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.

vii

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

viii

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

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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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