where some values are
zero (Shenstone and Hyndman, 2005), can be considered as a special class of count
series. However, there is no empirical evidence on the performance of INARMA
models in this area.
Accurate demand forecasting is a key to better inventory management. Although this
M.Mohammadipour, 2009, Chapter 9 203
research only focuses on forecasting, an improvement in accuracy of forecasts can be
translated to fewer inventories, less obsolescence and better customer service.
This PhD thesis has suggested using INARMA models to forecast intermittent
demand. The performance of these models has been compared to that of the
benchmark methods of Croston, SBA and SBJ in Chapter 8. However, it was
assumed that data were produced by one of the four INARMA models:
INARMA(0,0), INAR(1), INMA(1) or INARMA(1,1). In this chapter, the model
assumptions are relaxed by testing the results on empirical data.
This chapter is organized as follows. The purposes of empirical analysis are
explained in section 9.2. The series for this study consist of the demand data of
16,000 Royal Air Force (RAF) SKUs, some of which are highly lumpy, and 3,000
data series from the automotive industry. The filtering mechanism applied to the
demand data for INARMA forecasting is discussed in section 9.3. Details of
empirical analysis design are provided in section 9.4. The accuracy of INARMA
forecasts and those of the benchmark methods are compared in section 9.5. In this
section, the results of identification are compared with treating all as INAR(1) and
INARMA(1,1). The ?step ahead forecasts and lead time forecasts are also
presented. The sensitivity of the results to the length of history has been tested.
Finally, the conclusions of the empirical analysis are given in section 9.6.
9.2 Rationale for Empirical Analysis
The main purpose of empirical analysis is to validate the theoretical and simulation
findings on real data. The results of simulation show that when data is produced by
INAR(1) or INARMA(1,1) and the autoregressive parameter is high, INARMA
forecasting methods produce much more accurate forecasts than benchmark
methods. But when data is produced by INARMA(0,0) or INMA(1), the
improvement by using INARMA over benchmark methods is small. We are
interested in finding out whether the INARMA forecasting approach still
outperforms the benchmark methods for real intermittent demand data.
In the simulation chapter we also looked at the effect of various factors on forecast
M.Mohammadipour, 2009, Chapter 9 204
accuracy. The effect of YW and CLS (and CML for INAR(1)) estimates on the
accuracy of INARMA forecasts was studied. The results of identification or using the
most general INARMA model in the class were compared. The sample size effect on
the accuracy of forecasts was also tested. The empirical analysis will assess the effect
of these factors and enable us to validate the simulation results.
9.3 Demand Data Series
The real demand data series for this research consists of the Royal Air Force (RAF)
individual demand histories of 16,000 SKUs over a period of 6 years (monthly
observations). We have also used another data set which consists of 3,000 real
intermittent demand data series from the automotive industry1 (from Syntetos and
Boylan, 2005) which, unlike the previous one, has more occurrences of positive
demand than zeros. This data series consists of demand histories of 3,000 SKUs over
a period of 2 years (24 months). These two data sets are called 16,000 and 3,000
series from now on.
The 16,000 series are useful in assessing the effect of length of history on the
accuracy of forecasts because it has longer history. However, it is can be categorized
as a set of slower intermittent series because it has many periods of no demand. On
the other hand, the 3,000 series has a very short history but it contains faster
intermittent series with more positive demands.
As previously mentioned, this research has focused on INARMA processes with
Poisson innovations. Although some of the theoretical results are not based on a
distributional assumption, whenever a specific distribution was needed, such as for
estimation of parameters, a Poisson distribution was assumed.
Out of the four processes of this study, three of them have a Poisson distribution
when the marginal distribution is Poisson. The only exception is the INARMA(1,1)
process where:
??=?????1 +??+?????1
Equation 91
1 This data set is available from: http://www.forecasters.org/ijf/data/Empirical%20Data.xls
M.Mohammadipour, 2009, Chapter 9 205
?????=
1 +?
1??
?
Equation 92
var????=
1
1??2
[1 +?+?+ 3??]?
Equation 93
var????
?????
=
1 +?+?+ 3??
1 +?+?+??
?1.5 for 0???1, 0???1
Equation 94
In order to remove the data series with highly variable demands, a Poisson dispersion
test (also called the variance test) is needed for all processes except INARMA(1,1).
Under the null hypothesis that ?1,?,?? are Poisson distributioned, the test statistic:
???=?
(????)
2
?
?
?=1
Equation 95
has a chisquare distribution with (??1) degrees of freedom. Therefore, ?0 is
rejected if ???>???1;1??
2 .
A revised statistic is used to allow for the difference between the mean and variance
of an INARMA(1,1) process. The new test statistic is given by:
????=
???
1.5
Equation 96
The new statistic also has a chisquare distribution with (??1) degrees of freedom.
Further filtering of data was performed for series with fewer than two nonzero
demands. As previously mentioned in Chapter 8, the benchmark methods need at
least two nonzero observations for initialization.
Out of the 16,000 series, 5,168 series met the above criteria and therefore are used
for empirical analysis. The filtering of the 3,000 series results in 1,943 series. It can
be seen that although a substantial number of series has the potential to benefit from
PoINARMA models, for a large number of series these models are not appropriate.
M.Mohammadipour, 2009, Chapter 9 206
Other distributional assumptions would obviously result in different number of
filtered series, which can be pursued as a further study.
9.4 Design of Empirical Analysis
The design of the empirical analysis follows the detailed simulation design of
Chapter 7. As discussed in section 7.3.3.4, two fixed values have been used for the
smoothing parameter of Croston, SBA, and SBJ methods (?= 0.2, 0.5). The
initialization for these methods is based on using the first interdemand interval as
the first smoothed interdemand interval and the average of the first two nonzero
observations as the first smoothed size.
The data series is divided into two parts: ?estimation period? for initialization and
estimation of parameters and ?performance period? for assessing the accuracy of
forecasts. If at least two nonzero demands are observed in the estimation period, the
first half of the observations is assigned for the estimation period and the other half
for the performance period. However, if fewer than two nonzero demands are
observed in the estimation period, this period will be extended until the second non
zero demand is observed. When the effect of length of history is tested, the
performance period is fixed and the estimation period varies.
It is assumed that there are four possible INARMA models to use for forecasting.
Therefore, identification is undertaken among these models. This is done by applying
both twostage and onestage identification procedures (see section 4.6). The former
first uses the LjungBox test to distinguish between INARMA(0,0) and the other
three models. Then, the AIC is used to select among the other models. The latter uses
only the AIC to select among all INARMA models.
Based on the simulation results of chapter 8, we suggested that general models
(INAR(1) and INARMA(1,1) were tested in chapter 8) can be used as alternatives to
identification. This is also tested on empirical data.
YW and CLS (and CML only for the INAR(1) model) have been used to estimate the
parameters of INARMA models. The simulation results show that these methods
M.Mohammadipour, 2009, Chapter 9 207
result in similar forecasts when the length of history is high. But for short history and
high autoregressive parameters, CLS generally produces more accurate onestep
ahead forecasts in terms of MSE and MASE. These estimation methods have been
used for empirical analysis to test the validity of the simulation findings. It is worth
mentioning that the INARMA forecasts are minimum mean square (MMSE).
Therefore, we expect an improvement in terms of MSE but not necessarily in terms
of MASE.
The accuracy measures used in empirical analysis are discussed in chapter 2 in detail.
In addition to the measures used in the simulation experiment (ME, MSE, and
MASE), two relativetoanothermethod measures have been used. As previously
mentioned in section 2.4, the percent better (PB) determines how often a method is
better than another method. However, it does not show how much the improvement
is. The relative geometric rootmeansquare error (RGRMSE) is used to calculate the
magnitude of improvement of one method over another.
9.5 INARMA vs Benchmark Methods
In this section, the results of comparing INARMA forecasts with those of
benchmarks are presented. First, we use an allINAR(1) approach assuming that all
the series are in fact INAR(1) processes. The different estimation methods used in
the simulation chapter for estimation of INAR(1) parameters are also used here. An
allINARMA(1,1) approach is then used and the results are compared to both
benchmark methods and allINAR(1) results. As previously mentioned in sections
8.6.2.2 and 8.6.2.4, these approaches especially perform well compared to
identification for highly autocorrelated data and short data histories. To assess their
performance for empirical data, the forecast accuracy results based on identification
are also presented.
The results of identification show what INARMA models each series follows. Those
series that follow a specific INARMA model are then separated and each one is
forecasted with the corresponding INARMA method (either INARMA(0,0),
INAR(1), INMA(1), or INARMA(1,1)). The accuracy of these INARMA forecasts is
compared to the corresponding simulation results. The results include onestep,
M.Mohammadipour, 2009, Chapter 9 208
threestep and sixstepahead forecasts.
The sensitivity of INARMA forecasts to the length of history is also tested for the
16,000 series. Finally, the results of lead time forecasts for the INAR(1) model are
presented for both the 16,000 and 3,000 series.
9.5.1 AllINAR(1)
We suggest in chapter 8 that using a general INARMA model produces forecasts
comparable to those based on identification. As we will show in section 9.5.3, most
of the series in both 16,000 and 3,000 series are identified as INARMA(0,0) or
INAR(1). Therefore, using INAR(1) to forecast seems to be a promising approach
for these datasets. The results of INAR(1) and benchmarks for all points in time and
issue points are shown in Table 91 and Table 92 for 16,000 series.
Recall from chapter 5 that, as pointed out by AlOsh and Alzaid (1987), for small
sample sizes (??75) and small autoregressive parameter ??= 0.1?, because the
sample contains many zero values, CML is not as good as YW in terms of bias and
MSE of the estimates. The simulation results in section 8.3 show that for
corresponding parameters (see section 9.5.4), YW yields better forecasts than CLS
and CML using MSE and MASE. We test this by presenting CLS, YW and CML
based forecasts. The empirical results confirm the corresponding simulation results.
Table ?91 Comparing INAR(1) with benchmarks for all points in time (16000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INAR(1)
CLS
INAR(1)
YW
INAR(1)
CML
ME 0.0719 0.0933 0.0402 0.0086 0.0367 0.0196 0.0261 0.0283 0.0282
MSE 0.3910 0.4205 0.3802 0.3859 0.3793 0.3846 0.3609 0.3527 0.3555
MASE 2.8594 2.8852 2.7051 2.4925 2.6881 2.3640 1.9789 1.9124 1.9432
PB of MASE
INAR(1)CLS/Benchmark
0.6830 0.6952 0.6302 0.5624 0.6236 0.5116
PB of MASE
INAR(1)YW/Benchmark
0.6742 0.6868 0.6216 0.5555 0.6150 0.5050
PB of MASE
INAR(1)CML/Benchmark
0.6651 0.6878 0.6116 0.5545 0.6050 0.5050
RGRMSE
INAR(1)CLS/Benchmark
0.7812 0.7401 0.8272 0.9024 0.8352 0.9927
RGRMSE
INAR(1)YW/Benchmark
0.7898 0.7493 0.8382 0.9157 0.8463 1.0077
RGRMSE
INAR(1)CML/Benchmark
0.7894 0.7494 0.8367 0.9152 0.8449 1.0075
M.Mohammadipour, 2009, Chapter 9 209
Table ?92 Comparing INAR(1) with benchmarks for issue points (16000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INAR(1)
CLS
INAR(1)
YW
INAR(1)
CML
ME 0.0266 0.0441 0.0028 0.0339 0.0061 0.0598 0.0598 0.0324 0.0197
MSE 0.5329 0.5666 0.5237 0.5344 0.5231 0.5359 0.5792 0.5218 0.5420
MASE 0.3309 0.3375 0.3188 0.3048 0.3175 0.2951 0.3679 0.3473 0.3444
PB of MASE
INAR(1)CLS/Benchmark
0.4590 0.4509 0.4016 0.3219 0.3949 0.2900
PB of MASE
INAR(1)YW/Benchmark
0.4584 0.4493 0.4026 0.3273 0.3956 0.2941
PB of MASE
INAR(1)CML/Benchmark
0.4826 0.4785 0.4234 0.3534 0.4166 0.3196
RGRMSE
INAR(1)CLS/Benchmark
1.2416 1.3295 1.3402 1.6694 1.3531 1.8393
RGRMSE
INAR(1)YW/Benchmark
1.1734 1.2477 1.2728 1.5709 1.2847 1.7301
RGRMSE
INAR(1)CML/Benchmark
1.1060 1.2158 1.1989 1.5367 1.2105 1.6960
The results of Table 91 show that INAR(1) produces better results than benchmarks
regardless of the estimation method. The results also show that YW based INAR(1)
is the best estimation method among the three methods, outperforming the best
benchmark method (SBJ 0.2 for MSE and SBJ 0.5 for MASE) by 7 percent in terms
of MSE and 19 percent in terms of MASE. The results of PB and RGRMSE also
show superior performance of INARMA compared to the benchmark methods.
As can be seen from Table 92, when only issue points are considered, the INARMA
forecasts are biased, agreeing with the simulation results. We find that this is true for
all the cases that only issue points are considered so we do not discuss this again.
It can be seen from Table 91 that the MASE of all methods is very high which, as
explained in section 2.4, suggests that all of these methods are worse than na?ve. The
reason is that because the data series contain many zeros in the estimation period, the
error of na?ve in most of the periods is zero. Therefore the insample MAE is very
small and the MASE is very large.
However, when only issue points are considered, because it is likely that a nonzero
demand is followed with a zero demand, the absolute error of na?ve and therefore the
insample MAE is large. As a result, the MASE of the forecasting methods is smaller
compared to the all points in time case. The results of Table 92 confirm this.
As a result, for highly intermittent data, MASE does not provide reliable results for
M.Mohammadipour, 2009, Chapter 9 210
all points in time and issue points.
The simulation results of section 8.6.1 show the superiority of INARMA(0,0)
forecasts compared to the benchmarks for ?= 0.3,?= 96. It will be seen in section
9.5.4 that the majority of 16,000 series are identified as INARMA(0,0) with ? close
to 0.3 (see appendix 9.A) so the results for 16,000 series for all points in time agree
with the simulation results.
The results for 3,000 series are shown in Table 93 and Table 94. It can be seen that
YW again results in more accurate forecasts than both CLS and CML. CLS and
CML results are worse than benchmarks and YW results only improve MSE by 0.5
percent compared to the best benchmark which is SBA 0.2. However, the MASE of
SBJ 0.2 is better than that of INAR(1)YW by one percent. As mentioned in chapter
6, the INARMA forecasts provide the minimum MSE and not MASE.
The results of Table 93 also show the superior performance of INAR(1)YW to the
benchmark methods in terms of PB and RGRMSE except for SBA and SBJ 0.2.
Based on the results of Table 94, INAR(1)YW is better than the best benchmark in
terms of MSE although the INARMA forecasts are biased when only issue points are
considered. However, the MASE is still slightly worse.
Table ?93 Comparing INAR(1) with benchmarks for all points in time (3000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INAR(1)
CLS
INAR(1)
YW
INAR(1)
CML
ME 0.0419 0.1016 0.1662 0.4336 0.1894 0.6120 0.1016 0.0136 0.0145
MSE 3.2574 3.7054 3.2483 3.6470 3.2550 3.8249 3.3540 3.2319 3.2640
MASE 0.8694 0.9277 0.8543 0.8848 0.8535 0.8914 0.8757 0.8636 0.8720
PB of MASE
INAR(1)CLS/Benchmark
0.5097 0.5528 0.4885 0.5211 0.4886 0.5329
PB of MASE
INAR(1)YW/Benchmark
0.5171 0.5551 0.4842 0.5211 0.4839 0.5295
PB of MASE
INAR(1)CML/Benchmark
0.5133 0.5522 0.4828 0.5136 0.4812 0.5259
RGRMSE
INAR(1)CLS/Benchmark
0.5329 0.9814 0.9370 1.0027 1.0037 1.0067
RGRMSE
INAR(1)YW/Benchmark
0.9471 0.9175 0.9765 0.9868 0.9826 0.9857
RGRMSE
INAR(1)CML/Benchmark
0.9476 0.9207 0.9791 0.9898 0.9847 0.9894
M.Mohammadipour, 2009, Chapter 9 211
Table ?94 Comparing INAR(1) with benchmarks for issue points (3000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INAR(1)
CLS
INAR(1)
YW
INAR(1)
CML
ME 0.0518 0.1058 0.1544 0.4233 0.1773 0.5996 0.0037 0.0650 0.0732
MSE 3.3807 3.8368 3.3685 3.7724 3.3749 3.9483 3.4687 3.3563 3.3865
MASE 0.9083 0.9686 0.8918 0.9210 0.8907 0.9260 0.9164 0.9054 0.9130
PB of MASE
INAR(1)CLS/Benchmark
0.5087 0.5555 0.4861 0.5157 0.4857 0.5231
PB of MASE
INAR(1)YW/Benchmark
0.5117 0.5539 0.4756 0.5127 0.4744 0.5161
PB of MASE
INAR(1)CML/Benchmark
0.5097 0.5519 0.4741 0.5076 0.4723 0.5162
RGRMSE
INAR(1)CLS/Benchmark
1.0187 0.9738 1.0354 1.0442 1.0434 1.0503
RGRMSE
INAR(1)YW/Benchmark
0.9927 0.9618 1.0200 1.0311 1.0293 1.0377
RGRMSE
INAR(1)CML/Benchmark
0.9953 0.9686 1.0291 1.0434 1.0394 1.0495
The simulation results of chapter 8 suggest that, with the presence of a high
autocorrelation, INAR(1) outperforms the benchmarks. Also, the simulation results
show that for ?= 0.1 and ?= 1, 3, when the number of observations is small
(?= 24), SBA and SBJ (with smoothing parameter 0.2) are better than INAR(1) in
terms of both MSE and MASE. But for higher number of observations, INAR(1)
starts to slightly perform better.
It will be seen in section 9.5.4 that the estimated autoregressive parameter is generally
close to 0.1. Therefore, the results for 3000 series agree with the simulation results.
9.5.2 AllINARMA(1,1)
In this section it is assumed that all series follow the INARMA(1,1) model. The
results for 16,000 series are presented in Table 95 and Table 96.
The simulation results show that for INARMA(1,1) processes, CLS produces better
results than YW. We test this for empirical data by presenting both CLS and YW
based forecasts. As explained in chapter 5, the CML estimates have only been
obtained for INAR(p) models and therefore these estimates are not presented for the
INARMA(1,1) model.
M.Mohammadipour, 2009, Chapter 9 212
The results of Table 95 show that INARMA improves the accuracy of forecast by 3
percent in terms of MSE and 19 percent in terms of MASE compared to the best
benchmark (SBJ 0.2 for MSE and SBJ 0.5 for MASE). The RGRMSE results
confirm the MSE results. It can be seen that the YWbased INARMA(1,1) forecasts
are worse than CLSbased forecasts, agreeing with simulation results.
Table ?95 Comparing INARMA(1,1) with benchmarks for all points in time (16000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INARMA
CLS
INARMA
YW
ME 0.0719 0.0933 0.0402 0.0086 0.0367 0.0196 0.0258 0.0039
MSE 0.3910 0.4205 0.3802 0.3859 0.3793 0.3846 0.3668 0.4833
MASE 2.8594 2.8852 2.7051 2.4925 2.6881 2.3640 1.9128 2.3085
PB of MASE
INARMACLS/Benchmark
0.6692 0.6842 0.6161 0.5508 0.6095 0.5004
PB of MASE
INARMAYW/Benchmark
0.7359 0.7413 0.7150 0.6922 0.7122 0.6691
RGRMSE
INARMACLS/Benchmark
0.7653 0.7294 0.8128 0.8918 0.8205 0.9818
RGRMSE
INARMAYW/Benchmark
0.7843 0.7464 0.8200 0.8960 0.8285 0.9881
Table ?96 Comparing INARMA(1,1) with benchmarks for issue points (16000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INARMA
CLS
INARMA
YW
ME 0.0266 0.0441 0.0028 0.0339 0.0061 0.0598 0.0326 0.5620
MSE 0.5329 0.5666 0.5237 0.5344 0.5231 0.5359 0.5766 1.5057
MASE 0.3309 0.3375 0.3188 0.3048 0.3175 0.2951 0.3509 0.7313
PB of MASE
INARMACLS/Benchmark
0.4958 0.4797 0.4349 0.3507 0.4275 0.3162
PB of MASE
INARMAYW/Benchmark
0.3004 0.2923 0.2779 0.2375 0.2750 0.2185
RGRMSE
INARMACLS/Benchmark
1.0833 1.1712 1.1751 1.4768 1.1862 1.6282
RGRMSE
INARMAYW/Benchmark
2.1539 2.3492 2.4021 2.9709 2.4289 3.3006
The results of comparing INARMA(1,1) and benchmarks for all points in time and
issue points for 3,000 series are presented in Table 97 and Table 98 for both CLS
and YW estimates. The results confirm that INARMA(1,1) forecasts based on CLS
estimates are better than those based on YW estimates.
The results of Table 97 show that both CLS and YW based INARMA(1,1) forecasts
are worse than benchmarks. In order to understand the reason, we need to identify
M.Mohammadipour, 2009, Chapter 9 213
the autoregressive and moving average order of the data series. This will be
examined in the next section.
Table ?97 Comparing INARMA(1,1) with benchmarks for all points in time (3000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INARMA
CLS
INARMA
YW
ME 0.0419 0.1016 0.1662 0.4336 0.1894 0.6120 0.0461 0.0091
MSE 3.2574 3.7054 3.2483 3.6470 3.2550 3.8249 3.4038 4.0726
MASE 0.8694 0.9277 0.8543 0.8848 0.8535 0.8914 0.8869 0.9548
PB of MASE
INARMACLS/Benchmark
0.5063 0.5401 0.4777 0.5125 0.4767 0.5209
PB of MASE
INARMAYW/Benchmark
0.4734 0.5018 0.4517 0.4782 0.4522 0.4899
RGRMSE
INARMACLS/Benchmark
0.9893 0.9536 1.0231 1.0293 1.0289 1.0276
RGRMSE
INARMAYW/Benchmark
0.7435 0.7078 0.7675 0.7680 0.7723 0.7669
Table ?98 Comparing INARMA(1,1) with benchmarks for issue points (3000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INARMA
CLS
INARMA
YW
ME 0.0518 0.1058 0.1544 0.4233 0.1773 0.5996 0.1366 0.1972
MSE 3.3807 3.8368 3.3685 3.7724 3.3749 3.9483 3.5569 4.1304
MASE 0.9083 0.9686 0.8918 0.9210 0.8907 0.9260 0.9327 0.9902
PB of MASE
INARMACLS/Benchmark
0.5006 0.5378 0.4691 0.5027 0.4677 0.5097
PB of MASE
INARMAYW/Benchmark
0.4805 0.5108 0.4558 0.4844 0.4570 0.4961
RGRMSE
INARMACLS/Benchmark
1.0343 0.9940 1.0629 1.0707 1.0712 1.0772
RGRMSE
INARMAYW/Benchmark
0.8166 0.7749 0.8417 0.8469 0.8486 0.8527
The simulation results for the corresponding parameter set (?= 0.1, 0.3, ? close to
zero, and ?= 1,3) show that when the number of observations is small (?= 24),
INARMA(1,1) performance is poor. But for a higher autoregressive parameter and
number of observations, INARMA(1,1) performance improves greatly.
Comparing the results of this section with those of the previous section shows that
treating all as INAR(1) produces better results than treating all as INARMA(1,1)
which confirms the simulation results (see section 8.6.2.4). A possible explanation is
that the number of parameters to be estimated and therefore the estimation error is
less for allINAR(1) compared to allINARMA(1,1). The results of identification in
M.Mohammadipour, 2009, Chapter 9 214
the next section will show that 98.78 percent of 16,000 series (78.43 for 3,000 series)
are identified as INAR(0,0) or INAR(1). This could also justify the superior
performance of allINAR(1) because only 1.22 percent of the series (21.57 for 3000
series) are either INMA(1) or INARMA(1,1). Although more series are identified as
INMA(1) and INARMA(1,1) in 3,000 series, their moving average parameter is
generally between zero and 0.1 and these processes are close to INARMA(0,0) and
INAR(1). This could also help to understand why allINAR(1) works better than all
INARMA(1,1) for 3,000 series.
9.5.3 Identification among four Processes
In this section, the appropriate INARMA model is identified among the four possible
candidates. Both onestage and twostage identification procedures are tested (see
section 4.6 for details). For both identification methods, the results for the case where
all the INARMA forecasts are based on CLS and YW estimates are presented.
The accuracy of INARMA forecasts based on identification and treating all as
INAR(1) or INARMA(1,1) for 16,000 series are compared in Table 99 and Table
910. As expected, identification (twostage) produces slightly better results in terms
of MSE and MASE, (except for MSE of allINAR(1) for all points in time) but the
results are generally close.
Table ?99 The effect of identification on INARMA forecasts for all points in time (16000 series)
The
identification
method
ME MSE MASE
PB of MASE
(INARMA/Benchmark)
RGRMSE
(INARMA/Benchmark)
Crost
0.2
Crost
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Crost
0.2
Crost
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Twostage
identification (CLS)
0.0334 0.3529 1.8576 0.6659 0.6818 0.6099 0.5440 0.6030 0.4927 0.8004 0.7614 0.8514 0.9328 0.8595 1.0264
Twostage
identification (YW)
0.0321 0.3521 1.8647 0.6655 0.6811 0.6096 0.5440 0.6027 0.4924 0.7968 0.7577 0.8474 0.9286 0.8555 1.0217
Onestage
identification (CLS)
0.0307 0.3581 1.9008 0.6665 0.6832 0.6123 0.5481 0.6054 0.4963 0.7948 0.7549 0.8443 0.9232 0.8524 1.0165
Onestage
identification (YW)
0.0309 0.3533 1.9085 0.6359 0.6413 0.6150 0.5422 0.6022 0.4961 0.7945 0.7557 0.8436 0.9221 0.8529 1.0151
AllINAR(1) (YW) 0.0283 0.3527 1.9124 0.6742 0.6868 0.6216 0.5555 0.6150 0.5050 0.7898 0.7493 0.8382 0.9157 0.8463 1.0077
AllINARMA(1,1)
(CLS)
0.0258 0.3668 1.9128 0.6692 0.6842 0.6161 0.5508 0.6095 0.5004 0.7653 0.7294 0.8128 0.8918 0.8205 0.9818
M.Mohammadipour, 2009, Chapter 9 215
Table ?910 The effect of identification on INARMA forecasts for issue points (16000 series)
The
identification
method
ME MSE MASE
PB of MASE
(INARMA/Benchmark)
RGRMSE
(INARMA/Benchmark)
Crost
0.2
Crost
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Crost
0.2
Crost
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Twostage
identification (CLS)
0.0351 0.5045 0.3055 0.5562 0.5246 0.4903 0.3836 0.4817 0.3448 0.9315 0.9659 1.0093 1.2102 1.0186 1.3311
Twostage
identification (YW)
0.0256 0.5187 0.3093 0.5514 0.5196 0.4856 0.3795 0.4768 0.3406 0.9587 0.9910 1.0392 1.2435 1.0488 1.3688
Onestage
identification (CLS)
0.0012 0.5288 0.3293 0.5211 0.5014 0.4581 0.3682 0.4497 0.3308 1.0188 1.0941 1.1053 1.3783 1.1158 1.5188
Onestage
identification (YW)
0.0220 0.5257 0.3313 0.5204 0.5023 04579 0.3575 0.4450 0.3185 1.0145 1.0722 1.0951 1.3723 1.1116 1.5252
AllINAR(1) (YW) 0.0324 0.5218 0.3473 0.4584 0.4493 0.4026 0.3273 0.3956 0.2941 1.1734 1.2477 1.2728 1.5709 1.2847 1.7301
AllINARMA(1,1)
(CLS)
0.0326 0.5766 0.3509 0.4958 0.4797 0.4349 0.3507 0.4275 0.3162 1.0833 1.1712 1.1751 1.4768 1.1862 1.6282
The results of Table 99 and Table 910 also show that the twostage identification
procedure provides better results than the onestage. Since the majority of series are
identified as INARMA(0,0), this agrees with the simulation results of section 8.6.2.3.
It can also be seen that the CLS and YW yield close results for both twostage and
onestage identification methods. In order to be consistent with the simulation
analysis (see conclusion of section 8.4), we focus on the CLSbased twostage
identification results.
For 16,000 time series, out of the 5,168 series, 98.12 percent were identified as
INARMA(0,0), 0.66 percent as INAR(1), 1.04 percent as INMA(1), and 0.17 percent
were identified as INARMA(1,1).
As can be seen, the majority of the series are identified as INARMA(0,0). The
simulation results show that when data is in fact INARMA(0,0), the all
INARMA(1,1) forecasts are close to INARMA(0,0) forecasts. The simulation results
also show that when the order is known to be (0,0), allINARMA(1,1) produces
better forecasts than the best benchmark method. The results of Table 95 and Table
96 show that allINARMA(1,1) for 16,000 series performs better than benchmarks,
agreeing with the simulation results.
Now, the accuracy of INARMA forecasts based on identification and treating all as
INAR(1) or INARMA(1,1) for 3,000 series are compared in Table 911 and Table
912. Here again, the twostage identification produces slightly better results not only
M.Mohammadipour, 2009, Chapter 9 216
in terms of MSE and MASE, but also in terms of PB of MASE and RGRMSE (with
the exception of MSE of allINAR(1)). It should be mentioned that these results are
still not as good as SBA and SBJ with smoothing parameter 0.2.
Table ?911 The effect of identification on INARMA forecasts for all points in time (3000 series)
The
identification
method
ME MSE MASE
PB of MASE
(INARMA/Benchmark)
RGRMSE
(INARMA/Benchmark)
Crost
0.2
Crost
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Crost
0.2
Crost
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Twostage
identification (CLS)
0.0813 3.2925 0.8608 0.5238 0.5557 0.4973 0.5286 0.4974 0.5358 0.9376 0.9050 0.9599 0.9710 0.9647 0.9707
Twostage
identification (YW)
0.0191 3.2840 0.8603 0.5229 0.5514 0.4895 0.5244 0.4893 0.5315 0.9135 0.8875 0.9412 0.9544 0.9475 0.9500
Onestage
identification (CLS)
0.0276 3.3088 0.8697 0.5156 0.5497 0.4852 0.5191 0.4854 0.5276 0.9469 0.9145 0.9747 0.9829 0.9803 0.9823
Onestage
identification (YW)
0.0091 3.3726 0.8748 0.4734 0.5018 0.4517 0.4782 0.4522 0.4899 0.9475 0.9178 0.9679 0.9980 0.9823 0.9969
AllINAR(1) (YW) 0.0136 3.2319 0.8636 0.5171 0.5551 0.4842 0.5211 0.4839 0.5295 0.9471 0.9175 0.9765 0.9868 0.9826 0.9857
AllINARMA(1,1)
(CLS)
0.0461 3.4038 0.8869 0.5063 0.5401 0.4777 0.5125 0.4767 0.5209 0.9893 0.9536 1.0231 1.0293 1.0289 1.0276
Table ?912 The effect of identification on INARMA forecasts for issue points (3000 series)
The
identification
method
ME MSE MASE
PB of MASE
(INARMA/Benchmark)
RGRMSE
(INARMA/Benchmark)
Crost
0.2
Crost
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Crost
0.2
Crost
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Twostage
identification (CLS)
0.0099 3.4099 0.9013 0.5202 0.5544 0.4888 0.5225 0.4883 0.5256 0.9907 0.9560 1.0076 1.0214 1.0160 1.0277
Twostage
identification (YW)
0.0303 3.3835 0.8995 0.5190 0.5501 0.4827 0.5191 0.4814 0.5225 0.9653 0.9365 0.9879 1.0004 0.9973 1.0039
Onestage
identification (CLS)
0.0490 3.4327 0.9107 0.5107 0.5486 0.4783 0.5127 0.4780 0.5174 0.9961 0.9618 1.0188 1.0301 1.0282 1.0371
Onestage
identification (YW)
0.0572 3.5304 0.9202 0.5005 0.5408 0.4658 0.5144 0.4670 0.5161 0.9966 0.9749 1.0217 1.0469 1.0486 1.0527
AllINAR(1) (YW) 0.0650 3.3563 0.9054 0.5117 0.5539 0.4756 0.5127 0.4744 0.5161 0.9927 0.9618 1.0200 1.0311 1.0293 1.0377
AllINARMA(1,1)
(CLS)
0.1366 3.5569 0.9327 0.5006 0.5378 0.4691 0.5027 0.4677 0.5097 1.0343 0.9940 1.0629 1.0707 1.0712 1.0772
The results of Table 911 and Table 912 also show that the twostage identification
procedure provides better results than the onestage method. This agrees with the
corresponding simulation results of section 8.6.2.3 for each of INARMA(0,0),
INAR(1), INMA(1) and INARMA(1,1) models with similar parameters to those
estimated for 3,000 series (see section 9.5.4 to find the corresponding parameters).
The CLSbased twostage identification results for 3,000 series show that out of
1,943 filtered series, 54.55 percent were identified as INARMA(0,0), 23.88 percent
M.Mohammadipour, 2009, Chapter 9 217
as INAR(1), 17.96 percent as INMA(1), and 3.60 percent were identified as
INARMA(1,1).
Now, if we go back to the results of the previous section, the results show that all
INARMA(1,1) results are worse than the benchmark methods. The simulation results
also show that for the INAR(1) and INMA(1) processes with the corresponding
parameters (the most common INARMA parameters of the 1,943 series are provided
in the next section), the INARMA forecast accuracy is very close or slightly worse
than that of benchmark methods. This could explain the 4 percent superiority (in
terms of MSE) of the best benchmark forecasts (SBA 0.2) over the all
INARMA(1,1) forecasts.
Based on the results of this section, identification (twostage) results in better
INARMA forecasts than using an allINAR(1) or allINARMA(1,1) approach but the
accuracy benefits are small. As also mentioned in the simulation chapter, the all
INAR(1) and allINARMA(1,1) approaches are especially useful when the
autoregressive parameter is high, but this is not the case for our empirical data.
9.5.4 INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1) Series
As previously mentioned, a considerable percentage of series among 1,943 filtered
series (of 3,000 series) were identified as INARMA(0,0), INAR(1) or INMA(1) and
a few series were identified as INARMA(1,1). In this section, we separate the series
according to the models identified and study them individually.
Because most of the series in the 16,000 data set are identified as INARMA(0,0) and
a small number identified as other models, we do not present the results of each
model here. Similar results for 16,000 series can be found in Appendix 9.A.
As the results of identification in the previous section suggest, 54.55 percent of 1943
series are identified as INARMA(0,0). Now, we assume that the order of these 1,060
series is taken to be (0,0) and the INARMA(0,0) forecasting method, i.e. using the
average of all the previous observations as the forecast for the next period, is used.
The results for all points in time and issue points are presented in Table 913 and
Table 914.
M.Mohammadipour, 2009, Chapter 9 218
Table ?913 Only INARMA(0,0) series for all points in time (3000 series)
Accuracy
measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INARMA
ME 0.0288 0.0843 0.1719 0.3017 0.1878 0.4303 0.1420
MSE 2.0878 2.3629 2.0998 2.2830 2.1043 2.3483 2.0749
MASE 0.9139 0.9819 0.9021 0.9352 0.9016 0.9382 0.8978
PB of MASE
(INARMA/Benchmark)
0.5329 0.5624 0.5080 0.5453 0.5085 0.5491 
RGRMSE
(INARMA/Benchmark)
0.8992 0.8543 0.9111 0.8922 0.9152 0.8798 
Table ?914 Only INARMA(0,0) series for issue points (3000 series)
Accuracy
measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INARMA
ME 0.0071 0.0959 0.1482 0.2827 0.1639 0.4089 0.0830
MSE 2.1982 2.4911 2.2013 2.3901 2.2049 2.4482 2.1704
MASE 0.9472 1.0182 0.9327 0.9635 0.9318 0.9634 0.9298
PB of MASE
(INARMA/Benchmark)
0.5328 0.5642 0.5023 0.5388 0.5013 0.5362 
RGRMSE
(INARMA/Benchmark)
0.9743 0.9149 0.9763 0.9549 0.9836 0.9450 
Investigating the estimated parameter of the INARMA(0,0) process (?), we found
that in general ? is close to 1 (the average is 1.2641 and 73.87 percent are between
0.5 and 1.5).
The results of Table 913 agree with the corresponding simulation results (?= 1 and
?= 24). The simulation results show that INARMA produces the best results and
the empirical results also show a very narrow improvement over the best benchmark
method. The PB of MASE results confirm the results of MASE for both all points in
time and issue points.
The results of identification show that 23.88 percent of 1,943 series are identified as
INAR(1). Now, we assume that the order of these 464 series is taken to be (1,0) and
the INAR(1) forecasting method is used. We also investigate the effect of using
different estimation methods on forecast accuracy. For this reason, the forecasts
based on the three estimation methods of CLS, YW and CML are presented in Table
915 and Table 916.
M.Mohammadipour, 2009, Chapter 9 219
In general, the estimated autoregressive parameter of the INAR(1) process, ?, is
close to 0.1 (the average is 0.1234 and 50.65 percent are between 0.05 and 0.15) and
the estimated innovation parameter, ?, is around 2 (the average is 2.5972 and 40.52
percent are between 1 and 3). The simulation results for the corresponding
parameters (similar values of ?, ?, and ?) show that YW yields better results than
CLS and CML. The simulation results also show that the YW results are slightly
better than those of SBA and SBJ with smoothing parameter 0.2.
Table ?915 Only INAR(1) series for all points in time (3000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INAR(1)
CLS
INAR(1)
YW
INAR(1)
CML
ME 0.1557 0.1498 0.1365 0.5791 0.1689 0.8221 0.1963 0.1411 0.1257
MSE 4.8743 5.6076 4.8235 5.5217 4.8319 5.8396 4.9049 4.7921 4.8386
MASE 0.9168 0.9721 0.8923 0.9218 0.8906 0.9312 0.9203 0.9088 0.9177
PB of MASE
INAR(1)CLS/Benchmark
0.4989 0.5408 0.4596 0.4919 0.4605 0.5068
PB of MASE
INAR(1)YW/Benchmark
0.5119 0.5458 0.4682 0.5038 0.4662 0.5135
PB of MASE
INAR(1)CML/Benchmark
0.5083 0.5462 0.4657 0.4928 0.4653 0.5070
RGRMSE
INAR(1)CLS/Benchmark
0.9940 0.9733 1.0499 1.0924 1.0529 1.1180
RGRMSE
INAR(1)YW/Benchmark
0.9726 0.9554 1.0275 1.0703 1.0311 1.0950
RGRMSE
INAR(1)CML/Benchmark
0.9887 0.9672 1.0443 1.0835 1.0457 1.1087
Table ?916 Only INAR(1) series for issue points (3000 series)
Accuracy measure
Forecasting method
Croston
0.2
Croston
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INAR(1)
CLS
INAR(1)
YW
INAR(1)
CML
ME 0.1439 0.1390 0.1465 0.5856 0.1787 0.8272 0.2755 0.2165 0.2114
MSE 5.0350 5.7658 4.9933 5.7037 5.0027 6.0297 5.1167 4.9863 5.0296
MASE 0.9600 1.0166 0.9362 0.9669 0.9346 0.9764 0.9743 0.9609 0.9685
PB of MASE
INAR(1)CLS/Benchmark
0.4850 0.5330 0.4464 0.4837 0.4473 0.4962
PB of MASE
INAR(1)YW/Benchmark
0.4988 0.5411 0.4579 0.4954 0.4551 0.5024
PB of MASE
INAR(1)CML/Benchmark
0.5005 0.5413 0.4544 0.4872 0.4552 0.5001
RGRMSE
INAR(1)CLS/Benchmark
1.0272 1.0087 1.0864 1.1242 1.0913 1.1743
RGRMSE
INAR(1)YW/Benchmark
1.0041 0.9888 1.0625 1.1012 1.0682 1.1496
RGRMSE
INAR(1)CML/Benchmark
1.0177 0.9981 1.0731 1.1107 1.0778 1.1582
The results of Table 915 agree with the simulation results in that YW results are
M.Mohammadipour, 2009, Chapter 9 220
better than CLS and CML. The YWbased INAR(1) forecasts are also better than
benchmarks by one percent in terms of MSE. However, the MASE of INAR(1) is
worse than that of the best benchmark by one percent.
The results of Table 916 also show a narrow improvement of INAR(1)YW
compared to the best benchmark (0.15 percent in terms of MSE) although the
INARMA forecasts are biased when only issue points are considered.
Based on the results of identification, 17.96 percent of 1,943 series are identified as
INMA(1). We assume that the order of these 349 series is taken to be (0,1) and the
INMA(1) forecasting method is used. We also investigate the effect of using
different estimation methods (CLS and YW) on the forecasting accuracy. The results
are given in Table 917 and Table 918.
Table ?917 Only INMA(1) series for all points in time (3000 series)
Accuracy measure
Forecasting method
Croston
0.2
Crosto
n 0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INMA(1)
CLS
INMA(1)
YW
ME 0.0869 0.0875 0.1907 0.6066 0.2215 0.8380 0.1041 0.0936
MSE 4.3556 4.9339 4.3470 4.9673 4.3581 5.2745 4.3291 4.3464
MASE 0.6681 0.7022 0.6556 0.6778 0.6549 0.6904 0.6676 0.6668
PB of MASE
INMA(1)CLS/Benchmark
0.5093 0.5418 0.4728 0.5033 0.4702 0.5150
PB of MASE
INMA(1)YW/Benchmark
0.5150 0.5461 0.4742 0.5062 0.4723 0.5177
RGRMSE
INMA(1)CLS/Benchmark
0.9546 0.9687 0.9829 1.0729 0.9960 1.0516
RGRMSE
INMA(1)YW/Benchmark
0.9345 0.9527 0.9642 1.0524 0.9771 1.0316
Table ?918 Only INMA(1) series for issue points (3000 series)
Accuracy measure
Forecasting method
Croston
0.2
Crosto
n 0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INMA(1)
CLS
INMA(1)
YW
ME 0.0870 0.0846 0.1889 0.6043 0.2195 0.8340 0.1593 0.1444
MSE 4.4429 5.0046 4.4358 5.0503 4.4471 5.3611 4.4548 4.4742
MASE 0.7177 0.7512 0.7045 0.7255 0.7039 0.7391 0.7256 0.7235
PB of MASE
INMA(1)CLS/Benchmark
0.4924 0.5304 0.4575 0.4891 0.4539 0.5019
PB of MASE
INMA(1)YW/Benchmark
0.5026 0.5361 0.4616 0.4945 0.4594 0.5064
RGRMSE
INMA(1)CLS/Benchmark
0.9891 1.0325 1.0230 1.1233 1.0399 1.1033
RGRMSE
INMA(1)YW/Benchmark
0.9654 1.0118 0.9987 1.0970 1.0152 1.0773
M.Mohammadipour, 2009, Chapter 9 221
Looking at the estimated parameters of the INMA(1) process (?,?) reveals that in
general, ? is close to zero (the average is 0.0374 and 79.94 percent are between 0
and 0.05) and ? is between 2 and 3 (the average is 2.7357 and 43.55 percent are
between 2 and 3).
The simulation results for the corresponding parameters (similar values of ?, ?, and
?) show that YW yields slightly better results than CLS and also both CLS and YW
results are close to SBA and SBJ with smoothing parameter 0.2 (with INMA slightly
better in terms of MSE). The results of Table 917 show that YW forecasts are better
than CLS in terms of MASE (only by 0.1 percent) but worse in terms of MSE (only
by 0.5 percent). The best INMA(1) is better than the best benchmark by only 0.4
percent using MSE and worse than the best benchmark by 1.8 percent using MASE.
Finally, the results of identification show that 3.60 percent of 1,943 series are
identified as INARMA(1,1). We assume that the order of these 70 series is taken to
be (1,1) and the INARMA(1,1) forecasting method is used. We also investigate the
effect of using different estimation methods (CLS and YW) on the forecasting
accuracy. The results are given in Table 919 and Table 920.
Table ?919 Only INARMA(1,1) series for all points in time (3000 series)
Accuracy measure
Forecasting method
Croston
0.2
Crosto
n 0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
INMA(1)
CLS
INMA(1)
YW
ME 0.1340 0.1141 0.1556 0.6049 0.1878 0.8446 0.2016 0.0997
MSE 4.7747 5.3016 4.7204 5.2908 4.7275 5.6017 5.0449 5.8052
MASE 0.8853 0.9369 0.8695 0.9076 0.8690 0.9192 0.9178 0.9849
PB of MASE
INMA(1)CLS/Benchmark
0.4607 0.5226 0.4452 0.4821 0.4512 0.5036
PB of MASE
INMA(1)YW/Benchmark
0.4417 0.4881 0.4369 0.4702 0.4429 0.4905
RGRMSE
INMA(1)CLS/Benchmark
1.0854 1.0284 1.1681 1.0737 1.1793 1.1495
RGRMSE
INMA(1)YW/Benchmark
0.8169 0.7752 0.8747 0.7943 0.8910 0.8146
In general, the estimated autoregressive parameter of an INARMA(1,1) process is in
the range 0.1 ?
?
Equation 104
where ??
?? is the crosscovariance at lag ? given by Equation 103. Therefore, as the
second finding of this research, the autocorrelation function of an INARMA(p,q)
process is as follows:
??=?
?1???1 +?2???2 +?+??????+???0
??+??+1?1
??+?+??????
??
?0
???
?1???1 +?2???2 +?+?????? ?>?
?
Equation 105
where ?0 is the unconditional variance of the process and is given by Equation 102.
It is shown that if ?= 0, the Equation 105 reduces to the ACF of an INAR(p)
process given by Du and Li (1991). Also if ?= 0, the Equation 105 provides the
ACF of an INMA(q) process which is given by Br?nn?s and Hall (2001) (see section
3.3.8.2).
It is also shown is section 4.3.1 that the ACF structure of the INARMA(p,q) process
is analogous to that of an ARMA(p,q) process. Also, based on the discussion in
section 4.3.2.3, the partial autocorrelation function of an INARMA(p,q) process is
the analogue of that of an ARMA process, is infinite and behaves like the PACF of a
pure integer moving average process.
M.Mohammadipour, 2009, Chapter 10 246
10.3.3 The YW Estimators of an INARMA(1,1) Model
Based on Equation 102 and Equation 105, we have found the YW estimators for an
INARMA(1,1) process to be:
?=
?2
?1
=
? ??????(???2??)
?
?=3
? ??????(???1??)
?
?=2
Equation 106
?=
?1 +??(???1)
?1?1 + 3???1???2?2
Equation 107
where ?? is the estimate of the autocorrelation at lag ?, ??. The innovation parameter
is then estimated from the expected value of the process
?=
1??
1 +?
? ??
?
?=1
?
Equation 108
We have also derived the YW estimators for an INARMA(2,2) process for
presentational purposes (see section 5.8).
10.3.4 Lead Time Forecasting of an INARMA(p,q) Model
It is shown in chapter 6 that aggregation of an INARMA(p,q) process over a lead
time results in an INARMA(p,q) process. The aggregated process has the same
autoregressive and moving average parameters but a different innovation parameter.
When the innovations of the original process are ??~???(?) , the innovations of the
aggregated process are ??~???(??+ 1??).
It is also found that the aggregated process can be written in terms of the last ?
observations as follows:
M.Mohammadipour, 2009, Chapter 10 247
? ??+?
?+1
?=1
=? ? ??
1 ???
??
1
?=1
?+1
?=1
+? ? ??
2 ????1
??
2
?=1
?+1
?=1
+?
+? ? ??
??????+1
??
?
?=1
?+1
?=1
+? ? ??
?+1???+??
??
?+1
?=1
?+1
?=1
Equation 109
where all the parameters are defined in Table 64. The above result is then used to
find the conditional expected value of the aggregated process which is:
??? ??+?
?+1
?=1
????+1,?,???1 ,???=?? ? ??
1
??
1
?=1
?+1
?=1
???+?? ? ??
2
??
2
?=1
?+1
?=1
????1 +?
+?? ? ??
?
??
?
?=1
?+1
?=1
?????+1 +?? ? ??
?+1
??
?+1
?=1
?+1
?=1
??
Equation 1010
Based on the above results, the lead time forecasts for the specific INARMA
processes used in this research are derived as follows:
??? ??+?
?+1
?=1
???=
?
?
?
?
?
?
?
?
?
(?+ 1)? for INARMA(0,0)
?(1???+1)
1??
??+
?
1??
???+ 1??? ??
?+1
?=1
? for INAR(1)
??+ 1?(1 +?)? for INMA(1)
?(1???+1)
1??
??+
?(1 +?)
1??
???+ 1??? ??
?+1
?=1
? for INARMA(1,1)
?
Equation 1011
10.4 Conclusions from the Simulation Part of the Thesis
In this section, the main findings of the simulation part of this research are
summarized. Four INARMA models are used in this thesis which are:
INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1). The range of parameters
chosen for these processes is shown in Table 71.
M.Mohammadipour, 2009, Chapter 10 248
10.4.1 The Performance of Different Estimation Methods
The performance of YW, CLS, and CML (only for INAR(1) process) is compared.
The simulation results show that, for an INAR(1) process, the MSE of estimates
produced by CML is generally less than that of YW and CLS. However, the accuracy
of forecasts produce by CML is not very far from those by YW and CLS. The YW
and CLS estimators are close when the number of observations is high. But, for small
samples, the difference is high when the autoregressive parameter is high. In such
cases, CLS produces much better estimates for both ? and ? in terms of MSE than
YW. On the other hand, for small values of ?, YW results in better estimates. This is
also true for the accuracy of forecasts produced by these estimation methods.
For an INMA(1) process, for small number of observations, CLS generally has
smaller MSEs than YW except for very high values of ?. For large number of
observations, YW has smaller MSEs than CLS for high values of ?. However, this
does not have a great effect on the accuracy of forecasts produced by each method.
The MSE of ? for both YW and CLS estimates increases with an increase in ? but the
same is not necessarily true for the MSE of ?.
Finally, for an INARMA(1,1) process, CLS generally produces better estimates
especially when the number of observations is small and the autoregressive
parameter is high. This is also true for the accuracy of forecasts produced by CLS
compared to those by YW.
The results of threestep and sixstep ahead forecasts show that the forecasts based
on YW and CLS estimation methods are generally very close for all of the three
INARMA processes, but YWbased results are slightly better in many cases.
10.4.2 The CrostonSBA Categorization
Syntetos et al. (2005) provide a categorization scheme for Croston and SBA based on
MSE to establish the areas that each method should be used over the other. They use
the squared coefficient of variation (??2) of demand size and the average inter
demand interval (?) to identify different regions shown in Figure 81. Because this
M.Mohammadipour, 2009, Chapter 10 249
categorization is based on the assumption that demand occurs as an i.i.d. Bernoulli
process, it is worth testing if it also holds for an i.i.d. Poisson (an INARMA(0,0))
process. The simulation results show that not only the CrostonSBA categorization
based on MSE holds for i.i.d. Poisson demand, but it also generally holds using
MASE.
The simulation results also show that when the number of observation increases, the
advantage of SBA over Croston decreases. This is also shown by direct mathematical
calculations in section 8.5.
MSESBA ?MSECroston ???1?
?
2
?
2
?1??
?+ 2(1??)2?+1
2??
??
(??1)2?2
?4
+
?2
?2
?
Equation 1012
Based on the Equation 1012, when the smoothing parameter is small (?= 0.2), the
above coefficient decreases when ? increases; therefore the difference between MSE
of Croston and SBA also decreases. However, because the above coefficient reaches
a limit of ?
?2
4
????
?
2??
?, the advantage of SBA over Croston does not change
noticeably when the number of observations is high. On the other hand, when the
smoothing parameter is high (?= 0.5), the difference between the MSE of Croston
and SBA changes little with changes in ?.
Although the CrostonSBA categorization is for i.i.d. demand, we have also tested it
when demand is produced by an INAR(1), INMA(1) or an INARMA(1,1) process.
The results show that the CrostonSBA categorization generally holds for all of these
processes.
10.4.3 Identification in INARMA Models
We test the two identification approaches mentioned in section 10.2. The simulation
results show that when the data is produced by an INARMA(0,0) process, the two
stage method produces better results than the onestage method. This is true in terms
of the percentage of time that the model is identified correctly and also in terms of
the accuracy of forecasts. However, when data is produced by INAR(1), INMA(1) or
M.Mohammadipour, 2009, Chapter 10 250
INARMA(1,1) processes, the onestage method outperforms the twostage method.
For INAR(1) and INARMA(1,1) processes, when the autoregressive parameter is
low, the process is misidentified in most cases for both identification methods. But
when the autoregressive parameter is high, the performance of both identification
methods improves. Obviously, when more observations are available, the percentage
of correct identification increases for both methods.
The INMA(1) process is misidentified in most of the cases. However, the results
show that it does not affect the forecasting accuracy to a great extent. In general, the
performance of both identification methods improves for higher moving average
parameters and longer length of history.
The simulation results also show that, for INAR(1) and INARMA(1,1) processes,
misidentification has a great impact on the accuracy of forecasts when the
autoregressive parameter is high. When the autoregressive parameter is small, or the
process is an INMA(1) process, the effect of misidentification on forecasting
accuracy is small.
We also test the case that the identification step is ignored and an INAR(1) or an
INARMA(1,1) method is used for forecasting. The results show that when the
number of observations is small, this approach produces better forecasts for INAR(1)
and INMA(1) series. The results also show that the allINAR(1) approach produces
better forecasts than the allINARMA(1,1) approach for all four INARMA series.
10.4.4 Comparing INARMA with the Benchmark Methods
The results show that when the order of the INARMA model is known, the
INARMA method almost always produces the lowest MSE when all points in time is
considered. When only issue points are considered, the INARMA forecasts are
biased and therefore are not always better than the benchmark methods.
For INARMA(0,0) and INMA(1) processes, the improvement over benchmarks is
not considerable. However, for the INAR(1) and INARMA(1,1) processes, the
improvement is considerable when the autoregressive parameter is high. This is true
M.Mohammadipour, 2009, Chapter 10 251
for both MSE and MASE of forecasts.
The degree of improvement (by using INARMA over the benchmark methods) for
the ?step ahead forecasts does not change for INARMA(0,0) process. But for an
INMA(1) process, the performance of INARMA compared to benchmark methods is
improved for hstep ahead forecasts compared to onestep ahead forecasts. For
INAR(1) and INARMA(1,1) processes, the performance of INARMA over the
benchmark methods is improved compared to the onestep ahead case when the
autoregressive parameter is low. But when the autoregressive parameter is high, the
fact that the forecasts converge to the mean of the process results in poor forecasts
compared to the onestep ahead case.
All of the above results were for the case that the order of the INARMA model is
known. This is obviously not true in practice and the INARMA model needs to be
identified. The results of identification show that treating all as an INAR(1) model is a
promising approach especially for high autoregressive parameters and short length of
history. Therefore, we compare an allINAR(1) method with the benchmark methods.
The results show that for INARMA(0,0) and INMA(1) processes, the benchmark
methods outperform INARMA especially for more sparse demand. For INARMA(1,1)
series, when the autoregressive parameter is high, INARMA is considerably better
than the benchmark methods in terms of MSE and MASE. Obviously, the results for
INAR(1) series are the same as the case that the order is known.
We also compare the lead time forecasts produced by an allINAR(1) method with
the benchmark methods. The results show that, for INARMA(0,0) and INMA(1)
series, the allINAR(1) forecasts are better than the best benchmark in most of the
cases. For INAR(1) and INARMA(1,1) series, when the autoregressive parameter is
high, the improvement of INARMA over the best benchmark is narrow for short
length of history. However with more observations, the improvement also increases.
For small autoregressive parameters, the INAR(1) method always outperforms the
benchmark methods. Again, the improvement increases with an increase in the length
of history.
M.Mohammadipour, 2009, Chapter 10 252
10.5 Conclusions from the Empirical Part of the Thesis
The simulation analysis of this research compared the accuracy of forecasts (in terms
of MSE and MASE) by an INARMA method with those produced by benchmark
methods when the data is generated by an INARMA model. Empirical analysis is
performed to validate the theoretical and simulation results for real data. Two data
sets used in this research are the Royal Air Force (RAF) individual demand histories
of 16,000 SKUs over a period of 6 years (monthly observations) and demand history
of 3,000 SKUs from the automotive industry over a period of 2 years (24 months).
Both data sets are filtered to eliminate lumpy series, since the INARMA models with
Poisson marginal distribution are not appropriate for lumpy demand. The PB of
MASE and the RGRMSE are used to measure the forecast error in addition to ME,
MSE and MASE.
10.5.1 Identification in INARMA Models
The two identification methods (twostage and onestage) are compared. The
empirical results for both 16,000 and 3,000 series show that the twostage method
produces more accurate forecasts. This is not surprising because the identification
shows that the majority of series in both data sets are identified as INARMA(0,0) and
the simulation results show that for INARMA(0,0) series the twostage method is
better than the onestage method.
The empirical results also confirm the simulation results that ignoring the
identification step, and forecasting with an INARMA(1,1) or an INAR(1) process, is
a promising approach. It is also confirmed that the latter outperforms the former.
10.5.2 The Performance of Different Estimation Methods
After the appropriate INARMA model is identified for all the series in each data set,
different INARMA series are separated and forecasted by the corresponding
INARMA method. The performance of different estimation methods in terms of the
accuracy of forecasts are tested for each INARMA model.
M.Mohammadipour, 2009, Chapter 10 253
The empirical results agree with the simulation results for all three INARMA
models. For INAR(1) series (with ? around 0.1 and ? around 2), the YW produces
better forecasts than CLS and CML agreeing with the simulation results for similar
parameters. For INMA(1) series, YW produces slightly better results than CLS.
However, for INARMA(1,1) series, CLS outperforms YW, which again confirms the
simulation results.
10.5.3 Comparing INARMA with the Benchmark Methods
The forecast accuracy of INARMA (based on all identification methods) is compared
to that of the benchmark methods for both 16,000 and 3,000 series. The results show
that, for the former data set, there is an improvement by using INARMA methods for
onestep ahead forecasts for all accuracy measures. The improvement is narrow for
3,000 series which is expected because of the short length of history and low
autocorrelation. This slight improvement is based on MSE and, in fact, the MASE of
INARMA is worse than the benchmarks for 3,000 series.
The accuracy of ?step ahead INARMA forecasts are even better than the
benchmarks compared to the onestep ahead forecasts. This is true for both data sets.
The lead time forecasts are obtained based on the conditional expected value of the
aggregated INARMA process and also from the cumulative ?step ahead forecasts.
The empirical results show that for INAR(1) series of both data sets, both of these
methods produce considerably better forecasts than the benchmark methods with the
former outperforming the latter. The same is true for INMA(1) series of both data
sets but not for INARMA(1,1) series. For INARMA(1,1) series, the cumulative ?
step ahead forecasts are better than those based on conditional expected value of the
aggregated process. It should be borne in mind that the number of INARMA(1,1) for
both data sets are very small.
10.5.4 The Problem with MASE
The empirical results show that when the data series is highly intermittent (which is
M.Mohammadipour, 2009, Chapter 10 254
the case for most of the 16,000 series), the MASE of all forecasting methods is very
high. This would suggest that all methods are worse than na?ve. Because the data
series contain many zeros in the estimation period, the error of na?ve in most of the
time periods is zero. Therefore the insample MAE is very small and the MASE is
very large. This is true when all points in time are considered.
However, for the case of issue points, because it is more likely that a nonzero
demand is followed with a zero demand, the absolute error of na?ve and therefore the
insample MAE is large. As a result, the MASE of the forecasting methods is smaller
compared to the all points in time case. The empirical results confirm this. As a
result, for highly intermittent data, MASE does not provide reliable results.
10.6 Practical and Software implications
In this thesis we have applied an INARMA method to model and forecast nonerratic
intermittent demand (see the classification by Boylan et al., 2008). Four models have
been assumed to include autoregressive, moving average, mixed models and also an
i.i.d. Poisson process. All simulation and empirical results are based on the following
assumptions:
? The intermittent demand can be modelled with either an INARMA(0,0),
INAR(1), INMA(1) or an INARMA(1,1) process with Poisson marginal
? The benchmark methods to compete against INARMA method are: Croston,
SBA and SBJ
? The forecast accuracy measures are: ME, MSE, MASE, RGRMSE, and PB,
with the last two only used for empirical analysis
It has been shown that INARMA performs best when data has high autocorrelation.
Even with low autocorrelation INARMA outperforms the benchmarks when the
length of history is large. However, we cannot claim that when the above
assumptions are violated, these results still hold. For example, the empirical data of
this research were filtered to comply with the first assumption.
Therefore, in order to use the INARMA method, the distribution that fits the demand
M.Mohammadipour, 2009, Chapter 10 255
data of the organization should first be found. All discrete selfdecomposable
distributions can be used as marginal distributions of INARMA models. This
includes: Poisson, generalized Poisson, and negative binomial distributions. The
generalized Poisson allows for both under and overdispersion (Br?nn?s, 1994), and
the negative binomial allows for overdispersion (McCabe and Martin, 2005).
The results of this research suggest that although identification of the autoregressive
and moving average orders of the INARMA models results in more accurate
forecasts, using simple models such as an INAR(1) model also produces good
forecasts. This is an especially useful method when the length of history is not long
enough for identification. However, it should be mentioned that the simulation
results of this research were based on limited INARMA models and the empirical
data did not support higher order models. Therefore, for higher order models, the
performance of using a simple model instead of identification needs to be tested first
through simulation.
The YW and CLS estimation methods are not based on distributional assumptions
and therefore can be used for all cases. However, the maximum likelihood estimators
for the corresponding distribution should be obtained. The results of this research can
be used, at least for the specific INARMA models and parameters, as a guide to
when one estimation method should be used over another.
Although the lead time forecasts in this thesis are restricted to Poisson, the findings
can be easily amended for other distributions.
The conventional ARIMA models are offered by standard forecasting programmes
such as Autobox (Automatic Forecasting Systems) and Forecast Pro (Business
Forecast Systems). These programmes provide automated identification, estimation
and forecasting for ARIMA models. For instance, Forecast Pro uses the Bayesian
information criterion (BIC) along with some other rules for identification and
unconditional least squares for estimation (Goodrich, 2000). Autobox matches the
SACF with theoretical ACF for some starting ARIMA models and then uses AIC to
select the best model (Reilly, 2000).
Although INARMA models have had applications in many areas (see section 3.3.9),
there is currently no software based on these models. This research suggested a new
M.Mohammadipour, 2009, Chapter 10 256
application area for INARMA models in intermittent demand forecasting. The
improvement in forecast accuracy by using INARMA over the benchmark methods
such as Croston (which is offered in some forecasting programmes such as Forecast
Pro), also urges the need to develop forecasting software for these models.
In doing so, this research can be considered a starting point. We have addressed
issues of whether AIC produces satisfactory results, which estimation method is best
in what case, and ?step ahead and lead time forecasting for a limited number of
INARMA models.
10.7 Limitations and Further Research
Throughout this research we have made assumptions that can be relaxed in future
studies. Although most of the theoretical findings are not based on any marginal
distribution, for simulation and empirical analyses we restricted the research to a
specific distribution. As previously discussed in section 7.2, the Poisson distribution
was selected due to its interesting properties. However, it does not allow for
overdispersion in data and in such cases other distributions should be assumed. For
example, Alzaid and AlOsh (1993) show that the Generalized Poisson INAR(1)
process has more variability than PoINAR(1) because its variance is twice that of
PoINAR(1). This could be useful for modelling moderately lumpy demand.
Other adaptations of INARMA models to take into account nonstationarity,
seasonality and trend can also be used. For example the signed binomial thinning
models introduced by Kim and Park (2008) allow for negative values and negative
autocorrelations and also use the differencing operator to remove the trend and
seasonality.
As mentioned in section 7.3.1, the simulation and empirical results of this research
have been based on INARMA models with ?,??1. A natural extension would be to
use higher order models. As seen in chapter 9, our empirical data did not support
such models; however, other data sets may be better fitted by higher order INARMA
models.
M.Mohammadipour, 2009, Chapter 10 257
This research has only focused on aggregation of INARMA models over lead time.
The temporal and crosssectional aggregation in INARMA models were briefly
reviewed in chapter 3. A problem is identified with the result of temporal
aggregation of an INAR(1) process provided by Br?nn?s et al. (2002). More studies
should be carried out in both temporal and crosssectional aggregation fields.
As mentioned in section 4.5, the complexity of the likelihood function has restricted
the use of penalty functions for INARMA models. We have used the AIC based on
the likelihood function of ARIMA models and shown that the results are satisfactory.
However, there is still a need for finding the AIC based on INARMA models.
As confirmed in chapters 8 and 9, the INARMA forecasts are biased when only issue
points are considered. The estimates, and as a result, forecasts, can be revised to
reduce such bias. For example, the CLS of an INAR(1) process is based on the
following criterion:
?????=? [??????????1?]
2
?
?=1
Equation 1013
where ?= (?,?)?. Now, when only issue points are considered, the estimates of the
parameters will be updated after observing a positive demand. Therefore, in Equation
1013 the last observation which is ?? is definitely positive and its conditional
expected value should be ???????1,??> 0? instead of ???????1?. As a result, the
new least squares criterion will be:
??
????=? [??????????1?]
2
??1
?=1
+ [??????????1,??> 0?]
2
Equation 1014
The revised CLS estimates of parameters can be obtained by minimization of the
above criterion. Numerical methods are needed to find these estimates. The new
estimates can be compared to CLS estimates in terms of their impact on forecast
accuracy when only issue points are considered. Other estimation methods that allow
for such revisions such as maximum likelihood could also be considered.
Another limitation of this study is that we have used the conditional expectation to
produce forecasts. This enabled us to compare the MMSE INARMA forecasts to
M.Mohammadipour, 2009, Chapter 10 258
point forecasts from benchmark methods. The natural next step is to forecast the
whole distribution instead. This has been done in a number of studies in the
INARMA literature (Freeland and McCabe, 2004b; McCabe and Martin, 2005; Bu
and McCabe, 2008). This will then enable us to compare INARMA method with
bootstrapping methods in an IDF context.
Incorporating explanatory variables in INARMA models has been the subject of
some studies (Br?nn?s, 1995; Br?nn?s and Quoreshi, 2004). As discussed in section
2.3.3, causal models for IDF have not yet been well developed in the literature and
the integration of these models with INARMA models would be an interesting line of
research.
The benchmark methods of this research were Croston, SBA and SBJ. We have used
two arbitrary values for the smoothing parameter of these methods. This could be
replaced by the optimum smoothing parameter. Also, different smoothing parameters
could be used for updating demand size and interarrival time.
Although this study has only focused on forecasting and not inventory control, it is
expected that the improvement in the mean (forecasts) would translate to better
percentile estimates and better inventory results. Obviously, this could be tested in
future studies to find whether using INARMA method would result in better
inventory measures such as service level and inventory level that have been
suggested by Teunter and Duncan (2009).
Different inventory models need different estimation of parameters, i.e. mean,
percentiles, or estimates of demand sizes and interdemand invervals. Syntetos et al.
(2008) develop a modified periodic orderuptolevel inventory policy for
intermittent demand which relies upon demand sizes and interdemand intervals.
Depending on what inventory model is applied, INARMA methods can provide all of
these estimates. For example, in the case where demand follows an INAR(1) model:
??demand size?=
?/(1??)
1????/(1??)
Equation 1015
??demand interval?=
1??
?
Equation 1016
M.Mohammadipour, 2009, Chapter 10 259
Finally, the empirical results of this research have been restricted to two data sets.
More empirical analyses can be done to further assess the sensitivity of results to the
length of history and the level of intermittence.
M.Mohammadipour, 2009, References 260
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M.Mohammadipour, 2009, Appendix 3.A 272
Appendix 3.A Autocorrelation Function of an INARMA(1,1) Model
In this appendix, we show how to obtain the ACF of an INARMA(1,1) process of
??=?????1 +??+?????1. The unconditional variance of the INARMA(1,1)
process can be obtained from:
var????= var??????1?+ var????+ var??????1?+ 2cov(?????1,?????1)
Considering the fact that cov????,????=??cov(?,?), the above equation can
be written as:
var????=?
2var????1?+??1????????1?+??
2 +?2??
2 +??1?????
+2cov???2????2 +?????1 +??????2?,?????1?
=?2var????1?+??1???
??(1 +?)
1??
+??
2 +?2??
2 +??1?????
+ 2????
2
Hence, the unconditional variance of an INARMA(1,1) process is:
var????=
1
1??2
[??+?+????2???+ (1 +?
2 + 2??)??
2]
Equation 3.A1
If a Poisson distribution is assumed for innovations (??=??
2 =?), the above result
can be simplified to:
var????=
?
1??2
[1 +?+?+ 3??]
Equation 3.A2
The INARMA(1,1) process, ??, can be written in terms of ???? as follows:
??=?
??????+? ?
??????
??1
?=0
+? ?????????1
??1
?=0
Equation 3.A3
The autocovariance at lag ? is defined as:
??= cov(??,????)
Equation 3.A4
M.Mohammadipour, 2009, Appendix 3.A 273
Therefore, the autocovariance at lag ? is:
??= cov?????,??
??????+? ?
??????
??1
?=0
+? ?????????1
??1
?=0
??
=??var??????+ cov?????,? ?
??????
??1
?=0
?+ cov?????,? ?
????????1
??1
?=0
?
=??var??????+ cov?????,?
??1???????
=??var??????+ cov????????1 +????+???????1,?
??1???????
By substitution from Equation 3.A1:
??=
??
1??2
???+?+????2???+?1 +?
2 + 2?????
2?+???1???
2
=???1??
?(?+?+????2)
1??2
???+?
?(1 +?2 + 2??)
1??2
+????
2?
As a result:
??=?
??1??
?2 +??+?2????2
1??2
???+?
?+??2 +?2?+?
1??2
???
2?
Equation 3.A5
Consequently, the ACF of an INARMA(1,1) is:
??=?
??2 +??+?2????2???+ (?+??
2 +?2?+?)??
2
??+?+????2???+ (1 +?2 + 2??)??
2 for ?= 1
????1 for ?> 1
?
Equation 3.A6
For a PoINARMA(1,1) with ??=??
2 =?, the ACF would be:
??=?
?+?+??+?2 + 2?2?
1 +?+?+ 3??
for ?= 1
????1 for ?> 1
?
Equation 3.A7
M.Mohammadipour, 2009, Appendix 3.B 274
Appendix 3.B The Unconditional Variance of an INARMA(p,q)
Model
The unconditional variance of an INARMA(p,q) process of ??=? ???????
?
?=1 +
??+? ???????
?
?=1 , can be written as follows, where all the covariance terms, to be
found later, have been summarized using the expression ??cov? terms?:
var????= var??1????1?+?+ var?????????+ var????
+var??1????1?+?+ var?????????+?cov? terms
Equation 3.B1
var????=?1
2var????1?+?1?1??1??????1?+?+??
2var??????
+???1???????????+??
2 +?1
2??
2 +?1?1??1???+?+??
2??
2
+???1??????+?cov? terms
Equation 3.B2
Hence, assuming stationarity of the process, we have:
var????=
1
1?? ??
2?
?=1
??
1 +? ??
?
?=1
1?? ??
?
?=1
? ??1???
?
?=1
+? ??1???
?
?=1
????
?+?1 +? ??
2
?
?=1
???
2 +?cov? terms?
Equation 3.B3
Next, we focus on finding the covariance terms. There are two types of covariance:
1. The covariance between {??}s at different lags: cov(??,??) for ???.
2. The covariance between {??}s and {??}s: cov(??,??) for ???.
For ?>?, the random disturbance terms are independent of previous
observations.
M.Mohammadipour, 2009, Appendix 3.B 275
The first group of covariance terms is given by:
first Cov = 2cov??1????1,?2????2?+ 2cov??1????1,?3????3?+?
+2cov??1????1,????????
+2cov??2????2,?3????3?+ 2cov??2????2,?4????4?+?
+ 2cov??2????2,????????+?
+ 2cov????2?????+2,???1?????+1?+ 2cov????2?????+2,????????
+ 2cov????1?????+1,????????
first Cov = 2?1?2?1 + 2?1?3?2 +?+ 2?1?????1
+2?2?3?1 + 2?2?4?2 +?+ 2?2?????2 +?
+2 ???2???1?1 + 2???2???2
+2???1???1
first Cov = 2? ???+1?1
??1
?=1
+ 2? ???+2?2
??2
?=1
+?+ 2 ? ???+???1???1
?????1?
?=1
first Cov = 2? ? ???+???
???
?=1
??1
?=1
Equation 3.B4
Now we focus on the second type of covariance which is cov(??,??) for ???. Here
there are three cases:
1. ??
Each of these cases is discussed here separately.
M.Mohammadipour, 2009, Appendix 3.B 276
1. cov(??,??) when ??
Figure 3.B3 in an INARMA(p,q) process when
Second Cov 3 = 2cov??1????1,?1????1?+ 2cov??1????1,?2????2?+?
+2cov??1????1,????????
+2cov??2????2,?2????2?+ 2cov??2????2,?3????3?+?
+ 2cov??2????2,????????+?
+2cov????1?????+1,???1?????+1?
+ 2cov????1?????+1,????????
+2cov????????,????????
Then, the first terms in each of the rows are summed vertically, with the remaining
terms in the rows being summed horizontally (see Figure 3.B3).
Second Cov 3 = 2??
2? ???
?
?=1
+ 2? ?1?????1
??
?
?=2
+ 2? ?2?????2
??
?
?=3
+?
+2? ???1?????(??1)
??
?
?=?
),cov( ji ZY qp ?
???1
???1
???1
???2
?
???1
????+1
???1
????
???2
???2
?
???2
????+1
???2
????
????
????
? ?
M.Mohammadipour, 2009, Appendix 3.B 279
Second Cov 3 = 2??
2? ???
?
?=1
+ 2? ? ???????
??
?
?=?+1
??1
?=1
Equation 3.B7
The Equation 3.B7 can be written as 2??
2? ???
?
?=1 + 2? ? ???????
???
?=?+1
?
?=1 for
generalization purposes because for ?=? the second summation would be (? )??=?+1
which has zero terms.
Therefore, based on the results of Equation 3.B5, Equation 3.B6 and Equation 3.B
7, the second group of covariance is given by:
Second Cov = 2 ? ???
min (?,?)
?=1
??
2 + 2 ? ? ???????
??
?
?=?+1
min (?,?)
?=1
Equation 3.B8
Finally, from the Equation 3.B3, Equation 3.B4, and Equation 3.B8, the
unconditional variance of an INARMA(p,q) process is:
var????=
1
1?? ??
2?
?=1
??
1 +? ??
?
?=1
1?? ??
?
?=1
? ??1???
?
?=1
+? ??1???
?
?=1
????
+?1 +? ??
2
?
?=1
???
2 + 2? ? ???+???
???
?=1
??1
?=1
?+2 ? ???
min (?,?)
?=1
??
2 + 2 ? ? ???????
??
?
?=?+1
min (?,?)
?=1
var????=
??
1?? ??
2?
?=1
?
1 +? ??
?
?=1
1?? ??
?
?=1
? ??1???
?
?=1
+? ??1???
?
?=1
?
+
??
2
1?? ??
2?
?=1
?1 +? ??
2
?
?=1
+ 2 ? ???
min (?,?)
?=1
+
2? ? ???+???
???
?=1
??1
?=1 + 2? ? ???????
???
?=?+1
min (?,?)
?=1
1?? ??
2?
?=1
Equation 3.B9
M.Mohammadipour, 2009, Appendix 3.C 280
Appendix 3.C The CrossCovariance Function between Y and Z for
an INARMA(p,q) Model
The crosscovariance function, ??
??, is the covariance between ? and ? at lag ? and
is defined by ??
??=?(??????). Therefore, the crosscovariance at lag zero is given
by:
?0
??= cov???,???= cov?? ???????
?
?=1
+??+? ???????
?
?=1
,???
Considering the fact that the innovation terms are independent of previous
observations, we have:
?0
??= cov???,???=???????=??
2
The crosscovariance at lag one can be obtained from:
?1
??= cov???,???1?
= cov??1????1 +?+???????+??+?1????1 +?+???????,???1?
= cov??1????1,???1?+ cov??1????1,???1?=?1??
2 +?1??
2
The crosscovariance at lag ? (0????) is given by:
??
??= cov???,?????
= cov??1????1 +?+???????+??+?1????1 +?+???????,?????
= cov??1????1,?????+?+ cov????1???????1?,?????
+cov????????,?????+????
2
=????
2 +?1???1
?? +?+???1?1
??+????
2
??
??=????
2 +? ?????
??
?
?=1
Equation 3.C1
M.Mohammadipour, 2009, Appendix 3.D 281
Appendix 3.D Over Lead Time Aggregation of an INAR(1) Model
In this appendix, we show how to derive the conditional first and second moments of
a lead time aggregated PoINAR(1) process (??=??
2 =?). The aggregated process
over lead time can be written as:
? ??+?
?+1
?=1
=?????+??+1?+?????+1 +??+2?+?+?????+?+??+?+1?
=?????+??+1?+????????+??+1?+??+2?+?+?????+?+??+?+1?
=?????+??+1?+??
2???+????+1 +??+2?+?
+???+1???+?
????+1 +?
??1???+2 +?+????+?+??+?+1?
It can be simplified as
? ??+?
?+1
?=1
=?????+?
2???+?+?
?+1????
+???+1 +????+1 +?+?
????+1?
+???+2 +????+2 +?+?
??1???+2?+?+???+?+????+? +??+?+1
Equation 3.D1
The conditional expected value of the aggregated process is given by:
??? ??+?
?+1
?=1
???=????+?
2??+?+?
?+1???+??+??+?+?
???
+??+??+?+???1??+?+??+???+?
??? ??+?
?+1
?=1
???=
?(1???+1)
1??
??+
?
1??
??1???+1?+?1??? +?+?1????
Therefore, we have:
??? ??+?
?+1
?=1
???=
?(1???+1)
1??
??+
?
1??
???+ 1??? ??
?+1
?=1
?
Equation 3.D2
Now we want to find the conditional variance of the aggregated process. We know
that:
M.Mohammadipour, 2009, Appendix 3.D 282
cov?????,?????=???????2??????????????=????var(?)
Equation 3.D3
Hence, we have cov??????,?
?????=?
????.
The variance of the Equation 3.D1 given ?? is:
var?? ??+?
?+1
?=1
???= var??????+ var??
2????+?+ var??
?+1????
+2cov?????,?
2????+ 2cov?????,?
3????+?+ 2cov?????,?
?+1????
+ 2cov??2???,?
3????+ 2cov??
2???,?
4????+?+ 2cov??
2???,?
?+1????
+?+ 2cov????1???,?
?????+ 2cov??
??1???,?
?+1????
+2cov??????,?
?+1????
+var???+1?+ var?????+1?+ var??
2???+1?+?+ var??
????+1?
+2cov???+1,????+1?+ 2cov???+1 ,?
2???+1?+?+ 2cov???+1 ,?
????+1?
+2cov?????+1 ,?
2???+1?+ 2cov?????+1 ,?
3???+1?+?
+ 2cov?????+1 ,?
????+1?
+?+ 2cov????2???+1 ,?
??1???+1?+ 2cov??
??2???+1 ,?
????+1?
+2cov????1???+1 ,?
????+1?
+var???+2?+ var?????+2?+?+ var??
??1???+2?
+2cov???+2,????+2?+ 2cov???+2 ,?
2???+2?+?+ 2cov???+2 ,?
??1???+2?
+2cov?????+2 ,?
2???+2?+ 2cov?????+2 ,?
3???+2?+?
+ 2cov?????+2 ,?
??1???+2?
+?+ 2cov????3???+2 ,?
??2???+2?+ 2cov??
??3???+2 ,?
??1???+2?
+2cov????2???+2 ,?
??1???+2?+?
+var???+??1?+ var?????+??1?+ var??
2???+??1?
+2cov???+??1 ,????+??1?+ 2cov???+??1 ,?
2???+??1?
+ 2cov?????+??1 ,?
2???+??1?
+var???+?+ var?????+?
+2cov???+?,????+?
+var???+?+1?
Since ?? is fixed, cov??
????,?
?????=?
???var????= 0, using Equation 3.D3.
M.Mohammadipour, 2009, Appendix 3.D 283
var?? ??+?
?+1
?=1
???=?(1??)?????+?
2(1??2)?????+?+?
?+1(1???+1)?????
+?+??2?+??1?????+??4?+?2?1??2???+?+??2??+???1??????
+2??+?2 +?+???+ 2??3 +?4 +?+??+1??+?+ 2??2??3 +?2??2??
+ 2??2??1??
+?+??2?+??1?????+??4?+?2?1??2???+?
+??2??2?+???1?1????1???
+2??+?2 +?+???1??+ 2??3 +?4 +?+???+?+ 2??2??5 +?2??4??
+ 2??2??3??+?
+?+??2?+??1?????+??4?+?2?1??2???
+2??+?2??+ 2[?3]?
+?+??2?+??1?????
+2??
+?
The above result can be summarized to:
var?? ??+?
?+1
?=1
???=?(1??)??+?
2(1??2)??+?+?
?+1(1???+1)??
+?+????+??2??+?+?????
+2??+?2 +?+???+ 2??3 +?4 +?+??+1??+?+ 2??2??3 +?2??2??
+ 2??2??1??
+?+????+??2??+?+????1??
+2??+?2 +?+???1??+ 2??3 +?4 +?+???+?+ 2??2??5 +?2??4??
+ 2??2??3??+?
+?+????+??2??
+2??+?2??+ 2[?3]?
+?+????
+2??
+?
var?? ??+?
?+1
?=1
???=?(1??)??+?
2(1??2)??+?+?
?+1(1???+1)??
+??1 +?+?+???+??1 +?+?+???1?+?+??1 +?+?2?+??1 +??
+?
+2??+?2 +?+???+ 2??3 +?4 +?+??+1??+?+ 2??2??3 +?2??2??
+ 2??2??1??
+2??+?2 +?+???1??+ 2??3 +?4 +?+???+?+ 2??2??5 +?2??4??
+ 2??2??3??+?
+2??+?2??+ 2[?3]?
+2??
M.Mohammadipour, 2009, Appendix 3.D 284
var?? ??+?
?+1
?=1
???=??? ?
??1????
?+1
?=1
+
?
1??
???+ 1??? ??
?+1
?=1
?
+2???1 +?+?+??1?+ 2?3??1 +?+?+??2?+?
+2?2??3??1 +??+ 2?2??1?
+2???1 +?+?+??2?+ 2?3??1 +?+?+??3?+?
+2?2??5??1 +??+ 2?2??3?+?
+2???1 +??+ 2?3?
+2??
So
var?? ??+?
?+1
?=1
???=??? ?
??1????
?+1
?=1
+
?
1??
???+ 1??? ??
?+1
?=1
?
+
2??
1??
?1????+
2?3?
1??
?1???1?+?+
2?2??3?
1??
?1??2?+
2?2??1?
1??
[1??]
+
2??
1??
?1???1?+
2?3?
1??
?1???2?+?+
2?2??5?
1??
?1??2?
+
2?2??3?
1??
?1???
+?
+
2??
1??
?1??2?+
2?3?
1??
?1???
+
2??
1??
?1???
Summing the above expressions vertically results in:
var?? ??+?
?+1
?=1
???=??? ?
??1????
?+1
?=1
+
?
1??
???+ 1??? ??
?+1
?=1
?
+
2??
1??
???(?+?2 +?+??)?+
2?3?
1??
????1??(?+?2 +?+??1)?
+?+
2?2??3?
1??
?2?(?+?2)?+
2?2??1?
1??
[1??]
Finally, the conditional variance of the aggregated process is:
M.Mohammadipour, 2009, Appendix 3.D 285
var?? ??+?
?+1
?=1
???=??? ?
??1????
?+1
?=1
+
?
1??
???+ 1??? ??
?+1
?=1
?
+
2?
1??
? ?2??1?????+ 1??
?(1?????+1)
1??
?
?
?=1
Equation 3.D4
M.Mohammadipour, 2009, Appendix 4.A 286
Appendix 4.A Infinite Autoregressive Representation of an
INARMA(p,q) Model
It will be shown in this appendix that the Infinite AutoRegressive Representation
(IARR) of an INARMA(p,q) process (??=? ???????
?
?=1 +??+? ???????
?
?=1 ) is
as follows:
??=??+? ? ???????
??
?=1
?
?=1
Equation 4.A1
where
??=
?
?
?
?
?
?? ????
?
?=1
+ 1 0 ?
?
Equation 4.A2
Then ?? can be expressed in terms of {????}?=0
? as:
??=???? ? ???????
??
?=1
?
?=1
Equation 4.A3
where ?? is given by Equation 4.A2.
When expressing the INARMA(p,q) process in terms of {????}?=1
? , it should be noted
that because ???+????(?+?)??, these coefficients cannot be added.
Therefore, another summation over the number of ???? terms has been used (? )
??
?=1 .
First, it is shown that the expression for ?? is correct when ?>?. An example will
motivate the general case. Consider an INARMA(2,1) process of:
??=?1????1 +?2????2 +??+?1????1
M.Mohammadipour, 2009, Appendix 4.A 287
We are interested to find the number of ???3 terms in the IARR of this process. It can
be seen that:
???1 =???1??1????2??2????3??1????2
???2 =???2??1????3??2????4??1????3
???3 =???3??1????4??2????5??1????4
The INARMA(2,1) process can be written as:
??=?1????1 +?2????2 +??+?1?????1??1????2??2????3??1????2?
=?1????1 +?2????2 +??+?1????1??1?1????2??2?1????3??1
2
?(???2??1????3??2????4??1????3)
=?1????1 +?2????2 +??+?1????1??1?1????2??2?1????3??1
2????2
+?1?1
2????3 +?2?1
2????4 +?1
3?(???3??1????4??2????5??1????4)
It can be seen that ???3 terms come from ???1, ???2, and ???3 terms. It can be easily
shown that ???4 terms come from ???2, ???3, and ???4 terms. Using the same
argument, it can be shown that when ?>?, ???? terms come from {????+?}?=0
?
terms,
with each of them producing one ????.
Now, we want to check if, for ?>?, ?? is in fact ? ????
?
?=1 (??=???1 +???2 +?+
????). We know that ?? is the number of ???? in IARR of an INARMA(p,q) process,
so:
???1 =
the number of ????+1
???2 =
the number of ????+2
?
????=
the number of ????+?
which come from which come from which come from
? ? ? ?
{????+1+?}?=0
?
{????+2+?}?=0
?
? {????+?+?}?=0
?
The above terms can be written as follows:
M.Mohammadipour, 2009, Appendix 4.A 288
????+1 ????+2 ? ????+?
????+2 ????+3 ? ????+?+1
????+3 ????+4 ? ????+?+2
? ? ?
????+?+1 ????+?+2 ? ????+?+?
If we look at the elements that produce ???? (which is the set {????+?}?=0
? =
{????,????+1,?,????+?}) we can see that:
1. the number of ???? comes from {????+1,????+2,?,????+?} and as it can be
seen in the above table, these are shown by the first rectangle.
2. the number of ????+1 comes from {????+2,????+3,?,????+?+1} and as it can
be seen in the above table, these are shown by the second rectangle.
3. the number of ????+2 comes from {????+3,????+4,?,????+?+2} and as it can
be seen in the above table, these are shown by the third rectangle.
4. the number of ????+? comes from {????+?+1,????+?+2,?,????+?+?} and as
it can be seen in the above table, these are shown by the last rectangle.
Therefore, ??=???1 +???2 +?+????.
For the case of 0 2. As previously mentioned, when ??2, the number of ??
in ??+? is equal to the number of ?? in the two previous terms plus one.
The corresponding coefficient for ?? in each of {??+?}?=1
?+1 (say ??+3) is obtained from
?1 thinned the coefficient of ?? in the previous term (in this case ??+2) and ?2
thinned the coefficient of ?? in the next previous term (in this case ??+1). These
coefficients are shown in Table 6.A1.
Table 6.A1 Coefficients of in each of for an INAR(2) model
?= 1,?= 1 ?11
1 =?1
?= 2,?= 1,2
?12
1 =?1
2
?22
1 =?2
? ?
?=?+ 1,?= 1,?,??+1
1
?1(?+1)
1 =?1
?+1
?2(?+1)
1 =?1
??1?2
?
The number of ???1 in each of {??+?}?=1
?+1 is also equal to the sum of the number of ??
in the two previous terms (??+??1,??+??2). Therefore, ??
2 =???1
2 +???2
2 for ?> 1.
But, for ??1 this is equal to the number of ???1 in the two previous terms plus one.
tY 11
?
??
l
jjtY }{
M.Mohammadipour, 2009, Appendix 6.A 295
The corresponding coefficient for ???1 in each of {??+?}?=1
?+1 (say ??+3) is obtained
from ?1 thinned the coefficient of ???1 in the previous term (in this case ??+2) and ?2
thinned the coefficient of ???1 in the next previous term (in this case ??+1). These
coefficients are shown in Table 6.A2.
Table 6.A2 Coefficients of in each of for an INAR(2) model
?= 1,?= 1 ?11
2 =?2
?= 2,?= 1 ?12
2 =?1?2
?= 3,?= 1,2
?13
2 =?1
2?2
?23
2 =?1?2
? ?
?=?+ 1,?= 1,?,??+1
2
?1(?+1)
2 =?1
??2
?2(?+1)
2 =?1
??1?2
?
It can be seen from Equation 6.A2 that because the process is an autoregressive
process of order two, the number of ??+?? increases in each of {??+?}?=1
?+1 . For the
INAR(2) case, this number, shown by ??
3, can be obtained from the number of ??+??
in the two previous terms plus one because each of {??+?}?=1
?+1 has one ??+? as well.
The corresponding coefficient for each ??+??, shown by ??
3 , is ?1 thinned the
coefficients of ??+?? in the previous term (??+??1) and ?2 thinned the coefficients of
??+?? in the next previous term (??+??2). The coefficient for ??+? in each of
{??+?}?=1
?+1 is one. ?+?? is the subscripts of innovation terms in each of {??+?}?=1
?+1
which from Equation 6.A2 it can be seen that ?? is given by:
??=?
??(??2) for 1??????2
3
??(??1) for ???2
3 2, ?? equals to all the subscripts included in the two
1?tY 11
?
??
l
jjtY }{
M.Mohammadipour, 2009, Appendix 6.A 296
previous {??+?}?=1
?+1 plus ?. This means that ?? is as shown in Equation 6.A4.
The corresponding coefficient for each ??+?? and the subscript of innovation terms,
?+?? are shown in Table 6.A3.
Based on Equation 6.A3, the conditional expected value of the aggregated process
is:
??? ??+?
?+1
?=1
???=?? ? ??
1
??
1
?=1
?+1
?=1
???+?? ? ??
2
??
2
?=1
?+1
?=1
????1 +?? ? ??
3
??
3
?=1
?+1
?=1
??
Equation 6.A5
Table 6.A3 Coefficients of in each of for an INAR(2) model
?= 1
?= 1,?,?1
3
where ?1
3 = 1
?11
3 = 1 ?11 = 1
?= 2
?= 1,?,?2
3
where ?2
3 = 2
?12
3 =?1
?22
3 = 1
?12 = 1
?22 = 2
?= 3
?= 1,?,?3
3
where ?3
3 = 4
?13
3 =?1
2
?23
3 =?1
?33
3 =?2
?43
3 = 1
?13 = 1
?23 = 2
?33 = 1
?43 = 3
? ? ?
?=?+ 1
?= 1,?,??+1
3
?1(?+1)
3 =?1
?
?2(?+1)
3 =?1
??1
?
?1(?+1) = 1
?2??+1?= 2
?
ijkt
Z ?
1
1
?
??
l
jjtY }{
M.Mohammadipour, 2009, Appendix 6.B 297
Appendix 6.B Lead Time Forecasting for an INARMA(1,2) Model
For the INARMA(1,2) process of ??=?????1 +??+?1????1 +?2????2, the
cumulative ? over lead time ? is given by:
? ??+?
?+1
?=1
=??+1 +??+2 +?+??+?+1 =?????+??+1 +?1???+?2????1?
+?????+1 +??+2 +?1???+1 +?2????+?+
+?????+?+??+?+1 +?1???+?+?2???+?1?
Equation 6.B1
In order to find the conditional expectation of the aggregated process, Equation 6.B1
should be expressed in terms of ??.
? ??+?
?+1
?=1
=?????+??+1 +?1???+?2????1?
+??2???+????+1 +??1???+??2????1 +??+2 +?1???+1 +?2????
+??3???+?
2???+1 +?
2?1???+?
2?2????1 +????+2 +??1???+1
+??2???+??+3 +?1???+2 +?2???+1?+?
Equation 6.B2
The above expression is not an infinite series but a finite series where the remaining
terms can be obtained by repeated substitution. The above equation can be written as:
? ??+?
?+1
?=1
=? ? ??
1 ???
??
1
?=1
?+1
?=1
+? ? ??
2 ???+??
??
2
?=1
?+1
?=1
Equation 6.B3
where ??
1 is the number of ?? terms in each of {??+?}?=1
?+1 and ??
1 is the corresponding
coefficient for each ??. ??
2 is the number of ??+?? terms in each of {??+?}?=1
?+1 , ??
2 is
the corresponding coefficient for each ??+??. Each of these terms is explained below.
It can be seen from Equation 6.B2 that because the process is an integer
autoregressive of order one, each of {??+?}?=1
?+1 only has one ?? and therefore ??
1 = 1
(because the number of ?? in each of {??+?}?=1
?+1 is equal to the number of ?? in the
previous term (??+??1)). Therefore, the corresponding coefficient for ?? in each of
M.Mohammadipour, 2009, Appendix 6.B 298
{??+?}?=1
?+1 (say ??+2) is obtained from ? thinned the coefficient of ?? in the previous
term (in this case ??+1). Therefore, ??
1 =??. These coefficients are shown in Table
6.B1.
Table 6.B1 Coefficients of in each of for an INARMA(1,2) model
?= 1,?= 1 ?11
1 =?
?= 2,?= 1 ?12
1 =?2
? ?
?=?+ 1,?= 1 ?1(?+1)
1 =??+1
It can be seen from Equation 6.B2 that because the process has an autoregressive
component of order one and also a moving average component of order two, the
number of ??+?? increases in each of {??+?}?=1
?+1 . Each of {??+?}?=1
?+1 has three ??+??
and also all the ??+?? terms of the previous ? element (??+??1). Therefore, this
number, shown by ??
2, can be obtained from ???1
2 + 3. The same argument applies
for the INARMA(p,q) case where the number of ??+?? is equal to the number of
them in the ? previous terms plus ?+ 1 (see section 6.3.4). ?+?? is the subscripts
of innovation terms in each of {??+?}?=1
?+1 which from Equation 6.B2 it can be seen
that ?? is given by:
??=?
??(??1) for ?= 1,?,???1
2
?,??1,??2 for ?=???1
2 + 1,?,??
2
? for ?= 1,?,?+ 1
Equation 6.B4
For example, for ?= 1, the subscript of innovation terms is ??1 =?
1 ?= 1
0 ?= 2
?1 ?= 3
?,
because as can be seen from Equation 6.B1, there are three innovation terms in the
expression for ??+1 which are {??+1,??,???1}. For ?> 1, the ??+??s come from the
innovation terms included in the previous {??+?}?=1
?+1 plus {?,??1,??2}. This means
that ?? is as shown in Equation 6.B4.
The corresponding coefficient for each ??+?? and the subscript of innovation terms,
?+?? are shown in Table 6.B2.
tY 11
?
??
l
jjtY }{
M.Mohammadipour, 2009, Appendix 6.B 299
Based on Equation 6.B3, the conditional expected value of the aggregated process
is:
??? ??+?
?+1
?=1
???=?? ? ??
1
??
1
?=1
?+1
?=1
???+?? ? ??
2
??
2
?=1
?+1
?=1
??
Equation 6.B5
Table 6.B2 Coefficients of in each of for an INARMA(1,2) model
?= 1
?= 1,?,?1
2
where ?1
2 = 3
?11
2 = 1
?21
2 =?1
?31
2 =?2
?11 = 1
?21 = 0
?31 =?1
?= 2
?= 1,?,?2
2
where ?2
2 = 6
?12
2 =?
?22
2 =??1
?32
2 =??2
?42
2 = 1
?52
2 =?1
?62
2 =?2
?12 = 1
?22 = 0
?32 =?1
?42 = 2
?52 = 1
?62 = 0
?= 3
?= 1,?,?3
2
where ?3
2 = 9
?13
2 =?2
?23
2 =?2?1
?33
2 =?2?2
?43
2 =?
?53
2 =??1
?63
2 =??2
?73
2 = 1
?83
2 =?1
?93
2 =?2
?13 = 1
?23 = 0
?33 =?1
?43 = 2
?53 = 1
?63 = 0
?73 = 3
?83 = 2
?93 = 1
? ? ?
ijkt
Z ?
1
1
?
??
l
jjtY }{
M.Mohammadipour, 2009, Appendix 6.C 300
Appendix 6.C Lead Time Forecasting for an INARMA(p,q) Model
In order to find the conditional mean of the overleadtimeaggregated process, we
need to express the aggregated INARMA(p,q) process in terms of the last ?
observations (????+1,????+2,?,???1,??). The aggregated process is given by:
? ??+?
?+1
?=1
=??+1 +??+2 +?+??+?+1
Equation 6.C1
Each of the {??+?}?=1
?+1 in the RHS of the above equation needs to be expressed in
terms of {????+1}?=1
?
by repeated substitution of ??+? in Equation 350. Because the
autoregressive order of the process is ?, ??+? can be expressed in terms of ? previous
observations by ?1???+??1 +?+?????+???. Now, if ????(??1), as
mentioned in Appendix 6.A, there is one ???(??1) when we express the ?th
observation in the RHS of the Equation 6.C1 (??+?) without any need for repeated
substitution. Repeated substitution of (??+1,?,??+?????1?) by their ? previous
observations would result in obtaining more ???(??1). Therefore, in total, the number
of ???(??1) in each of {??+?}?=1
?+1 when ????(??1) is equal to the number of
???(??1) in its ? previous observations plus one.
However, as explained in Appendix 6.A, when ?>??(??1), each ??+? from
Equation 6.C1 should be substituted by Equation 350 in order to reach ???(??1),
and the number of ???(??1) in each of {??+?}?=1
?+1 would be equal to the number of
???(??1) in its ? previous observations.
For ????(??1), the corresponding coefficient of ???(??1) in the ?th observation
in the RHS of the Equation 6.C1 (??+?) is ??+(??1) because:
??+?=?1???+??1 +?+??+(??1)????(??1) +?+?????+???+??+?+? ?????+???
?
?=1
For other ???(??1) the coefficient in each of {??+?}?=1
?+1 is ?? thinned the coefficient of
???(??1) in the ?th previous observation for ?= 1,?,?.
M.Mohammadipour, 2009, Appendix 6.C 301
For ?>??(??1), again, the coefficient of ???(??1) in each of {??+?}?=1
?+1 is ??
thinned the coefficient of ???(??1) in the ?th previous observation for ?= 1,?,? (the
difference with the previous case is that we do not have ??+(??1)).
Now we come back to Equation 6.C1 to find the ? terms in each of {??+?}?=1
?+1 in the
RHS of the equation when they expressed in terms of {????+1}?=1
?
. As the process
has a moving average component of order ?, each {??+?}?=1
?+1 has ?+ 1 innovation
terms {??+?,??+??1,?,??+???}. However, as mentioned in Appendix 6.B, by
repeated substitution each {??+?}?=1
?+1 can be expressed in terms of ? previous
observations, each also with ?+ 1 innovation terms.
Therefore, the total number of innovation terms in each of {??+?}?=1
?+1 is equal to the
number of innovation terms in the ? previous observations, plus ?+ 1. The
corresponding coefficients for the ?+ 1 terms {??+?,??+??1,?,??+???} are
{1,?1,?,??}, respectively. For the innovation terms that come from the ? previous
observations, coefficients would be ?? thinned the coefficient of ??+?? in the ?th
previous observation for ?= 1,?,?.
?+?? denotes the subscript of ? for each ?,? (?= 1,?,?+ 1 and ?= 1,?,??
?+1
).
As previously mentioned, each {??+?}?=1
?+1 has ?+ 1 innovation terms
{??+?,??+??1,?,??+???}. Therefore, the subscripts for the last ?+ 1 innovation
terms in each {??+?}?=1
?+1 are {???,??1,?,?}. This is shown in Table 6.C1 by
?=???1
?+1 + 1,?,???1
?+1 +??
?+1
.
The other subscripts of innovation terms in each of {??+?}?=1
?+1 simply are the
subscripts of the innovation terms of ? previous observations.
As a result, the aggregated process can be expressed as Equation 626 with the
associated parameters as defined in Table 6.C1.
M.Mohammadipour, 2009, Appendix 6.C 302
Table 6.C1 Parameters of the overleadtimeaggregated INARMA(p,q) model
fo
r
?
=
1
,?
,?
??
?=?
?? ????
??
?=1 + 1 ????(??1)
? ????
??
?=1 ?>??(??1)
? ??
?=
?
?
?
?
?
?
?
?
?
?
?
?
????(???)
? ?= 1,?,????
?
? ?
?1??(??1)
? ?=???2
? + 1,?,???2
? +???1
?
??+(??1) ?=???1
? + 1
?????(??1)
?
????(???)
? ?= 1,?,????
?
? ?
?1??(??1)
? ?=???2
? + 1,?,???2
? +???1
?
??>??(??1)
?
??
?+1 =?? ????
?+1?
?=1 ?+ (?+ 1)
??
?+1 =
?
?
?
?
?
????(???)
?+1 ?= 1,?,????
?+1
? ?
?1??(??1)
?+1 ?=???2
?+1 + 1,?,???2
?+1 +???1
?+1
??,?,?1 , 1 ?=???1
?+1 + 1,?,???1
?+1 +??
?+1
?
??=
?
?
?
?
?
{??(???)} ?= 1,?,????
?+1
? ?
{??(??1)} ?=? ????
?+1?
?=2 + 1,?, (? ????
?+1?
?=2 ) +???1
?+1
???,?,??1,? ?=? ????
?+1?
?=1 + 1,?,??
?+1
?
M.Mohammadipour, 2009, Appendix 8.A 303
Appendix 8.A The MSE of YW and CLS Estimates for INAR(1),
INMA(1) and INARMA(1,1) Processes
In this appendix, the two estimation methods used in this research are compared
using MSE. Earlier in chapter 8, these methods have been compared in terms of their
impact on forecast accuracy. The results for ?= 24, 36, 48, 96, 500 are presented as
follows.
Table 8.A1 MSE of YW and CLS estimates of
for an INAR(1) process
Parameters
YW CLS
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.5 0.0184 0.0167 0.0144 0.0097 0.0031 0.0219 0.0186 0.0154 0.0100 0.0031
?= 0.5,?= 0.5 0.0742 0.0495 0.0359 0.0165 0.0030 0.0721 0.0481 0.0352 0.0161 0.0030
?= 0.9,?= 0.5 0.1460 0.0700 0.0435 0.0129 0.0009 0.1120 0.0519 0.0333 0.0104 0.0008
?= 0.1,?= 1 0.0196 0.0154 0.0138 0.0094 0.0026 0.0230 0.0167 0.0147 0.0097 0.0026
?= 0.5,?= 1 0.0725 0.0450 0.0333 0.0144 0.0028 0.0707 0.0440 0.0324 0.0140 0.0028
?= 0.9,?= 1 0.1546 0.0756 0.0456 0.0134 0.0008 0.1145 0.0582 0.0350 0.0109 0.0007
?= 0.1,?= 3 0.0192 0.0152 0.0136 0.0084 0.0026 0.0223 0.0167 0.0145 0.0087 0.0026
?= 0.5,?= 3 0.0679 0.0445 0.0319 0.0137 0.0021 0.0658 0.0430 0.0312 0.0135 0.0021
?= 0.9,?= 3 0.1468 0.0725 0.0442 0.0119 0.0008 0.1105 0.0555 0.0339 0.0095 0.0007
?= 0.1,?= 5 0.0192 0.0155 0.0129 0.0088 0.0027 0.0222 0.0170 0.0137 0.0091 0.0027
?= 0.5,?= 5 0.0709 0.0446 0.0320 0.0142 0.0022 0.0689 0.0429 0.0313 0.0139 0.0022
?= 0.9,?= 5 0.1455 0.0712 0.0437 0.0125 0.0008 0.1095 0.0550 0.0333 0.0101 0.0007
Table 8.A2 Comparison of YW and CLS estimates of
for an INAR(1) process
Parameters
YW CLS
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.5 0.0375 0.0243 0.0179 0.0100 0.0023 0.0380 0.0245 0.0181 0.0100 0.0023
?= 0.5,?= 0.5 0.1072 0.0712 0.0473 0.0224 0.0034 0.1030 0.0692 0.0464 0.0220 0.0034
?= 0.9,?= 0.5 3.9136 1.8446 1.1551 0.3370 0.0224 2.9900 1.3812 0.8930 0.2709 0.0206
?= 0.1,?= 1 0.0867 0.0608 0.0452 0.0272 0.0060 0.0898 0.0625 0.0463 0.0275 0.0060
?= 0.5,?= 1 0.3934 0.2350 0.1580 0.0713 0.0118 0.3852 0.2271 0.1545 0.0697 0.0118
?= 0.9,?= 1 16.5122 7.6779 4.7735 1.3540 0.0790 12.1860 5.8605 3.6707 1.1016 0.0721
?= 0.1,?= 3 0.3945 0.3001 0.2438 0.1413 0.0370 0.4275 0.3185 0.2528 0.1441 0.0372
?= 0.5,?= 3 2.8535 1.6850 1.2493 0.5244 0.0824 2.7658 1.6303 1.2121 0.5149 0.0822
?= 0.9,?= 3 132.6483 66.7297 40.2510 10.8786 0.6995 99.9406 51.2825 30.9385 8.7135 0.6386
?= 0.1,?= 5 0.8928 0.6958 0.5521 0.3577 0.0968 0.9817 0.7426 0.5772 0.3659 0.0972
?= 0.5,?= 5 7.4580 4.6569 3.4168 1.4914 0.2219 7.2026 4.4779 3.3367 1.4616 0.2204
?= 0.9,?= 5 368.6249 180.0181 111.2582 31.8673 2.0634 278.5108 139.5061 84.8648 25.7175 1.8826
?
?
M.Mohammadipour, 2009, Appendix 8.A 304
Table 8.A3 Comparison of YW and CLS estimates of for an INMA(1) process
Parameters
YW CLS
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.5 0.0516 0.0421 0.0349 0.0193 0.0039 0.0251 0.0216 0.0171 0.0097 0.0028
?= 0.5,?= 0.5 0.1349 0.1008 0.0843 0.0493 0.0110 0.1147 0.0781 0.0721 0.0436 0.0211
?= 0.9,?= 0.5 0.2161 0.1374 0.1024 0.0604 0.0121 0.2683 0.2008 0.1647 0.1131 0.0518
?= 0.1,?= 1 0.0459 0.0364 0.0310 0.0166 0.0040 0.0236 0.0179 0.0152 0.0096 0.0029
?= 0.5,?= 1 0.1233 0.0939 0.0822 0.0469 0.0106 0.1038 0.0805 0.0673 0.0417 0.0198
?= 0.9,?= 1 0.2010 0.1485 0.1019 0.0548 0.0117 0.2752 0.2180 0.1726 0.1182 0.0517
?= 0.1,?= 3 0.0468 0.0376 0.0297 0.0179 0.0038 0.0205 0.0172 0.0139 0.0096 0.0026
?= 0.5,?= 3 0.1245 0.0960 0.0784 0.0474 0.0102 0.1110 0.0822 0.0696 0.0436 0.0207
?= 0.9,?= 3 0.2141 0.1446 0.1156 0.0511 0.0113 0.3103 0.2391 0.2041 0.1321 0.0592
?= 0.1,?= 5 0.0517 0.0400 0.0343 0.0154 0.0038 0.0214 0.0165 0.0151 0.0082 0.0027
?= 0.5,?= 5 0.1222 0.0972 0.0829 0.0458 0.0095 0.1154 0.0902 0.0734 0.0427 0.0205
?= 0.9,?= 5 0.2081 0.1388 0.0996 0.0563 0.0120 0.3434 0.2682 0.2226 0.1598 0.0695
Table 8.A4 Comparison of YW and CLS estimates of for an INMA(1) process
Parameters
YW CLS
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.5 0.0364 0.0240 0.0194 0.0103 0.0021 0.0370 0.0238 0.0189 0.0098 0.0020
?= 0.5,?= 0.5 0.0599 0.0419 0.0280 0.0137 0.0025 0.0665 0.0456 0.0310 0.0160 0.0041
?= 0.9,?= 0.5 0.0739 0.0461 0.0305 0.0153 0.0025 0.0781 0.0541 0.0376 0.0193 0.0050
?= 0.1,?= 1 0.0830 0.0576 0.0467 0.0240 0.0057 0.0834 0.0562 0.0444 0.0220 0.0052
?= 0.5,?= 1 0.1637 0.1042 0.0851 0.0393 0.0074 0.1831 0.1152 0.0948 0.0448 0.0124
?= 0.9,?= 1 0.2442 0.1441 0.0993 0.0397 0.0065 0.2823 0.1737 0.1229 0.0540 0.0162
?= 0.1,?= 3 0.3812 0.2835 0.2296 0.1407 0.0371 0.3530 0.2541 0.1971 0.1166 0.0314
?= 0.5,?= 3 1.0334 0.7039 0.5304 0.2635 0.0507 1.1603 0.7754 0.5886 0.3137 0.1004
?= 0.9,?= 3 1.5474 0.9868 0.7012 0.2655 0.0430 1.9331 1.3063 0.9528 0.4364 0.1303
?= 0.1,?= 5 0.8853 0.6958 0.5688 0.3405 0.0880 0.7923 0.5717 0.4578 0.2725 0.0737
?= 0.5,?= 5 2.6253 1.8145 1.4464 0.6367 0.1212 3.0335 2.1586 1.6302 0.7749 0.2682
?= 0.9,?= 5 4.1058 2.4764 1.5930 0.7347 0.1130 5.8606 3.9471 2.7201 1.4349 0.4352
?
?
M.Mohammadipour, 2009, Appendix 8.A 305
Table 8.A5 Comparison of YW and CLS estimates of for an INARMA(1,1) process
Parameters
YW CLS
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.1,?= 0.5 0.2421 0.2299 0.2029 0.1588 0.0474 0.0498 0.0420 0.0334 0.0225 0.0136
?= 0.1,?= 0.9,?= 0.5 0.0590 0.0423 0.0326 0.0225 0.0085 0.0685 0.0576 0.0484 0.0329 0.0087
?= 0.5,?= 0.5,?= 0.5 0.1426 0.1129 0.0904 0.0458 0.0065 0.1035 0.0743 0.0586 0.0308 0.0056
?= 0.9,?= 0.1,?= 0.5 0.2587 0.1328 0.0806 0.0197 0.0011 0.1228 0.0654 0.0425 0.0129 0.0009
?= 0.1,?= 0.1,?= 1 0.2440 0.2409 0.1968 0.1470 0.0492 0.0490 0.0380 0.0299 0.0246 0.0136
?= 0.1,?= 0.9,?= 1 0.0666 0.0381 0.0336 0.0212 0.0081 0.0808 0.0643 0.0546 0.0337 0.0093
?= 0.5,?= 0.5,?= 1 0.1444 0.1108 0.0829 0.0412 0.0062 0.0850 0.0621 0.0478 0.0257 0.0051
?= 0.9,?= 0.1,?= 1 0.2593 0.1266 0.0744 0.0196 0.0011 0.1093 0.0571 0.0372 0.0111 0.0008
?= 0.1,?= 0.1,?= 5 0.2295 0.2187 0.2025 0.1606 0.0428 0.0434 0.0334 0.0275 0.0200 0.0127
?= 0.1,?= 0.9,?= 5 0.0606 0.0418 0.0317 0.0219 0.0082 0.1121 0.0965 0.0858 0.0631 0.0194
?= 0.5,?= 0.5,?= 5 0.1373 0.1032 0.0843 0.0392 0.0061 0.0618 0.0441 0.0327 0.0183 0.0058
?= 0.9,?= 0.1,?= 5 0.2611 0.1321 0.0758 0.0184 0.0012 0.1033 0.0554 0.0343 0.0101 0.0008
Table 8.A6 Comparison of YW and CLS estimates of
for an INARMA(1,1) process
Parameters
YW CLS
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.1,?= 0.5 0.0470 0.0407 0.0353 0.0226 0.0109 0.0266 0.0205 0.0178 0.0118 0.0067
?= 0.1,?= 0.9,?= 0.5 0.2646 0.1976 0.1597 0.0864 0.0158 0.4152 0.3694 0.3470 0.2715 0.1502
?= 0.5,?= 0.5,?= 0.5 0.1520 0.1420 0.1405 0.1259 0.0495 0.1197 0.1128 0.1042 0.0948 0.0831
?= 0.9,?= 0.1,?= 0.5 0.3237 0.2744 0.2342 0.1288 0.0189 0.0123 0.0100 0.0081 0.0063 0.0049
?= 0.1,?= 0.1,?= 1 0.0492 0.0333 0.0305 0.0214 0.0106 0.0243 0.0181 0.0173 0.0119 0.0068
?= 0.1,?= 0.9,?= 1 0.2876 0.2014 0.1632 0.0855 0.0163 0.4813 0.4255 0.3819 0.2880 0.1609
?= 0.5,?= 0.5,?= 1 0.1488 0.1407 0.1386 0.1196 0.0460 0.1468 0.1280 0.1240 0.1146 0.0895
?= 0.9,?= 0.1,?= 1 0.3443 0.2641 0.2224 0.1259 0.0164 0.0080 0.0075 0.0071 0.0063 0.0057
?= 0.1,?= 0.1,?= 5 0.0455 0.0356 0.0296 0.0213 0.0118 0.0181 0.0152 0.0137 0.0108 0.0070
?= 0.1,?= 0.9,?= 5 0.2685 0.2019 0.1575 0.0872 0.0149 0.6413 0.5912 0.5619 0.4657 0.2561
?= 0.5,?= 0.5,?= 5 0.1492 0.1453 0.1421 0.1215 0.0483 0.1797 0.1752 0.1676 0.1621 0.1180
?= 0.9,?= 0.1,?= 5 0.3464 0.2849 0.2332 0.1281 0.0184 0.0076 0.0075 0.0074 0.0074 0.0076
Table 8.A7 Comparison of YW and CLS estimates of for an INARMA(1,1) process
Parameters
YW CLS
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.1,?= 0.5 0.1007 0.0847 0.0731 0.0534 0.0124 0.0486 0.0315 0.0234 0.0136 0.0033
?= 0.1,?= 0.9,?= 0.5 0.0993 0.0634 0.0466 0.0213 0.0049 0.0953 0.0643 0.0509 0.0289 0.0151
?= 0.5,?= 0.5,?= 0.5 0.1929 0.1195 0.0785 0.0312 0.0045 0.1869 0.1295 0.0942 0.0485 0.0179
?= 0.9,?= 0.1,?= 0.5 3.2137 1.4902 0.8934 0.2528 0.0200 2.6707 1.3440 0.8699 0.2635 0.0230
?= 0.1,?= 0.1,?= 1 0.3580 0.3299 0.2642 0.1875 0.0484 0.1302 0.0933 0.0670 0.0396 0.0104
?= 0.1,?= 0.9,?= 1 0.3386 0.2362 0.1530 0.0743 0.0174 0.3246 0.2378 0.1694 0.1023 0.0597
?= 0.5,?= 0.5,?= 1 0.7878 0.4347 0.2709 0.1015 0.0160 0.7621 0.4629 0.3235 0.1661 0.0654
?= 0.9,?= 0.1,?= 1 14.3119 6.4062 3.7534 1.0360 0.0795 12.9766 6.1504 3.9367 1.1341 0.0898
?= 0.1,?= 0.1,?= 5 7.4699 6.9473 6.3751 4.8796 0.9735 1.6871 1.2283 0.9607 0.5643 0.1710
?= 0.1,?= 0.9,?= 5 6.4788 4.2740 2.8742 1.5911 0.3502 5.7193 4.0864 2.9201 2.0613 1.3153
?= 0.5,?= 0.5,?= 5 15.0720 9.3556 6.3524 2.3258 0.3487 15.1315 9.4722 6.8444 3.3042 1.3018
?= 0.9,?= 0.1,?= 5 325.6348 154.4483 89.1639 25.0749 2.1841 300.4573 163.8322 101.8301 30.2243 2.5295
?
?
?
M.Mohammadipour, 2009, Appendix 8.B 306
Appendix 8.B Impact of YW and CLS Estimates on Accuracy of
Forecasts using MASE
The comparison of YW and CLS estimates for INAR(1), INMA(1) and
INARMA(1,1) in terms of their impact on forecast accuracy using MASE is
presented in this appendix.
Table 8.B1 Forecast error comparison (YW and CLS) for INAR(1) series
Parameters
??????/??????? ??????/???????
?= 24 ?= 36 ?= 48 ?= 96 ?= 500 ?= 24
?= 0.1,?= 0.5 0.9888 0.9935 0.9975 0.9991 1.0000 0.9893
?= 0.5,?= 0.5 0.9983 1.0009 1.0001 1.0008 1.0002 1.0116
?= 0.9,?= 0.5 1.0837 1.0663 1.0483 1.0247 1.0038 1.1998
?= 0.1,?= 1 0.9858 0.9955 0.9973 0.9989 0.9999 0.9810
?= 0.5,?= 1 0.9942 0.9999 0.9993 1.0002 1.0001 1.0031
?= 0.9,?= 1 1.0593 1.0461 1.0364 1.0149 1.0012 1.1401
?= 0.1,?= 3 0.9916 0.9964 0.9979 0.9992 1.0000 0.9781
?= 0.5,?= 3 0.9955 1.0007 1.0015 0.9998 1.0000 0.9931
?= 0.9,?= 3 1.0462 1.0393 1.0265 1.0116 1.0005 1.1241
?= 0.1,?= 5 0.9914 0.9955 0.9979 0.9992 1.0000 0.9825
?= 0.5,?= 5 0.9956 0.9996 1.0001 1.0001 1.0000 1.0039
?= 0.9,?= 5 1.0500 1.0411 1.0292 1.0120 1.0006 
Table 8.B2 Forecast error comparison (YW and CLS) for INMA(1) series
Parameters
??????/???????
?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.5 1.0029 1.0012 1.0013 1.0004 1.0001
?= 0.5,?= 0.5 1.0043 1.0053 1.0030 1.0016 1.0001
?= 0.9,?= 0.5 1.0060 1.0074 1.0050 1.0027 0.9975
?= 0.1,?= 1 1.0011 1.0019 1.0011 1.0009 1.0005
?= 0.5,?= 1 1.0092 1.0075 1.0062 1.0053 1.0041
?= 0.9,?= 1 1.0099 1.0077 1.0083 1.0025 0.9971
?= 0.1,?= 3 1.0004 1.0010 1.0015 1.0008 1.0003
?= 0.5,?= 3 1.0025 1.0049 1.0032 1.0048 1.0020
?= 0.9,?= 3 1.0001 0.9991 1.0006 1.0035 1.0007
?= 0.1,?= 5 0.9993 1.0005 1.0006 1.0005 1.0001
?= 0.5,?= 5 1.0004 1.0001 1.0003 1.0020 1.0019
?= 0.9,?= 5 0.9933 0.9960 0.9972 0.9980 1.0018
M.Mohammadipour, 2009, Appendix 8.B 307
Table 8.B3 Comparison error comparison (YW and CLS) for INARMA(1,1) series
Parameters
??????/???????
?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.1,?= 0.5 0.9956 0.9984 1.0069 1.0072 1.0023
?= 0.1,?= 0.9,?= 0.5 1.0093 1.0250 1.0251 1.0212 1.0072
?= 0.5,?= 0.5,?= 0.5 1.0233 1.0419 1.0386 1.0246 1.0049
?= 0.9,?= 0.1,?= 0.5 1.1859 1.1255 1.0907 1.0368 1.0065
?= 0.1,?= 0.1,?= 1 1.0289 1.0316 1.0336 1.0293 1.0121
?= 0.1,?= 0.9,?= 1 1.0245 1.0364 1.0361 1.0283 1.0095
?= 0.5,?= 0.5,?= 1 1.0617 1.0558 1.0478 1.0281 1.0066
?= 0.9,?= 0.1,?= 1 1.1955 1.1250 1.0889 1.0295 1.0030
?= 0.1,?= 0.1,?= 5 1.0504 1.0474 1.0474 1.0473 1.0109
?= 0.1,?= 0.9,?= 5 1.0577 1.0643 1.0662 1.0625 1.0351
?= 0.5,?= 0.5,?= 5 1.0845 1.0751 1.0749 1.0415 1.0121
?= 0.9,?= 0.1,?= 5 1.3024 1.1553 1.1013 1.0309 1.0022
M.Mohammadipour, 2009, Appendix 8.C 308
Appendix 8.C CrostonSBA Categorization for INAR(1), INMA(1)
and INARMA(1,1)
The following tables show that the CrostonSBA categorization generally holds for
INAR(1), INMA(1), and INARMA(1,1) processes although it originally developed
for i.i.d. processes.
For INAR(1), when ?= 0.1,?= 0.5, ?= 0.5,?= 0.5, and ?= 0.1,?= 1, SBA
should outperform Croston based on the corresponding pvalue.
Table 8.C1 MSE of Croston and SBA with smoothing parameter 0.2 for INAR(1) series
Parameters
?????????? ??????
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.5 0.6782 0.6370 0.6204 0.6059 0.5888 0.6592 0.6248 0.6122 0.5986 0.5811
?= 0.5,?= 0.5 1.1889 1.1333 1.1098 1.0575 1.0550 1.1558 1.1027 1.0866 1.0333 1.0302
?= 0.9,?= 0.5 2.0661 2.1545 2.0588 2.0672 2.0163 2.3534 2.4621 2.3079 2.3119 2.2578
?= 0.1,?= 1 1.2918 1.2387 1.2419 1.2364 1.2016 1.2686 1.2244 1.2305 1.2236 1.1879
?= 0.5,?= 1 2.1306 2.0827 2.0617 2.0071 1.9784 2.1198 2.0755 2.0559 1.9977 1.9689
?= 0.9,?= 1 4.3286 4.1355 4.1565 4.0094 3.9999 5.4842 5.2874 5.2252 5.0125 4.9994
?= 0.1,?= 3 3.9531 3.8147 3.8414 3.7243 3.6340 4.0050 3.8476 3.8845 3.7627 3.6735
?= 0.5,?= 3 6.0937 5.8541 5.7937 5.7145 5.6107 6.3761 6.1316 6.0812 5.9983 5.8763
?= 0.9,?= 3 12.6744 12.5479 12.3444 12.0857 12.0343 21.6630 21.7462 21.4541 21.1452 21.0575
?= 0.1,?= 5 6.7050 6.3945 6.3156 6.1690 6.0777 6.9397 6.6147 6.5153 6.3704 6.2710
?= 0.5,?= 5 10.1928 9.6910 9.7351 9.3875 9.2467 11.1322 10.5665 10.6531 10.2523 10.0885
?= 0.9,?= 5 20.4023 20.8507 20.9265 20.1313 19.9123 48.8346 47.8702 46.9309 46.1660 44.8889
Table 8.C2 MSE of Croston and SBA with smoothing parameter 0.5 for INAR(1) series
Parameters
?????????? ??????
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.5 0.7352 0.7163 0.7050 0.6894 0.6711 0.6719 0.6559 0.6474 0.6334 0.6152
?= 0.5,?= 0.5 1.1981 1.1722 1.1599 1.1137 1.1063 1.1030 1.0661 1.0646 1.0190 1.0096
?= 0.9,?= 0.5 1.3146 1.3137 1.2846 1.2601 1.2360 3.1301 3.1514 2.9995 2.9740 2.9217
?= 0.1,?= 1 1.4453 1.4006 1.3999 1.4021 1.3642 1.3495 1.3189 1.3243 1.3199 1.2818
?= 0.5,?= 1 2.0781 2.0287 2.0136 1.9739 1.9389 2.1293 2.0815 2.0602 2.0147 1.9791
?= 0.9,?= 1 2.7022 2.5724 2.5546 2.4636 2.4352 9.9577 9.5920 9.5507 9.0077 9.0156
?= 0.1,?= 3 4.5453 4.4165 4.4184 4.2813 4.1936 4.8183 4.6422 4.6675 4.5205 4.4314
?= 0.5,?= 3 5.8520 5.6100 5.5535 5.4866 5.3744 7.7889 7.4908 7.4554 7.3408 7.1728
?= 0.9,?= 3 8.1248 7.7172 7.5610 7.3578 7.3587 68.5686 67.3196 66.2151 65.3240 64.8094
?= 0.1,?= 5 7.7797 7.4433 7.3601 7.1823 7.0591 9.0522 8.7136 8.5879 8.3941 8.2367
?= 0.5,?= 5 9.7481 9.2986 9.3367 9.0251 8.8548 15.7458 15.0617 15.1057 14.6590 14.3696
?= 0.9,?= 5 12.8141 12.8380 12.6929 12.3732 12.1335 186.8114 180.7881 178.0403 175.0332 170.3159
M.Mohammadipour, 2009, Appendix 8.C 309
For INMA(1), when ?= 0.1,?= 0.5, ?= 0.5,?= 0.5, ?= 0.9,?= 0.5, and ?=
0.1,?= 1, SBA should outperform Croston based on the corresponding pvalue.
Table 8.C3 MSE of Croston and SBA with smoothing parameter 0.2 for INMA(1) series
Parameters
?????????? ??????
?=?? ?=?? ?=?? ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.5 0.6682 0.6264 0.6169 0.5950 0.5826 0.6468 0.6148 0.6095 0.5876 0.5754
?= 0.5,?= 0.5 0.9637 0.9103 0.8507 0.8495 0.8195 0.9285 0.8867 0.8324 0.8315 0.8030
?= 0.9,?= 0.5 1.2077 1.1498 1.1031 1.0834 1.0584 1.1547 1.1164 1.0764 1.0562 1.0325
?= 0.1,?= 1 1.3089 1.2502 1.2256 1.2002 1.1860 1.2915 1.2344 1.2120 1.1869 1.1729
?= 0.5,?= 1 1.7781 1.6990 1.6832 1.6235 1.6365 1.7507 1.6803 1.6615 1.6031 1.6171
?= 0.9,?= 1 2.2710 2.1981 2.1295 2.0714 2.0319 2.2540 2.1770 2.1101 2.0549 2.0131
?= 0.1,?= 3 4.0167 3.8200 3.7213 3.6679 3.6364 4.0619 3.8592 3.7651 3.7018 3.6722
?= 0.5,?= 3 5.1518 4.9393 4.9167 4.7959 4.6883 5.2691 5.0550 5.0298 4.9097 4.8008
?= 0.9,?= 3 6.2893 6.1923 6.0164 5.8912 5.8008 6.5575 6.4307 6.2320 6.1184 6.0191
?= 0.1,?= 5 6.5944 6.5473 6.2777 6.1716 6.0266 6.8133 6.7225 6.4675 6.3628 6.2156
?= 0.5,?= 5 8.4587 8.1070 8.0842 7.9465 7.7757 8.9173 8.5772 8.5611 8.3625 0.5754
?= 0.9,?= 5 10.6487 10.1183 10.1251 9.8462 9.6391 11.4393 10.8493 10.8578 10.5863 0.8030
Table 8.C4 MSE of Croston and SBA with smoothing parameter 0.5 for INMA(1) series
Parameters
?????????? ??????
?=?? ?=?? ?= 48 ?=?? ?=??? ?=?? ?=?? ?=?? ?=?? ?=???
?= 0.1,?= 0.5 0.7150 0.7091 0.6930 0.6787 0.6620 0.6528 0.6465 0.6407 0.6232 0.6088
?= 0.5,?= 0.5 1.0583 1.0341 0.9870 0.9845 0.9506 0.9453 0.9228 0.8816 0.8793 0.8504
?= 0.9,?= 0.5 1.3103 1.2889 1.2476 1.2304 1.2033 1.1635 1.1542 1.1219 1.0987 1.0740
?= 0.1,?= 1 1.4580 1.4187 1.3924 1.3648 1.3498 1.3746 1.3317 1.3083 1.2818 1.2684
?= 0.5,?= 1 1.9305 1.8791 1.8557 1.7775 1.7988 1.8414 1.7931 1.7662 1.6996 1.7163
?= 0.9,?= 1 2.3641 2.3151 2.2397 2.1726 2.1380 2.3223 2.2708 2.2024 2.1439 2.1031
?= 0.1,?= 3 4.6389 4.4053 4.2831 4.2405 4.2051 4.8967 4.6412 4.5360 4.4624 4.4243
?= 0.5,?= 3 5.5047 5.3033 5.2630 5.1602 5.0283 6.3117 6.1039 6.0345 5.9233 5.7804
?= 0.9,?= 3 6.4306 6.2870 6.1180 5.9501 5.8768 8.0736 7.8664 7.6395 7.4564 7.3493
?= 0.1,?= 5 7.6307 7.6316 7.3059 7.1697 7.0108 8.8930 8.7858 8.4906 8.3299 8.1573
?= 0.5,?= 5 9.0566 8.6844 8.6661 8.4742 8.3162 11.9868 11.6011 11.5405 11.2015 11.0221
?= 0.9,?= 5 10.8019 10.1784 10.2150 9.9558 9.7360 15.8961 15.1546 15.1366 14.7591 14.4960
Finally, for INARMA(1,1), when ?= 0.1,?= 0.1,?= 0.5, ?= 0.1,?= 0.9,?=
0.5, ?= 0.5,?= 0.5,?= 0.5, and ?= 0.1,?= 0.1,?= 1, SBA should outperform
Croston and the opposite is true for the rest.
M.Mohammadipour, 2009, Appendix 8.C 310
Table 8.C5 MSE of Croston and SBA with smoothing parameter 0.2 for INARMA(1,1) series
Parameters
?????????? ??????
?=?? ?=?? ?= 48 ?=?? ?=??? ?=?? ?=?? ?= 48 ?=?? ?=???
?= 0.1,?= 0.1,?= 0.5 0.7572 0.7285 0.6975 0.6797 0.6672 0.7344 0.7133 0.6858 0.6691 0.6570
?= 0.1,?= 0.9,?= 0.5 1.4658 1.3971 1.2976 1.2855 1.2758 1.4061 1.3529 1.2653 1.2548 1.2440
?= 0.5,?= 0.5,?= 0.5 2.0666 1.9352 1.9751 1.8888 1.8780 2.0127 1.8915 1.9394 1.8472 1.8370
?= 0.9,?= 0.1,?= 0.5 2.4911 2.4166 2.4541 2.3439 2.3694 2.9636 2.7476 2.7803 2.6402 2.6659
?= 0.1,?= 0.1,?= 1 1.4875 1.4329 1.3832 1.3637 1.3459 1.4650 1.4128 1.3656 1.3466 1.3300
?= 0.1,?= 0.9,?= 1 2.5633 2.6071 2.5802 2.4522 2.4170 2.5445 2.5750 2.5632 2.4316 2.3971
?= 0.5,?= 0.5,?= 1 3.8122 3.5193 3.5507 3.4400 3.3913 3.8405 3.5307 3.5720 3.4446 3.4046
?= 0.9,?= 0.1,?= 1 4.9377 4.9217 4.8844 4.9002 4.6510 6.4337 6.3152 6.2141 6.1640 5.8469
?= 0.1,?= 0.1,?= 5 7.3304 7.0908 6.9961 6.8049 6.6737 7.6299 7.3524 7.2568 7.0638 6.9243
?= 0.1,?= 0.9,?= 5 12.2581 11.8649 11.8106 11.4788 11.3795 13.3116 12.8960 12.7726 12.3983 12.2912
?= 0.5,?= 0.5,?= 5 17.0179 16.6291 16.8486 16.3143 16.1399 19.2541 18.8518 18.9279 18.3672 18.1258
?= 0.9,?= 0.1,?= 5 25.9712 23.8322 24.3298 24.2873 23.3176 57.2282 56.2972 55.9089 54.7003 53.6411
Table 8.C6 MSE of Croston and SBA with smoothing parameter 0.5 for INARMA(1,1) series
Parameters
?????????? ??????
?=?? ?=?? ?= 48 ?=?? ?=??? ?=?? ?=?? ?= 48 ?=?? ?=???
?= 0.1,?= 0.1,?= 0.5 0.8278 0.8255 0.7996 0.7832 0.7660 0.7468 0.7457 0.7237 0.7103 0.6956
?= 0.1,?= 0.9,?= 0.5 1.5852 1.5845 1.4591 1.4463 1.4452 1.4111 1.3976 1.3085 1.2966 1.2900
?= 0.5,?= 0.5,?= 0.5 1.9839 1.8595 1.9022 1.8264 1.8119 1.8514 1.7625 1.8132 1.7249 1.7129
?= 0.9,?= 0.1,?= 0.5 1.4987 1.4757 1.4697 1.4072 1.4056 3.7822 3.5755 3.5684 3.4579 3.4478
?= 0.1,?= 0.1,?= 1 1.6598 1.6111 1.5495 1.5359 1.5132 1.5622 1.5098 1.4599 1.4443 1.4251
?= 0.1,?= 0.9,?= 1 2.6260 2.7332 2.6587 2.5459 2.5066 2.6141 2.6703 2.6471 2.5198 2.4867
?= 0.5,?= 0.5,?= 1 3.3841 3.1385 3.1717 3.0929 3.0331 3.7330 3.4326 3.4858 3.3582 3.3188
?= 0.9,?= 0.1,?= 1 3.0666 2.9571 2.9202 2.8722 2.7725 12.0092 11.4371 11.2427 10.9966 10.6574
?= 0.1,?= 0.1,?= 5 8.2959 7.9922 7.8299 7.6759 7.5434 10.0345 9.6419 9.4752 9.2596 9.0982
?= 0.1,?= 0.9,?= 5 12.2479 11.7657 11.7342 11.4066 11.3330 18.7567 18.1540 17.8894 17.4457 17.2565
?= 0.5,?= 0.5,?= 5 14.8220 14.5978 14.7661 14.3685 14.1216 29.0148 28.5701 28.2932 27.7230 27.1435
?= 0.9,?= 0.1,?= 5 15.6201 14.5525 14.5893 14.4396 13.9223 220.0655 216.2880 212.1152 207.3633 205.4631
M.Mohammadipour, 2009, Appendix 8.D 311
Appendix 8.D Comparing the Accuracy of INARMA Forecasts for all points in time and issue points
In this appendix, the forecast accuracy of INARMA methods when all points in time are considered is compared to the case of issue points.
Table 8.D1 Forecast accuracy for all points in time and issue points for INAR(1) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
All points in time Issue Points All points in time Issue Points All points in time Issue Points All points in time Issue Points
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0026 0.6614 1.1418 0.1034 0.8816 0.8592 0.0089 0.6286 1.0555 0.0840 0.7817 0.8115 0.0052 0.6047 1.0237 0.0491 0.7214 0.7881 0.0045 0.5822 0.9822 0.0247 0.6703 0.7545
?= 0.5,?= 0.5 0.0026 0.9466 1.3290 0.0241 1.2146 1.2289 0.0089 0.8743 1.1807 0.0316 1.0876 1.1100 0.0044 0.8532 1.1570 0.0153 1.0373 1.0915 0.0054 0.7834 1.0534 0.0107 0.9473 0.9871
?= 0.9,?= 0.5 0.0017 1.2082 1.3901 0.0028 1.2116 1.3893 0.0042 1.1489 1.2894 0.0013 1.1517 1.2895 0.0043 1.0811 1.2176 0.0058 1.0841 1.2172 0.0033 1.0164 1.1486 0.0047 1.0200 1.1492
?= 0.1,?= 1 0.0149 1.2880 0.9376 0.0962 1.4564 0.9781 0.0059 1.2072 0.8496 0.0582 1.2943 0.8580 0.0052 1.1967 0.8382 0.0333 1.2759 0.8390 0.0029 1.1663 0.8057 0.0132 1.2258 0.8090
?= 0.5,?= 1 0.0195 1.8846 1.0837 0.0428 2.0093 1.1169 0.0018 1.7244 1.0202 0.0126 1.8430 1.0457 0.0045 1.6745 1.0059 0.0104 1.7796 1.0307 0.0026 1.5905 0.9618 0.0002 1.6749 0.9852
?= 0.9,?= 1 0.0121 2.5216 1.2401 0.0125 2.5205 1.2400 0.0054 2.2655 1.1594 0.0053 2.2655 1.1594 0.0146 2.1925 1.1262 0.0147 2.1926 1.1262 0.0105 2.0319 1.0686 0.0107 2.0319 1.0686
?= 0.1,?= 3 0.0100 3.9224 0.8544 0.0101 3.9851 0.8781 0.0176 3.6703 0.8228 0.0242 3.7034 0.8448 0.0010 3.6641 0.8136 0.0101 3.7006 0.8353 0.0085 3.4701 0.7846 0.0050 3.4883 0.8019
?= 0.5,?= 3 0.0308 5.6926 1.0126 0.0325 5.6986 1.0139 0.0035 5.1557 0.9491 0.0057 5.1609 0.9514 0.0078 4.9742 0.9331 0.0086 4.9801 0.9350 0.0143 4.7749 0.9160 0.0135 4.7793 0.9177
?= 0.9,?= 3 0.0906 7.5862 1.1509 0.0906 7.5862 1.1509 0.0298 6.7494 1.1026 0.0298 6.7494 1.1026 0.0243 6.4376 1.0658 0.0243 6.4376 1.0658 0.0093 6.0230 1.0205 0.0093 6.0230 1.0205
?= 0.1,?= 5 0.0371 6.6926 0.8565 0.0335 6.6920 0.8580 0.0209 6.1869 0.8143 0.0179 6.1929 0.8176 0.0282 6.0095 0.7973 0.0260 6.0176 0.8016 0.0139 5.7529 0.7742 0.0133 5.7586 0.7778
?= 0.5,?= 5 0.0123 9.3467 0.9840 0.0121 9.3479 0.9843 0.0082 8.6581 0.9653 0.0082 8.6597 0.9655 0.0116 8.3583 0.9392 0.0116 8.3583 0.9393 0.0034 7.8699 0.9011 0.0035 7.8699 0.9012
?= 0.9,?= 5 0.0013 11.9986 1.1483 0.0013 11.9986 1.1483 0.0126 11.2051 1.0914 0.0126 11.2051 1.0914 0.0467 10.8985 1.0720 0.0467 10.8985 1.0720 0.0185 10.1102 1.0272 0.0185 10.1102 1.0272
M.Mohammadipour, 2009, Appendix 8.D 312
Table 8.D2 Forecast accuracy for all points in time and issue points for INMA(1) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
All points in time Issue Points All points in time Issue Points All points in time Issue Points All points in time Issue Points
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0031 0.6295 1.1297 0.0392 0.7823 0.8062 0.0027 0.6035 1.0473 0.0117 0.7134 0.7866 0.0114 0.5993 1.0493 0.0249 0.6832 0.7875 0.0007 0.5702 0.9736 0.0480 0.6570 0.7351
?= 0.5,?= 0.5 0.0395 0.8793 1.3407 0.1612 1.1706 1.1189 0.0279 0.8552 1.2527 0.1975 1.1200 1.0541 0.0229 0.7997 1.1494 0.1969 1.0123 0.9901 0.0031 0.7885 1.0948 0.2116 0.9847 0.9574
?= 0.9,?= 0.5 0.0767 1.1019 1.5472 0.2912 1.4609 1.3594 0.0671 1.0609 1.3632 0.2997 1.3175 1.2363 0.0628 1.0229 1.2606 0.3007 1.2400 1.1211 0.0445 0.9878 1.1797 0.3103 1.1874 1.0683
?= 0.1,?= 1 0.0234 1.2748 0.9163 0.0114 1.3964 0.9335 0.0000 1.2038 0.8547 0.0050 1.2923 0.8731 0.0082 1.1724 0.8365 0.0223 1.2557 0.8414 0.0048 1.1313 0.7957 0.0391 1.2009 0.7956
?= 0.5,?= 1 0.0347 1.7554 1.0666 0.1211 1.9307 1.1132 0.0440 1.6455 1.0247 0.1414 1.7795 1.0523 0.0310 1.6074 1.0005 0.1429 1.7315 1.0136 0.0204 1.5302 0.9648 0.1512 1.6235 0.9718
?= 0.9,?= 1 0.1249 2.2869 1.2485 0.2374 2.4432 1.2942 0.1025 2.1650 1.1687 0.2334 2.2741 1.1922 0.1148 2.0762 1.1223 0.2549 2.1785 1.1295 0.1116 1.9944 1.0920 0.2551 2.0664 1.0959
?= 0.1,?= 3 0.0452 3.9039 0.8547 0.0376 3.9375 0.8748 0.0402 3.6622 0.8158 0.0408 3.6950 0.8347 0.0150 3.5353 0.7960 0.0174 3.5579 0.8157 0.0062 3.4237 0.7767 0.0129 3.4386 0.7954
?= 0.5,?= 3 0.1260 5.2415 0.9971 0.1379 5.2750 1.0039 0.0781 4.9565 0.9450 0.0896 4.9647 0.9540 0.0929 4.8942 0.9424 0.1089 4.9067 0.9486 0.0608 4.6821 0.9036 0.0760 4.6843 0.9094
?= 0.9,?= 3 0.2970 6.7422 1.1268 0.3041 6.7525 1.1288 0.3209 6.5993 1.1023 0.3271 6.6089 1.1045 0.2796 6.2599 1.0587 0.2873 6.2574 1.0596 0.2718 6.0678 1.0364 0.2801 6.0598 1.0369
?= 0.1,?= 5 0.0838 6.4549 0.8431 0.0830 6.4710 0.8465 0.0132 6.2194 0.8182 0.0129 6.2335 0.8213 0.0226 5.9292 0.7937 0.0239 5.9364 0.7972 0.0111 5.7446 0.7806 0.0138 5.7500 0.7836
?= 0.5,?= 5 0.1407 8.7278 0.9676 0.1413 8.7273 0.9679 0.1984 8.3020 0.9417 0.1991 8.3003 0.9416 0.1738 8.0993 0.9245 0.1751 8.1031 0.9250 0.1269 7.8404 0.9063 0.1281 7.8395 0.9067
?= 0.9,?= 5 0.4387 11.6802 1.1864 0.4387 11.6802 1.1865 0.4081 10.9042 1.0713 0.4083 10.9074 1.0714 0.4804 10.7677 1.0895 0.4807 10.7679 1.0895 0.4479 10.1863 1.0242 0.4481 10.1860 1.0243
M.Mohammadipour, 2009, Appendix 8.D 313
Table 8.D3 Forecast accuracy for all points in time and issue points for INARMA(1,1) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
All points in time Issue Points All points in time Issue Points All points in time Issue Points All points in time Issue Points
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.1,
?= 0.5
0.0007 0.7482 1.2163 0.0700 1.0393 0.9354 0.0066 0.7138 1.1186 0.0382 0.9399 0.9074 0.0019 0.6777 1.0787 0.0039 0.8785 0.8779 0.0047 0.6468 1.0076 0.0172 0.7936 0.8191
?= 0.1,?= 0.9,
?= 0.5
0.0401 1.2112 1.4173 0.1570 1.5961 1.3111 0.0336 1.1340 1.2974 0.1877 1.4754 1.1898 0.0267 1.0734 1.1958 0.1686 1.3287 1.0998 0.0163 1.0429 1.1248 0.2050 1.2677 1.0215
?= 0.5,?= 0.5,
?= 0.5
0.0485 1.5791 1.4864 0.1213 1.9469 1.3908 0.0209 1.4070 1.2666 0.0948 1.6876 1.2107 0.0445 1.3825 1.1924 0.1280 1.6513 1.1484 0.0261 1.2261 1.1137 0.1174 1.4511 1.0591
?= 0.9,?= 0.1,
?= 0.5
0.0920 1.4003 1.4228 0.0903 1.4066 1.4232 0.0532 1.2843 1.3142 0.0514 1.2898 1.3152 0.0487 1.2226 1.2608 0.0482 1.2273 1.2634 0.0324 1.1110 1.1489 0.0317 1.1134 1.1492
?= 0.1,?= 0.1,
?= 1
0.0041 1.4908 0.9689 0.0435 1.6925 1.0017 0.0108 1.4146 0.9145 0.0350 1.5629 0.9390 0.0129 1.3358 0.8925 0.0158 1.4591 0.9120 0.0119 1.2833 0.8575 0.0018 1.3808 0.8725
?= 0.1,?= 0.9,
?= 1
0.0192 2.3265 1.1462 0.0618 2.5072 1.1789 0.0185 2.2569 1.0930 0.0507 2.4059 1.1171 0.0368 2.2224 1.0713 0.1105 2.3482 1.0744 0.0149 2.0900 1.0210 0.1107 2.1709 1.0172
?= 0.5,?= 0.5,
?= 1
0.0296 3.1322 1.1914 0.0338 3.2529 1.2063 0.0250 2.7101 1.1033 0.0443 2.8128 1.1132 0.0409 2.6748 1.0798 0.0617 2.7683 1.0860 0.0220 2.4363 1.0245 0.0523 2.5181 1.0290
?= 0.9,?= 0.1,
?= 1
0.0665 2.8324 1.2617 0.0665 2.8324 1.2617 0.0494 2.5623 1.1748 0.0492 2.5623 1.1747 0.0640 2.4301 1.1219 0.0640 2.4301 1.1219 0.0413 2.2588 1.0786 0.0413 2.2587 1.0786
?= 0.1,?= 0.1,
?= 5
0.0279 7.4165 0.8824 0.0270 7.4271 0.8849 0.0070 6.9501 0.8502 0.0086 6.9563 0.8519 0.0011 6.7534 0.8307 0.0008 6.7600 0.8323 0.0006 6.4253 0.8079 0.0011 6.4307 0.8101
?= 0.1,?= 0.9,
?= 5
0.0436 11.2147 1.0506 0.0436 11.2147 1.0506 0.0580 10.5085 0.9941 0.0580 10.5096 0.9942 0.0521 10.1101 0.9589 0.0521 10.1101 0.9589 0.0606 9.7366 0.9338 0.0606 9.7366 0.9338
?= 0.5,?= 0.5,
?= 5
0.1082 14.0754 1.0425 0.1082 14.0754 1.0425 0.1239 13.0227 1.0240 0.1239 13.0227 1.0240 0.1074 12.7060 0.9952 0.1074 12.7060 0.9952 0.1065 11.8259 0.9556 0.1065 11.8261 0.9556
?= 0.9,?= 0.1,
?= 5
0.1214 13.8720 1.1795 0.1214 13.8720 1.1795 0.1895 12.4636 1.0932 0.1895 12.4636 1.0932 0.1569 12.0389 1.0813 0.1569 12.0389 1.0813 0.0906 11.2782 1.0311 0.0906 11.2782 1.0311
M.Mohammadipour, 2009, Appendix 8.E 314
Appendix 8.E Comparison of MASE of INARMA (known order)
with Benchmarks
In this appendix, the degree of improvement by INARMA over benchmarks, using the
MASE measure, is presented. The results are for the case where all points in time are
taken into account.
Table 8.E1 for INARMA(0,0) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.8905 0.9259 0.9350 0.9269 0.9610 0.9658 0.9499 0.9841 0.9882 0.9670 1.0007 1.0046
?= 0.5 0.9622 0.9837 0.9868 0.9795 0.9987 1.0010 0.9852 1.0033 1.0053 0.9866 1.0053 1.0074
?= 0.7
0.9836 0.9954 0.9961 0.9873 0.9955 0.9961 0.9894 0.9969 0.9974 0.9881 0.9960 0.9966
?= 1 0.9773 0.9944 0.9953 0.9698 0.9851 0.9858 0.9591 0.9750 0.9758 0.9448 0.9605 0.9613
?= 3 0.9766 0.9853 0.9844 0.9669 0.9760 0.9753 0.9631 0.9722 0.9715 0.9538 0.9622 0.9614
?= 5 0.9756 0.9753 0.9726 0.9649 0.9623 0.9593 0.9631 0.9620 0.9591 0.9544 0.9535 0.9508
?= 20 0.9792 0.9146 0.8987 0.9652 0.9056 0.8906 0.9633 0.9011 0.8857 0.9564 0.8957 0.8803
Table 8.E2 for INMA(1) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9606 0.9810 0.9847 0.9775 0.9938 0.9956 0.9826 0.9981 0.9998 0.9867 1.0024 1.0041
?= 0.5,?= 0.5 0.9459 0.9640 0.9665 0.9601 0.9751 0.9762 0.9660 0.9795 0.9805 0.9627 0.9772 0.9784
?= 0.9,?= 0.5 0.9500 0.9717 0.9733 0.9453 0.9657 0.9679 0.9433 0.9617 0.9630 0.9243 0.9438 0.9453
?= 0.1,?= 1 0.9757 0.9941 0.9953 0.9694 0.9900 0.9914 0.9603 0.9796 0.9808 0.9453 0.9657 0.9671
?= 0.5,?= 1 0.9826 1.0020 1.0032 0.9828 1.0021 1.0034 0.9812 1.0010 1.0024 0.9796 1.0009 1.0026
?= 0.9,?= 1 0.9955 1.0093 1.0098 0.9830 0.9974 0.9980 0.9786 0.9930 0.9935 0.9719 0.9864 0.9870
?= 0.1,?= 3 0.9870 0.9934 0.9923 0.9740 0.9812 0.9802 0.9714 0.9796 0.9785 0.9675 0.9753 0.9743
?= 0.5,?= 3 1.0073 1.0087 1.0060 0.9990 1.0010 0.9985 0.9977 0.9971 0.9943 0.9868 0.9878 0.9853
?= 0.9,?= 3 1.0303 1.0191 1.0142 1.0296 1.0210 1.0164 1.0148 1.0087 1.0046 1.0089 1.0017 0.9973
?= 0.1,?= 5 0.9883 0.9809 0.9771 0.9758 0.9756 0.9724 0.9690 0.9665 0.9632 0.9666 0.9625 0.9590
?= 0.5,?= 5 1.0126 0.9957 0.9895 1.0091 0.9920 0.9855 0.9996 0.9825 0.9762 0.9906 0.9760 0.9700
?= 0.9,?= 5 1.0493 1.0174 1.0084 1.0361 1.0119 1.0033 1.0252 1.0016 0.9933 1.0123 0.9875 0.9792
BenchmarkINARMA MASEMASE /
BenchmarkINARMA MASEMASE /
M.Mohammadipour, 2009, Appendix 8.E 315
Table 8.E3 for INAR(1) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
?= 0.1,?= 0.5 0.9787 0.9505 0.9965 1.0121 0.3777 0.4554 0.9825 0.9381 0.9987 1.0027 1.0003 1.0329 0.9778 0.9312 0.9933 0.9957 0.9950 1.0273 0.9805 0.9342 0.9957 0.9979 0.9974 1.0279
?= 0.5,?= 0.5 0.8685 0.8610 0.8837 0.9055 0.8851 0.8811 0.8619 0.8470 0.8772 0.8956 0.8784 0.8728 0.8595 0.8365 0.8745 0.8857 0.8755 0.8672 0.8435 0.8218 0.8598 0.8714 0.8610 0.8552
?= 0.9,?= 0.5 0.7325 0.9403 0.6868 0.5770 0.6771 0.4419 0.7060 0.9252 0.6606 0.5598 0.6510 0.4228 0.6921 0.8938 0.6521 0.5511 0.6429 0.4191 0.6694 0.8741 0.6350 0.5394 0.6265 0.4093
?= 0.1,?= 1 0.9966 0.9365 1.0161 0.9858 1.0175 1.0012 0.9812 0.9161 1.0012 0.9632 1.0025 0.9828 0.9710 0.9071 0.9895 0.9545 0.9907 0.9708 0.9570 0.8934 0.9773 0.9415 0.9787 0.9612
?= 0.5,?= 1 0.9349 0.9482 0.9451 0.9571 0.9450 0.8936 0.9028 0.9227 0.9123 0.9256 0.9121 0.8622 0.8942 0.9079 0.9059 0.9180 0.9060 0.8574 0.8805 0.8954 0.8928 0.9039 0.8929 0.8467
?= 0.9,?= 1 0.7581 0.9606 0.6670 0.4577 0.6496 0.3447 0.7313 0.9358 0.6460 0.4430 0.6295 0.3328 0.7165 0.9215 0.6402 0.4357 0.6239 0.3283 0.7004 0.9012 0.6266 0.4306 0.6111 0.3247
?= 0.1,?= 3 0.9924 0.9297 0.9980 0.9202 0.9966 0.8996 0.9817 0.9136 0.9895 0.9117 0.9886 0.8982 0.9775 0.9152 0.9845 0.9078 0.9834 0.8899 0.9657 0.9028 0.9738 0.8988 0.9727 0.8794
?= 0.5,?= 3 0.9629 0.9835 0.9480 0.8547 0.9427 0.7377 0.9354 0.9586 0.9263 0.8362 0.9214 0.7231 0.9233 0.9463 0.9104 0.8199 0.9053 0.7095 0.9121 0.9333 0.9007 0.8129 0.8958 0.7042
?= 0.9,?= 3 0.7737 0.9664 0.5807 0.2813 0.5483 0.2114 0.7292 0.9333 0.5477 0.2692 0.5179 0.2021 0.7170 0.9195 0.5328 0.2631 0.5037 0.1975 0.7035 0.9023 0.5219 0.2572 0.4933 0.1930
?= 0.1,?= 5 1.0011 0.9328 0.9919 0.8695 0.9879 0.8169 0.9861 0.9142 0.9803 0.8533 0.9764 0.8033 0.9765 0.9051 0.9729 0.8484 0.9692 0.7991 0.9651 0.8974 0.9627 0.8415 0.9594 0.7954
?= 0.5,?= 5 0.9530 0.9735 0.9213 0.7596 0.9118 0.6289 0.9439 0.9638 0.9153 0.7520 0.9062 0.6239 0.9249 0.9464 0.8917 0.7361 0.8826 0.6100 0.9159 0.9355 0.8855 0.7255 0.8765 0.6026
?= 0.9,?= 5 0.7668 0.9666 0.4736 0.2105 0.4388 0.1575 0.7357 0.9335 0.4633 0.2066 0.4301 0.1548 0.7213 0.9263 0.4623 0.2057 0.4294 0.1541 0.7078 0.9048 0.4473 0.1990 0.4151 0.1491
BenchmarkINARMA MASEMASE /
M.Mohammadipour, 2009, Appendix 8.E 316
Table 8.E4 for INARMA(1,1) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
?= 0.1,?= 0.1,
?= 0.5
0.9814 0.9497 1.0001 1.0118 1.0013 1.0305 0.9758 0.9275 0.9914 0.9938 0.9930 1.0190 0.9760 0.9256 0.9898 0.9893 0.9912 1.0136 0.9734 0.9184 0.9879 0.9850 0.9894 1.0135
?= 0.1,?= 0.9,
?= 0.5
0.8861 0.8480 0.9101 0.9091 0.9105 0.9243 0.8839 0.8263 0.9048 0.8898 0.9065 0.9123 0.8869 0.8315 0.9066 0.8927 0.9082 0.9102 0.8753 0.8185 0.8963 0.8816 0.8980 0.9013
?= 0.5,?= 0.5,
?= 0.5
0.8580 0.8754 0.8728 0.9157 0.8726 0.8639 0.8286 0.8488 0.8451 0.8887 0.8463 0.8393 0.8177 0.8307 0.8341 0.8720 0.8353 0.8269 0.7894 0.8027 0.8086 0.8470 0.8101 0.8075
?= 0.9,?= 0.1,
?= 0.5
0.7329 0.9608 0.6755 0.5760 0.6648 0.4308 0.7079 0.9226 0.6649 0.5641 0.6555 0.4252 0.6793 0.8920 0.6412 0.5470 0.6324 0.4125 0.6602 0.8677 0.6275 0.5265 0.6189 0.3984
?= 0.1,?= 0.1,
?= 1
0.9943 0.9404 1.0155 0.9913 1.0172 1.0070 0.9830 0.9236 1.0034 0.9746 1.0049 0.9951 0.9755 0.9196 0.9971 0.9694 0.9987 0.9868 0.9638 0.9072 0.9856 0.9579 0.9872 0.9777
?= 0.1,?= 0.9,
?= 1
0.9421 0.9341 0.9531 0.9523 0.9530 0.9069 0.9172 0.9057 0.9325 0.9316 0.9332 0.9008 0.9145 0.9051 0.9283 0.9274 0.9287 0.8919 0.9127 0.9001 0.9282 0.9256 0.9288 0.8931
?= 0.5,?= 0.5,
?= 1
0.8946 0.9514 0.8971 0.9190 0.8959 0.8149 0.8692 0.9220 0.8757 0.8987 0.8749 0.7999 0.8575 0.9107 0.8651 0.8892 0.8645 0.7934 0.8356 0.8852 0.8460 0.8682 0.8456 0.7767
?= 0.9,?= 0.1,
?= 1
0.7502 0.9519 0.6610 0.4423 0.6431 0.3323 0.7118 0.9250 0.6280 0.4284 0.6115 0.3217 0.6924 0.9060 0.6162 0.4199 0.6005 0.3153 0.6693 0.8805 0.6004 0.4114 0.5854 0.3093
?= 0.1,?= 0.1,
?= 5
1.0032 0.9442 0.9930 0.8617 0.9884 0.7962 0.9894 0.9338 0.9838 0.8610 0.9798 0.7976 0.9832 0.9325 0.9775 0.8558 0.9734 0.7890 0.9693 0.9142 0.9626 0.8407 0.9585 0.7795
?= 0.1,?= 0.9,
?= 5
0.9532 0.9557 0.9246 0.7662 0.9161 0.6458 0.9347 0.9393 0.9065 0.7504 0.8981 0.6349 0.9226 0.9268 0.8982 0.7448 0.8902 0.6319 0.9176 0.9228 0.8923 0.7385 0.8843 0.6263
?= 0.5,?= 0.5,
?= 5
0.9041 0.9718 0.8526 0.6689 0.8404 0.5331 0.8796 0.9427 0.8318 0.6528 0.8201 0.5215 0.8640 0.9250 0.8255 0.6533 0.8149 0.5235 0.8477 0.9035 0.8073 0.6340 0.7963 0.5079
?= 0.9,?= 0.1,
?= 5
0.7328 0.9408 0.4704 0.2077 0.4368 0.1559 0.7219 0.9226 0.4469 0.1978 0.4142 0.1482 0.7002 0.9074 0.4446 0.1971 0.4125 0.1478 0.6810 0.6836 0.2741 0.6155 0.2058 0.1446
BenchmarkINARMA MASEMASE /
M.Mohammadipour, 2009, Appendix 8.F 317
Appendix 8.F Comparison of sixstep ahead MSE of INARMA
(known order) with Benchmarks
In this appendix the MSE of sixstep ahead INARMA forecasts (using YW to
estimate the parameters) is compared to the MSE of benchmark methods.
Table 8.F1 Sixstep ahead for INARMA(0,0) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.8313 0.8826 0.8879 0.9378 0.9650 0.9675 0.9727 0.9891 0.9906 0.9789 0.9866 0.9873
?= 0.5 0.9364 0.9695 0.9723 0.9747 0.9907 0.9917 0.9782 0.9891 0.9896 0.9648 0.9744 0.9747
?= 0.7
0.9637 0.9882 0.9901 0.9751 0.9877 0.9880 0.9685 0.9787 0.9790 0.9581 0.9673 0.9674
?= 1 0.9735 0.9894 0.9898 0.9675 0.9782 0.9780 0.9659 0.9740 0.9736 0.9468 0.9552 0.9549
?= 3 0.9649 0.9548 0.9505 0.9478 0.9421 0.9382 0.9347 0.9295 0.9257 0.9206 0.9139 0.9100
?= 5 0.9621 0.9455 0.9384 0.9419 0.9200 0.9125 0.9373 0.9146 0.9070 0.9143 0.8932 0.8860
?= 20 0.9573 0.8218 0.7942 0.9389 0.8066 0.7794 0.9239 0.7887 0.7619 0.9142 0.7881 0.7619
Table 8.F2 Sixstep ahead
for INMA(1) series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9173 0.9540 0.9572 0.9675 0.9856 0.9866 0.9669 0.9813 0.9821 0.9591 0.9705 0.9710
?= 0.5,?= 0.5 0.9211 0.9695 0.9737 0.9503 0.9794 0.9815 0.9420 0.9647 0.9660 0.9239 0.9456 0.9470
?= 0.9,?= 0.5 0.9292 0.9808 0.9853 0.9536 0.9849 0.9870 0.9432 0.9667 0.9681 0.9108 0.9346 0.9361
?= 0.1,?= 1 0.9682 0.9902 0.9912 0.9622 0.9762 0.9764 0.9499 0.9626 0.9626 0.9368 0.9496 0.9496
?= 0.5,?= 1 0.9668 0.9801 0.9797 0.9496 0.9675 0.9677 0.9342 0.9488 0.9487 0.9093 0.9231 0.9229
?= 0.9,?= 1 0.9688 0.9939 0.9944 0.9256 0.9393 0.9388 0.9006 0.9190 0.9189 0.8832 0.8986 0.8982
?= 0.1,?= 3 0.9577 0.9557 0.9519 0.9377 0.9248 0.9200 0.9216 0.9125 0.9081 0.9124 0.9060 0.9019
?= 0.5,?= 3 0.9567 0.9403 0.9342 0.9184 0.9130 0.9079 0.9028 0.8885 0.8827 0.8804 0.8729 0.8679
?= 0.9,?= 3 0.9476 0.9341 0.9269 0.9073 0.8973 0.8909 0.8862 0.8747 0.8683 0.8664 0.8512 0.8445
?= 0.1,?= 5 0.9542 0.9369 0.9292 0.9301 0.9093 0.9016 0.9177 0.8968 0.8893 0.8990 0.8739 0.8661
?= 0.5,?= 5 0.9579 0.9239 0.9130 0.9150 0.8823 0.8721 0.8984 0.8680 0.8580 0.8774 0.8470 0.8372
?= 0.9,?= 5 0.9455 0.8851 0.8710 0.9027 0.8651 0.8529 0.8832 0.8417 0.8293 0.8586 0.8193 0.8075
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Appendix 8.F 318
Table 8.F3 Sixstep ahead with smoothing parameter 0.2 for INAR(1)
series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9290 0.9705 0.9745 0.9732 0.9921 0.9933 0.9714 0.9835 0.9841 0.9528 0.9655 0.9661
?= 0.5,?= 0.5 1.0424 1.0900 1.0939 0.9670 0.9962 0.9981 0.9315 0.9587 0.9605 0.8780 0.9073 0.9094
?= 0.9,?= 0.5 1.1764 1.1710 1.1607 1.0385 1.0363 1.0283 0.9826 0.9879 0.9810 0.8998 0.9066 0.9005
?= 0.1,?= 1 0.9826 1.0046 1.0055 0.9705 0.9836 0.9836 0.9520 0.9622 0.9619 0.9356 0.9485 0.9486
?= 0.5,?= 1 1.0263 1.0491 1.0492 0.9406 0.9667 0.9673 0.9170 0.9371 0.9371 0.8620 0.8837 0.8841
?= 0.9,?= 1 1.1586 1.0883 1.0665 1.0187 0.9664 0.9480 0.9825 0.9356 0.9185 0.8863 0.8386 0.8223
?= 0.1,?= 3 0.9748 0.9704 0.9664 0.9371 0.9315 0.9274 0.9255 0.9186 0.9143 0.9078 0.9008 0.8967
?= 0.5,?= 3 0.9864 0.9747 0.9676 0.9131 0.9114 0.9055 0.8737 0.8627 0.8564 0.8393 0.8310 0.8251
?= 0.9,?= 3 1.1322 0.8947 0.8454 1.0627 0.8301 0.7841 0.9961 0.7865 0.7450 0.8885 0.6957 0.6589
?= 0.1,?= 5 0.9460 0.9188 0.9108 0.9225 0.8997 0.8919 0.9174 0.8917 0.8837 0.9026 0.8786 0.8708
?= 0.5,?= 5 1.0105 0.9777 0.9643 0.9255 0.8918 0.8795 0.8862 0.8513 0.8394 0.8337 0.7990 0.7877
?= 0.9,?= 5 1.1459 0.7273 0.6712 1.0230 0.6764 0.6242 0.9755 0.6680 0.6178 0.8781 0.6001 0.5553
Table 8.F4 Sixstep ahead with smoothing parameter 0.5 for INAR(1)
series (known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.8777 0.9762 0.9872 0.8689 0.9483 0.9536 0.8531 0.9259 0.9297 0.8261 0.9035 0.9095
?= 0.5,?= 0.5 0.9080 1.0391 1.0506 0.8214 0.9280 0.9327 0.7790 0.8829 0.8889 0.7437 0.8469 0.8534
?= 0.9,?= 0.5 1.1361 0.9462 0.7806 1.0338 0.8704 0.7239 0.9877 0.8391 0.6981 0.9112 0.7704 0.6399
?= 0.1,?= 1 0.8618 0.9320 0.9216 0.8344 0.8948 0.8832 0.8227 0.8788 0.8664 0.8092 0.8698 0.8589
?= 0.5,?= 1 0.8658 0.9195 0.8916 0.7825 0.8503 0.8275 0.7727 0.8259 0.8007 0.7184 0.7748 0.7540
?= 0.9,?= 1 1.1299 0.7420 0.5580 1.0106 0.6642 0.4991 0.9828 0.6461 0.4879 0.8961 0.5772 0.4317
?= 0.1,?= 3 0.8300 0.8018 0.7387 0.7874 0.7614 0.7013 0.7713 0.7441 0.6846 0.7603 0.7329 0.6749
?= 0.5,?= 3 0.8072 0.7399 0.6503 0.7424 0.6872 0.6053 0.7155 0.6538 0.5757 0.6844 0.6282 0.5531
?= 0.9,?= 3 1.1316 0.3808 0.2443 1.0441 0.3545 0.2283 1.0112 0.3477 0.2253 0.9042 0.3097 0.2011
?= 0.1,?= 5 0.7875 0.6921 0.6079 0.7670 0.6760 0.5899 0.7581 0.6655 0.5807 0.7516 0.6612 0.5766
?= 0.5,?= 5 0.8134 0.6490 0.5302 0.7560 0.6022 0.4922 0.7214 0.5764 0.4725 0.6793 0.5413 0.4441
?= 0.9,?= 5 1.1350 0.2570 0.1604 1.0247 0.2351 0.1454 0.9831 0.2347 0.1447 0.8907 0.2120 0.1311
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Appendix 8.F 319
Table 8.F5 Sixstep ahead with smoothing parameter 0.2 for INARMA(1,1) series
(known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.9512 0.9912 0.9943 0.9824 1.0024 1.0038 0.9602 0.9795 0.9808 0.9486 0.9642 0.9650
?= 0.1,?= 0.9,?= 0.5 0.9665 1.0085 1.0095 0.9345 0.9697 0.9722 0.9384 0.9654 0.9671 0.9016 0.9281 0.9299
?= 0.5,?= 0.5,?= 0.5 0.9867 1.0228 1.0250 0.9133 0.9481 0.9506 0.8961 0.9309 0.9334 0.8562 0.8870 0.8890
?= 0.9,?= 0.1,?= 0.5 1.0800 1.0416 1.0297 1.2237 1.2449 1.2379 1.1860 1.1924 1.1841 1.1816 1.1881 1.1794
?= 0.1,?= 0.1,?= 1 1.0128 1.0355 1.0364 0.9989 1.0145 1.0147 0.9648 0.9791 0.9791 0.9254 0.9384 0.9384
?= 0.1,?= 0.9,?= 1 0.9784 0.9922 0.9915 0.9274 0.9484 0.9485 0.9126 0.9302 0.9300 0.8686 0.8839 0.8835
?= 0.5,?= 0.5,?= 1 0.9791 0.9998 0.9994 0.9318 0.9640 0.9648 0.8847 0.9072 0.9073 0.8367 0.8587 0.8588
?= 0.9,?= 0.1,?= 1 1.2035 1.1125 1.0885 1.2078 1.1521 1.1302 1.2134 1.1574 1.1359 1.2180 1.1413 1.1183
?= 0.1,?= 0.1,?= 5 0.9730 0.9560 0.9479 0.9283 0.9102 0.9024 0.9171 0.8995 0.8919 0.8933 0.8683 0.8601
?= 0.1,?= 0.9,?= 5 0.9675 0.9382 0.9261 0.8961 0.8517 0.8392 0.8742 0.8333 0.8212 0.8486 0.8094 0.7974
?= 0.5,?= 0.5,?= 5 0.9692 0.9024 0.8857 0.8892 0.8508 0.8372 0.8585 0.8164 0.8030 0.8123 0.7740 0.7614
?= 0.9,?= 0.1,?= 5 1.3878 0.8918 0.8210 1.2591 0.8477 0.7841 1.3093 0.8560 0.7887 1.2348 0.8392 0.7758
Table 8.F6 Sixstep ahead with smoothing parameter 0.5 for INARMA(1,1) series
(known order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.8927 0.9881 1.0203 0.8549 0.9433 0.9929 0.8271 0.9201 0.9682 0.8163 0.9023 0.9468
?= 0.1,?= 0.9,?= 0.5 0.8253 0.9380 1.0095 0.7848 0.9007 0.9620 0.7870 0.8911 0.9425 0.7546 0.8594 0.9099
?= 0.5,?= 0.5,?= 0.5 0.8752 0.9799 1.0064 0.7694 0.8795 0.9363 0.7577 0.8631 0.9183 0.7199 0.8149 0.8674
?= 0.9,?= 0.1,?= 0.5 1.0323 0.8420 0.6669 1.2382 1.0821 0.8688 1.1947 1.0184 0.8128 1.1831 0.9947 0.7984
?= 0.1,?= 0.1,?= 1 0.8833 0.9587 0.9878 0.8632 0.9266 0.9517 0.8292 0.8911 0.9180 0.7970 0.8552 0.8796
?= 0.1,?= 0.9,?= 1 0.8156 0.8712 0.8932 0.7655 0.8308 0.8640 0.7558 0.8083 0.8436 0.7166 0.7668 0.7999
?= 0.5,?= 0.5,?= 1 0.8055 0.8536 0.8870 0.7681 0.8332 0.8714 0.7397 0.7893 0.8140 0.6923 0.7404 0.7682
?= 0.9,?= 0.1,?= 1 1.1949 0.7480 0.5465 1.2245 0.7934 0.5744 1.2326 0.8056 0.5853 1.2341 0.7862 0.5707
?= 0.1,?= 0.1,?= 5 0.8034 0.7148 0.6574 0.7576 0.6714 0.6247 0.7533 0.6665 0.6185 0.7360 0.6413 0.5896
?= 0.1,?= 0.9,?= 5 0.7717 0.6323 0.5574 0.7140 0.5658 0.4981 0.6934 0.5564 0.4883 0.6811 0.5392 0.4676
?= 0.5,?= 0.5,?= 5 0.7894 0.5789 0.4843 0.7187 0.5541 0.4678 0.6910 0.5299 0.4482 0.6545 0.5015 0.4242
?= 0.9,?= 0.1,?= 5 1.3669 0.3068 0.1871 1.2699 0.3015 0.1846 1.3210 0.2945 0.1798 1.2527 0.2946 0.1802
BenchmarkINARMA MSEMSE /
BenchmarkINARMA MSEMSE /
M.Mohammadipour, 2009, Appendix 8.G 320
Appendix 8.G Comparison of allINAR(1) and allINARMA(1,1)
In this appendix, the forecast accuracy of allINAR(1) and allINARMA(1,1) is compared. The results are presented for the case of INAR(1),
INMA(1) and INARMA(1,1) series. The corresponding results for an INARMA(0,0) series can be found from Table ?832 and Table ?843.
Table 8.G1 Accuracy of forecasts by allINAR(1) and allINARMA(1,1) approaches for INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
AllINAR(1) AllINARMA(1,1) AllINAR(1) AllINARMA(1,1) AllINAR(1) AllINARMA(1,1) AllINAR(1) AllINARMA(1,1)
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0026 0.6614 1.1418 0.0008 0.6699 1.1275 0.0089 0.6286 1.0555 0.0049 0.6447 1.0710 0.0052 0.6047 1.0237 0.0111 0.6389 1.0340 0.0045 0.5822 0.9822 0.0010 0.5857 0.9845
?= 0.5,?= 0.5 0.0026 0.9466 1.3290 0.0130 1.0056 1.3278 0.0089 0.8743 1.1807 0.0039 0.9436 1.2602 0.0044 0.8532 1.1570 0.0041 0.9131 1.1864 0.0054 0.7834 1.0534 0.0033 0.8186 1.0727
?= 0.9,?= 0.5 0.0017 1.2082 1.3901 0.0347 1.2681 1.4405 0.0042 1.1489 1.2894 0.0344 1.1681 1.3181 0.0043 1.0811 1.2176 0.0066 1.0919 1.2150 0.0033 1.0164 1.1486 0.0206 1.0235 1.1707
?= 0.1,?= 1 0.0149 1.2880 0.9376 0.0152 1.3755 0.9455 0.0059 1.2072 0.8496 0.0028 1.2778 0.8819 0.0052 1.1967 0.8382 0.0066 1.1947 0.8427 0.0029 1.1663 0.8057 0.0045 1.1875 0.8235
?= 0.5,?= 1 0.0195 1.8846 1.0837 0.0092 1.9908 1.1188 0.0018 1.7244 1.0202 0.0050 1.8491 1.0640 0.0045 1.6745 1.0059 0.0046 1.7735 1.0251 0.0026 1.5905 0.9618 0.0052 1.6469 0.9944
?= 0.9,?= 1 0.0121 2.5216 1.2401 0.0544 2.4325 1.2304 0.0054 2.2655 1.1594 0.0378 2.3301 1.1748 0.0146 2.1925 1.1262 0.0484 2.1853 1.1423 0.0105 2.0319 1.0686 0.0312 2.0343 1.0747
?= 0.1,?= 3 0.0100 3.9224 0.8544 0.0275 4.1292 0.8818 0.0176 3.6703 0.8228 0.0177 3.7457 0.8236 0.0010 3.6641 0.8136 0.0144 3.6442 0.8115 0.0085 3.4701 0.7846 0.0222 3.4929 0.7828
?= 0.5,?= 3 0.0308 5.6926 1.0126 0.0237 5.8020 1.0279 0.0035 5.1557 0.9491 0.0291 5.4831 0.9848 0.0078 4.9742 0.9331 0.0027 5.2166 0.9710 0.0143 4.7749 0.9160 0.0268 4.9748 0.9319
?= 0.9,?= 3 0.0906 7.5862 1.1509 0.1097 7.3168 1.1422 0.0298 6.7494 1.1026 0.1077 6.9143 1.1046 0.0243 6.4376 1.0658 0.0627 6.5204 1.0742 0.0093 6.0230 1.0205 0.0612 6.1440 1.0290
?= 0.1,?= 5 0.0371 6.6926 0.8565 0.0124 6.5975 0.8522 0.0209 6.1869 0.8143 0.0016 6.3370 0.8206 0.0282 6.0095 0.7973 0.0078 6.2455 0.8197 0.0139 5.7529 0.7742 0.0063 5.8121 0.7805
?= 0.5,?= 5 0.0123 9.3467 0.9840 0.0798 9.5203 1.0114 0.0082 8.6581 0.9653 0.0050 8.8544 0.9603 0.0116 8.3583 0.9392 0.0325 8.4735 0.9460 0.0034 7.8699 0.9011 0.0003 7.9638 0.9095
?= 0.9,?= 5 0.0013 11.9986 1.1483 0.0773 12.2386 1.1385 0.0126 11.2051 1.0914 0.1196 11.3739 1.0820 0.0467 10.8985 1.0720 0.1021 10.9289 1.0679 0.0185 10.1102 1.0272 0.0956 10.1452 1.0232
M.Mohammadipour, 2009, Appendix 8.G 321
Table 8.G2 Accuracy of forecasts by allINAR(1) and allINARMA(1,1) approaches for INMA(1) series
Parameters
?=?? ?=?? ?=?? ?=??
AllINAR(1) AllINARMA(1,1) AllINAR(1) AllINARMA(1,1) AllINAR(1) AllINARMA(1,1) AllINAR(1) AllINARMA(1,1)
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.5 0.0102 0.6916 1.1405 0.0056 0.6500 1.1831 0.0016 0.6215 1.0690 0.0038 0.6155 1.0589 0.0084 0.5981 1.0149 0.0071 0.6070 1.0025 0.00007 0.5687 0.9692 0.0063 0.5733 0.9809
?= 0.5,?= 0.5 0.0058 0.8285 1.2415 0.0205 0.8681 1.3428 0.0085 0.7828 1.1608 0.0151 0.8047 1.1628 0.0033 0.7599 1.1155 0.0011 0.7815 1.1069 0.0038 0.6959 1.0381 0.0038 0.7423 1.0637
?= 0.9,?= 0.5 0.0324 0.9442 1.2536 0.0430 1.0089 1.3941 0.0052 0.8721 1.2436 0.0352 0.9383 1.2649 0.0089 0.8149 1.1243 0.0122 0.9436 1.1917 0.0063 0.7802 1.0677 0.0250 0.9208 1.1448
?= 0.1,?= 1 0.0005 1.3118 0.8950 0.0202 1.3447 0.9356 0.0078 1.2120 0.8617 0.0154 1.2458 0.8487 0.0034 1.1882 0.8419 0.0017 1.1956 0.8443 0.0013 1.1519 0.8002 0.0016 1.1525 0.8117
?= 0.5,?= 1 0.0271 1.6581 1.0424 0.0060 1.6743 1.0374 0.0235 1.5574 0.9788 0.0128 1.5905 0.9873 0.0039 1.4772 0.9546 0.0030 1.5879 0.9894 0.0047 1.4121 0.9168 0.0077 1.4733 0.9413
?= 0.9,?= 1 0.0443 1.9014 1.1317 0.0145 1.9996 1.1350 0.0203 1.7164 1.0562 0.0180 1.9324 1.0960 0.0292 1.6602 1.0051 0.0165 1.8242 1.0545 0.0127 1.5469 0.9562 0.0184 1.7825 1.0406
?= 0.1,?= 3 0.0169 3.8869 0.8397 0.0153 3.8953 0.8642 0.0190 3.6392 0.8194 0.0022 3.7206 0.8211 0.0208 3.4924 0.7922 0.0041 3.6216 0.7983 0.0136 3.4409 0.7838 0.0215 3.5003 0.7873
?= 0.5,?= 3 0.0436 4.8555 0.9562 0.0306 4.8945 0.9557 0.0302 4.6371 0.9278 0.0137 4.7351 0.9278 0.0170 4.4562 0.8843 0.0013 4.6193 0.9143 0.0039 4.2607 0.8655 0.0061 4.3870 0.8783
?= 0.9,?= 3 0.1126 5.4157 1.0076 0.0763 5.8039 1.0545 0.0603 5.1071 0.9687 0.0192 5.5915 1.0116 0.0481 4.8830 0.9420 0.0118 5.2638 0.9726 0.0213 4.7140 0.9137 0.0439 5.1879 0.9612
?= 0.1,?= 5 0.0330 6.3471 0.8369 0.0108 6.5867 0.8570 0.0025 6.1832 0.8223 0.0108 6.2327 0.8241 0.0221 6.0512 0.7981 0.0045 6.0357 0.7906 0.0054 5.6957 0.7776 0.0103 5.7236 0.7712
?= 0.5,?= 5 0.0071 8.0533 0.9474 0.0335 8.3565 0.9718 0.0317 7.5846 0.8934 0.0231 7.7461 0.9228 0.0207 7.4066 0.8805 0.0136 7.6924 0.8954 0.0166 6.8773 0.8510 0.0160 7.2509 0.8698
?= 0.9,?= 5 0.1464 9.3220 1.0338 0.0927 9.8826 1.0531 0.0607 8.7192 0.9763 0.0643 8.8871 0.9798 0.0626 8.1938 0.9370 0.0683 8.9059 0.9673 0.0218 7.7859 0.9012 0.0524 8.4545 0.9443
M.Mohammadipour, 2009, Appendix 8.G 322
Table 8.G3 Accuracy of forecasts by allINAR(1) and allINARMA(1,1) approaches for INARMA(1,1) series
Parameters
?=?? ?=?? ?=?? ?=??
AllINAR(1) AllINARMA(1,1) AllINAR(1) AllINARMA(1,1) AllINAR(1) AllINARMA(1,1) AllINAR(1) AllINARMA(1,1)
ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE ME MSE MASE
?= 0.1,?= 0.1, ?= 0.5 0.0078 0.7157 1.1853 0.0007 0.7482 1.2163 0.0050 0.6776 1.0914 0.0066 0.7138 1.1186 0.0037 0.6789 1.0606 0.0019 0.6777 1.0787 0.0034 0.6319 1.0015 0.0047 0.6468 1.0076
?= 0.1,?= 0.9, ?= 0.5 0.0336 1.1234 1.3333 0.0401 1.2112 1.4173 0.0166 1.0234 1.1892 0.0336 1.1340 1.2974 0.0108 0.9883 1.1387 0.0267 1.0734 1.1958 0.0082 0.9030 1.0507 0.0163 1.0429 1.1248
?= 0.5,?= 0.5, ?= 0.5 0.0397 1.4167 1.3403 0.0485 1.5791 1.4864 0.0108 1.3213 1.2244 0.0209 1.4070 1.2666 0.0025 1.2331 1.1576 0.0445 1.3825 1.1924 0.0109 1.1697 1.0806 0.0261 1.2261 1.1137
?= 0.9,?= 0.1, ?= 0.5 0.0022 1.3138 1.3528 0.0920 1.4003 1.4228 0.0171 1.2480 1.2641 0.0532 1.2843 1.3142 0.0192 1.2192 1.2223 0.0487 1.2226 1.2608 0.0009 1.1111 1.1593 0.0324 1.1110 1.1489
?= 0.1,?= 0.1, ?= 1 0.0092 1.4535 0.9748 0.0041 1.4908 0.9689 0.0050 1.3592 0.8997 0.0108 1.4146 0.9145 0.0022 1.3239 0.8900 0.0129 1.3358 0.8925 0.0043 1.2502 0.8470 0.0119 1.2833 0.8575
?= 0.1,?= 0.9, ?= 1 0.0421 2.2562 1.1278 0.0192 2.3265 1.1462 0.0379 1.9904 1.0257 0.0185 2.2569 1.0930 0.0301 1.9530 1.0063 0.0368 2.2224 1.0713 0.0113 1.8374 0.9588 0.0149 2.0900 1.0210
?= 0.5,?= 0.5, ?= 1 0.0791 2.9429 1.1710 0.0296 3.1322 1.1914 0.0416 2.6021 1.0770 0.0250 2.7101 1.1033 0.0334 2.4496 1.0386 0.0409 2.6748 1.0798 0.0145 2.3081 0.9957 0.0220 2.4363 1.0245
?= 0.9,?= 0.1, ?= 1 0.0718 2.7062 1.2374 0.0665 2.8324 1.2617 0.0625 2.4894 1.1512 0.0494 2.5623 1.1748 0.0466 2.3962 1.1273 0.0640 2.4301 1.1219 0.0156 2.2173 1.0745 0.0413 2.2588 1.0786
?= 0.1,?= 0.1, ?= 5 0.0116 7.2720 0.8837 0.0279 7.4165 0.8824 0.0119 6.7888 0.8526 0.0070 6.9501 0.8502 0.0285 6.6017 0.8402 0.0011 6.7534 0.8307 0.0169 6.2483 0.8011 0.0006 6.4253 0.8079
?= 0.1,?= 0.9, ?= 5 0.1451 11.3162 1.0522 0.0436 11.2147 1.0506 0.1048 10.1377 0.9675 0.0580 10.5085 0.9941 0.0779 9.7491 0.9463 0.0521 10.1101 0.9589 0.0438 9.1660 0.9021 0.0606 9.7366 0.9338
?= 0.5,?= 0.5, ?= 5 0.1845 14.2594 1.0796 0.1082 14.0754 1.0425 0.1399 12.7866 1.0133 0.1239 13.0227 1.0240 0.0804 12.2174 0.9767 0.1074 12.7060 0.9952 0.0521 11.5310 0.9499 0.1065 11.8259 0.9556
?= 0.9,?= 0.1, ?= 5 0.3143 14.2380 1.1727 0.1214 13.8720 1.1795 0.1687 12.8863 1.1125 0.1895 12.4636 1.0932 0.0959 12.1338 1.0723 0.1569 12.0389 1.0813 0.0613 11.0139 1.0149 0.0906 11.2782 1.0311
M.Mohammadipour, 2009, Appendix 8.H 323
Appendix 8.H Comparison of MASE of INARMA (unknown order)
with Benchmarks
In this appendix, the degree of improvement by using allINAR(1) over benchmarks,
in terms of the MASE is presented. The results are for the case where all points in
time are taken into account. The results for INAR(1) series are the same as the case
where the order is known (Table 8.E3).
Table 8.H1 for INARMA(0,0) series (unknown order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.8989 0.9346 0.9438 0.9497 0.9846 0.9895 0.9533 0.9877 0.9918 0.9647 0.9983 1.0023
?= 0.5 0.9777 0.9995 1.0027 0.9877 1.0071 1.0094 1.0007 1.0192 1.0212 1.0002 1.0192 1.0213
?= 0.7
0.9992 1.0112 1.0119 0.9941 1.0024 1.0030 1.0107 1.0185 1.0189 0.9813 0.9892 0.9898
?= 1 1.0102 1.0278 1.0287 0.9808 0.9963 0.9970 0.9691 0.9851 0.9860 0.9644 0.9804 0.9813
?= 3 0.9967 1.0056 1.0047 0.9643 0.9734 0.9727 0.9577 0.9668 0.9660 0.9472 0.9555 0.9548
?= 5 0.9688 0.9684 0.9658 0.9527 0.9502 0.9472 0.9745 0.9734 0.9704 0.9655 0.9646 0.9618
?= 20 1.0024 0.9362 0.9199 0.9682 0.9083 0.8932 0.9815 0.9182 0.9025 0.9589 0.8980 0.8826
Table 8.H2 for INMA(1) series (unknown order)
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9698 0.9904 0.9941 0.9978 1.0144 1.0163 0.9504 0.9654 0.9670 0.9823 0.9978 0.9996
?= 0.5,?= 0.5 0.8759 0.8927 0.8950 0.8897 0.9036 0.9045 0.9375 0.9506 0.9516 0.9129 0.9266 0.9277
?= 0.9,?= 0.5 0.7697 0.7873 0.7886 0.8624 0.8810 0.8830 0.8413 0.8577 0.8589 0.8366 0.8542 0.8556
?= 0.1,?= 1 0.9530 0.9710 0.9722 0.9773 0.9981 0.9995 0.9665 0.9859 0.9871 0.9507 0.9711 0.9725
?= 0.5,?= 1 0.9603 0.9792 0.9804 0.9388 0.9572 0.9585 0.9362 0.9551 0.9564 0.9309 0.9511 0.9527
?= 0.9,?= 1 0.9024 0.9149 0.9153 0.8884 0.9014 0.9019 0.8764 0.8893 0.8898 0.8510 0.8637 0.8642
?= 0.1,?= 3 0.9696 0.9759 0.9749 0.9783 0.9856 0.9845 0.9668 0.9749 0.9738 0.9763 0.9842 0.9832
?= 0.5,?= 3 0.9660 0.9673 0.9647 0.9809 0.9827 0.9803 0.9362 0.9357 0.9330 0.9452 0.9461 0.9437
?= 0.9,?= 3 0.9213 0.9113 0.9069 0.9048 0.8973 0.8932 0.9029 0.8975 0.8938 0.8894 0.8831 0.8792
?= 0.1,?= 5 0.9810 0.9737 0.9699 0.9807 0.9804 0.9773 0.9744 0.9719 0.9686 0.9629 0.9588 0.9553
?= 0.5,?= 5 0.9914 0.9749 0.9688 0.9574 0.9411 0.9349 0.9520 0.9357 0.9298 0.9302 0.9164 0.9108
?= 0.9,?= 5 0.9143 0.8865 0.8787 0.9442 0.9222 0.9143 0.8817 0.8614 0.8543 0.8907 0.8689 0.8616
BenchmarkINARMA MASEMASE /
BenchmarkINARMA MASEMASE /
M.Mohammadipour, 2009, Appendix 8.H 324
Table 8.H3 for INARMA(1,1) series (unknown order)
Parameters
?=?? ?=?? ?=?? ?=??
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
Cros
0.2
Cros
0.5
SBA
0.2
SBA
0.5
SBJ
0.2
SBJ
0.5
?= 0.1,?= 0.1,?= 0.5 0.9564 0.9255 0.9746 0.9860 0.9758 1.0042 0.9520 0.9050 0.9673 0.9696 0.9688 0.9943 0.9596 0.9101 0.9732 0.9727 0.9745 0.9966 0.9675 0.9129 0.9820 0.9791 0.9834 1.0073
?= 0.1,?= 0.9,?= 0.5 0.8336 0.7977 0.8562 0.8552 0.8565 0.8695 0.8102 0.7574 0.8293 0.8156 0.8309 0.8362 0.8445 0.7918 0.8633 0.8501 0.8648 0.8667 0.8176 0.7645 0.8372 0.8235 0.8389 0.8419
?= 0.5,?= 0.5,?= 0.5 0.7737 0.7894 0.7870 0.8257 0.7868 0.7790 0.8010 0.8205 0.8170 0.8591 0.8181 0.8113 0.7938 0.8065 0.8098 0.8465 0.8109 0.8028 0.7659 0.7789 0.7845 0.8219 0.7860 0.7835
?= 0.9,?= 0.1,?= 0.5 0.6969 0.9135 0.6423 0.5477 0.6321 0.4096 0.6809 0.8874 0.6396 0.5426 0.6305 0.4090 0.6586 0.8647 0.6216 0.5303 0.6131 0.3999 0.6662 0.8756 0.6332 0.5312 0.6245 0.4020
?= 0.1,?= 0.1,?= 1 1.0003 0.9461 1.0217 0.9973 1.0234 1.0131 0.9671 0.9087 0.9872 0.9589 0.9887 0.9790 0.9728 0.9171 0.9943 0.9667 0.9959 0.9841 0.9520 0.8961 0.9736 0.9462 0.9751 0.9657
?= 0.1,?= 0.9,?= 1 0.9270 0.9192 0.9378 0.9370 0.9377 0.8924 0.8607 0.8499 0.8751 0.8742 0.8757 0.8454 0.8591 0.8502 0.8720 0.8711 0.8724 0.8377 0.8571 0.8453 0.8716 0.8692 0.8722 0.8387
?= 0.5,?= 0.5,?= 1 0.8793 0.9351 0.8818 0.9033 0.8806 0.8009 0.8485 0.9001 0.8548 0.8773 0.8540 0.7808 0.8247 0.8759 0.8321 0.8553 0.8315 0.7631 0.8122 0.8603 0.8222 0.8438 0.8218 0.7548
?= 0.9,?= 0.1,?= 1 0.7358 0.9336 0.6482 0.4337 0.6307 0.3259 0.6975 0.9064 0.6154 0.4198 0.5992 0.3153 0.6958 0.9104 0.6192 0.4219 0.6034 0.3168 0.6667 0.8771 0.5981 0.4099 0.5832 0.3081
?= 0.1,?= 0.1,?= 5 1.0047 0.9456 0.9945 0.8630 0.9898 0.7973 0.9922 0.9364 0.9866 0.8635 0.9826 0.7998 0.9944 0.9432 0.9887 0.8656 0.9845 0.7980 0.9611 0.9065 0.9545 0.8336 0.9504 0.7730
?= 0.1,?= 0.9,?= 5 0.9546 0.9572 0.9260 0.7674 0.9175 0.6468 0.9096 0.9142 0.8823 0.7304 0.8741 0.6179 0.9105 0.9147 0.8864 0.7350 0.8785 0.6236 0.8864 0.8915 0.8620 0.7134 0.8543 0.6051
?= 0.5,?= 0.5,?= 5 0.9363 1.0064 0.8830 0.6927 0.8703 0.5520 0.8704 0.9329 0.8232 0.6460 0.8115 0.5161 0.8479 0.9078 0.8102 0.6411 0.7997 0.5138 0.8426 0.8981 0.8025 0.6302 0.7916 0.5048
?= 0.9,?= 0.1,?= 5 0.7286 0.9354 0.4676 0.2065 0.4342 0.1550 0.7346 0.9389 0.4548 0.2012 0.4215 0.1508 0.6944 0.8999 0.4409 0.1955 0.4090 0.1465 0.6703 0.8707 0.4273 0.1897 0.3968 0.1423
BenchmarkINARMA MASEMASE /
M.Mohammadipour, 2009, Appendix 8.I 325
Appendix 8.I Comparison of MASE of INARMA with Benchmarks
for Lead Time Forecasts
In this appendix, the lead time forecasts of an allINAR(1) method are compared to
those of the benchmarks methods in terms of MASE. The results are presented for
INARMA(0,0), INAR(1), INMA(1), and INARMA(1,1) series and include both
cases of ?= 3 and ?= 6.
Table 8.I1 of leadtime forecasts for INARMA(0,0) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.8692 0.9131 0.9178 0.9420 0.9748 0.9779 0.9692 0.9899 0.9917 0.9603 0.9773 0.9785
?= 0.5 0.9484 0.9840 0.9874 0.9739 0.9993 1.0013 0.9742 0.9990 1.0010 0.9559 0.9827 0.9849
?= 0.7
0.9755 1.0034 1.0054 0.9722 0.9968 0.9983 0.9622 0.9846 0.9860 0.9364 0.9581 0.9593
?= 1 0.9824 0.9999 1.0000 0.9629 0.9819 0.9824 0.9516 0.9714 0.9719 0.9247 0.9481 0.9491
?= 3 0.9624 0.9560 0.9512 0.9303 0.9263 0.9218 0.9121 0.9100 0.9058 0.8984 0.8963 0.8920
?= 5 0.9567 0.9282 0.9184 0.9299 0.9056 0.8967 0.9112 0.8863 0.8775 0.8923 0.8710 0.8625
?= 20 0.9552 0.7915 0.7606 0.9198 0.7657 0.7352 0.9044 0.7575 0.7286 0.8822 0.7413 0.7129
Table 8.I2 of leadtime forecasts for INMA(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9463 0.9858 0.9897 0.9682 0.9978 1.0003 0.9644 0.9914 0.9937 0.9506 0.9789 0.9812
?= 0.5,?= 0.5 0.9748 1.0102 1.0134 0.9535 0.9838 0.9864 0.9443 0.9735 0.9759 0.9173 0.9462 0.9486
?= 0.9,?= 0.5 0.9756 1.0186 1.0225 0.9587 0.9907 0.9933 0.9426 0.9727 0.9751 0.9066 0.9363 0.9388
?= 0.1,?= 1 0.9854 1.0073 1.0082 0.9612 0.9833 0.9843 0.9455 0.9688 0.9699 0.9249 0.9493 0.9505
?= 0.5,?= 1 0.9848 1.0053 1.0061 0.9577 0.9769 0.9774 0.9437 0.9646 0.9654 0.9120 0.9354 0.9364
?= 0.9,?= 1 0.9974 1.0177 1.0183 0.9619 0.9824 0.9830 0.9447 0.9630 0.9632 0.9163 0.9376 0.9382
?= 0.1,?= 3 0.9688 0.9579 0.9523 0.9391 0.9423 0.9386 0.9241 0.9209 0.9166 0.9010 0.9011 0.8972
?= 0.5,?= 3 0.9849 0.9757 0.9701 0.9481 0.9412 0.9358 0.9369 0.9330 0.9281 0.9062 0.9027 0.8980
?= 0.9,?= 3 0.9993 0.9771 0.9698 0.9592 0.9415 0.9344 0.9437 0.9346 0.9286 0.9103 0.9009 0.8949
?= 0.1,?= 5 0.9686 0.9468 0.9378 0.9397 0.9124 0.9029 0.9151 0.8965 0.8882 0.9009 0.8794 0.8708
?= 0.5,?= 5 0.9822 0.9528 0.9423 0.9451 0.9115 0.9014 0.9356 0.9060 0.8959 0.9077 0.8796 0.8701
?= 0.9,?= 5 0.9935 0.9527 0.9404 0.9514 0.9110 0.8990 0.9358 0.9018 0.8906 0.9120 0.8763 0.8653
BenchmarkINARMA MASEMASE / )( 3?l
BenchmarkINARMA MASEMASE / )( 3?l
M.Mohammadipour, 2009, Appendix 8.I 326
Table 8.I3 of leadtime forecasts with smoothing parameter 0.2
for INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9691 1.0016 1.0046 0.9605 0.9901 0.9927 0.9635 0.9896 0.9918 0.9461 0.9730 0.9753
?= 0.5,?= 0.5 0.9392 0.9627 0.9644 0.9229 0.9469 0.9489 0.9013 0.9267 0.9288 0.8690 0.8978 0.9002
?= 0.9,?= 0.5 0.8990 0.8433 0.8308 0.8421 0.8083 0.7986 0.8296 0.7975 0.7880 0.7843 0.7552 0.7468
?= 0.1,?= 1 0.9926 1.0157 1.0165 0.9528 0.9751 0.9760 0.9498 0.9743 0.9755 0.9222 0.9469 0.9481
?= 0.5,?= 1 0.9731 0.9895 0.9895 0.9413 0.9572 0.9572 0.9302 0.9459 0.9458 0.8970 0.9150 0.9152
?= 0.9,?= 1 0.9012 0.8115 0.7908 0.8522 0.7722 0.7543 0.8198 0.7565 0.7402 0.7881 0.7223 0.7063
?= 0.1,?= 3 0.9695 0.9666 0.9621 0.9391 0.9385 0.9343 0.9215 0.9226 0.9188 0.8995 0.8993 0.8953
?= 0.5,?= 3 0.9916 0.9694 0.9617 0.9533 0.9368 0.9297 0.9340 0.9227 0.9164 0.9070 0.8965 0.8905
?= 0.9,?= 3 0.8966 0.6667 0.6190 0.8333 0.6410 0.5952 0.8571 0.6316 0.6000 0.7975 0.6063 0.5748
?= 0.1,?= 5 0.9660 0.9418 0.9321 0.9221 0.9014 0.8931 0.9247 0.9028 0.8941 0.8955 0.8747 0.8664
?= 0.5,?= 5 0.9913 0.9520 0.9390 0.9533 0.9072 0.8942 0.9410 0.9081 0.8963 0.9097 0.8701 0.8580
?= 0.9,?= 5 0.8966 0.5778 0.5306 0.8571 0.5455 0.5000 0.8214 0.5349 0.5000 0.7857 0.5238 0.4889
Table 8.I4 of leadtime forecasts with smoothing parameter 0.5
for INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9137 0.9997 1.0076 0.8495 0.9461 0.9612 0.8498 0.9392 0.9513 0.8287 0.9197 0.9326
?= 0.5,?= 0.5 0.9021 0.9735 0.9739 0.8617 0.9443 0.9504 0.8390 0.9220 0.9271 0.8090 0.8950 0.9021
?= 0.9,?= 0.5 1.0239 0.6822 0.5505 1.0005 0.6886 0.5592 0.9720 0.6675 0.5411 0.9348 0.6491 0.5276
?= 0.1,?= 1 0.8763 0.9355 0.9231 0.8292 0.8940 0.8841 0.8287 0.8959 0.8863 0.8015 0.8664 0.8583
?= 0.5,?= 1 0.9283 0.9531 0.9171 0.8947 0.9200 0.8878 0.8806 0.9039 0.8682 0.8519 0.8800 0.8473
?= 0.9,?= 1 1.0284 0.5483 0.4237 1.0036 0.5384 0.4173 0.9687 0.5301 0.4097 0.9332 0.5023 0.3872
?= 0.1,?= 3 0.8164 0.7827 0.7180 0.7918 0.7595 0.6955 0.7806 0.7501 0.6889 0.7590 0.7305 0.6706
?= 0.5,?= 3 0.9383 0.8022 0.6979 0.8934 0.7717 0.6728 0.8767 0.7636 0.6670 0.8472 0.7448 0.6519
?= 0.9,?= 3 1.0400 0.3210 0.2407 1.0000 0.3165 0.2404 1.0000 0.3200 0.2400 0.9341 0.3033 0.2277
?= 0.1,?= 5 0.8048 0.7006 0.6078 0.7681 0.6762 0.5918 0.7750 0.6764 0.5888 0.7502 0.6559 0.5726
?= 0.5,?= 5 0.9238 0.7035 0.5795 0.8844 0.6729 0.5568 0.8755 0.6731 0.5566 0.8480 0.6448 0.5321
?= 0.9,?= 5 1.0400 0.2549 0.1926 1.0000 0.2449 0.1832 0.9583 0.2421 0.1811 0.9167 0.2366 0.1774
Table 8.I5 of leadtime forecasts with smoothing parameter 0.2 for
INARMA(1,1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.9599 0.9975 1.0010 0.9618 0.9911 0.9936 0.9600 0.9855 0.9876 0.9372 0.9661 0.9687
?= 0.1,?= 0.9,?= 0.5 0.9729 1.0118 1.0154 0.9548 0.9863 0.9890 0.9307 0.9603 0.9629 0.9014 0.9314 0.9339
?= 0.5,?= 0.5,?= 0.5 0.9638 0.9822 0.9833 0.9093 0.9333 0.9351 0.8896 0.9166 0.9186 0.8623 0.8886 0.8906
?= 0.9,?= 0.1,?= 0.5 0.8905 0.8351 0.8234 0.8429 0.7986 0.7881 0.8129 0.7814 0.7721 0.7725 0.7529 0.7452
?= 0.1,?= 0.1,?= 1 0.9896 1.0122 1.0133 0.9582 0.9806 0.9816 0.9460 0.9691 0.9701 0.9205 0.9448 0.9461
?= 0.1,?= 0.9,?= 1 1.0038 1.0170 1.0168 0.9593 0.9787 0.9790 0.9374 0.9554 0.9557 0.9109 0.9314 0.9320
?= 0.5,?= 0.5,?= 1 0.9807 0.9911 0.9900 0.9378 0.9517 0.9510 0.9083 0.9248 0.9246 0.8828 0.8981 0.8979
?= 0.9,?= 0.1,?= 1 0.8947 0.7926 0.7720 0.8531 0.7740 0.7558 0.8164 0.7451 0.7283 0.7833 0.7197 0.7038
?= 0.1,?= 0.1,?= 5 0.9732 0.9490 0.9400 0.9484 0.9245 0.9152 0.9223 0.8989 0.8900 0.9046 0.8841 0.8756
?= 0.1,?= 0.9,?= 5 0.9920 0.9504 0.9376 0.9518 0.9100 0.8984 0.9261 0.8909 0.8798 0.9080 0.8755 0.8648
?= 0.5,?= 0.5,?= 5 0.9881 0.9263 0.9112 0.9412 0.8922 0.8781 0.9200 0.8737 0.8606 0.8924 0.8488 0.8359
?= 0.9,?= 0.1,?= 5 0.9310 0.5625 0.5294 0.8621 0.5435 0.5102 0.8276 0.5333 0.5000 0.7857 0.5116 0.4783
BenchmarkINARMA MASEMASE / )( 3?l
BenchmarkINARMA MASEMASE / )( 3?l
BenchmarkINARMA MASEMASE / )( 3?l
M.Mohammadipour, 2009, Appendix 8.I 327
Table 8.I6 of leadtime forecasts with smoothing parameter 0.5 for
INARMA(1,1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.8966 0.9951 1.0095 0.8618 0.9532 0.9666 0.8410 0.9323 0.9458 0.8194 0.9166 0.9328
?= 0.1,?= 0.9,?= 0.5 0.8946 0.9929 1.0052 0.8521 0.9468 0.9584 0.8340 0.9284 0.9387 0.8031 0.8964 0.9079
?= 0.5,?= 0.5,?= 0.5 0.9370 0.9940 0.9853 0.8716 0.9462 0.9462 0.8430 0.9201 0.9190 0.8263 0.8974 0.8935
?= 0.9,?= 0.1,?= 0.5 1.0438 0.6973 0.5664 0.9928 0.6744 0.5470 0.9687 0.6647 0.5385 0.9267 0.6488 0.5259
?= 0.1,?= 0.1,?= 1 0.8902 0.9511 0.9374 0.8435 0.9077 0.8970 0.8267 0.8905 0.8809 0.8091 0.8737 0.8649
?= 0.1,?= 0.9,?= 1 0.9096 0.9439 0.9177 0.8608 0.9032 0.8812 0.8485 0.8884 0.8635 0.8222 0.8629 0.8402
?= 0.5,?= 0.5,?= 1 0.9459 0.9530 0.9039 0.8983 0.9055 0.8584 0.8810 0.8928 0.8493 0.8492 0.8602 0.8193
?= 0.9,?= 0.1,?= 1 1.0289 0.5346 0.4123 0.9976 0.5324 0.4109 0.9667 0.5169 0.3985 0.9323 0.5015 0.3862
?= 0.1,?= 0.1,?= 5 0.8252 0.7119 0.6190 0.8066 0.6912 0.5989 0.7825 0.6771 0.5887 0.7676 0.6695 0.5832
?= 0.1,?= 0.9,?= 5 0.8871 0.6957 0.5817 0.8536 0.6805 0.5728 0.8240 0.6598 0.5556 0.8071 0.6483 0.5468
?= 0.5,?= 0.5,?= 5 0.9490 0.6764 0.5492 0.8956 0.6520 0.5311 0.8804 0.6434 0.5246 0.8565 0.6216 0.5054
?= 0.9,?= 0.1,?= 5 1.0385 0.2523 0.1901 1.0000 0.2475 0.1852 1.0000 0.2424 0.1818 0.9167 0.2292 0.1732
Table 8.I7 of leadtime forecasts for INARMA(0,0) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.3 0.9262 0.9793 0.9846 0.8890 0.9311 0.9352 0.9549 0.9871 0.9898 0.9510 0.9776 0.9798
?= 0.5 0.9864 1.0281 1.0313 0.9468 0.9776 0.9796 0.9496 0.9780 0.9796 0.9138 0.9451 0.9471
?= 0.7
0.9598 1.0038 1.0066 0.9614 0.9907 0.9919 0.9425 0.9644 0.9648 0.8939 0.9233 0.9247
?= 1 0.9710 1.0065 1.0077 0.9321 0.9571 0.9572 0.9145 0.9361 0.9356 0.8751 0.8965 0.8962
?= 3 0.9355 0.9117 0.9022 0.8895 0.8703 0.8619 0.8704 0.8593 0.8518 0.8250 0.8159 0.8091
?= 5 0.9362 0.8819 0.8666 0.8871 0.8368 0.8226 0.8589 0.8142 0.8004 0.8114 0.7758 0.7635
?= 20 0.9167 0.7097 0.6667 0.8750 0.6774 0.6563 0.8696 0.6667 0.6250 0.7826 0.6000 0.5806
Table 8.I8 of leadtime forecasts for INMA(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9733 1.0270 1.0317 0.9555 0.9873 0.9895 0.9460 0.9688 0.9699 0.9049 0.9360 0.9382
?= 0.5,?= 0.5 0.9523 1.0027 1.0072 0.9524 0.9850 0.9874 0.9383 0.9763 0.9792 0.8847 0.9200 0.9227
?= 0.9,?= 0.5 0.9755 1.0382 1.0438 0.9576 0.9981 1.0012 0.9222 0.9570 0.9594 0.8849 0.9195 0.9219
?= 0.1,?= 1 0.9687 0.9911 0.9911 0.9277 0.9523 0.9526 0.9085 0.9300 0.9298 0.8799 0.9028 0.9028
?= 0.5,?= 1 0.9863 1.0066 1.0061 0.9355 0.9576 0.9575 0.9192 0.9416 0.9414 0.8598 0.8835 0.8837
?= 0.9,?= 1 1.0103 1.0168 1.0145 0.9382 0.9550 0.9541 0.9144 0.9405 0.9405 0.8626 0.8844 0.8842
?= 0.1,?= 3 0.9560 0.9298 0.9205 0.8952 0.8736 0.8655 0.8626 0.8468 0.8394 0.8220 0.8114 0.8048
?= 0.5,?= 3 0.9711 0.9415 0.9315 0.9095 0.8958 0.8877 0.8734 0.8574 0.8500 0.8389 0.8249 0.8177
?= 0.9,?= 3 0.9943 0.9640 0.9526 0.9231 0.9063 0.8971 0.8925 0.8704 0.8609 0.8399 0.8205 0.8119
?= 0.1,?= 5 0.9580 0.9208 0.9069 0.8882 0.8398 0.8256 0.8529 0.8147 0.8021 0.8188 0.7821 0.7696
?= 0.5,?= 5 0.9740 0.9160 0.8988 0.9034 0.8376 0.8217 0.8770 0.8353 0.8218 0.8360 0.7921 0.7788
?= 0.9,?= 5 0.9902 0.9220 0.9034 0.9071 0.8420 0.8254 0.8766 0.8191 0.8032 0.8424 0.7897 0.7751
BenchmarkINARMA MASEMASE / )( 3?l
BenchmarkINARMA MASEMASE / )( 6?l
BenchmarkINARMA MASEMASE / )( 6?l
M.Mohammadipour, 2009, Appendix 8.I 328
Table 8.I9 of leadtime forecasts with smoothing parameter 0.2
for INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.9080 0.9615 0.9663 0.9518 0.9839 0.9863 0.9521 0.9779 0.9794 0.9064 0.9400 0.9424
?= 0.5,?= 0.5 0.9824 1.0279 1.0320 0.9527 0.9852 0.9877 0.9140 0.9488 0.9514 0.8656 0.9005 0.9032
?= 0.9,?= 0.5 1.0092 0.9926 0.9827 0.9498 0.9300 0.9211 0.9263 0.9104 0.9027 0.8765 0.8602 0.8525
?= 0.1,?= 1 0.9865 1.0081 1.0078 0.9450 0.9617 0.9612 0.9151 0.9374 0.9374 0.8687 0.8929 0.8933
?= 0.5,?= 1 1.0050 1.0159 1.0145 0.9551 0.9701 0.9692 0.9125 0.9373 0.9378 0.8690 0.8939 0.8942
?= 0.9,?= 1 1.0143 0.9115 0.8898 0.9557 0.8725 0.8532 0.9240 0.8675 0.8519 0.8679 0.8054 0.7894
?= 0.1,?= 3 0.9546 0.9386 0.9306 0.9051 0.8913 0.8837 0.8688 0.8543 0.8468 0.8307 0.8222 0.8158
?= 0.5,?= 3 0.9990 0.9675 0.9572 0.9385 0.9149 0.9059 0.9088 0.8915 0.8833 0.8642 0.8515 0.8437
?= 0.9,?= 3 1.0204 0.7576 0.7246 0.9388 0.7419 0.7077 0.9375 0.7258 0.6923 0.8723 0.6949 0.6613
?= 0.1,?= 5 0.9512 0.9068 0.8926 0.8869 0.8411 0.8280 0.8570 0.8155 0.8028 0.8227 0.7847 0.7726
?= 0.5,?= 5 1.0021 0.9556 0.9397 0.9286 0.8686 0.8528 0.9026 0.8527 0.8380 0.8573 0.8159 0.8026
?= 0.9,?= 5 1.0000 0.7143 0.6250 1.0000 0.7143 0.6250 0.8000 0.5714 0.5000 0.8000 0.5714 0.5714
Table 8.I10 of leadtime forecasts with smoothing parameter 0.5
for INAR(1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.5 0.8561 0.9604 0.9661 0.8196 0.9151 0.9187 0.7849 0.8858 0.8931 0.7399 0.8464 0.8567
?= 0.5,?= 0.5 0.9043 1.0091 1.0141 0.8535 0.9527 0.9551 0.8062 0.9129 0.9190 0.7588 0.8601 0.8668
?= 0.9,?= 0.5 1.0597 0.8030 0.6621 1.0240 0.7841 0.6514 1.0063 0.7790 0.6469 0.9540 0.7399 0.6155
?= 0.1,?= 1 0.8545 0.9039 0.8707 0.7854 0.8350 0.8092 0.7494 0.8087 0.7858 0.7065 0.7673 0.7476
?= 0.5,?= 1 0.8989 0.9150 0.8729 0.8459 0.8734 0.8343 0.8072 0.8468 0.8139 0.7685 0.8075 0.7742
?= 0.9,?= 1 1.0649 0.6216 0.4873 1.0164 0.6048 0.4736 1.0082 0.6263 0.4909 0.9450 0.5712 0.4472
?= 0.1,?= 3 0.7493 0.6930 0.6140 0.7068 0.6516 0.5799 0.6679 0.6167 0.5491 0.6462 0.6053 0.5406
?= 0.5,?= 3 0.8756 0.7363 0.6295 0.8082 0.6997 0.6050 0.7869 0.6843 0.5917 0.7447 0.6508 0.5625
?= 0.9,?= 3 1.0417 0.3817 0.2874 1.0222 0.3770 0.2822 1.0000 0.3719 0.2795 0.9535 0.3596 0.2715
?= 0.1,?= 5 0.7364 0.6034 0.5083 0.6842 0.5666 0.4794 0.6585 0.5442 0.4584 0.6361 0.5287 0.4476
?= 0.5,?= 5 0.8716 0.6526 0.5337 0.7972 0.5955 0.4873 0.7758 0.5884 0.4812 0.7404 0.5633 0.4614
?= 0.9,?= 5 1.0000 0.2941 0.2273 1.0000 0.3125 0.2381 1.0000 0.2667 0.1905 1.0000 0.2667 0.2105
Table 8.I11 of leadtime forecasts with smoothing parameter 0.2 for
INARMA(1,1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.9190 0.9654 0.9696 0.9551 0.9896 0.9920 0.9387 0.9724 0.9748 0.8993 0.9321 0.9346
?= 0.1,?= 0.9,?= 0.5 1.0104 1.0541 1.0576 0.9574 0.9895 0.9918 0.9131 0.9504 0.9532 0.8746 0.9094 0.9120
?= 0.5,?= 0.5,?= 0.5 1.0190 1.0438 1.0452 0.9688 0.9964 0.9980 0.9232 0.9541 0.9562 0.8675 0.9024 0.9051
?= 0.9,?= 0.1,?= 0.5 1.0060 0.9521 0.9398 0.9600 0.9113 0.9004 0.9176 0.8894 0.8806 0.8607 0.8460 0.8388
?= 0.1,?= 0.1,?= 1 0.9892 1.0070 1.0065 0.8959 0.8533 0.8398 0.9082 0.9370 0.9377 0.8690 0.8946 0.8951
?= 0.1,?= 0.9,?= 1 1.0118 1.0259 1.0245 0.9382 0.9564 0.9557 0.9108 0.9328 0.9327 0.8610 0.8808 0.8805
?= 0.5,?= 0.5,?= 1 1.0204 1.0394 1.0385 0.9611 0.9754 0.9741 0.9162 0.9332 0.9325 0.8625 0.8835 0.8833
?= 0.9,?= 0.1,?= 1 1.0111 0.9258 0.9037 0.9531 0.8750 0.8556 0.9201 0.8685 0.8520 0.8585 0.8062 0.7913
?= 0.1,?= 0.1,?= 5 0.9550 0.9088 0.8948 0.8946 0.8574 0.8447 0.8537 0.8125 0.8000 0.8299 0.7949 0.7830
?= 0.1,?= 0.9,?= 5 0.9852 0.9053 0.8866 0.9208 0.8598 0.8429 0.8834 0.8362 0.8213 0.8365 0.7916 0.7778
?= 0.5,?= 0.5,?= 5 1.0278 0.9487 0.9250 0.9444 0.8718 0.8718 0.9167 0.8462 0.8250 0.8571 0.8108 0.7895
?= 0.9,?= 0.1,?= 5 1.0000 0.6250 0.6250 1.0000 0.7143 0.6250 0.8000 0.5714 0.5000 0.8000 0.5714 0.5714
BenchmarkINARMA MASEMASE / )( 6?l
BenchmarkINARMA MASEMASE / )( 6?l
BenchmarkINARMA MASEMASE / )( 6?l
M.Mohammadipour, 2009, Appendix 8.I 329
Table 8.I12 of leadtime forecasts with smoothing parameter 0.5 for
INARMA(1,1) series
Parameters
?=?? ?=?? ?=?? ?=??
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
Croston
?=?.?
SBA
?=?.?
SBJ
?=?.?
?= 0.1,?= 0.1,?= 0.5 0.8436 0.9460 0.9554 0.8118 0.9176 0.9250 0.7733 0.8817 0.8922 0.7361 0.8400 0.8521
?= 0.1,?= 0.9,?= 0.5 0.9077 1.0043 1.0022 0.8329 0.9281 0.9292 0.7716 0.8820 0.8923 0.7343 0.8368 0.8437
?= 0.5,?= 0.5,?= 0.5 0.9502 1.0183 1.0071 0.8870 0.9706 0.9628 0.8296 0.9157 0.9126 0.7871 0.8756 0.8742
?= 0.9,?= 0.1,?= 0.5 1.0448 0.7635 0.6331 1.0426 0.7649 0.6361 0.9998 0.7685 0.6390 0.9469 0.7337 0.6099
?= 0.1,?= 0.1,?= 1 0.8417 0.8799 0.8498 0.6944 0.5712 0.4790 0.7451 0.8140 0.7957 0.7078 0.7706 0.7513
?= 0.1,?= 0.9,?= 1 0.8727 0.9003 0.8607 0.7934 0.8331 0.8001 0.7713 0.8107 0.7805 0.7290 0.7629 0.7318
?= 0.5,?= 0.5,?= 1 0.9169 0.9273 0.8766 0.8656 0.8799 0.8314 0.8179 0.8400 0.7997 0.7738 0.7972 0.7581
?= 0.9,?= 0.1,?= 1 1.0504 0.6051 0.4690 1.0216 0.6003 0.4673 0.9949 0.6140 0.4789 0.9411 0.5804 0.4538
?= 0.1,?= 0.1,?= 5 0.7387 0.6121 0.5152 0.6994 0.5795 0.4883 0.6619 0.5486 0.4636 0.6480 0.5389 0.4545
?= 0.1,?= 0.9,?= 5 0.8082 0.5982 0.4868 0.7500 0.5670 0.4637 0.7260 0.5548 0.4544 0.6815 0.5287 0.4350
?= 0.5,?= 0.5,?= 5 0.9024 0.6379 0.5139 0.8293 0.5965 0.4789 0.8049 0.5789 0.4714 0.7500 0.5556 0.4478
?= 0.9,?= 0.1,?= 5 1.0000 0.2941 0.2174 1.0000 0.3125 0.2273 1.0000 0.2500 0.1905 1.0000 0.2667 0.2000
BenchmarkINARMA MASEMASE / )( 6?l
M.Mohammadipour, 2009, Appendix 9.A 330
Appendix 9.A INARMA(0,0), INAR(1), INMA(1) and INARMA(1,1)
Series of 16,000 Series
In this appendix, the identified INARMA series of 16,000 data set are separated and
forecasted with the known INARMA models.
Investigating the estimated parameter of the INARMA(0,0) process (?), we found
that in general ? is close to 0.1 (the average is 0.1953 and 69.26 percent are between
0 and 0.15).
The estimated autoregressive parameter of the INAR(1) process, ?, is close to 0.2
(the average is 0.2460 and 52.94 percent are between 0.1 and 0.3) and the estimated
innovation parameter, ?, is around 0.5 (the average is 0.3562 and 97.06 percent are
between 0 and 1).
Looking at the estimated parameter of the INMA(1) process (?,?) reveals that in
general, ? is close to zero (the average is 0.0898 and 46.29 percent are between 0
and 0.1) and ? is around 0.3 (the average is 0.3782 and 55.56 percent are between 0.2
and 0.4).
In general, the estimated autoregressive parameter of an INARMA(1,1) process is in
the range 0.1