IMPLEMENTATION OF GENETIC ALGORITHM IN MODEL
IDENTIFICATION OF BOX-JENKINS METHODOLOGY
MOHD ZULARIFFIN MD MAAROF
UNIVERSITI TEKNOLOGI MALAYSIA
IMPLEMENTATION OF GENETIC ALGORITHM IN MODEL
IDENTIFICATION OF BOX-JENKINS METHODOLOGY
MOHD ZULARIFFIN BIN MD MAAROF
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Master of Science (Mathematics)
Faculty Of Science
Universiti Teknologi Malaysia
JUN 2013
iii
Dedicated to:
My beloved parents,
Md Maarof Mardi, Ramlah Abdul Latif
My supportive siblings,
Zulfadli, Noradilah, Nor Rislah, Zulkhairi, Zul Amin
My dedicated lecturers,
My endless spirits
and all my friends.
This is for you.
iv
ACKNOWLEDGEMENT
In the name of Allah, the Most Gracious and Most Merciful, all praise to
Allah SWT, the Almighty, for His love has given me strength, perseverance,
diligence and satisfaction in completing this project.
First and foremost, I would like to express my deepest gratitude especially to
my supervisor, Prof. Dr. Zuhaimy Hj. Ismail who had taken a lot of effort to
meticulously go through my work and come out with helpful suggestions. Not
forgotten, million appreciations for my co-supervisor Dr. Norhisham Bakhary, and
Dr. Noor Hazarina Hashim for their valuable critics and advices.
I would like to express my appreciation to Ministry of Science Technology
and Innovation (MOSTI), MyBrain Programme and Research University Grant
(RUG) for supporting the scholarship along my study.
Besides that, I also would like to acknowledge my special thanks to my
supportive friends, Ezza Syuhada Sazali, Lili Ayu Wulandari and Siti Khadijah
Mariam for their suggestions, comment and moral support. Their efforts are much
appreciated. May Allah bless all of you.
Finally, I would like to express my greatest gratitude to my beloved family
for their unstinting support and prayer. Without the family members support and
prayer, this project would have been difficult at best. Thank You.
v
ABSTRACT
During the past several decades, a considerable amount of studies have been
carried out on time series and in particular the Box-Jenkins (BJ) method. As with all
techniques of statistical analysis, the conclusions of time series analysis are critically
dependent on the assumptions underlying the analysis and BJ is a commonly used
forecasting method that can yield highly accurate forecasts for certain types of
data. Genetic Algorithm (GA) is a heuristic method of optimization. This study
presents the study on developing an extrapolative BJ model with the use of GA
method to produce forecasting models using time series data. BJ method has a cycle
of four phases, the data transformation phase for model identification, parameter
estimation, model diagnostic checking or validation, and finally producing the
forecast. Although many researchers and practitioners have concentrated in the
parameter estimation part of BJ model, the most crucial stage in building the model
is in the data transformation and model identification where any false identification
will lead to assuming a wrong model and will increase in the cost of re-
identification. Hence, using GA a subset of artificial intelligence methods was
introduced into the process of BJ to solve the problem in the model identification and
parameter estimation phase. The data used in this study are the monthly data of
international tourists arrival into Malaysia from 1990 to 2011. This is a case study
in the implementation of GA-BJ model. The result from this study may be divided
into two main parts, namely the result for the in-sample data (fitted model) and out-
sample data (forecast model). The analysis shows that the out-sample values using
GA-BJ model gives better forecast accuracy than the out-sample values for BJ
model. This shows that the combination of BJ and GA methods gives a more
accurate model than using a single method for forecasting. This study concludes that
GA method can be an alternative way in identifying the right order of component in
BJ model.
vi
ABSTRAK
Dalam beberapa dekad yang lalu, sejumlah besar kajian telah dijalankan ke
atas siri masa dan khususnya kaedah Box-Jenkins (BJ). Seperti semua teknik analisis
statistik, kesimpulan analisis siri masa adalah amat bergantung kepada andaian
analisis. BJ adalah satu kaedah yang lazim digunakan yang boleh menghasilkan
ramalan yang sangat tepat untuk sesetengah jenis data. Kajian ini membentangkan
hasil kajian kaedah ekstrapolatif model Box-Jenkins (BJ) bagi menghasilkan model
Univariat dengan menggunakan data siri masa. Kaedah BJ mempunyai empat fasa
utama iaitu model identifikasi, model penaksiran, model pengesahan, dan model
peramalan. Walaupun banyak penyelidik dan pengamal telah tertumpu di bahagian
anggaran parameter model BJ, peringkat yang paling penting dalam membina model
adalah dalam transformasi data dan pengenalan model jika apa-apa pengenalan palsu
akan membawa kepada andaian model yang salah dan akan meningkatkan kos
semula membina model pengenalan. Oleh itu, dalam kajian ini, model algoritma
genetik (GA) adalah subset bagi kaedah kepintaran tiruan yang diperkenalkan untuk
menyelesaikan masalah yang dihadapi di fasa pertama dan kedua iaitu model
identifikasi dan model penaksiran. Data yang digunakan dalam kajian ini adalah data
bulanan pelancong antarabangsa melawat Malaysia mulai tahun 1990 sehingga 2011.
Ini adalah kajian kes dalam implementasi model GA-BJ. Hasil analisis kajian ini
dibahagikan kepada dua bahagian iaitu sampel dalam (model ujian) dan sampel luar
(model ramalan). Di akhir kajian ini, model GA-BJ bagi sampel luar lebih tepat dan
mempunyai ralat yang lebih kecil berbanding model asas iaitu model BJ bagi sampel
luar. Ini menunjukkan bahawa model kombinasi kaedah BJ dan GA menghasilkan
model ramalan yang lebih tepat berbanding menggunakan hanya satu model.
Kesimpulannya, kajian ini menunjukkan bahawa kaedah GA boleh menjadi kaedah
alternatif bagi mengenalpasti komponen model pengenalan BJ yang betul.
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TABLE OF CONTENTS
CHAPTER TITLE PAGE
DECLARATION ii
DEDICATION iii
ACKNOWLEDGMENT iv
ABSTRACT v
ABSTRAK vi
TABLE OF CONTENT vii
LIST OF TABLES xii
LIST OF FIGURES xiv
LIST OF SYMBOLS xvii
LIST OF ABBREVIATIONS xix
LIST OF APPENDIXES xxi
1 INTRODUCTION
1.1 Introduction 1
1.2 Background of Problem 2
1.3 Statement of Problem 3
1.4 Research Question 4
1.5 Objective of the Study 4
1.6 Scope of the Study 5
1.7 Research Contribution 5
1.8 Research Data 6
1.9 Thesis Plan 6
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2 LITERATURE REVIEW
2.1 Introduction 7
2.2 Review of Box-Jenkins Modelling in Tourism
Forecasting Demand 8
2.3 Review of Identification Method in Box-Jenkins
Procedure 10
2.4 Genetic Algorithm 11
2.5 Genetic Operator 12
2.6 Encoding 12
2.6.1 Binary Encoding 12
2.6.2 Real Number Encoding 13
2.6.3 Encoding Values 14
2.7 Fitness Function 14
2.8 Selection 15
2.9 Previous Works on Modelling Tourist arrival Using
Box-Jenkins Model 16
2.10 Recent Works on Modification of Box-Jenkins Model 19
2.11 Summary and Conclusion 21
3 RESEARCH METHODOLOGY
3.1 Introduction 22
3.2 Time Series Analysis : Box-Jenkins Model 22
3.3 Preparation Phase 24
3.3.1 Stationary of Mean 24
3.3.2 Stationary of Variance 25
3.3.3 F-test Variances 25
3.3.4 Seasonality 26
3.3.5 Portmanteau Tests 27
3.4 Model Identification 28
ix
3.4.1 Theoretical Autocorrelation Coefficient 29
3.4.2 Autocorrelation Function (ACF) 30
3.4.3 Partial Autocorrelation Function (PACF) 31
3.4.4 Standard Error 32
3.4.5 Test Hypothesis 32
3.5 Model Estimation 33
3.5.1 General Least Squares Algorithm 33
3.5.2 Alternative form for the Algorithm 35
3.6 Model Checking/ Diagnostic Model 36
3.6.1 Method for Test Normality 37
3.6.2 Methods for Uncorrelated Test (Using
Autocorrelation Test) 38
3.7 Forecasting 39
3.7.1 Backward Shift Operator 39
3.7.2 Differencing Operator 40
3.7.3 Autoregressive Model (AR) 40
3.7.3.1 Example of AR(1) One Step Ahead 41
3.7.4 Moving Average Model (MA) 42
3.7.4.1 Example of MA(1) One Step Ahead 42
3.7.5 Autoregressive Moving Average
Model (ARMA) 43
3.7.5.1 Example of ARMA(1,1) One Step Ahead 43
3.7.6 Seasonal Backward Shift Operator
Models 44
3.7.7 Seasonal Autoregressive Integrated
Moving Average (SARIMA) 45
3.7.8 Forecast Accuracy 46
3.8 Model Identification by Using Genetic Algorithms 47
3.9 Roulette Wheel Method 50
3.10 Crossover 51
3.10.1 Method of Single Point Crossover 52
3.11 Mutation 53
3.12 Elitism 53
3.13 Convergence Criterion 54
x
4 RESULTS AND DISCUSSION I
4.1 Introduction 55
4.1.1 Preliminary Stage : Data Analysis 56
4.1.2 Stationary of Variance 59
4.1.3 Stationary of the Mean 61
4.1.4 Seasonality Identification 64
4.2 Identification Phase 66
4.3 Estimation Phase 72
4.4 Diagnostic Phase 75
4.4.1 Follow Normal Distribution 75
4.4.2 Uncorrelated Errors 76
4.4.3 The Mean is Zero and Variance is Constant 78
4.5 Forecasting Phase 80
4.6 Summary 82
5 RESULTS AND DISCUSSION II
5.1 Introduction 83
5.2 Case Study for the Method of Proposed Model 84
5.3 Proposed Identification Order of SARIMA Model 85
5.4 General Proposed SARIMA Model 88
5.5 Proposed Method in Estimating Parameters 90
5.6 Objective Function 92
5.7 Chromosome Encoded 98
5.8 Genetic Algorithm Operator 99
5.8.1 Selection Calculation 102
5.8.2 Uniform Crossover and Mutation
Calculation 102
5.8.3 Convergence Criterion 104
5.9 Comparison Traditional Model and Proposed Model 104
5.10 Summary 111
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6 CONCLUSION
6.1 Summary 115
6.2 Conclusion 116
6.3 Suggestion for Further Research 116
REFERENCES 118
xii
LIST OF TABLES
TABLE NO. TITLE PAGE
2.1 Binary encoding 13
2.2 Real numbers encoding 13
2.3 Encoding value 14
3.2 Primary distinguishing characteristics of theoretical ACF's 29
and PACF's for stationary process
4.1 The international tourist arrival time series data to Malaysia 57
4.2 F-Test Two-Sample for Variances 60
4.3 First non-seasonal differenced series of monthly 63
international tourist arrival
4.4 Identified tentative model of monthly international . 72
tourist arrival to Malaysia
4.5 Tentative model coefficient 73
4.6 Tentative model for modeling monthly international tourist arrival 73
4.7 Normality of tentative ARIMA/ SARIMA Model 75
xiii
4.8 Uncorrelated error of tentative model 76
4.9 Forecast accuracy for fitted SARIMA models 81
4.10 Forecast accuracy for forecast SARIMA models 81
5.1 Maximum number of order MA and SMA models 86
5.2 Maximum number of order AR and SAR models 88
5.3 Maximum number of the order SARIMA component models 88
5.4 Mean square error of forecast Model 97
5.5 Fitness function of GA-SARIMA Model 98
5.6 Forecast accuracy for benchmark Fitted Model 104
5.7 Forecast accuracy for benchmark forecast model 106
5.8 Several MSE for proposed fitted mode 108
5.9 Several MSE for proposed forecast model 109
5.10 Comparison forecast accuracy for in-sample international tourist
arrival model 111
5.11 Comparison forecast accuracy for out-sample international 112
tourist arrival model
5.12 Comparison between BJ and GA-BJ procedures 114
xiv
LIST OF FIGURES
NO. OF FIGURES TITLE PAGE
3.1 The iterative BJ modelling strategy 23
3.4 Research flow of Genetic Algorithm procedure 49
3.5 Roulette Wheel selection 50
3.6 Single point crossover 52
4.1 Time series plot of international tourist arrival to Malaysia 57
4.2 ACF of Monthly international tourist arrival to Malaysia 58
4.3 Original series of international tourist arrival to Malaysia 59
4.4 First non-seasonal differenced series of monthly 62
tourist arrival data
4.5 ACF function for differenced series of 64
monthly tourist arrival
4.6 ACF for seasonal differenced series of monthly 65
tourist arrival to Malaysia
xv
4.7 ACF and PACF for non-seasonal differenced series 70
of monthly international tourist arrival to Malaysia,
ARIMA(0,1,0)
4.8 ACF and PACF for seasonal differenced series of 71
monthly international tourist arrival to Malaysia,
SARIMA(0,1,0)(0,1,0)12
4.9 Residual analysis of ACF and PACF plot 78
4.10 Residual analysis for SARIMA(1,1,1)(1,0,1)12
79
4.11 Residual analysis for SARIMA(1,1,2)(1,0,1)12
79
4.12 Forecast SARIMA model of monthly international 82
tourist arrival to Malaysia
5.1 ACF for first differenced international tourist arrival data 85
5.2 PACF for first differenced international tourist arrival data 87
5.3 Process flow of estimating parameters of SARIMA model 91
5.4 Genetic Algorithm Architecture for the Proposed 93
SARIMA-GA model
5.5 Chromosome encoded for proposed model 99
5.6 Fitted benchmark model of international tourist
arrival to Malaysia 105
5.7 Forecast benchmark model of international tourist arrival
to Malaysia 107
xvi
5.8 Fitted proposed model of international tourist arrival 109
to Malaysia
5.9 Forecast proposed model of international tourist arrival 110
to Malaysia
5.10 In-sample benchmark model and proposed international
tourist arrival model for monthly to Malaysia. 112
5.11 Out-sample benchmark and proposed model for monthly
international tourist arrival to Malaysia 113
xvii
LIST OF SYMBOLS
te - The residual
p - The order of autoregressive model
q - The order of moving average model
P - The order of seasonal autoregressive model
Q - The order of seasonal moving average model
t - time
xf - Value of fitness function
x - Number of chromosome
n - Number of observation in the time series
tz - The number of monthly international tourist arrival
tz - The estimate numbers of international tourist arrival
0H - Hypothesis one
1H - Hypothesis two
2
1s - Larger variance
2
2s - Smaller variance
r - Correlation coefficient
y - The mean of the time series
h - Maximum number of lag
k - the time lag
I - The difference of seasonal nor non-seasonal
kk - Partial autocorrelation coefficient
- population size
- Parameter for autoregressive model
- Parameter for moving average model
xviii
- Parameter for seasonal moving average model
- Parameter for seasonal moving average model
B - Backward shift operator
tw - The mean of difference time series data
tw - The difference of time series data
y
- Partial differential with respect to
y
- Partial differential with respect to
xix
LIST OF ABBREVIATIONS
ARIMA - Autoregressive integrated moving average
SARIMA - Seasonal autoregressive integrated moving
average
AR - Autoregressive model
MA - Moving average model
ARMA - Autoregressive moving average model
GA - Genetic algorithm
BJ - Box Jenkins
GA-BJ - Genetic algorithm- Box Jenkins model
MSE - Mean square error
MAPE - Mean absolute percentage error
MAE - Mean absolute error
GA-SARIMA - Genetic algorithm-seasonal integrated moving
average model
GA-ARIMA - Genetic algorithm- autoregressive integrated
moving average model
SE - Standard error
ARFIMA - Autoregressive fractionally integrated moving
average
ACF - Autocorrelation function
PACF - Partial autocorrelation function
FPE - Final prediction error
MEV - Minimum eigenvalue vector
MDL - Maximum distributed length
AIC - Akaike information criterion
AI - Artificial intelligence
ANN - Artificial neural network
VAR - Vector autoregressive model
ARDL - Autoregressive distributed lag
xx
STSM - Structural time series model
TVP - Time varying parameter
GFS - Genetic fuzzy system
xxi
LIST OF APPENDICES
APPENDIX TITLE PAGE
A 1) GA-BJ Code for identifying the order 121
2) GA-BJ Code for estimating the order 128
3) Crossover fitted codes 133
4) Mutation fitted codes 134
5) Roulette Wheel codes 135
6) Monthly international tourist arrival time series data 137
B First differenced series of monthly international tourist arrival 140
1
CHAPTER 1
INTRODUCTION
1.1 Introduction
Every definition of forecasting defined it as the process of predicting the
future by organizing the past information. It is also described as a method to predict
future events. Forecasting can be applied in areas as such, forecasting electrical
demand, water demand, sales of product demand, tourisms demand and also for
government policy making. For example, hotels management mostly use forecasting
to determine operational requirements. Furthermore, it can helps in reducing the risk
of decisions and the cost of expenditure for future planning.
The methods of forecasting can be categorized into two major groups which
are qualitative methods and quantitative methods. Qualitative method requires no
overt manipulation of data, and only the judgments of the forecaster were used.
Meanwhile, quantitative method is a technique that can be applied when there is
enough historical data. Most researchers and forecasters used quantitative method
because this method involves a mathematical analysis of the historical data in
developing a model for forecasting. Furthermore, quantitative methods can be
categorized into two types namely, time series and causal methods. The common
2
time series methods is Box-Jenkins (BJ) methods because it is one of the most
powerful and accurate forecasting techniques for short term forecast-specially for
univariate time series.
In this study, an heuristic method, genetic algorithm, (GA) is introduced and
applied to BJ method in determining the right order of component BJ model (p ,q, P,
Q), where p, q ,P, Q are the degree of autoregressive model, moving average model,
seasonal autoregressive model and seasonal moving average model respectively.
The implementation of GA is to improve the weakness of BJ procedure in
identifying the right order of BJ tentative model. Thus, to examine the effectiveness
of this combination method, a comparison study was conducted between BJ model
and the combination of GA-BJ model. Therefore, the following subsection describes
the background of the problem.
1.2 Background of Problem
The stationary of univariate time series data are mostly been analyzed by
using the application of Box and Jenkins [3] which introduced the autoregressive
integrated moving average (ARIMA) model. However, when the time series data set
contains seasonal effect, SARIMA(p,d,q)(P,D,Q) is applied. If there is no seasonal
effect, SARIMA(p,d,q)(P,D,Q) will be reduced to pure ARIMA(p,d,q) model, and
when the time series data set is stationary, a pure ARIMA(p,d,q) reduces to
ARMA(p,q).
3
There are four stages to construct BJ models which are model identification,
model estimation, model checking and forecasting. Although many researchers and
practitioners have concentrated in estimation part of BJ model, the most crucial stage
in building the model [4] is the identification part as the false identification will
contribute to the increment of the cost of re-identification.
During identifying the order of BJ model, the intervention of a human expert
is also required in order to identify the best model because it is also not fully
automatic [3]. The current approaches are focusing on model estimation. The used of
correloggram method is to identify the order of autoregressive model (p) or the order
of moving average (q) of ARIMA or SARIMA model is complicated and not easily
conducted. Hence, the identification model and the parameter estimation will be
overlapped at diagnostic model checking stage until the best fitted model is found.
1.3 Statement of Problem
As stated in the background problem, the first step in building ARIMA or
SARIMA models is by determining the fitting order for the model identification
stage.
Since GA could works effectively by finding the approximate optimum solution
in complex data set, this study seeks to use paradigms based on artificial intelligence
GA for solving the best order for p, q, P, and Q of BJ model. Therefore, the correct
parameter of BJ model can be properly estimated.
4
1.4 Research Question
Questions arise when developing GA model for ARIMA and SARIMA
model identification procedure. It can be summarized as follow:
i. How to identify the order of ARIMA/SARIMA model using GA?
ii. How to estimate the parameter of ARIMA/SARIMA model using GA?
iii. How to model international tourist arrival time series data using GA-BJ
model?
1.5 Objective of the Study
The main objective of this study can be categorized into three parts. The
objectives are stated as follow:
i. To design and develop GA method for model identification in the BJ model.
ii. To implement GA in estimating parameters of BJ model.
iii. To develop a monthly international model for tourist arrival to Malaysia and
forecast tourist arrival using GA-BJ model.
5
1.6 Scope of the Study
The scopes of this study are:
i. This study focuses only on Box-Jenkins methodology where the time series
data used is based on historical past value.
ii. Forecast accuracy in this study will be defined by measuring the lowest error
in term of mean square error (MSE)
iii. In genetic algorithm architecture, chromosomes are randomly generated using
genetic operator.
iv. The data used as the secondary data of tourist arrival to Malaysia.
v. Time series data used in GA-BJ model is the process data that has been
analysis in Chapter 4.
vi. GA-BJ model is used for modeling one type of case study which is k step
ahead only.
vii. Out-sample data analysis is focuses on forecasting 12 months international
tourist arrival to Malaysia.
viii. The analysis in Chapter 4 and Chapter 5 were based on off-line data only.
1.7 Research Contribution
The main contribution of this study is to develop an alternative way of using
GA in BJ model identification method and estimation phases. The second
contribution of this study is to the development of mathematical model combining
GA-BJ forecast model and fitted model for monthly international tourist arrival to
Malaysia. Last but not least, the third contribution of this study is to the forecast
accuracy of this combining GA-BJ model is better than methods proposed by Box
and Jenkins.
6
1.8 Research Data
The data used in this study are the secondary data provided by Malaysian
Tourism Promotion Board. The data is an annual time series data that covered the
period from 1990 to 2011.
1.9 Thesis Plan
This thesis was divided into six main chapters. Chapter 1 outlined the
introduction parts which are including general background of study and some
reviews from previous work and describes the problem statements which led to this
research, objectives, scope and the contribution of study. Chapter 2 will describe the
literature reviews, current knowledge and related theories regarding the identification
method in BJ procedure. Next, Chapter 3 will explain in details the BJ procedures
and GA-BJ procedures. The case study on forecasting monthly international tourist
arrival to Malaysia using BJ model will be described in Chapter 4. In Chapter 5, the
construction of forecasting model for monthly international tourists’ arrival to
Malaysia by using proposed GA-BJ model. Finally, Chapter 6 summarizes all the
experimental findings and conclusions and presented some suggestions for future
studies related to this research.
118
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