DETECTING ADDITIVE AND INNOVATIONAL OUTLIERS IN BL(p,0 ...

DETECTING ADDITIVE AND INNOVATIONAL OUTLIERS IN

BL(p,0,1,1) PROCESS

MOHD ISFAHANI BIN ISMAIL

INSTITUTE OF MATHEMATICAL SCIENCES

FACULTY OF SCIENCE

UNIVERSITY OF MALAYA

KUALA LUMPUR

2009

DETECTING ADDITIVE AND INNOVATIONAL OUTLIERS IN BL(p,0,1,1) PROCESS

MOHD ISFAHANI BIN ISMAIL

DISSERTATION SUBMITTED IN FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF

SCIENCE

INSTITUTE OF MATHEMATICAL SCIENCES FACULTY OF SCIENCE

UNIVERSITY OF MALAYA KUALA LUMPUR

2009

i

ABSTRAK

Kajian ini adalah berkenaan dengan proses mengesan nilai tersisih untuk model-

model BL(p,0,1,1), di mana 3,2,1=p . Dalam proses ini, model siri masa dibina dengan

menggunakan pendekatan Box-Jenkins. Pada peringkat penganggaran, penganggar untuk

parameter diperolehi menggunakan kaedah kuasa dua terkecil tak linear.

Dalam kajian ini, kewujudan nilai tersisih tambahan (AO) and nilai tersisih

inovasi (IO) dalam data dari model-model BL(p,0,1,1), 3,2,1=p , dikaji. Sifat-sifat nilai

tersisih ini telah dikaji supaya perbezaan pola bagi kedua-dua jenis nilai tersisih ini boleh

dikenali. Seterusnya, ukuran kekesanan nilai tersisih untuk AO and IO telah diterbitkan

menggunakan kaedah kuasa dua terkecil. Disebabkan oleh bentuk statistik yang

kompleks, kaedah bootstrap digunakan untuk mencari varians bagi statistik. Berdasarkan

sampel bootstrap, terdapat tiga formula untuk mengira varians. Formula-formula tersebut

ialah formula asas, purata trimmed (TM) dan MAD. Ujian kriteria and ujian statistik yang

sesuai untuk mengesan kewujudan nilai tersisih diperolehi dengan mempiawaikan

cerapan ω dari tiga proses bootstrap-asas di atas. Kemudian, proses in dibandingkan

dengan proses model-asas (MB).

Pengesanan nilai tersisih diterbitkan dengan memeriksa nilai maksimum dari

statistik piawai pada kesan nilai tersisih. Proses pengesanan nilai tersisih untuk mengenali

jenis nilai tersisih pada titik masa t telah dijalankan. Proses simulasi diterbitkan untuk

mengkaji keberkesanan proses pada model-model BL(p,0,1,1), 3,2,1=p . Secara amnya,

proses berkenaan berkesan dalam mengesan nilai tersisih. Sebagai contoh, keberkesanan

proses diaplikasikan pada data hujan dan data indeks kualiti udara.

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ABSTRACT

This study proposed an outlier detection procedure for the BL(p,0,1,1) models,

where 3,2,1=p . In this process, a time series was first fitted by the models using the

Box-Jenkins approach. In the estimation stage, the parameter estimates for the model

were found using the nonlinear least squares method.

The existence of additive outlier (AO) and innovational outlier (IO) in data from

the BL(p,0,1,1) models, 3,2,1=p , were considered in this study. Their features were

studied so that the different patterns caused by both type of outliers were distinguishable.

Further, the measure of outlier effect for AO and IO were derived using the least square

method. Due to the complexity of the statistics, bootstrapping is used to find the variance

of the statistics. Based on the bootstrap samples, three different formulae were used to

calculate the variance. These formulas are the standard formula, trimmed mean (TM) and

MAD. The appropriate test criteria and test statistics to identify the occurrence of outliers

were found by standardizing the observed ω giving three different bootstrap-based

procedures. These procedures are then compared to the model-based (MB) procedure.

The detection of outliers was carried out by examining the maximum value of the

standardized statistics of the outlier effects. The outlier detection procedure for

identifying the type of outlier at time point t was proposed. Simulation study was carried

out to study the performance of the procedure in BL(p,0,1,1) models, 3,2,1=p . It was

found out, in general, the proposed procedure performed well in detecting outliers. As for

illustration, the proposed procedure was applied on rainfall data and air quality index

data.

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ACKNOWLEDGEMENTS

The research recorded in this thesis was conducted under the excellent supervision of

Assoc. Prof. Dr. Ibrahim bin Mohamed and Prof. Dr. Mohd. Sahar Yahya at the Institute

of Mathematical Sciences (ISM), University of Malaya, Malaysia. I would like to

express my deepest appreciation to my supervisors for their guidance, continuous

encouragement, patience and help throughout this MSc program which contributed to the

completion of this thesis.

My heartfelt appreciation goes to my parents Hj. Ismail Ahmad and Hjh. Noraini Md.

Noor, my wife Siti Mariam Yahya, my sisters Busyra, Juhairah, Nurul Husna, Nurul

Asyiqin, Nurul Izzah, Nurul Hidayah, my brothers Hakimi, Naim and my family

members for their continuous love, inspiration, motivation, support and prayers for my

success.

I would like to record my appreciation to the staff of ISM and PASUM, University of

Malaya, especially Hidayah, Miss. Ng and Kak Budi, and my friends, especially Ali,

Akmal, Norli, Jedzry, Md Nor, Zaidi, Jat, Shauki, Zam, Syam, Pendi, Din, Mizi and

Kmal, for their assistance and support during my course of study.

I am most grateful to the Ministry of Science, Technology and Innovation (MOSTI) for

providing me with a scholarship and giving me the opportunity to fulfill my ambition.

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TABLE OF CONTENT

Page

ABSTRAK i

ABSTRACT ii

ACKNOWLEDGEMENT iii

LIST OF TABLES viii

LIST OF FIGURES x

LIST OF SYMBOLS AND ABBREVIATIONS xiii

CHAPTER ONE - INTRODUCTION

1.1 Background 1

1.2 Time series 4

1.2.1 Linear time series models 5

1.2.2 Nonlinear time series models 7

1.3 Outlier 9

1.4 Problem statement 12

1.5 Objectives 12

1.6 Thesis outline 13

CHAPTER TWO - BILINEAR MODEL

2.1 A review of bilinear model 15

2.2 General formulation 16

2.3 Properties of bilinear model 19

2.3.1 Stationary property 19

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2.3.2 Invertibility property 20

2.4 Nonlinearity tests 21

2.4.1 Keenan’s test 22

2.4.2 F-test 23

2.5 Comparison of bilinear model 25

2.5.1 Akaike’s information criterion (AIC) 25

2.5.2 Akaike’s Bayesian information criterion (BIC) 26

2.5.3 Schwarz’s criterion (SBIC) 27

2.6 Parameter estimation 28

2.6.1 Nonlinear least squares method 28

2.7 Simulation study 32

2.8 Summary 36

CHAPTER THREE – OUTLIERS

3.1 A review of outliers 37

3.2 Types of outliers 38

3.3 Causes outliers 39

3.3.1 Outliers from data errors 39

3.3.2 Outliers from intentional or motivated misreporting 40

3.3.3 Outliers from sampling error 40

3.4 Treatment of outliers 40

3.5 Outlier detection in time series 44

3.5.1 Bootstrap-based procedure 46

3.5.2 Model-based procedure 49

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3.6 Summary 53

CHAPTER FOUR – A STUDY ON NATURE OF OUTLIERS IN BL(p,q,r,s) MODEL

4.1 Model formulation of BL(p,q,r,s) 54

4.1.1 Formulation of AO effects on observations 55

4.1.2 Formulation of AO effects on residuals 56

4.1.3 Effect of IO on observations 60

4.1.4 Effect of IO on residuals 63

4.2 Illustration 68

4.3 Summary 75

CHAPTER FIVE – PROCEDURE FOR DETECTING SINGLE OUTLIER USING

BL(p,0,1,1) PROCESS

5.1 Nonlinear least squares method for BL(p,0,1,1) 76

5.2 Derivation of measure of outlier effect 79

5.2.1 Additive Outlier (AO) 79

5.2.2 Innovational Outlier (IO) 80

5.3 Variance of estimate of outlier effect 81

5.3.1 Other bootstrap-based procedure 81

5.4 A general single detection procedure to identify type of outlier 82

5.5 Illustration 84

5.6 Summary 86

CHAPTER SIX – SIMULATION STUDY

6.1 Sampling behavior of test statistics 87

6.2 Performance of test criteria 115

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6.3 Performance of outlier detection procedure 126

6.4 Summary 133

CHAPTER SEVEN – ANALYSIS OF DATA

7.1 Kampung Aring monthly rainfall data 134

7.2 Kuala Lumpur air quality data 136

7.3 Summary 137

CHAPTER EIGHT – CONCLUSION AND FUTHER REASEARCH

8.1 Summary of the study 139

8.2 Contributions 141

8.3 Further research 142

REFERENCES 143

APPENDIX 160

viii

LIST OF TABLES

Page

Table 2.1 Different parameters estimation methods of BL(p,q,r,s) models 31

Table 2.2 Parameter estimation for BL(1,0,1,1) 33



Table 6.1 List of model used for the determination of critical values for BL(1,0,1,1) 88



Table 6.4 List of cases considered in the performance study for BL(1,0,1,1) 117



Table 6.7 Proportion of correctly detecting AO using AO test criterion for BL(1,0,1,1) 120

Table 6.8 Proportion of correctly detecting IO using IO test criterion for

BL(1,0,1,1) 121



BL(2,0,1,1) 123



BL(3,0,1,1) 125

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Table 6.13 Proportion of correctly detecting AO the outlier detection procedure for BL(1,0,1,1) 127

Table 6.14 Proportion of correctly detecting IO the outlier detection procedure for BL(1,0,1,1) 128





Table 7.1 Summary of diagnostic results for the Kampung Aring rainfall data 135

Table 7.2 The test statistic value of outlier detection procedure on the Kampung Aring rainfall data 135

Table 7.3 Summary of diagnostic results for the Kuala Lumpur air quality data 137

Table 7.4 The test statistic value of outlier detection procedure on the Kuala Lumpur air quality data 137

x

LIST OF FIGURES

Page

Figure 4.1 Plot of simulated data 69

Figure 4.2 Plot of AO effect on observations, 1.01 =a , 1.0=b for BL(1,0,1,1) 70

Figure 4.3 Plot of AO effect on residuals, 1.01 =a , 1.0=b for BL(1,0,1,1) 70

Figure 4.4 Plot of IO effect on observations, 1.01 =a , 1.0=b for BL(1,0,1,1) 70

Figure 4.5 Plot of IO effect on observations, 3.01 =a , 3.0=b for BL(1,0,1,1) 71

Figure 4.6 Plot of IO effect on observations, 4.01 −=a , 4.0−=b for BL(1,0,1,1) 71

Figure 4.7 Plot of IO effect on residuals, 1.01 =a , 1.0=b for BL(1,0,1,1) 71

Figure 4.8 Plot of IO effect on residuals, 3.01 =a , 3.0=b for BL(1,0,1,1) 72

Figure 4.9 Plot of IO effect on residuals, 4.01 −=a , 4.0−=b for BL(1,0,1,1) 72

Figure 4.10 Plot of AO effect on observations, 1.01 =a , 1.02 =a , 1.0=b for BL(2,0,1,1) 72

Figure 4.11 Plot of AO effect on residuals, 1.01 =a , 1.02 =a , 1.0=b for

BL(2,0,1,1) 73

Figure 4.12 Plot of IO effect on observations, 1.01 =a , 1.02 =a , 1.0=b for BL(2,0,1,1) 73

Figure 4.13 Plot of IO effect on residuals, 1.01 =a , 1.02 =a , 1.0=b for

BL(2,0,1,1) 73

Figure 4.14 Plot of AO effect on observations, 1.01 =a , 1.02 =a , 1.03 =a , 1.0=b for BL(3,0,1,1) 74

Figure 4.15 Plot of AO effect on residuals, 1.01 =a , 1.02 =a , 1.03 =a , 1.0=b for BL(3,0,1,1) 74

xi

Figure 4.16 Plot of IO effect on observations, 1.01 =a , 1.02 =a , 1.03 =a , 1.0=b for BL(3,0,1,1) 74

Figure 4.17 Plot of IO effect on residuals, 1.01 =a , 1.02 =a , 1.03 =a , 1.0=b for BL(3,0,1,1) 75

Figure 5.1 Plot of simulated data 85

Figure 5.2 Plot of AOω , with AO, 1.01 =a , 1.0=b at 40=d 85

Figure 5.3 Plot of IOω , with IO, 1.01 =a , 1.0=b at 40=d 85

Figure 6.1 Plot of critical values of AO on standard procedure for BL(1,0,1,1) 91

Figure 6.2 Plot of critical values of AO on MAD procedure for BL(1,0,1,1) 92

Figure 6.3 Plot of critical values of AO on trimmed mean procedure for BL(1,0,1,1) 93

Figure 6.4 Plot of critical values of AO on model-based procedure for BL(1,0,1,1) 94

Figure 6.5 Plot of critical values of IO on standard procedure for BL(1,0,1,1) 95

Figure 6.6 Plot of critical values of IO on MAD procedure for BL(1,0,1,1) 96

Figure 6.7 Plot of critical values of IO on trimmed mean procedure for BL(1,0,1,1) 97

Figure 6.8 Plot of critical values of IO on model-based procedure for BL(1,0,1,1) 98







xii











Figure 7.1 Plot of the Kampung Aring rainfall data 134

Figure 7.2 Plot of the Kuala Lumpur air quality data 136

xiii

LIST OF SYMBOLS AND ABBREVIATIONS

tY Observation at time t

*tY Observation contaminated with outlier at time t

{ }tY A sequence of tY , for ,...2,1=t or ,...2,1,0 ±±=t

n Sample size

Y Column vector of observations ( )nYYY ,...,, 21

tY Column vector of observations ( )11,..., +− ptt YY

te Residual at time t

te Estimated residual at time t

{ }te A sequence of te , for ,...2,1=t or ,...2,1,0 ±±=t

e Column vector of residuals ( )neee ,...,, 21

e Column vector of estimated residuals ( )neee ˆ,...,ˆ,ˆ 21

tete

Column vector of ( )11,..., +− ptt ee

BL(p,q,r,s) Bilinear model with parameter p, q, r and s

1p ( )rp,max

1q ( )sq,max

1γ ( )srp ,,max

klji bca ,, Coefficients of bilinear model, pt ,...,2,1= , qj ,...,2,1= , rk ,...,2,1= and

sl ,...,2,1=

N Number of coefficients considered in the model

xiv

θ Set of possible bilinear coefficients defined accordingly on the model considered

nS Column vector of ( )nppn eeYYY +11

,...,,,...,, 21

tξ Column vector of ( )strtttpttt eYeYYYY −−−−−−− ,...,,,...,, 1121

H Hessian matrix

G Gradient matrix

i

Qθ∂∂ Partial differentiation of equation Q with respect to iθ

E(Y) Mean sample of ( )nYYY ,...,, 21

2σ Variance of population

2σ Estimated variance

2~σ Estimated bootstrap variance

τ Precision equal to 2

1σ

A Matrix

[ x y z ] or (x,y,z) Vector AIC Akaike’s information criteria

BIC Akaike’s Bayesian information criteria

SBIC Schwarz’s criterion

1

CHAPTER ONE

INTRODUCTION

1.1 BACKGROUND

Time series is defined as a collection of observations made sequentially in time.

In general, time series predictability is a measure of how well future values of a time

series can be forecasted, where a time series is a sequence of an observation

{ }NtYt ,...,2,1, = . Time series predictability indicates to what extent the past can be used

to determine the future in a time series. Time series analysis and its applications have

become increasingly important in various fields of research, such as business, economics,

engineering, medicine, social sciences and politics. This analysis can be used to carry out

different goals such as descriptive analysis, spectral analysis, forecasting, intervention

analysis and explanative analysis. Since Box and Jenkins [1970, 1976] published the

seminal book entitled Time Series Analysis: Forecasting and Control, a number of books

and a vast number of research papers have been published in this area. Brockwell and

Davis [1991] for instance, discuss the theory of time series in depth.

In the classical theory of time series analysis, one used to assume that the

structure of the series can be represented by linear time series models, for example, the

autoregressive model (AR), moving average model (MA) and autoregressive moving

average model (ARMA), and integrated autoregressive moving average model (ARIMA)

by taking into account the seasonality effect. A good account of this theory is available,

for example, Box and Jenkins [1976], Fuller [1976] and Chatfield [1996]. These models

2

have reasonably been successful for analysis and forecasting. However, ARMA model

also has some limitations as well as pointed out by Tong [1983].

On the other hand, not all linear models are adequate for time series data. The

Wolfer's sunspot data (refer Box and Jenkins [1976]) is a good example of a nonlinear

data set. By allowing for the time series plot of the sunspot data, one can observe that

“there exist a systematic periodic cycles with the downturn which is faster than the

upturn. This pattern will never be explained fully by any linear model”. That is one of

the characteristics of nonlinear time series data. For this particular data set, Granger and

Andersen [1978a] and Rao [1981] had shown that fitting nonlinear models such as the

bilinear model produce better results compared to linear model. Earlier monographs on

nonlinear time series include Priestley [1988] and Tong [1990]. Tong [1990] provides

inclusive coverage of parametric nonlinear time series analysis.

Theory in bilinear model started with the discussion by Ruberti et al. [1972] and

Mohler [1973] with the application on control theory. A real in-depth statistical study

began only after Granger and Anderson [1978a] published a manuscript on bilinear

models. Monographs on bilinear models include those by Subba-Rao and Gabr [1984]

and Terdik [1999]. They showed that bilinear model performs better compared to linear

model when applied on nonlinear data set such as the IBM daily common stock closing

prices available in Box and Jenkins [1976]. Another interesting feature of bilinear model

is the fact that it is an extension of the linear ARMA model as well as a simplified case of

nonlinear Volterra Series Expansion (Weiner [1958]).

One special characteristic of data sets generated from a bilinear process is that

there are high amplitude oscillations at certain time points of the data. It can be a single

3

spike or a collection of spikes at a time interval depending on the coefficient of the

bilinear process. These observations will be detected as outlier if linear model is fitted on

the data. The existence of observations that deviate markedly from the rest of the

observations occurs frequently in time series data. These observations are called with

different names such as "contaminants", "outliers" and "extreme values". In studying the

problem of outliers, the results can be used, among others, as a diagnostic tool to test the

strength and weakness of the model, to put up outliers in order to make inferences about

the parameter, to progress the model and to look at the influence of outliers.

Fox [1972] was the first to study outliers in time series data. Other researchers

considered various ARIMA cases with outlier effects (Chang [1982], Bell et al. [1983],

Tsay [1986b], Chang et al. [1988], Abraham and Chuang [1989], Pena and Maravall

[1991], Chan [1992], Wright and Booth [2001] and Choy [2001]). Nevertheless, thus far,

only a limited number of papers have been published on the occurrence of outliers in

bilinear process. Chen [1997] considered the existence of the additive outlier in general

BL(p,q,r,s) using Gibbs Sampling procedure. Earlier, Zaharim [1996] studied the outlier

detection procedure for all four types of outlier in simple bilinear case using least squares

method via nonlinear minimization function available in MATLAB. Zaharim et al.

[2006] extended the procedure to a BL(1,1,1,1) case by suggesting four explicit statistics

for measuring the outlier effects for additive outlier (AO), innovational outlier (IO),

temporary change (TC) and level change (LC) formulated by using the classical least

squares method. Battaglia and Orfei [2005] used model-based procedure for identifying

and estimating outliers in bilinear, threshold autoregressive and exponential

autoregressive models.

4

The modelling of bilinear model and the application of outlier detection procedure

proposed in this study will be applied to the environmental data, in particular the rainfall

data and the air quality data. Data used are secondary data obtained from the Department

of Meteorology and Department of Environment, Malaysia.

1.2 TIME SERIES

Time series analysis can be used to accomplish different goals. The first goal is a

descriptive analysis of time series data which determines the trends and patterns of a time

series by plotting or using more complex techniques. The most basic approach is to plot

the time series data and consider overall trends (increase, decrease, etc.), cyclic patterns

(seasonal effects, etc.), outliers (point of the data that may be erroneous) and turning

points (different trends within a data series). The second is the spectral analysis that is

analysis is carried out to explain the variation in a time series that may be accounted for

by cyclic components. This may also be referred to as “Frequency Domain”.

Consequently, an estimate of the spectrum over a range of frequencies can be achieved

and periodic components in a noisy environment can be separated out. The third is

forecasting. If a time series behaved in a certain way in the past, the future behavior can

be forecasted within certain confidence limits by building models for the data. The fourth

is the intervention analysis. This is used to describe changes in the time series caused by

certain event. The final type of analysis is the explanative analysis that uses one or more

variable time series to explain the mechanism that is the outcome of results in a

dependent time series.

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1.2.1 LINEAR TIME SERIES MODELS

In this section, basic ideas of linear time series models will be discussed with a

focus on simple autoregressive (AR), simple moving-average (MA) and autoregressive

moving-average (ARMA) models.

The very simplest form of time series is generated by a strict white noise process

and denoted by { }te . It is basically a sequence of independent random variables, and if

stationary, the mean µ and variance 2σ are equal for all time point t , while the

correlation between values at different time points is zero.

A simple autoregressive (AR) model of order 1 or simply an AR(1) model is

given by

ttt eYY ++= −110 φφ (1.1)

where { }te is assumed to be a white noise series with mean zero and variance 2eσ .

Suppose that te , the error term at time t, is normally distributed with mean zero and

constant variance 2eσ . If re and se are uncorrelated for r ≠ s, then the series te is called

a white noise process.

A basic generalization of the AR(1) model is the AR(p) model given by

tptptt eYYY ++++= −− φφφ ...110 (1.2)

where p is a non-negative integer.

Next, another class of simple linear models is the moving-average (MA) models.

6

The general form of an MA(1) model is

110 −−+= ttt eecY θ (1.3)

where 0c is a constant and { }te is a white noise series. An MA(q) model is given by

qtqtttt eeeecY −−− −−−−+= θθθ ...22110 (1.4)

where q > 0.

In some applications, the AR or MA models become cumbersome because one

may need a high-order model with many parameters to adequately describe the dynamic

structure of the data. To overcome this difficulty, the autoregressive moving-average

(ARMA) models are introduced (Box, Jenkins and Reinsel [1994]). Basically, an ARMA

model combines the ideas of AR and MA models into a compact form so that the number

of parameters used is kept small. The simplest ARMA(1,1) model is

11011 −− −+=− tttt eeYY θφφ (1.5)

where { }te is a white noise series. The left-hand side is the AR component of the model

and the right-hand side gives the MA component. The constant term is 0φ . For this model

to be significant, we need 11 θφ ≠ .

A general ARMA(p, q) model is in the form of

∑∑=

−=

− −++=q

iiti

p

ititit eeYY

110 θφφ (1.6)

where { }te is a white noise series and p and q are non-negative integers. The AR(p) and

MA(q) models are special cases of the ARMA(p, q) model.

7

1.2.2 NONLINEAR TIME SERIES MODELS

In this section, we discuss some nonlinear time series models. The pioneering

work in nonlinear time series modeling is due to Wiener [1958] who produced a general

class of nonlinear models called Volterra series expansion. The model is given by

∑ ∑ ∑∞

−∞=

∞

−∞=

∞

−∞=−−−−−− ++++=

i ji kjiktjtitijkjtitijitit eeebeebebY

, ,,...µ (1.7)

where µ is the mean of tY and te is a stationary process of independent and identically

distributed random variables, ∞<<∞− t . tY is nonlinear if at least one of the higher

order coefficients, ,..., ijkij bb is nonzero. The theoretical properties of (1.7) had been

discussed by a number of authors including Brillinger [1970]. The model contains too

many parameters to be estimated. Priestley [1978] noted that the estimation is only

possible if the sequence ,...,, ijkiji bbb possesses some form of “smoothness” properties

based on the frequency domain approach. However, the approach breaks down if more

than one term is considered.

There are several models for nonlinear time series data. They include; ARCH

models, state dependent models, threshold autoregressive models, nonparametric

autoregressive models, bilinear models and exponential autoregressive models. These

models can be generalized by the state dependent model (SDM) proposed by Priestley

[1980]. Given a single time series { }tY which is observed at time points ,...2,1,0 ±±=t , a

general relationship between tY and finitely many values of the past { }tY and { }te is

given by

( ) tqttpttt eeeYYhY += −−−− ,...,,,..., 11 (1.8)

8

where { }te is a sequence of independently and identically distributed random variables

with mean zero and variance 2σ , and h is a real-valued function. Expanding the right

hand side in a Taylor series about some fixed time point, Priestley [1980] stated that

equation (1.8) can be rewritten in the form

( ) ( ) ( )∑ ∑= =

−−−−− +++=p

i

q

jttjttjittit eZeZYZY

1 1111 µθφ (1.9)

where ( )'11 ,...,..., tpttqtt YYeeZ +−+−= . The unidentified parameters of the model are

( ) ( )11 , −− tjti ZZ θφ and ( )1−tZµ for pi ,...,1= and qj ,...,1= where all of them rely on the

‘state’ of the process at the time 1−t and the variance 2σ . A number of significant time

series models can be obtained from (1.9) by choosing the correct form of the parameters.

Bilinear model is a very important model to capture the nonlinear characteristic of

a time series data. This model is the simplest extension of the ARMA models by adding

nonlinear terms into the model. If ( ) 1−ti Zφ and ( )1−tZµ of SDM are chosen to be

constants and ( ) ∑=

−− +=s

kktjktj YZ

1j1 , βθθ then (1.9) becomes

∑ ∑ ∑∑= = = =

−−−− ++++=p

i

q

j

q

k

s

ltltktkljtjitit eeYeYY

1 1 1 1βθφµ

For convenient, let the upper limit of third term rq = , then the above equation becomes

∑ ∑ ∑∑= = = =

−−−− ++++=p

i

q

j

r

k

s

ltltktkljtjitit eeYeYY

1 1 1 1βθφµ (1.10)

Equation (1.10) is a general formula for bilinear models and denoted by BL(p,q,r,s),

where p,q,r,s are positive integers or zero.

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The equations (1.10) and (1.7) show that bilinear model is a special case of the

Volterra series expansion with smaller number of parameters. Further discussion on the

theory of bilinear model is given in Chapter 2.

1.3 OUTLIER

Observations that deviate markedly from the rest exist frequently in time series

data and are identified by various names such as, “discordant observations”, “extreme

values”, “outliers” and “contaminants”. Beckman and Cook [1983] defined discordant

observations as any observations that appear discrepant to the investigator, while a

contaminant is defined as any observation that is not a realization of the target

distribution.

On the other hand, Walfish [2006] defined extreme values as observations that

might have a low probability of occurrence but cannot be statistically shown to originate

from the different distribution than the rest of the data. Meanwhile, Olive [2007] defined

an outlier as observation that is far from the bulk of the data. The statistical definition of

an “outlier” depends on the underlying distribution of the variable in question.

Mendenhall et al. [1993] applied the term “outliers” to values “that lie very far from the

middle of the distribution in either direction”.

A different definition of outliers provided by Pyle [1999]: “An outlier is a single,

or very low frequency, occurrence of the value of a variable that is far away from the

bulk of the values of the variable”. A more general definition of an outlier is given in

Barnett and Lewis [1995]: “An observation (or subset of observations) which appears to

be inconsistent with the remainder of that set of data”. According to Hawkins [1980],

10

“An outlier is an observation that deviates so much from other observations so as to

arouse suspicion that it was generated by a different mechanism”. Han and Kamber

[2000] defined outliers as data objects that are grossly different from or inconsistent with

the rest of the data.

Outliers can provide constructive information about the process. The result of

studying outliers can be used as diagnostic tool to test the strength and weakness of a

model and also to decide whether to accommodate outliers in order to make inferences

about a parameter. Consequently, improvement on the model can be done by inspecting

their influence on the model. In general, there are four type of outliers, additive outlier

(AO), innovational outlier (IO), level change (LC) and temporary change (TC).

The AO describes an event that affects a time series at one particular time period

only. Typically, this is the only type of outlier considered in regression analysis. Unlike

the AO, the IO describes an event with its effect propagating according to the process. In

this manner, the IO affects the subsequent observations after its occurrence. The LC is an

event that affects a series at a given time, and its effect becomes permanent afterward.

Finally, the TC describes an event having an initial impact and then dies out

exponentially according to dampening factor, λ . Details regarding the mathematical

formulation and interpretation for the outliers can be found, inter alia, in Liu and Hudak

[1992] and Chen and Liu [1993a].

The earliest discussion on outliers was by Bernoulli [1777] in which the

assumption of identically distributed errors in regression problem was discussed. Later

Pierce [1852] and Chauvenet [1863] made and attempt to develop a specific criterion for

11

the rejection of outliers. Wright [1884] extended the work and established a rule in which

an outlier is rejected if it residual exceeds 3.37 times the standard deviation.

In the early years, visual inspection of data was used to deal with outliers. Barnett

[1978] suggested four different approaches for handling outliers; by accommodating

them using robust method, rejecting them, placing them within a homogenous probability

model setting so that no observations appear discordant, or enhancing their importance by

setting up a mixture model to explain their presence.

D’Agostino and Stephens [1986] suggested performing two set of analyses with

and without outlier. If the results are different, any conclusion from the model should be

used with care whether to accommodate or to reject the outlier. In this study, the scope is

limited to the detection of outliers.

So far, a large amount of studies have focused on the detection of outliers in linear

time series models. Among the first to study outliers in time series data was Fox [1972].

Others considered various ARIMA cases with outlier effects such as Chang [1982], Tsay

[1986b], Chang et al. [1988], Abraham and Chuang [1989], Pena et al. [1991], Chan

[1992], Wright and Booth [2001] and Choy [2001].

On the other hand, a number of papers has been published on the detection of

outliers in data from bilinear model; Zaharim [1996] studied the outlier detection

procedure for AO, IO, LC and TC in the bilinear model, BL(1,0,1,1), using least squares

method, Chen [1997] considered the existence of additive outlier in the general

BL(p,q,r,s) process using Gibbs sampling procedure, Battaglia and Orfei [2005] studied

the problem of identifying the time location and estimating the amplitude of outliers in

nonlinear time series using model-based method and Zaharim et al. [2006] considered

12

the performances of test statistics for single outlier detection in the BL(1,1,1,1) model.

They have shown that the outlier detection procedure for bilinear models developed

based on the least squares method perform well in detecting and identifying the type of

outlier considered in their study.

1.4 PROBLEM STATEMENT

A large number of studies on outlier detection procedures have been developed

for linear ARMA models. Procedures for detecting AO and IO have been developed

extensively for general ARMA models using least squares method. Battaglia and Orfei

[2005] suggested a procedure based on least squares method such that similar definition

of AO and IO as in linear case are used. Further, they used Taylor’s expansion to estimate

the effect of AO.

In this study, the effects of AO and IO according to the exact process of the

bilinear model are formulated. We propose new statistics based on the exact effect of AO

and IO for the BL(p,0,1,1) models, where p=1,2,3. The bootstrap procedures are used to

estimate the variance of the statistics and the process is carried out using the standard

deviation formula, trimmed mean and mean absolute median (MAD) procedures. Hence,

the performances of the detection procedure for AO and IO are compared.

1.5 OBJECTIVES

The objectives of this study are as follows:

a) To formulate the effect of AO and IO on observations generated from the

BL(p,q,r,s) process and residuals from the fitted BL(p,q,r,s) models, where p,q,r,s

are positive integers or zero.

b) To derive the statistics that measure the outliers effect for AO and IO in

BL(p,q,r,s) models.

13

c) To propose bootstrap-based outlier detection procedures for the BL(p,0,1,1)

models, where p=1,2,3.

d) To compare the performance of the bootstrap-based procedures with the model-

based procedure on identifying AO and IO in the BL(p,0,1,1) models, where

p=1,2,3.

e) To show that bilinear model can be an alternative model if compared to linear

model when fitted on environmental data system.

1.6 THESIS OUTLINE

Chapter 2 presents literature review on bilinear models. It includes the general

formulation and properties of bilinear model, and a discussion on parameter estimation

method with a special focus on nonlinear least squares method. We also perform the

simulation process for estimating the parameters on the BL(p,0,1,1) models, where

p=1,2,3.

Chapter 3 presents a literature review on outliers in time series. The main focus is

on existing outlier detection procedures in bilinear model. This chapter also discusses the

procedures used in the study to obtain standard deviation for the statistics to measure the

outlier effect.

Chapter 4 presents findings on the nature of AO and IO in BL(p,0,1,1) models,

p=1,2,3. The effects of AO and IO on observations and residuals are formulated, while

the statistics to measure the outlier effects are derived.

14

Chapter 5 presents the development of the bootstrap-based procedures for

detecting AO and IO. The procedures are expected to be able to detect outliers and to

identify the type of outlier.

Chapter 6, simulation studies are carried out to investigate the sampling behavior

of the test statistics. Extended simulation work is carried out to study the performance of

outlier detection procedures.

Chapter 7 illustrates the proposed procedures on two real data sets. The first is on

rainfall data collected from Kampung Aring weather station (Kelantan) and the second is

on the air quality index of Kuala Lumpur.

Finally, chapter 8 concludes the thesis with a summary of the study, a list of

contributions and suggestions for further research.

15

CHAPTER TWO

BILINEAR MODEL

2.1 A REVIEW OF BILINEAR MODEL

Linear time series models such as the autoregressive (ARMA) models have been

extensively and successfully used in many fields. The reasons are that these models can

be easily analyzed and provide fairly good approximations for the underlying chance

mechanisms of numerous real-life time series. However, in some particular situations one

may ask if there exist other models which can provide a better fit. This led us to consider

non-stationary or non-linear models. A simple class of non-linear model is the bilinear

model which has been found to be useful in many areas; for example, biological sciences,

ecology and engineering (see Mohler [1973], Bruni et. al [1974]). These models,

originally deterministic, have been transformed into stochastic models and were

subsequently studied by Granger and Andersen [1978a, b, c].

Over the past 20 years, a great deal of concentration has been paid to bilinear

class of nonlinear systems. One can mention the lecture notes of Subba Roa and Gabr

[1984] and the paper of Tuan [1993] by time series side and the works from Ruberti et. at

[1972] and Mohler and Kolodziej [1980] by system theory side. Papers discussing

bilinear time series include: Akamanam and Rao [1986], Brillinger [1990], Chanda

[1991], Gabr and Subba Rao [1984], Guegan and Ngatchou [1996], Guegan and Pham

[1989], Hannan [1982], Igloi and Terdik [1997], Jia and Huang [1992], Kim and Billard

[1990], Liu [1992], Liu and Brockwell [1988], Liu [1985], Priestley [1978, 1980, 1988],

16

Quinn [1982], Rao et. at [1983], Sesay and Subba Rao [1983, 1991, 1992], Terdik [1985,

1990, 1997], Tuan [1993], Turkman and Turkman [1997].

Theory in bilinear model started with the conversation in economics and

engineering by Zellner [1971], Ruberti et al. [1972] and Mohler [1973] with the

application on control theory. Later, others discussed the properties of the bilinear model

especially on the invertibility and stationarity of the bilinear model by Granger and

Andersen [1978a], Priestly [1991], Pham and Tran [1981], Rao [1981], Quinn [1982] and

Liu and Brockwell [1988]. In addition, a real in-depth statistical study began only after

Granger and Andersen [1978] published a manuscript on bilinear model. Granger and

Andersen [1978] showed that bilinear model performs better compared to linear model

when applied on nonlinear data set such as the IBM daily common stock closing prices

available in Box and Jenkins [1976]. Another fascinating feature of bilinear model is the

fact that bilinear model is an extension of the linear ARMA model and is also a

simplified case of nonlinear Volterra Series Expansion (see Weiner [1958]). In this

chapter we consider some basic preliminaries and important properties of bilinear model

including the methods that we used in this study are discussed.

2.2 GENERAL FORMULATION

The general bilinear model, denoted by BL(p,q,r,s), is given by

t

r

k

s

tktk

q

jjtj

p

iitit eeYbecYaY +++= ∑∑∑∑

= =−−

=−

=−

1 111 lll (2.1)

where ji c,a and lkb are any real numbers satisfying the stationary condition of the

model whereas tY and te are the observation and residual, respectively for t = 1, 2, 3,... .

17

The te 's are assumed to follow normal distribution with mean zero and precision τ, τ > 0

( ),0(~ 1−τNet ). The first component on the right-hand side of (2.1) are basically the

autoregressive(AR) model with parameter p and the second component represent moving

average(MA) with parameter q. The combinations of first and second component

represent the mixed autoregressive moving average (ARMA) model. The second last

component is nonlinear which helps to explain the nonlinearity characteristic of the data

being modelled. Thus, ARMA (p,q) is a special case of the BL(p,q,r,s) when .sr 0==

The bilinear model of equation (2.1) can be rewritten in vector form below:

∑=

−−− ++=s

jjttjttt eYBCeAYY

111

where

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+−−= 11ptY1tYtY L'tY is a 1× 1p vector,

⎥⎦

⎤⎢⎣

⎡−−= qte1tete L'

te is a 1× ( )1+q vector,

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

01000

000100000100aaa p21

LL

MOOM

MOOM

MOOM

LO

LL

LL

A is a 1p × 1p matrix,

18

( )

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

−

000000

00000000bbbb rjj1r2j1j

LL

OM

LL

LL

jB qj ,...,2,1= is a 1p × 1p matrix,

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

000

000ccc q10

L

MM

L

L

C is a 1p × ( )1+q matrix,

and ( )rpp ,max1 = .

Bilinear model can be divided into several cases.

Case 1 : When 0== sr , then BL(p,q,0,0) is exactly the same as ARMA(p,q).

Case 2 : When 0=q , then the model is homogeneous in the output of tY .

Case 3 : When 0≠q , and if any 0≠klb , then the model is said to have heterogeneous

error, that is, it does not have constant variance.

Case 4 : When 0== qp it is referred to as a completely bilinear model. In matrix form,

it is given by

trtttstttt eYYYeeeY += −−−−−−'

2121 ][][ LL 'β (2.2)

where

{ } rkbkl ,...,2,1, ==β and .,...,2,1 sl =

If 0=klb for all lk ≠ , then it is a diagonal model.

If 0=klb for all lk > , then it is a superdiagonal model.

If 0=klb for all lk < , then it is a subdiagonal model.

19

2.3 PROPERTIES OF BILINEAR MODEL

Studies on the stationarity and invertability properties of difference types of

bilinear models have been undertaken by several researchers since Granger and Andersen

[1978a]. They comprise Pham and Tran [1981], Rao [1981], Quinn [1982] and Liu and

Brockwell [1998].

2.3.1 Stationary property

Generally, a time series is said to be stationary if there is no systematic change in

mean, if there is no systematic change in variance and if strictly periodic variations have

been removed through filtering or differencing. Most of the probability theory of time

series applies only to stationary time series.

In many applications, the form of distribution functions is unknown. To overcome

this, a time series is instead defined to be weakly stationary if

a) )( tYE is constant for all t.

b) The covariance matrix of ),...,,(21 nttt YYY is the same as the covariance matrix of

),...,,(21 τττ +++ nttt YYY for all no empty finite sets of ),...,,( 21 nttt and all τ such that

),...,,,,...,,( 2121 τττ +++ nn tttttt are contained in the index set, that is, the

autocovariance function only depends on the lag.

There are a number of papers that propose the essential and adequate condition

for the existence of stationary processes for certain type of bilinear model. Pham and

Tran [1981] explained that for simple bilinear model BL(1,0,1,1),

20

111111 −−− ++= ttttt eYbeYaY (2.3)

there exist a unique strictly stationary process { }tY if

11 <a , 12211

21 <+ σba (2.4)

Further, if there exists a second-order stationary process { }tY with te admitting a finite

fourth moment, 12211

21 <+ σba then must necessarily hold.

In this thesis, we use BL(p,0,1,1) models, p=1,2,3 , given by

ttt

p

iitit eeYbYaY ++= −−

=−∑ 1111

1 (2.5)

where p=1,2,3. The essential and adequate condition for the existence of stationary

processes for our models is such that there exist a unique strictly stationary process { }tY

if

1<ia , ∑=

<+3

1

2211

2 1i

i ba σ (2.6)

If we consider BL(p,0,1,1) model, there exists a second-order stationary process { }tY

with te admitting a finite fourth moment if ∑=

<+3

1

2211

2 1i

i ba σ .

2.3.2 Invertibility property

The concept of invertibility is very useful for statistical applications, such as the

forecast of tY given its past, or the use of algorithms for computing estimates of the

parameters. Most of these conditions are based on the stationarity assumptions. For a

time series model to be useful for forecasting purposes, it is necessary that it should be

21

invertible. The invertibility of linear time series model has been discussed by Box and

Jenkins [1970].

Granger and Andersen [1978c] have provided another definition of invertibility

which can be applied to both linear and non-linear time series models. Granger and

Andersen [1978], Guiegan and Pham[1989], Pham and Tran [1981], Subba Rao and Gabr

[1984] and Liu [1989] had derived invertibility conditions for some particular stationary

bilinear models.

Just as in linear ARMA models, an invertible bilinear model are necessary if, say

for forecasting purposes, one is interested in associating present events with the past in a

unique manner. Althought different definitions of invertibility have been proposed, only

for a few simple cases have they actually been obtained (see Granger and Andersen

[1978c], Rao [1981], Quinn [1982] and Liu [1985]). Let tY be a discrete parameter time

series such that

{ } tjtjtt epjeYfY +== −− ,...,2,1,, (2.7)

where { }te are unobservable pure white noise.

Further, let te be an “estimate” of te and initial value of te equal to zero. Then

model (2.7) is said to be invertible if

{ } 0ˆlim 2 →−∞→ ttt

eeE (2.8)

when the model and the parameters are known completely.

2.4 NONLINEARITY TESTS

The objective of nonlinearity test is to statistically distinguish a linear time series

data from a nonlinear time series data. The reason is clear, that is, to make sure that the

22

model considered is adequate. Keenan [1985] and Tsay [1986a] introduced the Keenan’s

test and F-test, respectively, where the model tested is in Volterra series expansions form.

Bilinear model in fact is a special case of the Volterra series expansion. Next Keenan’s

test and F-test will be discussed in the following sections.

2.4.1 KEENAN’S TEST

Keenan [1985] adopted the idea of Tukey’s [1949] one degree of freedom test for

nonadditivity to derive a time-domain statistics. Assuming that a time series tY can be

adequately approximated by a second-order Volterra expansion series of the form

∑ ∑∞

−∞=

∞

−∞=−−− ++=

i jijtitijitit eececY

,µ (2.9)

The approximation will be linear if the last term on the right-hand side is zero. Equation

(2.9) takes a similar form of a linear regression model with an interaction term. Tukey’s

one degree of freedom test for nonadditivity is used to test whether the interaction term

equals to zero. Keenan’s test procedure is developed for the same purpose in time series

context to ensure that the last term on the right-hand side is zero. It works in three steps:

a) Regress tY on { }Mtt YY −− ,...,,1 1 and calculate the fitted values { }tY , the residuals,

{ }te , for ,,...,1 nMt += and the residual sum of squares, ∑>=< 2ˆˆ,ˆ seee .

b) Regress 2tY on { }Mtt YY −− ,...,,1 1 and calculate the residuals { }tξ , for .,...,1 nMt +=

c) Regress { }nM eee ˆ,...,ˆˆ 1+= on { }nM ξξξ ˆ,...,ˆˆ1+= and obtain η and F via

⎟⎠

⎞⎜⎝

⎛= ∑

+=

n

Mtt

1

20

ˆˆˆ ξηη

23

where 0η is the regression coefficient, and

( )2ˆˆ,ˆ22ˆˆ

ηη

−><−−

=ee

MnF (2.10)

follows approximately 22,1 −− MnF where the degree of freedom is associated with

>< e,e ˆˆ is ( ) 1−−− MMn .

Step (a) of the Keenan’s test requires a value chosen for M. It is chosen so that an

adequate autoregressive approximation can be obtained. Keenan’s test is developed based

on the argument that if any of ijc in (2.9) is non-zero, say 12c , then the nonlinearity

should be distributionally reflected in the diagnostics of the fitted linear model. It

happens if the residuals of the linear model are correlated with 21 −− tt YY . As in Tukey’s

[1949] non-additivity test, Keenan’s test uses the aggregated quantity 2tY , the square of

the fitted value of tY based on linear model, to obtain the quadratic term upon which the

residuals can be correlated.

2.4.2 F-TEST

Tsay [1986a] modified Keenan’s test by replacing the aggregated quantity 2tY

with the disaggregated variable MjiYY jtit ,...,1, =−− where M is as defined in Keenan’s

test. The F-test procedure is as follows:

a) Regress tY on { }Mtt YY −− ,...,,1 1 and calculate the residuals, ( )′= tnt2t1 e,...,e,e ˆˆˆet ,

for .,...,1 nMt += The regression model is denoted by

24

tt eY += φWt (2.11)

where { }Mtt YY −−= ,...,,1 1tW and ( )'10 ,...,, Mφφφ=φ .

b) Regress vector tZ on { }Mtt YY −− ,...,,1 1 and calculate the residuals,

( )′= tnt2t1 X,...,X,X ˆˆˆX t , for .,...,1 nMt += In this step, the multivariate

regression model is ttt XHWZ += where tZ is an ( )121

+= MMm

dimensional vector defined by ( )t't

't UUZ vech= with ( )Mtt YY −−= ,...,1tU and

‘vech’ denoting the half staking vector, and ( )'H M10 H,...,H,H= are the

coefficients of the model.

c) Fit

ttt Xe εβ += , nMt ,...,1+= (2.12)

and define

( )( ) ( )

1mMnε

mF

−−−∑

∑∑∑

=

−

2

1

t

t'tt

'tt

't

ˆ

eXXXeX

(2.13)

where the summation is over t from 1+M to n and tε is the least squares

residual from (2.12.). Here, F is asymptotically distributed as

( )1, −−− MmnmF .

The above procedure can be reduced to Keenan’s test if one aggregate tZ with

weights determined by the least squares estimate of (2.11) becomes a scalar variable.

25

2.5 COMPARISON OF BILINEAR MODELS

Whittle [1963] and Jenkins and Watts [1968] introduced an order selection

technique based on residual variance plots. In these technique, linear models such as the

AR(p) model are fitted to the data. If a sequence of a model of increasing order is fitted,

an 2εσ is evaluated in each case, then the plot 2

εσ against p is expected to decrease at first

and then “level out” at the point where p approaches the true order.

Akaike [1969] further refined the residual variance plot where it looks at the value

of p which minimizes the statistic called the “order selection criterion” based on the

estimated residual variance and the order p for AR(p). The criteria are also applicable on

MA and ARMA models and can be extended to include bilinear model. Three such order

selection criteria are described in this section. They are based on the derivations given in

DeLurgio [1998].

2.5.1 AKAIKE’S INFORMATION CRITERION (AIC)

Akaike [1969] first proposed the order criterion for AR(p) defined by

2ˆ)( epnpnpFPE σ

−+

= (2.14)

where n is the number of observations fitted and 2ˆeσ is the maximum likelihood estimate

of the variance of the residuals. Later, Akaike [1974] introduced the Akaike’s

information criteria (AIC) for statistical model identification, including bilinear model,

given by

AIC = - 2 log (maximum likelihood) + 2m (2.15)

26

where m is the number of terms estimated in the model. The first term of AIC can be

approximated using

- 2 log (maximum likelihood) + 2m ( )( ) 2ˆlog2log1 enn σπ ++≈ + 2m (2.16)

where 2ˆeσ is the variance of the residuals based on the fitted model and n is the number of

observations in the series. Thus, AIC can be found approximately using the formula

AIC ( )( ) mnn e 2ˆlog2log1 2 +++≈ σπ (2.17)

The order of models is determined by computing the AIC criterion over a selected grid of

values of p, q, r, s and choosing those values of p, q, r, s at which AIC attains its

minimum.

2.5.2 AKAIKE’S BAYESIAN INFORMATION CRITERION (BIC)

Akaike [1979] developed a Bayesian extension of the minimum AIC procedure called

BIC for AR(p) and the order selection criterion is defined by

BIC ( ) ( ) ⎟⎠⎞

⎜⎝⎛ −−−=

nppnn e 1logˆlog 2σ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−++ − 1

ˆˆ

loglog 2

21

e

Yppnpσσ

where 2ˆeσ is the estimate of the variance of residuals based on the p-th parameter model

and 2ˆYσ is the raw sample variance of the observations.

27

When p is small relative to n, the approximation ( ) pnppn ≈⎟⎠⎞

⎜⎝⎛ −−− 1log is used

so that

BIC ( )2ˆlog en σ= ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−+++ − 1

ˆˆ

loglog1 2

21

e

Yppnpσσ (2.18)

The last term of the right hand side of (2.18) is independent of n. Thus, the approximate

expression for BIC becomes

BIC ( )2ˆlog en σ≈ ( )np log1++ (2.19)

In general, p is taken as the number of parameters considered in the model. Hence, BIC

becomes

BIC ( )2ˆlog en σ≈ ( )nm log1++ (2.20)

where 2eσ and m are as defined previously.

2.5.3 SCHWARZ’S CRITERION (SBIC)

Schwarz [1978] suggests that, for AR(p), the order selection criterion is

S(p) ( )2ˆlog en σ≈ np log+ (2.21)

which is similar to Akaike’s BIC in terms of its dependence on log n. It can be

generalized into

SBIC ( )2ˆlog en σ≈ nm log+ (2.22)

In choosing the best model out of several competing models, the criteria above use the

information obtained from the data through the likelihood and then adjust them to include

a penalty for the number of terms in the model.

28

2.6 PARAMETER ESTIMATION

This section presents the method used to estimate parameters, namely, the

nonlinear least squares method. There are several methods for estimating the parameters

of BL(p,q,r,s): recursive estimation method, method of the moments, robust estimation

method, Bayesian’s conjugate family method, Bayesian’s Gibbs sampling method and

conditional least squares method. Priestley [1978] used the maximum likelihood

estimation method together with Newton-Raphson procedure for estimating the

parameters.

Meanwhile, Kim et al. [1988] used least squares and moment methods. Grahn

[1995] expressed the conditional least square approach to bilinear model estimation.

Then, Gabr [1998] used robust Monte-Carlo study on least squares method on bilinear

model to estimate the parameters. A number of studies used the Gibbs sampling approach

and Bayesian approach, including Chen [1992a, 1992b].

2.6.1 Nonlinear least squares method

A classical method of estimating parameters of bilinear models is the nonlinear

least squares method described by, for example, Goldfeld and Quandt [1972]. Granger

and Andersen [1978a] and Liu [1985] used the method to fit BL(1,0,1,1) and BL(2,1,1,1)

models respectively on the Wolfer’s sunspot data.

The nonlinear least squares method for BL(p,0,r,s) model as described in Priesley

[1991] is presented here. Say γYYYY ,...,,, 321 is known where },,max{ srp=γ . Let

29

( )'321 ,...,,, Nθθθθ=θ denote the complete set of parameters of ja and klb of BL(p,0,r,s)

model. The objective is to minimize the following equation:

( ) ∑+=

=n

tteQ

1

2

γ

θ (2.23)

where te ’s are obtained from equation below :

t

r

k

s

lltktkl

p

iitit eeYbYaY ++= ∑∑∑

= =−−

=−

1 11 (2.24)

It is achieved by using Newton-Raphson procedure:

( ) ( ) ( )( ) ( )( )ii1i1i θGθHθθ −+ −= (2.25)

where ( )iθ is vector of parameters estimated in the i-th iteration, G is Gradient vector and

H is Hessian matrix such that

⎭⎬⎫∂∂∂

⎩⎨⎧ ∂

=Nd

Qd

Qd

Qd

Qθθθθ

θ ,...,,,321

)G( (2.26)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂∂∂

=ji

Qθθ

θ2

)H( , i, j = 1, 2,… (2.27)

The partial derivatives of Q with respect to ( )}{ iθ are:

2=∂∂

i

Qθ ∑

+= ∂∂n

t i

tt

ee

1γ θ , i = 1, 2, …, N. (2.28)

22

=∂∂

∂

ji

Qθθ j

tn

t i

tt

eee

θθγ ∂∂

∂∂∑

+= 1

+ 2 ∑+=

n

tte

1γ ji

teθθ ∂∂

∂2

, i, j = 1, 2, …, N. (2.29)

The following initial conditions were assumed:

02

=∂∂

∂=

∂∂

=ji

t

i

tt

eee

θθθ (2.30)

30

for γ,...,2,1=t and Nji ,...,1, = together with an initial set of parameter values { }ia and

{ }klb . Vector G and H can then be evaluated and (2.25) implemented. The iteration will

continue until the following conditions are met, that is, to construct simui

2i

1i a,...,a,a and

simukl

2kl

1kl b,...,b,b for each parameter { }ia and { }klb until

<∈− −1simui

simui aa (2.31)

<∈− −1simukl

simukl bb (2.32)

for a tolerance 0∈> and simu is the simu-th iterated values.

A different factor that has to be considered in the iterative procedure is the

determination of initial values. Priestley [1991] suggested the following approach:

“If BL(p,0,1,1) is to be fitted, then the parameter estimates of AR(p) model

form initial estimates of paa ,...,1 while the initial estimate of 11b is taken to

be zero. For BL(p,0,2,1) and BL(p,0,1,2) models, the estimates with the

initial values of 21b and 12b are taken to be zero, respectively. For

BL(p,0,2,2) model, the parameter estimates of BL(p,0,2,1) or BL(p,0,1,2) are

used as the initial values and the initial value for 22b is taken to be zero. The

process then continues.”

Several other researchers including Granger and Andersen [1978a], Liu [1985] and Chen

[1992b] used nonlinear least squares method for bilinear model of higher order, but did

not state their approach of determining the initial values. The approach discussed here

was used in this study and detailed description of the procedure for BL(p,0,1,1) models is

presented in Chapter 5.

31

Herewith, a table of different parameters estimation method of BL(p,q,r,s) models

is provided for comparison. From Table 2.1, the first three methods have been used to

describe the most general model of BL(p,q,r,s). The second and third methods are

efficient to be used if the assumption that the prior distribution of the time series data is

normal-gamma is satisfied. The first method is a common parameters estimation method

where the procedure is simpler and easier to apply.

Table 2.1 Different parameters estimation methods of BL(p,q,r,s) models

Methods Highest order

studied

Statement

Nonlinear least

squares method

BL(p,q,r,s) Assume that the residuals follow ),0( 1−τN .

Simpler and easier to apply

Bayesian’s conjugate

family approach

BL(p,q,r,s) Assume that the prior distribution is

normal-gamma or Jeffreys non-informative

prior distribution only

Bayesian’s Gibbs

sampling approach

BL(p,q,r,s) Assume that the prior distribution for

parameters is multivariate normal

distribution and that for 2σ is inverse

gamma distribution

Robust method BL(p,0,r,s) Condition becomes complicated if the

model of the bilinear parameters increases

Recursive method BL(p,0,r,s) Requires a large computational

programming in estimating

Moment method BL(0,0,1,1) Difficulties to estimate higher-order

moment for the general bilinear model

Conditional least

squares method

Super-diagonal

model and

BL(p,0,p,1)

The existence of the higher order moments

of bilinear process and invertability of

linear part of the bilinear series

32

2.7 SIMULATION STUDY

In this section, we perform simulation study to investigate the performance of

nonlinear least squares method in estimating the coefficients of BL(p,0,1,1) models.

Below are the steps taken to look at the performance of the method:

a) We generate the BL(p,0,1,1) series of length n = 100 with known parameter

values of 1a , 2a , 3a and 11b for p=1,2,3. We assume that te follows standard

normal distribution.

b) We then obtain the estimates of 1a , 2a , 3a and 11b using nonlinear least squares

method. This estimation is repeated s times.

c) Let { }11321 ,,, baaa=χ , iχ be an estimate at step si ,...,1= and the mean of χ ,

∑=

=s

ii

1

χχ . The bias for each parameter is obtained.

d) Throughout our simulation, we use number of simulation s=1000.

Table 2.2 to Table 2.4 give the results for BL(1,0,1,1), BL(2,0,1,1) and

BL(3,0,1,1), respectively. The first column of each table gives the difference combination

of coefficients of the bilinear models. The parameter estimate and biases are given in the

subsequent column according to the number of parameters of each model. For example,

in Table 2.3 the second and third columns give the estimate of 1a and the bias of 1a

respectively, the fourth and fifth columns give the estimate of 2a and the bias of 2a

respectively, while the last two columns give the estimate of 11b and the bias of 11b .

33

By looking at the biases of the parameter estimates of BL(p,0,1,1) for p=1,2,3, it

can be seen that values of the biases are small for all models considered. It can be

concluded that the nonlinear least squares method performs well in estimating the

coefficients of BL(p,0,1,1), p=1,2,3. In addition, this method is also simple and easy to

apply.

Table 2.2 Parameter estimation for BL(1,0,1,1)

True value

( )111 ,ba 1a bias of 1a

11b bias of 11b

(0.1,0.2) 0.105 0.005 0.207 0.007

(0.1,0.4) 0.095 0.005 0.398 0.002

(0.2,0.2) 0.187 0.013 0.201 0.001

(0.3,0.3) 0.279 0.021 0.291 0.009

(0.4,0.1) 0.379 0.021 0.095 0.005

(0.4,0.3) 0.359 0.041 0.288 0.012

(0.5,0.1) 0.458 0.042 0.096 0.004

(-0.1,-0.1) -0.119 0.001 -0.101 0.001

(-0.1,-0.3) -0.088 0.012 -0.292 0.008

(-0.2,0.3) -0.199 0.001 0.298 0.002

(-0.3.0.1) -0.306 0.006 0.102 0.002

(-0.3,-0.3) -0.283 0.017 -0.299 0.001

(-0.4,-0.3) -0.377 0.023 -0.295 0.005

(-0.5,0.1) -0.471 0.029 0.096 0.004

34


True value

( )1121 ,, baa 1a bias of

1a 2a bias of 2a

11b bias of 11b

(0.1,0.1,0.1) 0.106 0.006 0.083 0.017 0.103 0.003

(0.1,0.3,0.3) 0.062 0.038 0.284 0.016 0.296 0.004

(0.2,0.1,0.1) 0.203 0.003 0.088 0.012 0.086 0.014

(0.2,0.2,0.2) 0.199 0.001 0.168 0.032 0.203 0.003

(0.3,0.1,0.1) 0.289 0.011 0.085 0.015 0.100 0.000

(0.3,0.2,0.2) 0.270 0.030 0.178 0.022 0.185 0.015

(0.4,0.1,0.1) 0.382 0.018 0.085 0.015 0.096 0.004

(0.4,0.3,0.1) 0.377 0.023 0.274 0.026 0.078 0.022

(0.5,0.1,0.1) 0.475 0.025 0.085 0.015 0.087 0.013

(0.5,0.3,0.1) 0.472 0.028 0.267 0.033 0.063 0.037

(-0.1,-0.1,-0.1) -0.109 0.009 -0.115 0.015 -0.101 0.001

(-0.1,0.3,-0.3) -0.108 0.008 0.286 0.014 -0.299 0.001

(-0.2,0.1,-0.2) -0.203 0.003 0.083 0.017 -0.199 0.001

(-0.2,-0.2,0.2) -0.199 0.001 -0.210 0.010 -0.199 0.001

(-0.3,-0.1,-0.1) -0.310 0.010 -0.109 0.009 -0.112 0.012

(-0.3,-0.3,-0.3) -0.290 0.010 -0.297 0.003 -0.300 0.000

(-0.4,-0.1,-0.1) -0.405 0.005 -0.105 0.005 -0.101 0.001

(-0.4,0.2,-0.2) -0.372 0.028 0.193 0.007 -0.190 0.010

(-0.5,-0.1,-0.1) -0.495 0.005 -0.104 0.004 -0.099 0.001

(-0.5,0.1,-0.3) -0.465 0.035 0.105 0.005 -0.300 0.000

35


True value

( )11321 ,,, baaa 1a bias

of 1a 2a bias

of 2a 3a bias

of 3a 11b Bias

of 11b

(0.1,0.1,0.1,0.1) 0.087 0.013 0.082 0.018 0.092 0.008 0.102 0.002

(0.1,0.1,0.3,0.5) 0.029 0.071 0.051 0.049 0.270 0.030 0.460 0.040

(0.2,0.2,0.2,0.2) 0.163 0.037 0.159 0.041 0.177 0.023 0.184 0.016

(0.2,0.1,0.4,0.2) 0.154 0.046 0.060 0.040 0.359 0.041 0.177 0.023

(0.3,0.1,0.4,0.1) 0.266 0.034 0.071 0.029 0.368 0.032 0.080 0.020

(0.3,0.3,0.3,0.3) 0.225 0.075 0.237 0.063 0.269 0.031 0.231 0.069

(0.4,0.1,0.1,0.1) 0.375 0.025 0.078 0.022 0.091 0.009 0.089 0.011

(0.4,0.2,0.3,0.1) 0.367 0.033 0.163 0.037 0.273 0.027 0.075 0.025

(0.5,0.1,0.3,0.1) 0.468 0.032 0.069 0.031 0.273 0.027 0.076 0.024

(0.5,0.1,0.4,0.2) 0.440 0.060 0.061 0.039 0.381 0.019 0.155 0.045

(-0.1,-0.1,-0.1,-0.1) -0.102 0.002 -0.108 0.008 -0.098 0.002 -0.102 0.002

(-0.1,-0.2,0.4,-0.4) -0.099 0.001 -0.193 0.007 0.379 0.021 -0.395 0.005

(-0.2,-0.2,-0.2,-0.2) -0.197 0.003 -0.202 0.002 -0.192 0.008 -0.204 0.004

(-0.2,0.4,-0.4,-0.1) -0.191 0.009 0.380 0.020 -0.389 0.011 -0.099 0.001

(-0.3,-0.1,-0.1,-0.1) -0.299 0.001 -0.109 0.009 -0.099 0.001 -0.103 0.003

(-0.3,0.3,-0.3,0.3) -0.288 0.012 0.290 0.010 -0.282 0.018 0.276 0.034

(-0.4,-0.1,-0.1,-0.4) -0.376 0.024 -0.097 0.003 -0.092 0.008 -0.405 0.005

(-0.4,-0.2,0.5,0.1) -0.403 0.003 -0.205 0.005 0.476 0.024 0.103 0.003

(-0.5,-0.1,-0.1,-0.5) -0.475 0.025 -0.085 0.015 -0.079 0.021 -0.456 0.044

(-0.5,-0.2,-0.2,-0.2) -0.483 0.017 -0.210 0.010 -0.193 0.007 -0.203 0.003

36

2.8 SUMMARY

In this chapter, the theory, definition and properties of bilinear model were

discussed. Different methods of parameters estimation were reviewed with a concise

description on the nonlinear least squares method. The simulated values of parameter

estimation were presented. It was found that the nonlinear least squares method can

perform well in estimating the coefficients of BL(p,0,1,1), p=1,2,3.

37

CHAPTER THREE

OUTLIERS

3.1 A REVIEW OF OUTLIERS

The occurrence of outliers provides interesting case studies for further

exploration. Their existence should be investigated and never be ignored. In any

scientific research, full disclosure of data modeling is required, including a disclosure

and discussion of outliers. There has been much debate in the literature regarding

what to do with the existence of outliers in data sets including time series data.

Studies had shown that outliers affect the performance of standard statistical

methodology in modelling, forecasting and diagnostic purposes. In some cases, the

effect is disastrous.

Various approaches of detecting and handling outliers have been considered

with the objective of improving the efficiency and adequacy of statistical analyses.

They range from how to detect outliers to whether they should be removed from the

data set or otherwise. Recent approaches in time series comprise an iterative process

of identifying the locations and types of outliers, removing their effect from the data,

and modeling the data until an outlier-free model is obtained. It has been shown to

work in linear time series problems and several nonlinear time series problems. In

this chapter, an overview of outliers relevant to the work of this study is presented

including their definition, sources, causes on statistical methodology, detection,

treatment and handling.

38

3.2 TYPES OF OUTLIERS

In time series, there are four basic types of outliers typically considered (Chang et

al. [1988] and Tsay [1988]). These are additive outlier (AO), innovational outlier (IO),

temporary change outlier (TC) and level change outlier (LC). Other types of outliers

usually can be expressed in combinations of these four basic types.

As the name implies, an AO affects only the value observed at the time of the

outlier. Hence, AO has no effect on future values of observations. Therefore, effect of

AO on observations does not persist. On the other hand, the effect of IO persists. That is,

there is an initial impact at the time the outlier occurs and its effect continues in a lagged

fashion with subsequent observations. Similarly, the effect of LC also persists. They have

the effect of either increasing or lowering the mean of the series starting at the time the

outlier occurs. This change in the mean is abrupt and permanent. The effect of TC also

persists. There is an abrupt change in the mean of the series at the time this outlier occurs.

The change gradually decays and eventually brings the mean of the series back to its

original value. The rate of this decay is modelled using an input parameter, δ . The

default value of 7.0=δ is recommended for general use by Chen and Liu [1993a]. The

nature of these outliers will be described further in Chapter 4.

The most extensively used type of outlier is the AO. The abovementioned authors

looked at the incidence of this outlier in their study. Others include Vogelsang [1999],

Berkoun et al. [2003], Perron and Rogriguez [2003] and Wright and Hu [2003]. This type

of outlier is deterministic in nature and is most likely caused by an isolated incident such

as recording error, earthquake and so on.

39

Another type of outlier is the IO mentioned by Fox [1972], Martin [1980], Choy

[2001] and Caroni and Karioti [2002, 2004]. Meanwhile, Box and Tiao [1965], Chen and

Tiao [1986], Chen and Liu [1993a], Balke [1993] and Lanne et al. [2002] were among

others who considered the existence of LC in time series. The effects of LC cause a

permanent shift in value after the occurrence of LC. In addition, Tsay [1986b] and Chen

and Liu [1993a] defined another type of outlier, the TC. The effect of TC will die out

eventually according to a dampening factor which takes values from 0 to 1.

3.3 CAUSES OF OUTLIERS

Outliers can occur due to several different mechanisms or causes. The occurrence

can be caused either by data recording or entry errors, motivated mis-reporting and

sampling errors. Anscombe [1960] sorted outliers into two major categories: those arising

from errors in the data and those arising from the inherent variability of the data. “Not all

outliers are illegitimate contaminants and not all illegitimate scores show up as outliers”

(see Barnett & Lewis [1995]). It is therefore important to consider the range of causes

that may be responsible for outliers in a given data set.

3.3.1 Outliers from data errors

Outliers are often caused by human error, for example, errors in data collection,

recording or entry. Data from an interview can be mistakenly recorded or miscued upon

data entry.

40

3.3.2 Outliers from intentional or motivated misreporting

It is possible that an outlier can come from motivated misreporting. There are

times when participants purposefully report mistaken data to experimenters or surveyors.

A participant may make a conscious effort to sabotage the research (Huck [2000]) or may

be acting from other motives. Environmental conditions can motivate over-reporting or

mis-reporting.

3.3.3 Outliers from sampling error

Another cause of outlier is due to sampling error. It is possible that a few

members of a sample were unintentionally drawn from a different population than the

rest of the sample. For instance, in education, unintentional sampling of the academically

gifted or mentally retarded students is possible and may provide undesirable outliers.

These cases should be removed as they do not reflect the target population.

3.4 TREATMENT OF OUTLIERS

There is a large pool of literatures on outliers that extends many years.

Consequently, one would expect that a brief definition of an outlier could be easily

provided but in fact this has turned out to be a difficult task. It is a complex matter to

precisely encapsulate what an outlier is. Many researchers have expressed a notion of

what an outlier is in a larger series of observations. However, providing an objective

statement that can be used to identify an outlier seems to be a great challenge.

41

While recognizing the subjective nature of outlier identification, once identified,

they can be treated in two broad ways as either values for rejection or values that point to

a phenomenon of interest. They are rejected if they can be considered as values that are

drawn from another population or values resulting from measurement error and suitable

for exclusion from the sample as it distorts the analysis. Alternatively, they can be

considered a phenomenon of interest that should not be excluded.

In the past, the approach for outlier detection placed emphasis on rejecting

outliers as values that were not part of the population being analyzed. Hence, more

statistically based approaches were used.

Outliers are of great interest to many researchers. For a given sample, the aim is

to find outliers that represent valid data that is significant but differs from most of the

sample. Applications include identifying unusual weather events, fraud, intrusion,

medical conditions and public health issues. A range of other approaches exist for

detection of outliers including model based techniques as well as techniques that make

use of class labels. If class labels are used and training set data is available, a supervised

learning approach could be employed instead. If no training data is available, an

unsupervised technique may be employed. Outliers of this type are often referred to as

noise and would typically be rejected from the sample as they would distort the analysis

being carried out. An excellent overview of all these approaches is by Tan et al. [2006].

Other literature on outliers includes Knorr and Ng [1997] who take an intuitive

notion of outliers and provide formalization. Liu [1998] addresses the problem of

distinguishing between outliers that should be rejected and those that should be retained

in the sample as phenomenon of interest. The criteria considered in making this decision

42

are the characteristics of the data also relevant domain knowledge. It suggested that

models noise and error processes and accepts outliers as phenomenon of interest if the

noise model cannot account for them.

One of the approaches to deal with the outliers in time series is an iterative

operation of identifying the location and types of outliers and adjusting their effects in

observations. Tsay [1986b] proposed an iterative procedure consisting of specification,

detection and removal cycles to reduce the outlier effects on model specification. The

proposed procedure uses least squares method. Similar approaches are used by Chang et

al. [1988] and Chen and Liu [1993a].

Abraham and Chuang [1989] modified the outlier detection procedure used in

regression analysis for application in time series analysis and then proposed a procedure

for modelling time series data in the presence of outliers. Later, Abraham and Chuang

[1993] applied the expectation-maximization algorithm to time series situations where

outliers may be present. Choy [2001] proposed Whittle-type estimators in developing the

iterative procedure spectrum-based outlier detection algorithm for any stationary AR or

ARMA processes while Baragona et al. [2001] considered the genetic algorithm for the

identification of outliers in time series.

Outliers can also be dealt with using robust estimation methods. The possible

outliers were first detected, for instance, by looking at plots of residuals. This approach is

used by, among others, Martin [1979, 1980], Martin and Thomson [1982] and De Luna

and Genton [2001]. To reduce the influence of outliers, the outlying observation is down

weighted through various types of ψ-functions in the estimation processes such as the

43

Huber’s monotone function and bi-square type. Examples of robust estimation methods

include M-estimation and Generalized M-estimation.

Several authors suggested other ways to deal with the outliers in a given sample.

Anscombe [1960] suggested that the outliers could be discarded if they were caused by

large measurement of execution error and when any possible rectification was not

possible. D’Agostino and Stephens [1986] gave several options to treat outliers.

First, outliers can be omitted and the reduce sample is treated as a “new” sample.

Second, the outlier can be omitted and treat the reduce sample as a censored sample.

Third, outliers are replaced with the nearest “good” observation in order to preserve the

measurement. Fourth, an additional observation is searched to replace the outlier. Fifth,

both analyses can be performed with and without outliers before reaching any conclusion.

The principle of outlier treatment is based on the premises of accommodation and

discordance. Knowing the causes of outliers can help in determining suitable outlier

treatments. Confirmed erroneous observations can simply be removed. However, it is not

always possible to know the exact cause of discrepancies in individual data observations.

As for the treatment of true observations, we need to perform statistical outlier tests to

determine whether the outlier in question belongs to the same group. The user can then

decide the treatment of the identified outlier as to be ‘‘removed’’, ‘‘retained’’ or

‘‘revised’’ (Grubbs and Beck [1972]). Another remedial suggestion on outlier(s)

treatment is, whenever possible, to increase the sample size to ensure the true nature of

the underlying population. This is encouraged if factors, such as affordable cost of

collecting data, are available.

44

3.5 OUTLIER DETECTION IN TIME SERIES

The earliest study on outliers in time series data was carried out by Fox [1972].

The study looked at detailed examination of the detection and testing of outliers in

stationary time series including a procedure of detecting the occurrence of AO and IO in

non-seasonal AR(p) processes. The likelihood ratio statistics were derived for both types

of outliers as the test statistics in the detection procedure. The critical values of the test

had been generated through simulation. Other studies followed afterward by, inter alia,

Chang [1982], Bell et al. [1983], Hillmer [1984], Tsay [1986a], Pena [1987], Chang et al.

[1988], Abraham and Chuang [1989], Chan [1992], Ljung [1993] and Atkinson et al.

[1997].

Various methods of detecting outliers in linear ARMA models have been

proposed. These include methods based on Bayesian approach (Box and Tiao [1968],

Gutmann [1973] and Dempster and Rosner [1975]), robust approach (Denby and Martin

[1979], Martin [1980], Abraham and Chuang [1989], and de Luna and Genton [2001]),

the test of hypothesis (Chang [1982], Hillmer et al. [1983], and Tsay [1988]), iterative

maximum likelihood approach (Chang [1982] and Caroni and Karioti [2004]), Lagrange

multiplier (L-M) test (Abraham and Yatawara [1988]), intervention approach (Atkinson

et al. [1997]), spectrum-based outlier detection (Choy [2001]), generic algorithm

(Baragona et al. [2001]) and wavelet transform (Struzik and Siebes [2002]).

A special case of multiple outliers is occurrence of outliers in patches. Tsay et al.

[2000] showed that multivariate innovative outlier in a vector time series can introduce

patch of additive outliers in univariate marginal time series. Justel et al. [2001] proposed

an adaptive Gibbs algorithm to detect patches of additive outliers in AR model. The

45

procedure identifies the beginning and end of possible outlier patches using Gibbs

sampling. The adaptive procedure with block interpolation is then performed. Sanchez

and Pena [2003] proposed a procedure that can distinguish innovative outliers and level

shifts which occur together. Besides, joint test for sequences of additive outliers were

proposed. Some studies were reported on outlier detection in time series nonlinear time

series.

For threshold model, Hau [1984] and Hau and Tong [1989] developed outlier

detection based on hat matrix for detecting AO. For bilinear model, outlier detection has

been studied by Zaharim [1996], Chen [1997], Mohamed [2005] and Zaharim et al.

[2006] using Gibbs sampling or least squares method. The least squares approach has

been adopted to detect outlier in GARCH models (see Franses and Ghijsels [1999] and

Amelie and Oliver [2004]).

The problem of detecting multiple outliers has also been investigated. Chen and

Liu [1993a] derived formulation for the occurrence of two additive outliers in ARMA

model. Knowing the location of the outliers, the effects of the outliers may be estimated

jointly. However, the statistics require the estimated values of outlier effect to be

obtained separately. That is, only a single outlier was assumed to be present at one time.

Consequently, the result depends on which outlier is being estimated first. Chen

and Liu [1993a] pointed out that, from a computational point of view, the only feasible

approach in dealing with multiple outliers is by detecting outliers one by one although the

most appropriate way is to estimate the effect jointly.

46

3.5.1 Bootstrap-based procedure

Bootstrap is a general technique for assessing uncertainty in estimation

procedures in which computer simulation through resampling data replaces mathematical

analysis. This method was introduced by Efron and Tibshirani [1986]. We will focus on

using bootstrap to find a standard error to an estimated parameter.

In a simple case, suppose that we are interested in estimating the mean from an

unknown population on the basis of randomly sampled data. The sample mean can be

used to estimate the parameter, while the standard error is used to measure the

uncertainty in this estimate. The standard error is basically the standard deviation of the

sampling distribution of the sample mean.

It is well known that, in statistical theory, standard error of a sample mean equals

the population standard deviation divided by the square root of the sample size. In cases

where the population size is unknown, we usually use the sample standard deviation

instead of the population standard deviation. For normal shaped populations or large

samples from non-normal populations, we may also conclude that the shape of the

sampling distribution is approximately normal. This enables us to compute the

confidence intervals of the parameter of interest.

Consider now the problem of estimating the population median. Again, the

sample median is a natural estimate. However, it may be troublesome to find the standard

error of median. Hence, bootstrapping can be used to find the estimate of standard error

without great mathematical difficulty.

In principle, the ideal way to estimate the standard error of a sample median

would be to take a very large number of samples of the original size from the population,

47

compute the sample median of each, and use the standard deviation of this large

collection of simulated sample medians as an estimate of the true standard error.

Unfortunately, we do not have the ability to sample repeatedly from the population. We

can, however, sample repeatedly from our original sample, which is in itself an estimate

of the population. This is how bootstrap works.

a) Take the original sample of size n from a population of interest.

b) Compute the desired sample statistic (such as the median).

c) From the original sample, resample with replacement a bootstrap sample of size n.

Some numbers in the original sample may be included several times in the

bootstrap sample. Others may be excluded. This creates a bootstrap data set of the

same size as the original.

d) Apply the estimation procedure to the bootstrap sample and store this value.

e) Repeat steps c) and d) for B times and store all results. The estimated standard

error is the standard deviation of the B separate estimates. For estimating a

standard error, a number like B = 200 is usually sufficiently high.

f) If a histogram of the bootstrap estimates is approximately normal in shape, we

may use normal theory to find confidence intervals for the unknown parameter. If

the shape is not normal, the sampling distribution is not normal and more

advanced techniques are needed to find a confidence interval. However, the

bootstrap-generated standard error is still an able measure of the variability in the

estimation procedure.

48

While for standard case, sampling is done on the observation, but in time series,

sampling is done on residuals. The process of drawing random samples with replacement

from residuals is described below:

(a) Let ( )neee ...,,, 21 be the original residuals. Sampling with replacement is

carried out from the original residuals giving a bootstrap sample of size n,

say, ( ) ) , , ,( *n

*2

*1 eeee …=1* . This is repeated a large number of times, say B

times, giving B sets of bootstrap samples ( ) ( ) ( )Beee *2*1* ...,, , .

(b) For each bootstrap sample ( )Me* , M = 1, 2, ..., B, we calculate Mω~ , the

statistic of interest.

(c) The sample standard deviation of ω~ is given by

( )( )

21

1

2

1

~~~

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

∑ −= =

B

B

MBSM

BS

ωωσ (3.1)

where

.~~1

1 ∑==

− B

MMBS B ωω

Efron and Tibshirani [1986] showed that as B → ∞, BSσ~ approaches ,σ the bootstrap

estimate of the standard deviation.

49

3.5.2 Model-based procedure

A model-based procedure is proposed by Battaglia and Orfei [2005] for detecting

the presence of additive outliers or innovational outliers when the series is generated by a

general nonlinear model. There models are considered; bilinear, threshold autoregressive

and exponential autoregressive models. The procedure works by first choosing a suitable

time series model. They argue that this step is very important when searching for outliers.

This is because a large residual variance caused by overall lack of fit would result in

under-identification of outliers. On the other hand, a model which is unable to explain

the local behavior of the series would yield single large residuals resulting in over-

identification. Battaglia and Orfei [2005] considered a first-order Taylor expansion about

( ) ( )( )1*1* , −− tAO

tAO eY to approximate the bilinear process, tY = ( ) ( )( ) t

tt eeYf +−− 11 ; , which is given

by

( ) ( )( ) ( ) ( )( ) ( ) ( )tYYeYfeYf j

p

jAOjtjt

tAO

tAO

tt λ∑=

−−−−−− −+≅

1

*,

1*1*11 ,;

( ) ( )∑=

−− −+s

jjAOjtjt tee

1

*, µ (3.2)

where

( ) ( )( ) ( )( )1*1*

*,

, −−

−∂∂

= tAO

tAO

AOjtj eYf

Ytλ , j=1,…,p ; ( ) 0=tjλ , j > p

( ) ( )( ) ( )( )1*1*

*,

, −−

−∂∂

= tAO

tAO

AOjtj eYf

etµ , j=1,…,s ; ( ) 0=tjµ , j > s

From (3.2) we obtain:

( ) ( ) ( )⎭⎬⎫

⎩⎨⎧

+−++−≅− ∑=

−+−+++ jdeejdee k

j

kkjdAOkjdjdjdAOjd µλω

1

*,

*, , for j=1,2,… (3.3)

50

Now, recalling that ddAOd ee ω=−*, , and defining the recursion 1=oc , we have

( ) ( )⎥⎦

⎤⎢⎣

⎡+++−= ∑

=− jdcjdc k

j

kkjjj µλ

1

, j=1,2…

we can write

djjdAOjd cee ω≅− ++*

, , j=0,1,2,… (3.4)

Thus ∑=

=n

tteS

1

2 can be rewritten as:

( )∑+=

≅n

tte

1

2

γ

( ) ( )∑ ∑−

+=

−

=+ −+

1

1 0

2*,

2*,

d

t

dn

jdjAOjdAOt cee

γ

ω

then minimized by

∑

∑−

=

+

−

=∧

= qn

jj

AOjd

dn

jj

AO

c

ec

0

2

*,

0ω (3.5)

and the variance of AOω is given by equation (3.6) as given in Battaglia and Orfei [2005].

⎭⎬⎫

⎩⎨⎧

−−

= ∑∑−

=

∧

+=

dn

jjAO

n

tttMBAO ce

n 0

22

1

2*2*,,

1 ωγ

σγ

(3.6)

For innovational outlier (IO), the sample variance is given by

( ) ( ) ( ) ( )γ

σ γ

−

+++++= +−+

∧

neeee ndd

dIO

**1

*1

*1

2

,......

(3.7)

This procedure adopts a similar strategy from the iterative frame work of Chen

and Liu [1993a], based on the following steps:

a) Derive initial estimates of model parameters.

51

b) Given the parameter values, for any t and for each type of outliers, assume that an

outlier has occurred at time t, and estimate its amplitude. If the largest absolute

estimated amplitude is significant (i.e. larger than an a priori fixed sensitivity

level, usually 3.5 or 4 times of its estimated standard error), then identify an

outlier of that type at that time; otherwise stop.

c) Remove the effect of the identified outlier by subtracting its estimated amplitude

from ty (and also correcting all subsequent observations according to the

estimated model in case of innovational outlier).

d) Estimate again the model parameters on the corrected series, and iterate step 2.

Battaglia and Orfei [2005] followed the above steps in this proposed model-based

procedure. Following Tsay [1986b] and Chang et al. [1988], let H0 denote the hypothesis

that 0=ω in the bilinear model considered. Let H1 denote the situations 0≠ω in

bilinear model with AO and IO, respectively, at time t. Test statistics can be derived for

testing one hypothesis versus another as follows:

For model-based procedure on AO:

H0 v H1 :

∑

∑−

=

−

=+

=tn

jjtMBAO

tn

jjtj

tMBAO

c

c

0

2*,,

0,,ˆ

σ

ητ , .,...,1 nt = (3.8)

where

⎭⎬⎫

⎩⎨⎧

−−

= ∑ ∑+=

−

=

n

t

tn

jjtktMBAO ce

n 1 0

222*2*,, ˆ1

γ

ωγ

σ

52

For model-based procedure on IO:

H0 v H1 : tMBIO

tIOtMBIO

e

,,

*,

,, ~ˆσ

τ = , .,...,1 nt = (3.9)

where

( ) ( ) ( ) ( )γ

σ γ

−

+++++= +−+

neeee ndd

dMBIO

**1

*1

*12

,,

......~

In general, the time point where an outlier occurs is unknown. Hence, the values

of the test statistics can be obtained at every time point t = 1, 2,... , n. The test for

identifying the type of outlier at a particular point t in BL(p,0,1,1), where p=1,2,3 models

begin with modeling the original time series Y by assuming that there is no outlier in the

data. The maximum values of the test statistics (3.8) and (3.9) are examined. The full

procedure is described below:

a) Compute the least squares estimates of BL(p,0,1,1) models, p=1,2,3 based on the

original data. Hence, obtain the residuals.

b) Compute tMBTPτ ,,ˆ for each t, t = 1, 2, ..., n, using the residuals obtained in Stage a).

(TP= AO or IO and MB=model-based)

c) Let { }tMBTPtMB ,,n1,2,...,t, ˆmax τη=

= . Given a pre-determined critical value C, if

,, Cη tTP > then there is a possibility of an AO or IO occurring at time t. (TP=AO

or IO and MB=model-based)

Through the suggested model-based procedure, the occurrence of AO or IO can be

detected at any time t.

53

3.6 SUMMARY

The benefits of time series outlier detection and estimation are not limited to

providing better model estimates theoretically. More importantly, as shown in this

example, outlier detection often leads to the discovery of events that may provide useful

information or knowledge. Additional interesting examples can be found in various

articles including Chang et al. [1988], Liu and Chen [1991] and Chen and Liu [1993b].

54

CHAPTER FOUR

A STUDY ON THE NATURE OF OUTLIERS IN BL(p,0,1,1) MODEL

We derive, in this chapter, the formulation of AO and IO effects on observations

and residuals of BL(p,q,r,s) models. We will, however, focus on BL(p,0,1,1) models in

studying the nature of AO and IO as they appear in the models. The study extends the

works on outliers in ARMA(p,q) models by Chen and Liu [1993a] and in BL(1,1,1,1)

models by Zaharim et al. [2006]. It is important to understand the nature of outliers so

that their occurrence can be initially detected and appreciated in their preliminary form

through visualization.

4.1 MODEL FORMULATION OF BL(p,q,r,s)

An outlier-free BL(p,q,r,s) model is given by:

∑∑∑∑=

−−=

−=

−=

+++=r

ktltkt

s

lkljt

q

jjit

p

iit eeYbecYaY

1 111 (4.1)

where a, b, and c are any real number satisfying the stationary condition of the model

whereas tY and te are outlier-free observation and residual respectively, ,...3,2,1=t .

They are referred as original observation and residual herewith. Equation (4.1) can be

rewritten as

tlt

q

llkt

p

rkk

r

iitjt

s

jijit eecYaYebaY ++++= −

=−

+==−−

=∑∑∑ ∑

111 1])[( δ (4.2)

where

55

⎩⎨⎧

>≤

=rpforrpfor

10

δ

and

*

1

*

11

**

1

** )( lt

q

llkt

p

rkk

r

iitjt

s

jijitt ecYaYebaYe −

=−

+==−−

=∑∑∑ ∑ −−+−= δ (4.3)

where *te is the contaminated residual obtained when an outlier exists in the data.

Equation (4.3) can be written as

∑ ∑∑∑ ∑=

−−=

−==

−−=

−−+−=r

iitjt

s

jijjt

s

jij

r

iitjt

s

jijitt YebebYebaYe

1

*

1

*

11

**

1

** )(])[(

*

1

*

1lt

q

llkt

p

rkk ecYa −

=−

+=∑∑ −−δ (4.4)

4.1.1 Formulation of AO effects on observations

Consider the AO case first. Let *,AOtY be the observed values from BL(p,q,r,s)

process with an AO occurs at time point dt = with magnitude ω and let tY be the

observations at time t that would have been obtained if there were no outliers in the data

and will be referred herewith as “original observation”, nt ,...,3,2,1= . For dt < , clearly

tAOt YY =*, and for dt ≥ , the values will be different and are given by the following

formulations:

⎩⎨⎧

=+≠

=dtforYdtforY

Yt

tAOt ω

*, (4.5)

Equation (4.5) suggests that shock caused by an AO affects the original observation at

dt = only with a magnitude ω and the rest remains unaffected.

56

4.1.2 Formulation of AO effects on residuals

For AO effects on residuals, let *,AOte be the resulting residual when BL(p,q,r,s) is

fitted on the contaminated data and let te be the residuals at time t that would have been

obtained if there were no outliers in the data and will be referred herewith as “original

residual”, nt ,...,3,2,1= . For dt < , clearly tAOt ee =*, and for dt ≥ and 0≥k , the

values will be different and are given by the following formulations:

For dt = :

∑ ∑∑∑ ∑=

−−=

−==

−−=

−−+−=r

iAOidjd

s

jijAOjd

s

jij

r

iAOidAOjd

s

jijiAOdAOd YebebYebaYe

1

*,

1

*,

11

*,

*,

1

*,

*, )(])[(

*,

1

*,

1AOld

q

llAOkd

p

rkk ecYa −

=−

+=∑∑ −−δ

∑ ∑∑∑ ∑=

−−=

−==

−−=

−−+−+=r

iidjd

s

jijjd

s

jij

r

iidjd

s

jijid YebebYebaY

1 111 1)(])[()( ω

ld

q

llkd

p

rkk ecYa −

=−

+=∑∑ −−

11δ

])[(111 1

ωδ +−−+−= −=

−+==

−−=

∑∑∑ ∑ ld

q

llkd

p

rkk

r

iidjd

s

jijid ecYaYebaY

)( ω−−= de

AOd Ae ,0ω−=

The last two steps are undertaken to simplify the derivation for general formulation of

outlier effect on the residuals. Similar approaches are also taken for the IO case

discussed later in the thesis.

57

For 1+= dt :

∑ ∑∑∑ ∑=

−+−+=

−+==

−+−+=

++ −−+−=r

iAOidjd

s

jijAOjd

s

jij

r

iAOidAOjd

s

jijiAOdAOd YebebYebaYe

1

*,11

1

*,1

11

*,1

*,1

1

*,1

*,1 )(])[(

*,1

1

*,1

1AOld

q

llAOkd

p

rkk ecYa −+

=−+

+=∑∑ −−δ

∑ ∑∑=

−+−+=

−+=

+ +−+−=r

iAOidAOjd

s

jijiAOdAOjd

s

jjd YebaYebaY

2

*,1

*,1

1

*,

*,1

1111 ])[()(

*,1

11

*,11

1

*,1

2

*,1 )( AOkd

p

rkk

r

iAOidjd

s

jijAOjd

s

jijAOdi YaYebebeb −+

+==−+−+

=−+

=∑∑ ∑∑ −−+− δ

*,1

2

*,1 AOld

q

llAOd ecec −+

=∑−−

ld

q

llkd

p

rkk

r

iidjd

s

jijid ecYaYebaY −+

=−+

+==−+−+

=+ ∑∑∑ ∑ −−+−= 1

11

1111

11 ])[( δ

AO

r

iAOidijd

s

jj AcYAbeba ,01

1

*,1011

111 )( ωωω +++− ∑∑

=−+−+

=

])()[(1

1,0

1

*,1011

1111 ∑ ∑∑

= =−+−+

=+ +−+−=

jAOj

r

iAOidijd

s

jjd AcYAbebae ω

AOd Ae ,11 ω−= +

For 2+= dt :

*,2

11

*,2

*,2

1

*,2

*,2 ])[( AOkd

p

rkk

r

iAOidAOjd

s

jijiAOdAOd YaYebaYe −+

+==−+−+

=++ ∑∑ ∑ −+−= δ

*,2

11

*,22

1

*,2

1)( AOld

q

ll

r

iAOidjd

s

jijAOjd

s

jij ecYebeb −+

==−+−+

=−+

=∑∑ ∑∑ −−−

58

∑ ∑∑≠=

−+−+=

−+=

+ +−++−=r

iiidjd

s

jijidjd

s

jjd YebaYebaY

2,122

12

1222 ])[())(( ω

∑ ∑∑=

−+=

+−++=

+−+−−−r

ijd

s

jijAOdiAOdiAOkd

p

rkk ebAebAebYa

12

3,02,111

*,2

1)()([ ωωδ

*,22

1] AOidjd

s

jij Yeb −+−+

=∑− ld

q

llAOdAOd ecAecAec −+

=+ ∑−−−−− 2

3,02,111 )()( ωω

])()[(3

1,2

1

*,22

1222 ∑ ∑∑

=−+

=−+−+

=+ +−+−=

jAOjdj

r

iAOidijjd

s

jjd AcYbebae ω

AOd Ae ,22 ω−= +

For 3+= dt :

*,3

11

*,3

*,3

1

*,3

*,3 ])[( AOkd

p

rkk

r

iAOidAOjd

s


+==−+−+

=++ ∑∑ ∑ −+−= δ

*,3

11

*,33

1

*,3

1)( AOld

q

ll

r

iAOidjd

s

jijAOjd

s

jij ecYebeb −+

==−+−+

=−+

=∑∑ ∑∑ −−−

∑ ∑∑≠=

−+−+=

−+=

+ +−+−=r

iiAOidjd

s

jijiAOdjd

s

jjd YebaYebaY

3,1

*,33

1

*,3

1333 ])[()(

∑ ∑∑=

−+=

++−++=

+++−−r

iAOjd

s

jijAOdiAOdiAOdiAOkd

p

rkk ebebebebYa

1

*,3

4

*,3

*,12

*,21

*,3

1[δ

*,3

4

*,3

*,12

*,21

*,33

1] AOld

q

llAOdAOdAOdAOidjd

s

jij ececececYeb −+

=++−+−+

=∑∑ −−−−−

∑∑=

−+−+=

+ +−+−=r

iAOAOidijd

s

jjd AcYbebae

1,21

*,313

1333 )()[( ωω

])()( ,031

*,3,03,12

1

*,302 AO

r

iAOidAOiAO

r

iAOidi AcYAbAcYAb ωωωω +−+− ∑∑

=−+

=−+

59

])()[(3

1,3

1

*,33

1333 ∑ ∑∑

=−+

=−+−+

=+ +−+−=

jAOjdj

r

iAOidijjd

s

jjd AcYbebae ω

AOd Ae ,33 ω−= +

For 4+= dt :

*,4

11

*,3

*,4

1

*,4

*,4 ])[( AOkd

p

rkk

r

iAOidAOjd

s


+==−+−+

=++ ∑∑ ∑ −+−= δ

*,4

11

*,44

1

*,4

1)( AOld

q

ll

r

iAOidjd

s

jijAOjd

s

jij ecYebeb −+

==−+−+

=−+

=∑∑ ∑∑ −−−

∑ ∑∑≠=

−+−+=

−+=

+ +−+−=r

iiAOidjd

s

jijiAOdjd

s

jjd YebaYebaY

4,1

*,44

1

*,4

1444 ])[()(

∑ ∑∑=

−+=

+++−++=

++++−−r

iAOjd

s

jijAOdiAOdiAOdiAOdiAOkd

p

rkk ebebebebebYa

1

*,4

5

*,4

*,13

*,22

*,31

*,4

1[δ

∑ ∑∑∑= =

−+=

−+−+−+=

+ −−+−=r

i jAOjdj

jAOidjdijjd

s

jjd AcYAbebae

1

4

1,4

4

1

*,444

1444 ])[( ωωω

∑ ∑∑=

−+=

−+−+=

+ +−+−=4

1,4

1

*,44

1444 ])()[(

jAOjdj

r

iAOidijjd

s

jjd AcYbebae ω

AOd Ae ,44 ω−= +

In general, for kdt += :

AOkkdAOkd Aee ,*

, ω−= ++ (4.6)

where

⎪⎩

⎪⎨⎧

≥+−+

=−= ∑ ∑∑

=−+

=−+

=−+ 1)()(

01

1,

1

*,

1

, kforAcYbeba

kforA k

jAOjkdj

r

iAOikdij

s

jjkdkjk

AOk

Several residuals from dt = onward should be affected as described in equation (4.6).

60

4.1.3 Effect of IO on observations

Using similar definition as AO, when IO occurs at time dt < , tIOt YY =*, . On the

other hand, for dt ≥ , the formulation of IO effects on observations is derived as follows:

For dt = :

∑∑∑ ∑=

−+=

−=

−=

− +++++=q

ldldl

p

rkIOkdk

r

iIOid

s

jjdijiIOd eecYaYebaY

11

*,

1

*,

1

*, )( ωδ

∑∑∑ ∑=

−+=

−=

−=

− +++++=q

ldldl

p

rkkdk

r

iid

s

jjdiji eecYaYeba

111 1)( ωδ

ω+= dY

IOd AY ,0ω+=

For 1+= dt :

∑∑∑ ∑=

+−++=

−+=

−+=

−++ ++++=q

ldldl

p

rkIOkdk

r

iIOid

s


111

1

*,1

1

*,1

11

*,1 )( δ

∑∑ ∑∑+=

−+=

−+=

−+=

−+ ++++=p

rkkdk

r

iid

s

jjdijiIOd

s

jjdij YaYebaYeba

11

21

11

*,

111 )()( δ

∑=

+−+ ++q

ldldl eec

111

∑ ∑∑=

−+=

−+=

−+ ++++=r

iid

s

jjdijid

s

jjdij YebaYeba

21

11

111 )())(( ω

∑∑=

+−++=

−+ +++q

ldldl

p

rkkdk eecYa

111

11δ

61

ω)(1

111 ∑=

−++ ++=s

jjdijd ebaY

∑ ∑=

−=

−++ ++=1

11

111 )(

mm

s

jjdmjmd AebaY ω

IOd AY ,11 ω+= +

For 2+= dt :

∑∑∑ ∑=

+−++=

−+=

−+=

−++ ++++=q

ldldl

p

rkIOkdk

r

iIOid

s


122

1

*,2

1

*,2

12

*,2 )( δ

∑ ∑∑∑=

−+=

−+=

−++=

−+ +++++=r

iid

s

jjdijiIOd

s

jjdjIOd

s

jjdij YebaYebaYeba

32

12

*,

1222

*,1

121 )()()(

∑∑=

+−++=

−+ +++q

ldldl

p

rkkdk eecYa

122

12δ

∑ ∑=

−=

−++ ++=2

1,2

122 )(

mIOm

s

jjdmjmd AebaY ω

IOd AY ,22 ω+= +

For 3+= dt :

∑∑∑ ∑=

+−++=

−+=

−+=

−++ ++++=q

ldldl

p

rkIOkdk

r

iIOid

s


133

1

*,3

1

*,3

13

*,3 )( δ

*,

1333

*,1

1322

*,3

131 )()()( IOd

s

jjdjIOd

s

jjdjIOd

s

jjdij YebaYebaYeba ∑∑∑

=−++

=−++

=−+ +++++=

∑∑∑ ∑=

+−++=

−+=

−+=

−+ +++++q

ldldl

p

rkkdk

r

iid

s

jjdiji eecYaYeba

133

13

43

13 )( δ

62

IO

s

jjdjIO

s

jjdijd AebaAebaY ,1

1322,2

1313 )()( ωω ∑∑

=−+

=−++ ++++=

IO

s

jjdj Aeba ,0

1333 )( ω∑

=−+++

∑ ∑=

−=

−++ ++=3

1,3

133 )(

mIOm

s

jjdmjmd AebaY ω

IOd AY ,33 ω+= +

For 4+= dt :

∑∑∑ ∑=

+−++=

−+=

−+=

−++ ++++=q

ldldl

p

rkIOkdk

r

iIOid

s


144

1

*,4

1

*,4

14

*,4 )( δ

*,1

1433

*,2

1422

*,3

141 )()()( IOd

s

jjdjIOd

s

jjdjIOd

s

jjdij YebaYebaYeba +

=−++

=−++

=−+ ∑∑∑ +++++=

∑ ∑∑=

−+=

−+=

−+ ++++r

iid

s

jjdijiIOd

s

jjdj YebaYeba

54

14

*,

1444 )()(

∑∑=

+−++=

−+ +++q

ldldl

p

rkIOkdk eecYa

144

1

*,4δ

IO

s

jjdjIO

s

jjdijd AebaAebaY ,2

1422,3

1414 )()( ωω ∑∑

=−+

=−++ ++++=

IO

s

jjdjIO

s

jjdj AebaAeba ,0

1444,1

1433 )()( ωω ∑∑

=−+

=−+ ++++

∑ ∑=

−=

−++ ++=4

1,4

144 )(

mIOm

s

jjdmjmd AebaY ω

IOd AY ,44 ω+= +

63

In general, for kdt += :

IOkkdIOkd AYY ,*

, ω+= ++ (4.7)

where

⎪⎩

⎪⎨⎧

≥+

== ∑ ∑

=−

=+ 1)(

01

1,

1

, kforAeba

kforA k

mIOmk

s

jkdmjm

IOk

It can be seen that IO will not only change the observation at dt = but also several

subsequent observations.

4.1.4 Effect of IO on residuals

Using similar definition as AO, for IO effects on residuals, when IO occurs at

time dt < , tIOt ee =*, . On the other hand, for dt ≥ and 0≥k , the formulation of IO

effects on observations is derived as follows:

For dt = :

∑ ∑∑∑ ∑=

−−=

−==

−−=

−−+−=r

iIOidjd

s

jijIOjd

s

jij

r

iIOidIOjd

s

jijiIOdIOd YebebYebaYe

1

*,

1

*,

11

*,

*,

1

*,

*, )()(

*,

1

*,

1IOld

q

llIOkd

p

rkk ecYa −

=−

+=∑∑ −−δ

∑ ∑∑∑ ∑=

−−=

−==

−−=

−−+−+=r

iidjd

s

jijjd

s

jij

r

iidjd

s

jijiIOd YebebYebaAY

1 111

*

1,0 )()()( ω

ld

q

llkd

p

rkk ecYa −

=−

+=∑∑ −−

11δ

64

∑ ∑∑∑=

−=

−+=

−−=

+−−+−=r

iIOld

q

llkd

p

rkkidjd

s

jijid AecYaYebaY

1,0

11

*

1)( ωδ

IOd Ae ,0ω+=

dd fe ω+=

For 1+= dt :

∑ ∑=

−+−+=

++ +−=r

iIOidIOjd

s

jijiIOdIOd YebaYe

1

*,1

*,1

1

*,1

*,1 )(

*,1

1

*,1

11

*,11

1

*,1

1)( IOld

q

llIOkd

p

rkk

r

iIOidjd

s

jijIOjd

s

jij ecYaYebeb −+

=−+

+==−+−+

=−+

=∑∑∑ ∑∑ −−−− δ

∑ ∑∑=

−+−+=

−+=

+ +−+−+=r

iIOidIOjd

s

jijiIOdIOjd

s

jjIOd YebaYebaAY

2

*,1

*,1

1

*,

*,1

111,11 )()()( ω

*,1

11

*,11

1

*,1

2

*,1 )( IOkd

p

rkk

r

iIOidjd

s

jijIOjd

s

jijIOdi YaYebebeb −+

+==−+−+

=−+

=∑∑ ∑∑ −−+− δ

*,1

2

*,1 IOld

q

llIOd ecec −+

=∑−−

ld

q

llkd

p

rkk

r

iidjd

s

jijid ecYaYebaY −+

=−+

+==−+−+

=+ ∑∑∑ ∑ −−+−= 1

11

1111

11 )( δ

IOIO

r

iIOidIOiIOjd

s

jj AAcYAbAeba ,1,01

1

*,1,01,01

111 )( ωωωω ++−+− ∑∑

=−+−+

=

)]([ 11

*,111

111,0,11 cYbebaAAe

r

iIOidijd

s

jjIOIOd +++−+= ∑∑

=−+−+

=+ ω

∑ ∑=

−−+=

+ +−+=1

1,11

1,11 )([

kIOkkd

s

jkjkIOd AebaAe ω

65

])(1

11

1

1

1

*,11 ∑∑ ∑

=−+

= =−+−+ −−

kkdk

r

i mIOidimmd fcYbf

11 ++ += dd fe ω

For 2+= dt :

*,2

11

*,2

*,2

1

*,2

*,2 )( IOkd

p

rkk

r

iIOidIOjd

s

jijiIOdIOd YaYebaYe −+

+==−+−+

=++ ∑∑ ∑ −+−= δ

*,2

11

*,22

1

*,2

1)( IOld

q

ll

r

iIOidjd

s

jijIOjd

s

jij ecYebeb −+

==−+−+

=−+

=∑∑ ∑∑ −−−

IOd

s

jjIOd

s

jjIOd AebaAebaAY ,0

122,11

111,22 )()()( ωωω ∑∑

=+

=+ +−+−+=

∑ ∑=

−+−+=

+−r

iidjd

s

jiji Yeba

122

1)(

∑ ∑∑=

−+−+=

−+=

+ −++−r

iIOidjd

s

jijIOjd

s

jijIOdiIOdi Yebebebeb

1

*,22

1

*,2

3

*,2

*,11 )(

*,2

1IOkd

p

rkkYa −+

+=∑−δ *

23

*2

*11 ld

q

lldd ececec −+

=+ ∑−−−

∑ ∑= =

−+++ −−+=r

i

r

iIOiddidiIOd YfbfbAe

1 1

*,2211,22 )([ω

])(2

12

2

1,22

1∑∑ ∑=

−+=

−−+=

−+−k

kdkk

IOkkd

s

jkjk fcAeba

∑ ∑=

−+=

−++ −+=r

iIOid

mmdimIOd YfbAe

1

*,2

2

12,22 )([ω

])(2

12

2

1,22

1∑∑ ∑=

−+=

−−+=

−+−k

kdkk

IOkkd

s

jkjk fcAeba

22 ++ += dd fe ω

66

For 3+= dt :

*,3

11

*,3

*,3

1

*,3

*,3 )( IOkd

p

rkk

r

iIOidIOjd

s


+==−+−+

=++ ∑∑ ∑ −+−= δ

*,3

11

*,33

1

*,3

1)( IOld

q

ll

r

iIOidjd

s

jijIOjd

s

jij ecYebeb −+

==−+−+

=−+

=∑∑ ∑∑ −−−

*,11

122

*,22

111,33 )()()( IOdd

s

jjIOdd

s

jjIOd YebaYebaAY ++

=++

=+ ∑∑ +−+−+= ω

∑ ∑∑=

−+−+=

−+=

+−+−r

iIOidjd

s

jijiIOdjd

s

jj YebaYeba

4

*,33

1

*,3

133 )()(

∑ ∑∑=

−+−+=

−+=

++ −+++−r

iIOidjd

s

jijIOjd

s

jijIOdiIOdiIOdi Yebebebebeb

1

*,33

1

*,3

4

*,3

*,12

*,21 )(

*,3

4

*,3

*,12

*,21

*,3

1IOld

q

llIOdIOdIOdIOkd

p

rkk ececececYa −+

=++−+

+=∑∑ −−−−−δ

∑=

−++++ ++−+=r

iIOiddididiIOd Yfbfbfbe

1

*,331221,33 )([A ω

])(3

1

3

13,33

1∑ ∑∑= =

−+−−+=

−+−k k

kdkIOkkd

s

jkjk fcAeba

∑ ∑=

−+=

−++ −+=r

iIOid

mmdimIOd Yfbe

1

*,3

3

13,33 )([A ω

])(3

1

3

13,33

1∑ ∑∑= =

−+−−+=

−+−k k

kdkIOkkd

s

jkjk fcAeba

33 ++ += dd fe ω

For 4+= dt :

*,4

11

*,4

*,4

1

*,4

*,4 )( IOkd

p

rkk

r

iIOidIOjd

s


+==−+−+

=++ ∑∑ ∑ −+−= δ

67

*,4

11

*,44

1

*,4

1)( IOld

q

ll

r

iIOidjd

s

jijIOjd

s

jij ecYebeb −+

==−+−+

=−+

=∑∑ ∑∑ −−−

*,22

122

*,33

111,44 )()()( IOdd

s

jjIOdd

s

jjIOd YebaYebaAY ++

=++

=+ ∑∑ +−+−+= ω

*,11

133 )( IOdd

s

jj Yeba ++

=∑+−

*,4

14

*,44

1

*,

144 )()( IOkd

p

rkk

r

iIOidjd

s

jijiIOdd

s

jj YaYebaYeba −+

+==−+−+

==∑∑ ∑∑ −+−+− δ

∑ ∑=

−+=

+++ ++++−r

iIOjd

s

jijIOdiIOdiIOdiIOdi ebebebebeb

1

*,4

5

*,4

*,13

*,22

*,31(

*,4

5

*,4

*,13

*,22

*,31

*,44

1) IOld

q

llIOdIOdIOdIOdIOidjd

s

jij ecececececYeb −+

=+++−+−+

=∑∑ −−−−−−

∑ ∑=

−+=

−++ −+=r

iIOid

mmdimIOd Yfbe

1

*,4

4

14,44 )([A ω

])(4

1

4

14,44

1∑ ∑∑= =

−+−−+=

−+−k k

kdkIOkkd

s

jkjk fcAeba

44 ++ += dd fe ω

In general, for hdt += :

hdhdIOhd fee +++ += ω*, (4.8)

where

⎪⎩

⎪⎨⎧

≥−+−−

== ∑ ∑ ∑∑∑

= = =−+−

=−+−+

=−+

+ 1)()(

0

1 1 1,

*,

1,

,0

hforfcAebaYfbA

hforAf r

i

h

k

h

kkhdkIOkh

s

ijkhdkjkIOihd

h

mmhdimIOh

IO

hd

It can be seen that IO will not only change the residual at dt = but also several

subsequent residuals.

68

4.2 ILLUSTRATION

For the purpose of illustration, we simulated data from BL(p,0,1,1) process using

S-PLUS package. The model is given by

tttit

p

iit eebYYaY ++= −−−

=∑ 11

1 , where .3,2,1=p (4.9)

where te is generated from normal distribution random generator rnorm of S-PLUS

statistical package with mean zero and variance unity, nt ,...,1= by letting 01 =Y . The

main purpose of this section is to show graphically the effect of AO and IO on

observations values and their respective residuals.

The plot of simulated data for BL(1,0,1,1) with parameter 1a =0.1, b=0.1 is given

in Figure 4.1. Figure 4.2 to Figure 4.9 give the plots for the BL(1,0,1,1) model with an

AO or IO of size 6=ω is introduced in the simulated data. For clarity, plot of

observations with/without outliers for 30=t until 50=t are shown in the plots. Figure

4.2 gives the plot of observation with/without AO. It can be seen that AO only changes

the observation at 40=t . The rest of observations are undisturbed. Figure 4.3 illustrates

the effect of an AO on residuals. In general, when an AO occurs at time 40=d , the

effect not only change the residual at 40=t but also the subsequent one before it dies

out. For IO, we generate several samples of size 100 with different values of 1a and b but

a single set of residual. We set the value of IO effect, ω , as 6 and let it occurs at time

40=d . The resulting plot for IO type of outliers on observed values are shown in Figure

4.4 by focusing on observations 30 to 50 for 1.01 =a and 1.0=b . It is clearly seen that

the effect of IO only started at time 40=d and affect several other observations

69

afterwards. The number of points affected after the occurrences of IO does not differ

much when ω is altered. As we increase the magnitude of 1a and b, the number of

changes of contaminated observation immediately increase after 40=d . Figures 4.5-4.6

show the effects as we change the magnitude of 1a and b from 0.1 to 0.4. When both

parameters change their signs to negative, the disturbed observations after 40=d are

“zig-zagging” around the original observations. The same effect on the residuals can be

observed as illustrated in Figure 4.7 to Figure 4.9. Several residual values after 40=t

have changed.

The plots for the BL(2,0,1,1) model is given in Figure 4.10 to Figure 4.13 and for

the BL(3,0,1,1) model is given in Figure 4.14 to Figure 4.17. The plots for both models

show similar pattern with the plots of the BL(1,0,1,1) model either in AO or IO. The

pattern of disturbance does not depend on the sign of 1a , 2a , 3a and b. However, the size

of b does affect the changes in residual quite significantly. The disturbance causes the

next residual values after 40=t to have larger magnitude compared to when we used a

smaller value of b.

Figure 4.1 Plot of simulated data

70

Figure 4.2 Plot of AO effect on observations, 1.01 =a , 1.0=b for BL(1,0,1,1)

Figure 4.3 Plot of AO effect on residuals, 1.01 =a , 1.0=b for BL(1,0,1,1)

Figure 4.4 Plot of IO effect on observations, 1.01 =a , 1.0=b for BL(1,0,1,1)

71

Figure 4.5 Plot of IO effect on observations, 3.01 =a , 3.0=b for BL(1,0,1,1)

Figure 4.6 Plot of IO effect on observations, 4.01 −=a , 4.0−=b for BL(1,0,1,1)

Figure 4.7 Plot of IO effect on residuals, 1.01 =a , 1.0=b for BL(1,0,1,1)

72

Figure 4.8 Plot of IO effect on residuals, 3.01 =a , 3.0=b for BL(1,0,1,1)

Figure 4.9 Plot of IO effect on residuals, 4.01 −=a , 4.0−=b for BL(1,0,1,1)

5 1 0 1 5 2 0T im e

-20

24

w i th o u t A Ow ith A O

Valu

e

Figure 4.10 Plot of AO effect on observations, 1.01 =a , 1.02 =a , 1.0=b for

BL(2,0,1,1)

73

1 0 2 0

-20

24

w ith o u t A Ow ith A O

T im e

Val

ue

Figure 4.11 Plot of AO effect on residuals, 1.01 =a , 1.02 =a , 1.0=b for BL(2,0,1,1)

5 1 0 1 5 2 0T im e

02

46

w i th o u t IOw ith IO

Valu

e

Figure 4.12 Plot of IO effect on observations, 1.01 =a , 1.02 =a , 1.0=b for BL(2,0,1,1)

1 0 2 0

-4-2

02

46

T im e

Val

ue

w ith o u t IOw ith IO

Figure 4.13 Plot of IO effect on residuals, 1.01 =a , 1.02 =a , 1.0=b for BL(2,0,1,1)

74

5 1 0 1 5 2 0

T im e

02

4

w ith o u t A Ow ith A O

Valu

e

Figure 4.14 Plot of AO effect on observations, 1.01 =a , 1.02 =a , 1.03 =a , 1.0=b for

BL(3,0,1,1)

1 0 2 0

02

4

w i t h o u t A Ow it h A O

T im e

Val

ue

Figure 4.15 Plot of AO effect on residuals, 1.01 =a , 1.02 =a , 1.03 =a , 1.0=b for

BL(3,0,1,1)

1 0 2 0

-20

24

6 w ith o u t IOw ith IO

T im e

Valu

e

Figure 4.16 Plot of IO effect on observations, 1.01 =a , 1.02 =a , 1.03 =a , 1.0=b for

BL(3,0,1,1)

75

1 0 2 0

-20

24

6 w ith o u t IOw ith IO

T im e

Val

ue

Figure 4.17 Plot of IO effect on residuals, 1.01 =a , 1.02 =a , 1.03 =a , 1.0=b for BL(3,0,1,1)

4.3 SUMMARY

In this chapter we have derived the formulation of AO and IO effect on

observations and residuals from BL(p,q,r,s) models. We have further investigated the

nature of outlier effect of AO and IO on the original observations and residuals in

BL(p,0,1,1).

76

CHAPTER FIVE

PROCEDURE FOR DETECTING SINGLE OUTLIER FOR BL(p,0,1,1)

PROCESS

In this chapter, a single outlier detection procedure for data generated from

BL(p,0,1,1), where p=1,2,3 is developed. The measure of outlier effect for AO and IO,

denoted by AOω and IOω , is derived. Then, test statistics are defined for classifying an

observation as an outlier of its respective type. Finally, a general single outlier detection

procedure is presented to distinguish a particular type of outlier at time point t.

5.1 NONLINEAR LEAST SQUARES METHOD FOR BL(p,0,1,1)

The general procedure of nonlinear least squares method has been presented in

section 2.4.1. The method is suggested by Priestley [1991] for BL(p,0,r,s) model. In this

section, the nonlinear least squares estimation method for BL(p,0,1,1) models, where

p=1,2,3, is described. The BL(p,0,1,1) models, where p=1,2,3, is given by

tttit

p


=∑ 11

1 (5.1)

Let '11 ),...,( += pθθθ denote the complete set of parameters of paa ,...,1 and b of the

BL(p,0,1,1) , where p=1,2,3. The objective here is to minimize the following equation:

∑=

=n

tteQ

2

2)(θ (5.2)

77

where { }te is obtained from (5.1). The minimization is achieved through Newton-

Raphson iterative procedure:

))G(θ(θHθθ (i)(i)1(i)1)(i −+ −= (5.3)

where (i)θ is the vector of parameter estimation from the i-th iteration, G is gradient

vector and H is Hessian matrix where

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ ∂∂

=+11

,...,pdQ

dQ

θθθ )G( (5.4)

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂∂∂

=ji

Qθθ

θ2

)H( 1,...,1, += pji (5.5)

The partial derivatives of Q with respect to { }iθ are

∑= ∂

∂=

∂∂ n

t i

tt

i

eeQ

22

θθ, Ni ,...,2,1= (5.6)

∑∑== ∂∂

∂+

∂∂

∂∂

=∂∂

∂ n

t ji

tt

n

t j

t

i

t

ji

ee

eeQ2

2

2

2

22θθθθθθ

, Nji ,...,2,1, = (5.7)

Based on equation (5.1), the following can be obtained:

j

ttjt

j

t

ae

bYYae

∂∂

−−=∂∂ −

−−1

1 , .3,2,1=j

b

ebYeY

be t

tttt

∂∂

−−=∂∂ −

−−−1

111

21

2

12

2

j

tt

j

t

ae

bYae

∂∂

−=∂∂ −

− , .3,2,1=j

21

2

11

12

2

2be

bYb

eY

be t

tt

tt

∂∂

−∂∂

−=∂∂ −

−−

−

ba

ebY

ae

Yba

e

j

tt

j

tt

j

t

∂∂∂

−∂∂

−=∂∂

∂ −−

−−

12

11

1

2

, .3,2,1=j

78

For simplicity, the most common choice, if no prior information is available, is to choose

the following conditions:

02

=∂∂

∂=

∂∂

=ji

t

i

tt

eee

θθθ for all 2and1=t and 3,2,1, =ji

The iterative procedure given by (5.3) can now be implemented. The iteration is stopped

when the following conditions are met, that is, Naaa ,...,, 21 and Nbbb ,...,, 21 which are

constructed for each parameter a and b until

<∈− −1Ni

Ni θθ

for )',(,2,1 bai == θ and tolerance ε . In this study, ε was chosen to 310− .

The Newton-Raphson procedure employed requires initial values for the

parameters. For that, the steps below are followed:

(a) Given a data set { } ntYt ,...,2,1, = , the AR(1) estimate for 1a , AR(2) estimates for

1a and 2a , and AR(3) estimates for 1a , 2a and 3a , say 1a , 2a and 3a are

obtained.

(b) 1a , 2a , 3a and 0 are used as the initial values of 1a , 2a , 3a and b, respectively,

in the Newton-Raphson procedure for estimating the parameters of BL(p,0,1,1),

p=1,2,3 which is given by

tttit

p


=∑ 11

1

Let the estimated values of 1a , 2a , 3a and b be 1a( , 2a( , 3a( and b(

, respectively.

Hence, the final estimates of 1a , 2a , 3a and b are obtained, that is 1a( , 2a( , 3a( and b(

.

79

5.2 DERIVATION OF MEASURE OF OUTLIER EFFECT

5.2.1 Additive Outlier (AO)

The derivation of the statistics to measure the magnitude of outlier effects for AO

is now developed. The statistics can be obtained using the least squares method by

minimizing the following equation:

∑=

=n

tteS

1

2 (5.8)

Assuming that AO occurs at dt = , equation (5.8) becomes

∑∑−

=+

−

=

−+=dn

kAOkAOkd

d

tt AeeS

0

2,

*,

1

1

2 )( ω (5.9)

Equation (5.9) is then minimized with respect to ω :

∑−

=+ −−=

dn

kAOkAOkAOkd AAe

ddS

0,,

*, ))((2 ω

ω

∑ ∑−

=

−

=+−=

dn

k

dn

kAOkkdAOk AeA

0 0,

*2, 22 ω

Thus, the least square estimate of ω at time dt = for AO is given by

∑

∑−

=

−

=+

= dn

kAOk

dn

kAOkAOkd

AO

A

Ae

0

2,

0,

*,

ω (5.10)

where

⎪⎩

⎪⎨⎧

≥+−+−

== ∑ ∑∑

= =−+

=−+ 1)()(

01

1 1

*,

1

, kforcYbeba

kforA k

j

r

ijAOjkdij

s

jjkdkjk

AOk

80

5.2.2 Innovational Outlier (IO)

By following similar steps, the process yields the following measure of outlier

effects for IO:

∑=

=n

tteS

1

2

∑∑−

=+

−

=

−+=dn

kIOkIOkd

d

tt Aee

0

2,

*,

1

1

2 )( ω (5.11)

Equation (5.11) is then minimized with respect to ω :

∑−

=+ −−=

dn

kIOkIOkIOkd AAe

ddS

0,,

*, ))((2 ω

ω

∑ ∑−

=

−

=+−=

dn

k

dn

kIOkkdIOk AeA

0 0,

*2, 22 ω (5.12)

Thus, the least square estimate of ω at time dt = for IO is given by

∑

∑−

=

−

=+

= dn

kIOk

dn

kIOkIOkd

IO

A

Ae

0

2,

0,

*,

ω (5.13)

where

⎪⎩

⎪⎨⎧

≥+−

== ∑ ∑

=−

=−+ 1)(

01

1,

1

*,

, kforAcYb

kforA k

mIOmk

r

imIOikdim

IOk

81

5.3 VARIANCE OF ESTIMATE OF OUTLIER EFFECT

In linear ARMA cases, an exact expression of ( )TPωVar can be derived.

However, in the current bilinear cases, the complexity of the formulae makes the

determination of an algebraic expression for ( )TPωVar insurmountable. As an alternative,

the bootstrap procedure is used to obtain the estimates of the variances of TPω . The

procedure has emerged as a powerful tool for constructing inferential procedures in

modern statistical analysis. For bootstrap procedure, it is carried out through the process

of drawing random samples with replacement from the observed sample.

The importance of bootstrap procedure in bringing new insights to some of the

difficult problems of data analysis has been highlighted. The bootstrap procedure has

been applied on time series, for example, Efron and Tibshirani [1986], Chen and Romano

[1999], Swensen [2003] and Pascual et al. [2004] and for model-based procedure,

Battaglia and Orfei [2005].

5.3.1 Other bootstrap-based procedure

In this study, the bootstrap-based procedure given by equation (3.1) as described

in Chapter Three is improved. That is, we use two other ways of calculating the variance;

the mean absolute deviance (MAD) and the trimmed mean (TM). The two formulae are

described as follows:

82

(a) The mean absolute deviance (MAD)

Instead of using equation (5.14) to calculate the standard deviation of ω , we

propose to use the procedure suggested by Hampel et al. [1986] in which the standard

deviation is computed using the following formula

⎭⎬⎫

⎩⎨⎧ −×= ωωσ ~ˆmedian483.1ˆ tMAD (5.15)

where ω~ is the median of the bootstrap estimates, Mω~ .

(b) The 5% trimmed mean (TM)

The calculation of standard deviation used the trimmed sample such that smallest

and largest 5% of Mω~ are removed from the calculation. Equation (5.14) is then used to

give the standard deviation, TMσ .

The use of these formulae has been shown to be able to improve the performance

detection procedure for ARIMA models as it is able to overcome the problem of

overestimation in the computation of standard deviation (Chen and Liu [1993]).

5.4 A GENERAL SINGLE DETECTION PROCEDURE TO IDENTIFY THE

TYPE OF OUTLIER

Following Tsay [1986b] and Chang et al. [1988], let H0 denote the hypothesis that

0=ω in the bilinear model considered. Let H1 denote the situations 0≠ω in bilinear

83

model with AO and IO, respectively, at time t. Test statistics can be derived for testing

one hypothesis versus another as follows:

For bootstrap-based procedures:

H0 v H1 : ( )

tTYPETP

tTYPETPtTPtTYPETP

,,

,,,,, ~

~ˆˆ

σωω

τ−

= , (5.16)

where .,...,1 nt = , TP =AO or IO, and TYPE=standard, MAD or TM.

In general, the time point where an outlier occurs is unknown. Hence, the values

of the test statistics can be obtained at every time point t = 1,2,...,n. The test for

identifying the type of outlier at a particular point t in BL(p,0,1,1), where p=1,2,3 models

begin with modelling the original time series Y by assuming that there is no outlier in the

data.

The maximum values of the test statistics (5.16) are examined. The procedure is

described below:

a) Compute the least squares estimates of BL(p,0,1,1) models, p=1,2,3 based on the

original data. Hence, obtain the residuals.

b) Compute tTYPETPτ ,,ˆ for each t, t = 1, 2,...,n, using the residuals obtained in part a).

(TP= AO or IO and TYPE=standard, MAD or TM)

c) Let { }tTYPETPtTYPE ,,n1,2,...,t, ˆmax τη=

= . Given a pre-determined critical value C, if

,, Cη tTP > then there is a possibility of an AO or IO occurring at time t. (TP=AO

or IO and TYPE=standard, MAD or TM)

84

Through the suggested procedure, the occurrence of AO or IO can be detected at any

time t.

5.5 ILLUSTRATION

The data is simulated by using S-PLUS package. For illustration, we generate

several samples of size 100. The main purpose of this section is to show graphically the

measure of outlier effects either on AO or IO. The plot of simulated data is given in

Figure 5.1 for 1.01 =a and 1.0=b . Further, an AO and IO of size 6=ω is introduced at

time 40=d .

For clarity, we plot the measure of outlier effects, TPω , TP=AO,IO, at time point

)100,...,1( =t when applied onto the simulated data. For AO, it can be seen that most

tAO,ω , lie in the interval [-2,2] except 40,ˆ AOω as shown in Figure 5.2. This value

corresponds to the AO introduced at time 40=t in the original simulated data. This

suggests that the derived statistics measuring AO effect given by equation (5.10) is able

to isolate time point at which AO occurs.

Figure 5.3 gives the plot of measure of IO effect for each time t, the value of

40,ˆ IOω is 4.1047, which is significantly different from the rest. The others lie in [-2.1483,

2.5721].

85

Figure 5.1 Plot of simulated data

Figure 5.2 Plot of AOω , with AO, 1.01 =a , 1.0=b at 40=d

Figure 5.3 Plot of IOω , with IO, 1.01 =a , 1.0=b at 40=d

86

5.6 SUMMARY

In this chapter, the measure of outlier effect has been derived using least squares

method for two types of outliers considered in this study. Three bootstrap-based

procedures are proposed depending on the formula used to estimate the variances of the

mean of outlier effect.

87

CHAPTER SIX

SIMULATION STUDY

In this chapter, simulation studies are carried out to investigate the sampling

behavior of test statistics, and performance of the criteria and outlier detection procedure

for three bootstrap-based procedures described in Chapter 5 and model-based procedure

described in Chapter 3.

6.1 SAMPLING BEHAVIOR OF TEST STATISTICS

An outlier detection procedure for BL(p,0,1,1) models, p=1,2,3, is developed

based on the maxima of the test statistics measuring the effects of AO and IO given by

equations in Chapter 5. The simulation study in this section is carried out in order to

investigate the sampling properties of the maxima of the outlier test statistics. It is

associated with the sample size, type of outlier and coefficients chosen for BL(p,0,1,1).

Models in Table 6.1, Table 6.2 and Table 6.3 are considered for BL(1,0,1,1),

BL(2,0,1,1) and BL(3,0,1,1), respectively. They represent a broad choice of coefficients

of BL(p,0,1,1) models, 3,2,1=p satisfying stationary condition of bilinear model. For

each model, three cases of samples are considered, 60=n , 100=n and 200=n . The

random errors, te ’s are assumed to follow the standard normal distribution. For each

model and each sample size, 100 series are generated. The test statistics for AO and IO

are calculated separately based on equations (5.16), (5.17) and (5.18), respectively, in

Chapter 5. The focus is to examine the sampling behavior of { }tTPTP ,n1,2,...,tˆmax τη

== where

88

TP can be AO or IO. In particular, the percentiles of the test statistics at the 1%, 5% and

10% levels are estimated when no outlier is present in the series. The plots are given in

Figures 6.1-6.24 consisting of the combination of two types of outliers, four types of

different methods and three significance levels. In each plot, percentiles for each are

displayed ( 200,100,60=n ). From this simulation study, we expect to obtain the cut

points used for detecting AO and IO type of outliers in the proposed outlier detection

procedure with an acceptable level of misdetection.

Table 6.1 List of model used for the determination of critical values for BL(1,0,1,1)

MODEL FULL MODEL 1 ttttt eeYYY ++= −−− 111 1.01.0 2 ttttt eeYYY ++= −−− 111 3.01.0 3 ttttt eeYYY ++= −−− 111 5.01.0 4 ttttt eeYYY ++= −−− 111 2.02.0 5 ttttt eeYYY ++= −−− 111 3.03.0 6 ttttt eeYYY ++= −−− 111 2.04.0 7 ttttt eeYYY ++= −−− 111 1.05.0 8 ttttt eeYYY +−−= −−− 111 1.01.0 9 ttttt eeYYY ++−= −−− 111 3.01.0 10 ttttt eeYYY +−−= −−− 111 2.02.0 11 ttttt eeYYY ++−= −−− 111 2.04.0 12 ttttt eeYYY ++−= −−− 111 1.03.0 13 ttttt eeYYY +−−= −−− 111 1.05.0

89

Table 6.2 List of models used for the determination of critical values for BL(2,0,1,1)

MODEL FULL MODEL 1 tttttt eeYYYY +++= −−−− 1121 1.01.01.0 2 tttttt eeYYYY +++= −−−− 1121 2.02.02.0 3 tttttt eeYYYY +++= −−−− 1121 1.01.03.0 4 tttttt eeYYYY +++= −−−− 1121 1.01.04.0 5 tttttt eeYYYY +++= −−−− 1121 3.01.04.0 6 tttttt eeYYYY +++= −−−− 1121 1.01.05.0 7 tttttt eeYYYY +−−−= −−−− 1121 1.01.01.0 8 tttttt eeYYYY +−−−= −−−− 1121 2.02.02.0 9 tttttt eeYYYY +++−= −−−− 1121 2.04.02.0 10 tttttt eeYYYY ++−−= −−−− 1121 1.05.03.0 11 tttttt eeYYYY +−+−= −−−− 1121 2.02.04.0 12 tttttt eeYYYY +−−−= −−−− 1121 4.04.04.0 13 tttttt eeYYYY +−−−= −−−− 1121 1.01.05.0

Table 6.3 List of models used for the determination of critical values for BL(3,0,1,1)

MODEL FULL MODEL 1 ttttttt eeYYYYY ++++= −−−−− 11321 1.01.01.01.0 2 ttttttt eeYYYYY ++++= −−−−− 11321 3.03.02.01.0 3 ttttttt eeYYYYY ++++= −−−−− 11321 2.02.02.02.0 4 ttttttt eeYYYYY ++++= −−−−− 11321 1.01.01.03.0 5 ttttttt eeYYYYY ++++= −−−−− 11321 1.04.01.04.0 6 ttttttt eeYYYYY ++++= −−−−− 11321 1.03.01.05.0 7 ttttttt eeYYYYY +−−−−= −−−−− 11321 1.01.01.01.0 8 ttttttt eeYYYYY ++−+−= −−−−− 11321 2.05.03.01.0 9 ttttttt eeYYYYY +−−−−= −−−−− 11321 2.02.02.02.0 10 ttttttt eeYYYYY +−−+−= −−−−− 11321 1.04.04.02.0 11 ttttttt eeYYYYY +−−−−= −−−−− 11321 1.01.01.03.0 12 ttttttt eeYYYYY ++−+−= −−−−− 11321 2.04.02.04.0 13 ttttttt eeYYYYY +−−−−= −−−−− 11321 5.01.01.05.0

90

The plots of percentiles of AO and IO for BL(1,0,1,1) cases are shown in Figure

6.1 to Figure 6.8. Results of AO and IO for the standard procedure are shown in Figure

6.1 and Figure 6.5, respectively, MAD procedure is shown in Figure 6.2 and Figure 6.6,

respectively, trimmed mean (TM) procedure in Figure 6.3 and Figure 6.7 respectively

and lastly for model-based (MB) procedure is shown in Figure 6.4 and Figure 6.8

respectively. From the figures, there is no clear pattern of increment or decrement of

values in sample size of n, n=60,100,200, for all 1%, 5% and 10% upper percentile

values. This suggests that the cut points does not depend on the sample size, for

n=60,100,200. For determining the range of cut points, we consider plots of the 5% upper

percentile. For standard, MAD and MB procedures of identifying AO and IO, most cut

point values lie between 3 to 4, while for TM procedure, the value lie is higher in the

range of 3.8 to 4.8.

Similar results for BL(2,0,1,1) and BL(3,0,1,1) are illustrated by Figure 6.9-6.16

and Figure 6.17 to Figure 6.24, respectively. That is, the cut points do not depend on

sample size n, n=60,100,200, for both AO and IO cases. Further, based on the 5% upper

percentile, the cut points lie between 3 to 4 for standard, MAD and MB procedures while

for TM procedure, the value lie higher between 3.5 to 4.8.

Based on the results, for standard, MAD and MB procedures, the critical values of

2.5 to 4.0 seem to be suitable choice for series if size between 60-200, while we may use

higher values between 3 to 4.5 for TM procedure. In practice, more than one critical

value is suggested for the analysis.

91

0 2 4 6 8 1 0 1 2 1 4M o d e l

2 .9

3 .0

3 .1

3 .2

3 .3

3 .4n = 6 0n = 1 0 0n = 2 0 0

1 0 % u p p e r p e rce n tile

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .0

3 .2

3 .4

3 .6

3 .8

5 % u p p e r p e rc e n tile

n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .1

3 .5

3 .9

4 .3


n = 6 0n = 1 0 0n = 2 0 0

Figure 6.1 Plot of critical values of AO on standard procedure for BL(1,0,1,1)

92

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .0

3 .4

3 .8

4 .2

1 0 % u p p e r p e rc e n tile

n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .1

3 .5

3 .9

4 .3


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .6

4 .1

4 .6

5 .1

5 .6


n = 6 0n = 1 0 0n = 2 0 0

Figure 6.2 Plot of critical values of AO on MAD procedure for BL(1,0,1,1)

93

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .7

3 .9

4 .1

4 .3


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .8

4 .0

4 .2

4 .4

4 .6

4 .8


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

4 .0

4 .4

4 .8

5 .2

5 .6


n = 6 0n = 1 0 0n = 2 0 0

Figure 6.3 Plot of critical values of AO on TM procedure for BL(1,0,1,1)

94

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .1

3 .2

3 .3

3 .4

3 .5

3 .6


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .2

3 .3

3 .4

3 .5

3 .6

3 .7

3 .8


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .5

3 .7

3 .9

4 .1

4 .3


n = 6 0n = 1 0 0n = 2 0 0

Figure 6.4 Plot of critical values of AO on MB procedure for BL(1,0,1,1)

95

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .0

3 .1

3 .2

3 .3


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .0

3 .2

3 .4

3 .6

3 .8

4 .0


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .4

3 .6

3 .8

4 .0

4 .2

4 .4


n = 6 0n = 1 0 0n = 2 0 0

Figure 6.5 Plot of critical values of IO on standard procedure for BL(1,0,1,1)

96

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .0

3 .2

3 .4

3 .6

3 .8


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .4

3 .6

3 .8

4 .0

4 .2

4 .4


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .5

4 .0

4 .5

5 .0

5 .5

6 .0

6 .5


n = 6 0n = 1 0 0n = 2 0 0

Figure 6.6 Plot of critical values of IO on MAD procedure for BL(1,0,1,1)

97

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .6

3 .8

4 .0

4 .2

4 .4


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .8

4 .0

4 .2

4 .4

4 .6

4 .8

5 .0


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

4 .0

4 .5

5 .0

5 .5

6 .0


n = 6 0n = 1 0 0n = 2 0 0

Figure 6.7 Plot of critical values of IO on TM procedure for BL(1,0,1,1)

98

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .1

3 .2

3 .3

3 .4

3 .5

1 0 % u p p e r p e rc e n t i le

n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .3

3 .4

3 .5

3 .6

3 .7

3 .8

5 % u p p e r p e rc e n t ile

n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M o d e l

3 .5

3 .7

3 .9

4 .1

4 .3


n = 6 0n = 1 0 0n = 2 0 0

Figure 6.8 Plot of critical values of IO on MB procedure for BL(1,0,1,1)

99

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .4

2 .6

2 .8

3 .0

3 .2


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .5

2 .7

2 .9

3 .1

3 .3

3 .5


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .5

3 .0

3 .5

4 .0


n = 6 0n = 1 0 0n = 2 0 0


100

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .5

2 .9

3 .3

3 .7


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .8

3 .0

3 .2

3 .4

3 .6

3 .8


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .5

4 .0

4 .5


n = 6 0n = 1 0 0n = 2 0 0


101

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .4

3 .8

4 .2


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .5

4 .0

4 .5

5 % u p p e r p e rc e n t i le

n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .7

4 .2

4 .7

5 .2


n = 6 0n = 1 0 0n = 2 0 0


102

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .9

3 .1

3 .3

3 .5

1 0 % u p p e r p e rc e n t ile

n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .2

3 .4

3 .6

3 .8


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .4

3 .6

3 .8

4 .0

4 .2

4 .4


n = 6 0n = 1 0 0n = 2 0 0


103

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .5

2 .7

2 .9

3 .1

3 .3

3 .5


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .8

3 .0

3 .2

3 .4

3 .6


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .1

3 .6

4 .1


n = 6 0n = 1 0 0n = 2 0 0


104

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .5

3 .0

3 .5

4 .0


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .5

4 .0

4 .5


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .5

4 .0

4 .5

5 .0


n = 6 0n = 1 0 0n = 2 0 0


105

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .5

4 .0

4 .5


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .5

3 .9

4 .3

4 .7


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .5

4 .0

4 .5

5 .0

5 .5

1 % u p p e r p e rce n tile

n = 6 0n = 1 0 0n = 2 0 0


106

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .8

3 .0

3 .2

3 .4

3 .6


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M 0 D E L

3 .0

3 .2

3 .4

3 .6

3 .8

5 % u p p e r p e rce n tile

n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .2

3 .6

4 .0

4 .4


n = 6 0n = 1 0 0n = 2 0 0


107

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .4

2 .6

2 .8

3 .0

3 .2


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M 0 D E L

2 .5

2 .9

3 .3

3 .7


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D A L

2 .5

3 .5

4 .5

5 .5

1 % u p p e r p e r c e n t i le

n = 6 0n = 1 0 0n = 2 0 0


108

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .4

2 .8

3 .2

3 .6


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .5

3 .0

3 .5

4 .0

5 % u p p e r p e r c e n t i le

n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3

4

5

6

7

8

9


n = 6 0n = 1 0 0n = 2 0 0


109

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .8

3 .2

3 .6

4 .0


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .5

4 .0

4 .5


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

4

6

8

1 0


n = 6 0n = 1 0 0n = 2 0 0


110

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .4

2 .6

2 .8

3 .0

3 .2

3 .4

3 .6


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .8

3 .0

3 .2

3 .4

3 .6


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .5

4 .0

4 .5

5 .0

5 .5

6 .0


n = 6 0n = 1 0 0n = 2 0 0


111

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .4

2 .6

2 .8

3 .0

3 .2

3 .4


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .8

3 .0

3 .2

3 .4

3 .6

3 .8


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3

4

5

6


n = 6 0n = 1 0 0n = 2 0 0


112

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .5

3 .0

3 .5

4 .0


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .5

3 .0

3 .5

4 .0

4 .5


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3

4

5

6

7


n = 6 0n = 1 0 0n = 2 0 0


113

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .0

3 .5

4 .0

4 .5


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .5

4 .0

4 .5

5 .0

5 .5


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

4

5

6

7

8


n = 6 0n = 1 0 0n = 2 0 0


114

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .7

2 .9

3 .1

3 .3

3 .5

1 0 % u p p e r p e rc e n t ile

n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

2 .8

3 .0

3 .2

3 .4

3 .6

3 .8

4 .0


n = 6 0n = 1 0 0n = 2 0 0

0 2 4 6 8 1 0 1 2 1 4M O D E L

3 .5

4 .0

4 .5

5 .0


n = 6 0n = 1 0 0n = 2 0 0


115

6.2 PERFORMANCE OF TEST CRITERIA

A simulation study was carried out to observe the performance of the test of

criteria for detecting AO and IO individually. The test criterion was applied to cases

characterized by a combination of the following factors:

a) Two types of outliers; AO and IO.

b) Three underlying models; BL(1,0,1,1), BL(2,0,1,1) and BL(3,0,1,1), with

different combinations of coefficients

c) A single outlier at 40=t in samples of size 100.

d) Two different values of outlier effect; 5,3=ω .

e) Three different levels of critical values; 2.5, 3.0, 3.5.

Overall we had 39 models, with 13 models each for BL(1,0,1,1), BL(2,0,1,1) and

BL(3,0,1,1), respectively. Series were generated to contain one of the outlier types. The

standard deviation of the noise process for each model was set to be unity. For the given

model, 500 series of length 100 were generated using rnorm procedure in S-Plus.

Model 1 to model 13 as given in Table 6.4 were considered for BL(1,0,1,1) cases.

Results for AO and IO cases on BL(1,0,1,1) were given in Table 6.7 and Table 6.8,

respectively. From Table 6.7, the results show that, with critical value 2.5 and 5=ω , the

four AO testing criterion performed well. The standard procedure performed better than

the others procedures. On the other hand, for 3=ω the proportion was rather small.

Further, for larger critical values, proportion of correct detection was higher for TM

compared to the other procedures. This may be due to the finding in Section 6.1 that the

cut point of the TM procedure was higher compared to the standard procedure. On the

other hand, both procedures performed better than the MAD and MB procedures.

116

Meanwhile, Table 6.8 shows the results of testing criterion performance for the IO case.

These results are almost similar with AO case such that, with critical value 2.5 and

5=ω , performed well , but not as well for smaller value of ω . For larger critical values,

TM procedure performed slightly better compared to the standard procedure. Further,

both procedures are preferable compared to MAD and model-based procedures.



respectively. Result from Table 6.9 shows that the proportion of correctly detecting AO

using all procedures were close to unity for almost all models with 5=ω . The table also

shows that MAD had consistently given a slightly lower proportion of correctly detecting

AO. On the other hand, standard procedure, TM procedure and MB procedure had almost

similar proportion but MB procedure was the best compared to the others. Overall, the

proportion of detecting AO for almost all models and methods are greater than 55%. The

performance of testing criterion for AO improves when a larger value of ω is considered.

The results in IO case are almost similar to AO case, but the proportion of correctly

detecting outlier for IO case was higher than the AO case. MAD procedure still gives the

lowest proportion of correctly detecting IO.



respectively. Overall, the proportions of detecting AO at 3=ω for all procedures were

almost greater than 50% and the proportion are close to unity for 5=ω . So we can say

that the performance of the testing criterion improves for larger values of ω . The results

of proportions for all procedures are almost the same but TM procedure was better than

117

others. The same scenario was seen for IO. Results are shown in Table 6.12. The

proportion of correctly detecting IO increased when larger ω was used.

Table 6.4 List of cases considered in the performance study for BL(1,0,1,1) MODEL FULL MODEL MAGNITUDE

1A ttttt eeYYY ++= −−− 111 1.01.0 3=ω 1B ttttt eeYYY ++= −−− 111 1.01.0 5=ω 2A ttttt eeYYY ++= −−− 111 3.01.0 3=ω 2B ttttt eeYYY ++= −−− 111 3.01.0 5=ω 3A ttttt eeYYY ++= −−− 111 5.01.0 3=ω 3B ttttt eeYYY ++= −−− 111 5.01.0 5=ω 4A ttttt eeYYY ++= −−− 111 2.02.0 3=ω 4B ttttt eeYYY ++= −−− 111 2.02.0 5=ω 5A ttttt eeYYY ++= −−− 111 3.03.0 3=ω 5B ttttt eeYYY ++= −−− 111 3.03.0 5=ω 6A ttttt eeYYY ++= −−− 111 2.04.0 3=ω 6B ttttt eeYYY ++= −−− 111 2.04.0 5=ω 7A ttttt eeYYY ++= −−− 111 1.05.0 3=ω 7B ttttt eeYYY ++= −−− 111 1.05.0 5=ω 8A ttttt eeYYY +−−= −−− 111 1.01.0 3=ω 8B ttttt eeYYY +−−= −−− 111 1.01.0 5=ω 9A ttttt eeYYY ++−= −−− 111 3.01.0 3=ω 9B ttttt eeYYY ++−= −−− 111 3.01.0 5=ω

10A ttttt eeYYY +−−= −−− 111 2.02.0 3=ω 10B ttttt eeYYY +−−= −−− 111 2.02.0 5=ω 11A ttttt eeYYY ++−= −−− 111 1.03.0 3=ω 11B ttttt eeYYY ++−= −−− 111 1.03.0 5=ω 12A ttttt eeYYY ++−= −−− 111 2.04.0 3=ω 12B ttttt eeYYY ++−= −−− 111 2.04.0 5=ω 13A ttttt eeYYY +−−= −−− 111 1.05.0 3=ω 13B ttttt eeYYY +−−= −−− 111 1.05.0 5=ω

118

Table 6.5 List of cases considered in the performance study for BL(2,0,1,1)

MODEL FULL MODEL MAGNITUDE

1A tttttt eeYYYY +++= −−−− 1121 1.01.01.0 3=ω 1B tttttt eeYYYY +++= −−−− 1121 1.01.01.0 5=ω 2A tttttt eeYYYY +++= −−−− 1121 2.02.02.0 3=ω 2B tttttt eeYYYY +++= −−−− 1121 2.02.02.0 5=ω 3A tttttt eeYYYY +++= −−−− 1121 1.01.03.0 3=ω 3B tttttt eeYYYY +++= −−−− 1121 1.01.03.0 5=ω 4A tttttt eeYYYY +++= −−−− 1121 1.01.04.0 3=ω 4B tttttt eeYYYY +++= −−−− 1121 1.01.04.0 5=ω 5A tttttt eeYYYY +++= −−−− 1121 3.01.04.0 3=ω 5B tttttt eeYYYY +++= −−−− 1121 3.01.04.0 5=ω 6A tttttt eeYYYY +++= −−−− 1121 1.01.05.0 3=ω 6B tttttt eeYYYY +++= −−−− 1121 1.01.05.0 5=ω 7A tttttt eeYYYY +−−−= −−−− 1121 1.01.01.0 3=ω 7B tttttt eeYYYY +−−−= −−−− 1121 1.01.01.0 5=ω 8A tttttt eeYYYY +−−−= −−−− 1121 2.02.02.0 3=ω 8B tttttt eeYYYY +−−−= −−−− 1121 2.02.02.0 5=ω 9A tttttt eeYYYY +++−= −−−− 1121 2.04.02.0 3=ω 9B tttttt eeYYYY +++−= −−−− 1121 2.04.02.0 5=ω

10A tttttt eeYYYY ++−−= −−−− 1121 1.05.03.0 3=ω 10B tttttt eeYYYY ++−−= −−−− 1121 1.05.03.0 5=ω 11A tttttt eeYYYY +−+−= −−−− 1121 2.02.04.0 3=ω 11B tttttt eeYYYY +−+−= −−−− 1121 2.02.04.0 5=ω 12A tttttt eeYYYY +−−−= −−−− 1121 4.04.04.0 3=ω 12B tttttt eeYYYY +−−−= −−−− 1121 4.04.04.0 5=ω 13A tttttt eeYYYY +−−−= −−−− 1121 1.01.05.0 3=ω 13B tttttt eeYYYY +−−−= −−−− 1121 1.01.05.0 5=ω

119

Table 6.6 List of cases considered in the performance study for BL(3,0,1,1) MODEL FULL MODEL MAGNITUDE

1A ttttttt eeYYYYY ++++= −−−−− 11321 1.01.01.01.0 3=ω 1B ttttttt eeYYYYY ++++= −−−−− 11321 1.01.01.01.0 5=ω 2A ttttttt eeYYYYY ++++= −−−−− 11321 3.03.02.01.0 3=ω 2B ttttttt eeYYYYY ++++= −−−−− 11321 3.03.02.01.0 5=ω 3A ttttttt eeYYYYY ++++= −−−−− 11321 2.02.02.02.0 3=ω 3B ttttttt eeYYYYY ++++= −−−−− 11321 2.02.02.02.0 5=ω 4A ttttttt eeYYYYY ++++= −−−−− 11321 1.01.01.03.0 3=ω 4B ttttttt eeYYYYY ++++= −−−−− 11321 1.01.01.03.0 5=ω 5A ttttttt eeYYYYY ++++= −−−−− 11321 1.04.01.04.0 3=ω 5B ttttttt eeYYYYY ++++= −−−−− 11321 1.04.01.04.0 5=ω 6A ttttttt eeYYYYY ++++= −−−−− 11321 1.03.01.05.0 3=ω 6B ttttttt eeYYYYY ++++= −−−−− 11321 1.03.01.05.0 5=ω 7A ttttttt eeYYYYY +−−−−= −−−−− 11321 1.01.01.01.0 3=ω 7B ttttttt eeYYYYY +−−−−= −−−−− 11321 1.01.01.01.0 5=ω 8A ttttttt eeYYYYY ++−+−= −−−−− 11321 2.05.03.01.0 3=ω 8B ttttttt eeYYYYY ++−+−= −−−−− 11321 2.05.03.01.0 5=ω 9A ttttttt eeYYYYY +−−−−= −−−−− 11321 2.02.02.02.0 3=ω 9B ttttttt eeYYYYY +−−−−= −−−−− 11321 2.02.02.02.0 5=ω

10A ttttttt eeYYYYY +−−+−= −−−−− 11321 1.04.04.02.0 3=ω 10B ttttttt eeYYYYY +−−+−= −−−−− 11321 1.04.04.02.0 5=ω 11A ttttttt eeYYYYY +−−−−= −−−−− 11321 1.01.01.03.0 3=ω 11B ttttttt eeYYYYY +−−−−= −−−−− 11321 1.01.01.03.0 5=ω 12A ttttttt eeYYYYY ++−+−= −−−−− 11321 2.04.02.04.0 3=ω 12B ttttttt eeYYYYY ++−+−= −−−−− 11321 2.04.02.04.0 5=ω 13A ttttttt eeYYYYY +−−−−= −−−−− 11321 5.01.01.05.0 3=ω 13B ttttttt eeYYYYY +−−−−= −−−−− 11321 5.01.01.05.0 5=ω

120

Table 6.7 Proportion of correctly detecting AO using AO test criterion for BL(1,0,1,1)

MODEL

PROCEDURES and CRITICAL VALUES STANDARD TM MAD MB

2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.69 0.58 0.27 0.72 0.66 0.60 0.67 0.63 0.39 0.65 0.54 0.33

1B 1.00 0.98 0.86 0.98 0.98 0.98 0.98 0.98 0.88 1.00 0.98 0.96

2A 0.85 0.47 0.28 0.83 0.81 0.54 0.80 0.57 0.33 0.42 0.31 0.25

2B 1.00 1.00 0.96 1.00 1.00 1.00 1.00 1.00 0.98 0.96 0.94 0.85

3A 0.41 0.28 0.13 0.39 0.36 0.36 0.31 0.25 0.19 0.36 0.33 0.30

3B 0.78 0.78 0.74 0.78 0.78 0.78 0.74 0.72 0.72 0.78 0.78 0.72

4A 0.80 0.50 0.32 0.77 0.71 0.56 0.67 0.59 0.37 0.46 0.40 0.29

4B 0.98 0.96 0.92 1.00 1.00 0.98 0.96 0.96 0.92 0.98 0.96 0.94

5A 0.74 0.59 0.30 0.72 0.69 0.62 0.64 0.54 0.25 0.62 0.62 0.55

5B 0.95 0.89 0.81 0.97 0.97 0.92 0.89 0.81 0.81 0.97 0.92 0.89

6A 0.67 0.50 0.33 0.64 0.59 0.56 0.59 0.35 0.30 0.62 0.56 0.41

6B 1.00 0.97 0.92 0.97 0.95 0.95 0.90 0.90 0.85 0.97 0.97 0.97

7A 0.83 0.57 0.33 0.77 0.75 0.64 0.67 0.60 0.40 0.77 0.67 0.51

7B 0.98 0.98 0.93 0.95 0.95 0.95 0.98 0.98 0.95 0.98 0.98 0.98

8A 0.67 0.50 0.17 0.66 0.62 0.54 0.63 0.51 0.33 0.59 0.51 0.27

8B 1.00 0.96 0.82 1.00 1.00 1.00 1.00 0.96 0.86 1.00 0.98 0.98

9A 0.79 0.56 0.40 0.80 0.80 0.66 0.75 0.56 0.42 0.44 0.36 0.26

9B 1.00 1.00 0.96 1.00 1.00 1.00 1.00 1.00 0.96 0.96 0.90 0.78

10A 0.70 0.55 0.28 0.69 0.67 0.63 0.73 0.55 0.27 0.60 0.47 0.23

10B 1.00 0.98 0.84 1.00 1.00 1.00 1.00 0.98 0.86 1.00 0.98 0.98

11A 0.65 0.57 0.23 0.68 0.64 0.64 0.69 0.58 0.35 0.52 0.40 0.22

11B 1.00 0.98 0.96 1.00 1.00 1.00 1.00 1.00 0.98 1.00 0.98 0.88

12A 0.60 0.40 0.24 0.59 0.57 0.48 0.64 0.52 0.39 0.33 0.28 0.20

12B 1.00 1.00 0.96 1.00 1.00 1.00 1.00 1.00 0.98 0.96 0.88 0.78

13A 0.57 0.48 0.23 0.61 0.59 0.52 0.63 0.52 0.28 0.69 0.56 0.40

13B 0.97 0.91 0.87 0.97 0.97 0.97 0.97 0.97 0.93 0.99 0.99 0.89

121

Table 6.8 Proportion of correctly detecting IO using IO test criterion for BL(1,0,1,1)

MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.67 0.57 0.29 0.68 0.64 0.58 0.57 0.51 0.33 0.61 0.53 0.35

1B 1.00 1.00 0.88 0.98 0.98 0.98 0.98 0.98 0.90 1.00 0.98 0.96

2A 0.75 0.41 0.26 0.67 0.65 0.52 0.70 0.50 0.35 0.46 0.40 0.31

2B 0.94 0.94 0.87 0.96 0.96 0.96 0.94 0.94 0.92 0.89 0.89 0.85

3A 0.47 0.22 0.07 0.33 0.33 0.27 0.43 0.29 0.21 0.53 0.53 0.47

3B 0.71 0.71 0.71 0.76 0.76 0.76 0.71 0.71 0.71 0.59 0.59 0.59

4A 0.61 0.43 0.27 0.60 0.56 0.46 0.58 0.47 0.33 0.46 0.40 0.31

4B 1.00 1.00 0.89 1.00 1.00 1.00 1.00 1.00 0.95 1.00 1.00 0.95

5A 0.80 0.70 0.40 0.70 0.70 0.70 0.50 0.50 0.00 0.50 0.50 0.20

5B 1.00 1.00 0.91 1.00 1.00 1.00 0.91 0.91 0.91 1.00 0.91 0.91

6A 0.46 0.23 0.08 0.50 0.43 0.29 0.36 0.14 0.14 0.29 0.14 0.07

6B 0.93 0.86 0.79 0.93 0.93 0.93 0.86 0.79 0.71 0.93 0.93 0.64

7A 0.50 0.33 0.22 0.50 0.50 0.39 0.44 0.33 0.28 0.56 0.50 0.33

7B 1.00 0.83 0.72 1.00 1.00 0.95 0.83 0.83 0.78 1.00 1.00 0.78

8A 0.42 0.28 0.15 0.40 0.34 0.30 0.37 0.33 0.14 0.45 0.40 0.20

8B 0.95 0.90 0.80 1.00 1.00 1.00 1.00 0.90 0.85 1.00 1.00 1.00

9A 0.61 0.50 0.27 0.60 0.56 0.45 0.58 0.35 0.18 0.45 0.40 0.31

9B 1.00 1.00 0.89 1.00 1.00 1.00 0.95 0.95 0.89 1.00 1.00 0.85

10A 0.65 0.35 0.15 0.60 0.60 0.45 0.30 0.15 0.15 0.53 0.33 0.18

10B 1.00 0.95 0.85 0.95 0.95 0.95 0.95 0.90 0.80 1.00 1.00 1.00

11A 0.55 0.40 0.25 0.55 0.55 0.45 0.45 0.35 0.30 0.65 0.45 0.20

11B 1.00 1.00 0.85 1.00 1.00 1.00 0.95 0.95 0.90 1.00 1.00 1.00

12A 0.61 0.50 0.22 0.65 0.60 0.45 0.53 0.35 0.18 0.45 0.40 0.20

12B 1.00 1.00 0.90 1.00 1.00 1.00 0.95 0.95 0.90 1.00 1.00 1.00

13A 0.47 0.47 0.13 0.60 0.53 0.53 0.50 0.43 0.21 0.47 0.47 0.27

13B 1.00 1.00 0.88 1.00 1.00 1.00 0.94 0.94 0.94 1.00 1.00 1.00

122


MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.58 0.50 0.24 0.58 0.58 0.52 0.59 0.52 0.38 0.64 0.56 0.34

1B 0.92 0.88 0.77 0.92 0.91 0.89 0.89 0.88 0.78 0.98 0.97 0.94

2A 0.63 0.42 0.21 0.64 0.57 0.45 0.55 0.38 0.21 0.62 0.52 0.31

2B 0.97 0.92 0.78 0.97 0.96 0.92 0.94 0.86 0.79 0.97 0.97 0.92

3A 0.57 0.48 0.21 0.59 0.54 0.43 0.43 0.39 0.33 0.64 0.57 0.40

3B 0.94 0.91 0.69 0.96 0.96 0.96 0.94 0.92 0.82 0.98 0.98 0.95

4A 0.48 0.30 0.20 0.51 0.49 0.40 0.47 0.38 0.27 0.56 0.56 0.33

4B 0.99 0.98 0.73 0.99 0.99 0.99 0.95 0.93 0.87 0.99 0.99 0.95

5A 0.66 0.45 0.21 0.67 0.64 0.54 0.58 0.45 0.34 0.64 0.59 0.46

5B 0.89 0.83 0.83 0.89 0.89 0.83 0.94 0.94 0.89 0.94 0.94 0.94

6A 0.66 0.51 0.27 0.66 0.63 0.57 0.57 0.51 0.39 0.67 0.61 0.45

6B 0.97 0.97 0.90 0.97 0.97 0.97 0.97 0.97 0.87 1.00 1.00 1.00

7A 0.56 0.40 0.16 0.52 0.48 0.44 0.50 0.42 0.32 0.62 0.52 0.28

7B 0.92 0.85 0.70 0.92 0.91 0.88 0.91 0.85 0.77 0.97 0.95 0.93

8A 0.56 0.34 0.12 0.54 0.52 0.42 0.46 0.38 0.22 0.56 0.48 0.30

8B 0.96 0.89 0.74 0.96 0.96 0.94 0.93 0.90 0.80 0.99 0.98 0.96

9A 0.49 0.37 0.20 0.49 0.47 0.40 0.46 0.39 0.24 0.36 0.29 0.15

9B 0.90 0.77 0.53 0.86 0.86 0.80 0.80 0.71 0.62 0.90 0.87 0.78

10A 0.66 0.52 0.38 0.62 0.58 0.54 0.58 0.50 0.40 0.52 0.48 0.26

10B 0.94 0.87 0.81 0.92 0.92 0.87 0.86 0.83 0.78 0.94 0.92 0.87

11A 0.59 0.50 0.22 0.56 0.56 0.47 0.55 0.45 0.24 0.71 0.62 0.38

11B 0.91 0.89 0.74 0.91 0.91 0.89 0.88 0.88 0.82 0.98 0.98 0.92

12A 0.55 0.43 0.23 0.60 0.56 0.51 0.56 0.48 0.32 0.60 0.56 0.47

12B 0.86 0.80 0.64 0.84 0.84 0.80 0.86 0.84 0.74 0.88 0.88 0.84

13A 0.60 0.48 0.25 0.58 0.58 0.52 0.54 0.45 0.34 0.66 0.53 0.36

13B 0.94 0.94 0.82 0.92 0.92 0.92 0.92 0.90 0.80 1.00 1.00 1.00

123


MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.67 0.55 0.33 0.68 0.66 0.60 0.60 0.58 0.36 0.60 0.52 0.35

1B 0.98 0.96 0.86 0.98 0.98 0.98 0.98 0.98 0.90 1.00 0.98 0.90

2A 0.74 0.55 0.48 0.72 0.72 0.59 0.51 0.41 0.32 0.46 0.36 0.26

2B 0.97 0.97 0.92 0.97 0.97 0.97 0.95 0.95 0.89 0.89 0.89 0.84

3A 0.50 0.36 0.20 0.53 0.51 0.49 0.40 0.40 0.31 0.49 0.33 0.24

3B 0.96 0.93 0.78 0.96 0.96 0.93 0.96 0.96 0.84 0.98 0.96 0.91

4A 0.65 0.53 0.37 0.64 0.64 0.57 0.59 0.55 0.36 0.59 0.52 0.41

4B 0.97 0.89 0.72 0.94 0.94 0.92 0.92 0.89 0.78 0.89 0.86 0.81

5A 0.88 0.65 0.47 0.78 0.78 0.67 0.67 0.67 0.50 0.39 0.39 0.28

5B 0.91 0.91 0.91 1.00 1.00 1.00 1.00 1.00 1.00 0.91 0.91 0.82

6A 0.37 0.33 0.19 0.37 0.37 0.37 0.33 0.26 0.22 0.33 0.30 0.22

6B 0.93 0.93 0.72 0.86 0.86 0.86 0.93 0.93 0.83 1.00 1.00 0.90

7A 0.61 0.47 0.22 0.56 0.54 0.50 0.53 0.45 0.35 0.58 0.46 0.28

7B 1.00 0.96 0.82 1.00 1.00 0.98 1.00 0.98 0.86 1.00 0.98 0.94

8A 0.68 0.52 0.28 0.64 0.62 0.58 0.58 0.44 0.34 0.54 0.46 0.28

8B 1.00 0.98 0.84 1.00 1.00 0.98 0.98 0.96 0.88 0.98 0.96 0.88

9A 0.67 0.55 0.36 0.64 0.64 0.57 0.64 0.57 0.38 0.52 0.43 0.26

9B 0.98 0.96 0.89 0.98 0.98 0.98 0.96 0.93 0.93 1.00 1.00 0.93

10A 0.70 0.56 0.32 0.70 0.68 0.60 0.62 0.60 0.42 0.56 0.54 0.32

10B 0.98 0.96 0.84 0.98 0.98 0.98 1.00 0.98 0.88 1.00 0.98 0.90

11A 0.77 0.67 0.47 0.81 0.81 0.74 0.77 0.63 0.53 0.74 0.55 0.45

11B 0.92 0.92 0.86 0.92 0.92 0.92 0.92 0.92 0.92 0.84 0.82 0.68

12A 0.74 0.67 0.56 0.76 0.76 0.67 0.64 0.57 0.45 0.51 0.33 0.20

12B 0.98 0.95 0.83 0.98 0.98 0.98 0.93 0.93 0.83 0.95 0.93 0.83

13A 0.49 0.34 0.15 0.45 0.45 0.38 0.43 0.39 0.28 0.54 0.35 0.11

13B 0.98 0.96 0.81 0.98 0.98 0.98 0.98 0.94 0.83 1.00 0.98 0.90

124


MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.45 0.35 0.15 0.45 0.45 0.35 0.45 0.30 0.15 0.45 0.40 0.30

1B 0.88 0.88 0.77 0.93 0.93 0.88 0.88 0.88 0.77 1.00 1.00 0.97

2A 0.48 0.43 0.34 0.48 0.48 0.48 0.48 0.45 0.36 0.45 0.39 0.34

2B 0.89 0.89 0.78 0.89 0.89 0.89 0.89 0.89 0.89 1.00 0.89 0.89

3A 0.66 0.51 0.28 0.68 0.66 0.55 0.62 0.57 0.43 0.66 0.62 0.40

3B 0.89 0.89 0.56 0.89 0.89 0.89 0.78 0.67 0.56 0.78 0.78 0.78

4A 0.56 0.44 0.26 0.56 0.51 0.44 0.53 0.42 0.33 0.62 0.60 0.42

4B 0.91 0.86 0.73 0.91 0.91 0.91 0.89 0.86 0.75 0.98 0.98 0.91

5A 0.58 0.47 0.21 0.63 0.63 0.63 0.53 0.47 0.37 0.74 0.68 0.47

5B 0.95 0.95 0.89 1.00 1.00 1.00 0.95 0.95 0.95 1.00 1.00 1.00

6A 0.58 0.53 0.21 0.53 0.53 0.53 0.53 0.47 0.37 0.74 0.68 0.53

6B 0.95 0.95 0.89 1.00 1.00 1.00 0.95 0.95 0.95 1.00 1.00 1.00

7A 0.52 0.38 0.18 0.50 0.50 0.42 0.50 0.44 0.30 0.60 0.54 0.28

7B 0.98 0.90 0.74 0.98 0.96 0.94 0.96 0.92 0.84 1.00 0.98 0.94

8A 0.50 0.47 0.35 0.50 0.50 0.50 0.50 0.50 0.38 0.50 0.50 0.29

8B 0.94 0.75 0.69 0.95 0.92 0.86 0.83 0.78 0.61 0.92 0.89 0.76

9A 0.54 0.38 0.14 0.58 0.50 0.40 0.44 0.34 0.24 0.56 0.54 0.32

9B 0.96 0.88 0.74 0.96 0.96 0.92 0.96 0.96 0.80 1.00 1.00 0.94

10A 0.67 0.54 0.33 0.65 0.65 0.65 0.61 0.59 0.37 0.48 0.37 0.22

10B 0.98 0.96 0.92 0.98 0.98 0.98 0.96 0.96 0.85 0.96 0.96 0.88

11A 0.60 0.44 0.20 0.62 0.60 0.48 0.46 0.40 0.28 0.66 0.62 0.42

11B 0.94 0.92 0.78 0.94 0.94 0.92 0.96 0.90 0.82 1.00 1.00 0.98

12A 0.60 0.46 0.23 0.60 0.57 0.46 0.50 0.37 0.17 0.40 0.31 0.17

12B 0.77 0.67 0.44 0.74 0.74 0.71 0.67 0.60 0.51 0.74 0.60 0.56

13A 0.46 0.42 0.12 0.52 0.48 0.44 0.42 0.42 0.27 0.63 0.59 0.56

13B 0.91 0.84 0.70 0.91 0.91 0.86 0.88 0.84 0.74 0.91 0.89 0.89

125


MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.63 0.47 0.21 0.53 0.53 0.47 0.53 0.32 0.32 0.53 0.42 0.21

1B 0.95 0.95 0.89 0.95 0.95 0.95 0.89 0.89 0.84 1.00 1.00 0.95

2A 0.56 0.50 0.22 0.67 0.67 0.61 0.72 0.67 0.33 0.39 0.33 0.17

2B 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.94 0.94 0.89

3A 0.54 0.41 0.33 0.56 0.56 0.41 0.49 0.38 0.36 0.56 0.46 0.33

3B 0.97 0.97 0.89 0.97 0.97 0.97 0.97 0.97 0.92 0.89 0.87 0.84

4A 0.50 0.45 0.30 0.50 0.48 0.48 0.48 0.45 0.39 0.55 0.48 0.30

4B 0.95 0.87 0.74 0.95 0.95 0.90 0.87 0.87 0.79 0.92 0.90 0.82

5A 0.75 0.42 0.25 0.75 0.75 0.75 0.67 0.42 0.25 0.67 0.50 0.25

5B 0.89 0.89 0.67 1.00 1.00 1.00 0.78 0.78 0.78 1.00 1.00 1.00

6A 0.60 0.60 0.40 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60 0.60

6B 0.78 0.78 0.78 0.89 0.89 0.89 0.89 0.89 0.78 0.89 0.89 0.89

7A 0.45 0.30 0.15 0.45 0.40 0.35 0.30 0.25 0.20 0.40 0.40 0.20

7B 1.00 0.95 0.85 1.00 1.00 0.95 1.00 0.95 0.90 1.00 1.00 1.00

8A 0.67 0.34 0.07 0.77 0.77 0.48 0.67 0.48 0.07 0.63 0.42 0.14

8B 1.00 1.00 0.92 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.92

9A 0.60 0.40 0.20 0.60 0.55 0.45 0.50 0.35 0.25 0.40 0.40 0.20

9B 1.00 0.95 0.80 1.00 1.00 0.95 0.90 0.90 0.80 0.95 0.95 0.85

10A 0.38 0.38 0.26 0.42 0.42 0.38 0.38 0.32 0.32 0.42 0.42 0.07

10B 0.95 0.89 0.84 0.95 0.95 0.89 0.95 0.89 0.84 0.95 0.95 0.84

11A 0.45 0.35 0.20 0.45 0.40 0.35 0.30 0.25 0.20 0.40 0.40 0.20

11B 0.95 0.90 0.85 1.00 1.00 0.95 0.95 0.90 0.80 1.00 1.00 0.95

12A 0.53 0.47 0.33 0.53 0.53 0.47 0.53 0.47 0.27 0.54 0.46 0.31

12B 0.88 0.88 0.69 0.94 0.94 0.94 0.88 0.88 0.88 0.88 0.88 0.75

13A 0.80 0.73 0.53 0.81 0.75 0.75 0.69 0.69 0.63 0.56 0.50 0.31

13B 0.88 0.88 0.88 0.88 0.88 0.88 0.94 0.94 0.94 0.83 0.83 0.83

126

6.3 PERFORMANCE OF OUTLIER DETECTION PROCEDURE

In this section, performance of outlier detection procedure to identify the type of

outlier was considered through simulation work. The same factors considered in section

6.2 were used. Proportion of correct detection was reported in Table 6.13 to Table 6.18.

Results for AO and IO cases on BL(1,0,1,1) were given in Table 6.13 and Table

6.14, respectively. From Table 6.13, the results show that, when 3=ω , the proportion of

correct detection was small compared to 5=ω for all procedures. Further, with critical

value 2.5 and 5=ω , the procedure performed well for all procedures. On the other hand,

standard and trimmed mean (TM) procedures perform better than the MAD and model-

based (MB) procedures. Meanwhile, Table 6.14 shows the result of performance of the

outlier detection procedure for IO case. These results were almost similar with AO case

such that, with critical value 2.5 and 5=ω the procedure performed well. For both cases,

3=ω is too small compared to the fluctuation of the time series.


6.16, respectively. Result from Table 6.15 show that the with critical value 2.5 and

5=ω , proportion of performance of the outlier detection procedure performed well.

Standard procedure was better compared to the other procedures. On the other hand,

results in IO case were almost similar to AO case. In this case, standard procedure is still

the best compared to the other procedures. Overall, the procedures performed well

especially with critical value 2.5 and 5=ω . For both cases, the proportion increases

when larger ω was used.

127

Table 6.13 Proportion of correctly detecting AO the outlier detection procedure for

BL(1,0,1,1)

MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.42 0.28 0.16 0.44 0.28 0.18 0.40 0.34 0.20 0.42 0.30 0.16

1B 0.50 0.50 0.48 0.48 0.48 0.46 0.56 0.54 0.52 0.44 0.44 0.38

2A 0.38 0.24 0.18 0.36 0.22 0.16 0.36 0.24 0.16 0.50 0.42 0.34

2B 0.48 0.46 0.46 0.52 0.50 0.50 0.58 0.56 0.54 0.62 0.62 0.60

3A 0.26 0.17 0.17 0.36 0.31 0.29 0.36 0.29 0.24 0.74 0.74 0.64

3B 0.47 0.47 0.44 0.45 0.45 0.43 0.43 0.43 0.40 0.60 0.60 0.57

4A 0.43 0.34 0.23 0.36 0.28 0.17 0.34 0.32 0.23 0.53 0.45 0.32

4B 0.67 0.67 0.60 0.67 0.67 0.60 0.52 0.52 0.52 0.60 0.60 0.56

5A 0.44 0.41 0.35 0.44 0.41 0.35 0.41 0.38 0.32 0.67 0.62 0.47

5B 0.67 0.67 0.67 0.67 0.67 0.67 0.57 0.57 0.57 0.67 0.67 0.67

6A 0.49 0.49 0.38 0.41 0.41 0.31 0.33 0.33 0.26 0.54 0.54 0.41

6B 0.73 0.73 0.73 0.65 0.65 0.65 0.60 0.60 0.60 0.70 0.70 0.70

7A 0.52 0.45 0.19 0.48 0.40 0.19 0.40 0.36 0.17 0.40 0.40 0.17

7B 0.78 0.78 0.78 0.75 0.75 0.73 0.68 0.68 0.68 0.48 0.48 0.45

8A 0.32 0.28 0.18 0.30 0.26 0.20 0.20 0.20 0.10 0.28 0.26 0.16

8B 0.54 0.52 0.50 0.48 0.46 0.44 0.56 0.54 0.52 0.32 0.32 0.30

9A 0.44 0.36 0.24 0.38 0.26 0.18 0.40 0.30 0.20 0.46 0.20 0.38

9B 0.48 0.48 0.48 0.50 0.50 0.50 0.50 0.50 0.48 0.60 0.60 0.56

10A 0.18 0.16 0.10 0.24 0.22 0.12 0.26 0.22 0.12 0.46 0.38 0.22

10B 0.54 0.54 0.48 0.56 0.56 0.50 0.50 0.50 0.46 0.60 0.60 0.54

11A 0.40 0.32 0.20 0.44 0.36 0.20 0.40 0.30 0.18 0. 18 0.16 0.08

11B 0.76 0.74 0.62 0.78 0.76 0.64 0.64 0.62 0.52 0. 32 0.28 0.14

12A 0.40 0.34 0.21 0.45 0.38 0.23 0.30 0.30 0.13 0.23 0.21 0.06

12B 0.79 0.77 0.72 0.83 0.81 0.77 0.68 0.68 0.64 0.30 0.30 0.30

13A 0.33 0.29 0.24 0.31 0.29 0.24 0.31 0.29 0.20 0.51 0.42 0.27

13B 0.82 0.82 0.76 0.84 0.84 0.78 0.76 0.76 0.70 0.56 0.56 0.40

128

Table 6.14 Proportion of correctly detecting IO the outlier detection procedure for

BL(1,0,1,1)

MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.36 0.30 0.28 0.38 0.32 0.30 0.32 0.28 0.26 0.40 0.32 0.28

1B 0.50 0.50 0.44 0.56 0.56 0.50 0.60 0.60 0.54 0.50 0.50 0.46

2A 0.54 0.48 0.42 0.46 0.40 0.36 0.40 0.36 0.30 0.32 0.28 0.24

2B 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.65 0.65 0.50 0.50 0.50

3A 0.53 0.53 0.38 0.53 0.53 0.38 0.41 0.41 0.29 0.29 0.29 0.12

3B 0.57 0.57 0.54 0.57 0.57 0.54 0.54 0.54 0.51 0.32 0.32 0.32

4A 0.49 0.43 0.25 0.47 0.43 0.25 0.51 0.43 0.35 0.43 0.39 0.25

4B 0.64 0.64 0.60 0.60 0.60 0.56 0.66 0.66 0.62 0.60 0.60 0.58

5A 0.52 0.52 0.52 0.52 0.52 0.52 0.58 0.58 0.58 0.38 0.38 0.38

5B 0.59 0.53 0.38 0.56 0.47 0.34 0.47 0.41 0.28 0.45 0.45 0.45

6A 0.46 0.34 0.23 0.46 0.31 0.20 0.34 0.26 0.20 0.29 0.23 0.17

6B 0.71 0.71 0.68 0.71 0.71 0.66 0.68 0.68 0.66 0.66 0.66 0.63

7A 0.38 0.38 0.30 0.41 0.41 0.30 0.38 0.35 0.30 0.49 0.41 0.30

7B 0.69 0.69 0.61 0.78 0.78 0.69 0.69 0.69 0.61 0.75 0.75 0.67

8A 0.34 0.24 0.12 0.30 0.22 0.10 0.36 0.26 0.10 0.36 0.22 0.08

8B 0.64 0.64 0.60 0.68 0.68 0.62 0.58 0.58 0.54 0.70 0.70 0.64

9A 0.62 0.56 0.46 0.64 0.58 0.46 0.58 0.52 0.40 0.24 0.24 0.16

9B 0.70 0.70 0.68 0.62 0.62 0.60 0.48 0.48 0.48 0.34 0.34 0.34

10A 0.42 0.26 0.08 0.42 0.26 0.08 0.32 0.20 0.06 0.28 0.18 0.04

10B 0.74 0.74 0.66 0.68 0.68 0.62 0.54 0.54 0.50 0.60 0.60 0.58

11A 0.42 0.32 0.20 0.44 0.34 0.22 0.32 0.28 0.20 0.38 0.32 0.26

11B 0.64 0.62 0.62 0.66 0.64 0.62 0.60 0.58 0.54 0.62 0.60 0.56

12A 0.40 0.36 0.30 0.36 0.32 0.28 0.50 0.42 0.38 0.32 0.30 0.26

12B 0.70 0.70 0.66 0.68 0.68 0.64 0.68 0.68 0.64 0.46 0.46 0.44

13A 0.38 0.31 0.21 0.42 0.36 0.21 0.31 0.26 0.17 0.36 0.29 0.21

13B 0.58 0.56 0.56 0.67 0.64 0.64 0.58 0.56 0.56 0.60 0.60 0.60

129


BL(2,0,1,1)

MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.24 0.22 0.14 0.18 0.14 0.08 0.24 0.20 0.14 0.20 0.12 0.08

1B 0.58 0.58 0.54 0.48 0.48 0.44 0.44 0.44 0.42 0.42 0.42 0.40

2A 0.44 0.36 0.20 0.40 0.33 0.20 0.40 0.33 0.24 0.44 0.42 0.24

2B 0.66 0.66 0.64 0.66 0.66 0.64 0.48 0.48 0.43 0.50 0.48 0.48

3A 0.53 0.47 0.37 0.49 0.43 0.35 0.43 0.41 0.29 0.49 0.41 0.31

3B 0.73 0.73 0.67 0.67 0.67 0.61 0.59 0.59 0.53 0.59 0.59 0.50

4A 0.48 0.45 0.39 0.45 0.43 0.39 0.43 0.41 0.36 0.48 0.45 0.33

4B 0.79 0.79 0.72 0.63 0.63 0.58 0.60 0.60 0.56 0.58 0.58 0.53

5A 0.65 0.61 0.52 0.61 0.61 0.52 0.57 0.52 0.48 0.87 0.78 0.61

5B 0.65 0.65 0.65 0.65 0.65 0.65 0.53 0.53 0.53 0.88 0.88 0.88

6A 0.69 0.63 0.38 0.66 0.59 0.34 0.56 0.53 0.31 0.52 0.50 0.31

6B 0.88 0.88 0.88 0.79 0.79 0.79 0.67 0.67 0.67 0.56 0.56 0.56

7A 0.22 0.14 0.12 0.26 0.16 0.14 0.20 0.16 0.16 0.30 0.26 0.22

7B 0.46 0.46 0.38 0.44 0.44 0.36 0.38 0.38 0.30 0.36 0.30 0.26

8A 0.28 0.26 0.16 0.28 0.26 0.16 0.36 0.32 0.18 0.40 0.36 0.24

8B 0.58 0.56 0.54 0.62 0.60 0.58 0.54 0.52 0.52 0.46 0.46 0.46

9A 0.37 0.35 0.28 0.40 0.40 0.30 0.37 0.37 0.30 0.33 0.30 0.26

9B 0.84 0.84 0.84 0.80 0.80 0.80 0.80 0.78 0.78 0.36 0.36 0.36

10A 0.40 0.36 0.20 0.38 0.36 0.22 0.38 0.36 0.22 0.18 0.18 0.08

10B 0.80 0.78 0.76 0.80 0.78 0.76 0.72 0.70 0.68 0.20 0.20 0.18

11A 0.58 0.48 0.20 0.55 0.45 0.23 0.53 0.48 0.23 0.68 0.65 0.35

11B 0.71 0.71 0.71 0.61 0.61 0.61 0.59 0.59 0.57 0.71 0.71 0.71

12A 0.53 0.49 0.44 0.47 0.42 0.37 0.35 0.30 0.28 0.65 0.63 0.71

12B 0.78 0.78 0.78 0.68 0.68 0.68 0.64 0.64 0.64 0.96 0.96 0.96

13A 0.40 0.38 0.24 0.38 0.38 0.24 0.40 0.40 0.22 0.50 0.46 0.24

13B 0.76 0.76 0.66 0.70 0.70 0.62 0.62 0.62 0.58 0.50 0.50 0.40

130


BL(2,0,1,1)

MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.32 0.22 0.12 0.34 0.24 0.14 0.20 0.14 0.06 0.30 0.18 0.12

1B 0.78 0.76 0.70 0.80 0.78 0.72 0.70 0.68 0.64 0.66 0.66 0.64

2A 0.44 0.40 0.30 0.52 0.46 0.32 0.38 0.34 0.26 0.44 0.40 0.34

2B 0.61 0.61 0.56 0.69 0.69 0.64 0.61 0.61 0.61 0.50 0.50 0.50

3A 0.45 0.40 0.30 0.47 0.43 0.33 0.45 0.41 0.33 0.53 0.49 0.37

3B 0.68 0.68 0.61 0.68 0.68 0.61 0.50 0.50 0.45 0.68 0.64 0.59

4A 0.36 0.32 0.28 0.32 0.28 0.22 0.28 0.22 0.20 0.34 0.30 0.26

4B 0.81 0.81 0.78 0.76 0.76 0.73 0.68 0.68 0.65 0.89 0.89 0.81

5A 0.50 0.44 0.31 0.50 0.44 0.31 0.50 0.38 0.25 0.25 0.25 0.19

5B 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.31 0.25 0.25

6A 0.34 0.28 0.16 0.28 0.22 0.16 0.31 0.25 0.16 0.44 0.28 0.16

6B 0.64 0.64 0.61 0.64 0.64 0.61 0.79 0.79 0.71 0.61 0.61 0.57

7A 0.60 0.44 0.28 0.50 0.40 0.24 0.38 0.32 0.22 0.48 0.38 0.22

7B 0.62 0.60 0.54 0.58 0.58 0.52 0.54 0.54 0.48 0.74 0.74 0.62

8A 0.52 0.44 0.24 0.56 0.50 0.26 0.40 0.34 0.22 0.28 0.26 0.18

8B 0.76 0.74 0.72 0.74 0.72 0.70 0.70 0.68 0.66 0.68 0.68 0.68

9A 0.41 0.28 0.26 0.46 0.31 0.26 0.46 0.31 0.26 0.28 0.23 0.18

9B 0.75 0.75 0.73 0.77 0.77 0.75 0.80 0.80 0.77 0.43 0.43 0.43

10A 0.32 0.30 0.10 0.34 0.32 0.10 0.28 0.26 0.10 0.32 0.28 0.08

10B 0.88 0.84 0.82 0.90 0.86 0.84 0.84 0.80 0.80 0.56 0.54 0.50

11A 0.43 0.43 0.23 0.51 0.51 0.31 0.49 0.49 0.31 0.34 0.34 0.23

11B 0.64 0.64 0.60 0.71 0.71 0.69 0.60 0.60 0.58 0.62 0.62 0.60

12A 0.40 0.35 0.23 0.37 0.33 0.23 0.44 0.40 0.33 0.09 0.09 0.05

12B 0.74 0.70 0.70 0.76 0.72 0.72 0.68 0.66 0.66 0.40 0.36 0.36

13A 0.32 0.26 0.12 0.30 0.26 0.14 0.28 0.26 0.14 0.34 0.32 0.14

13B 0.74 0.72 0.66 0.74 0.72 0.66 0.68 0.64 0.60 0.84 0.82 0.76

131


BL(3,0,1,1)

MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.30 0.24 0.14 0.32 0.24 0.16 0.28 0.26 0.14 0.22 0.20 0.10

1B 0.56 0.56 0.54 0.56 0.56 0.56 0.52 0.52 0.50 0.42 0.40 0.34

2A 0.50 0.40 0.40 0.50 0.40 0.40 0.30 0.20 0.20 0.33 0.33 0.33

2B 0.75 0.75 0.69 0.69 0.69 0.63 0.51 0.51 0.51 0.50 0.50 0.50

3A 0.55 0.55 0.27 0.59 0.59 0.32 0.32 0.32 0.19 0.50 0.50 0.32

3B 0.68 0.68 0.68 0.61 0.61 0.61 0.64 0.64 0.64 0.57 0.57 0.57

4A 0.53 0.44 0.35 0.49 0.42 0.33 0.42 0.37 0.30 0.33 0.33 0.33

4B 0.61 0.61 0.61 0.57 0.57 0.57 0.47 0.47 0.47 0.51 0.42 0.30

5A 0.42 0.42 0.33 0.33 0.33 0.25 0.33 0.33 0.33 0.30 0.30 0.17

5B 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.33 0.33 0.33

6A 0.50 0.50 0.33 0.50 0.50 0.33 0.50 0.50 0.33 0.42 0.42 0.42

6B 1.00 1.00 0.92 0.83 0.83 0.75 0.67 0.67 0.59 0.50 0.50 0.42

7A 0.28 0.24 0.12 0.24 0.22 0.10 0.28 0.22 0.10 0.22 0.20 0.10

7B 0.62 0.58 0.46 0.56 0.54 0.42 0.42 0.40 0.34 0.28 0.22 0.18

8A 0.49 0.49 0.38 0.46 0.43 0.33 0.38 0.33 0.21 0.36 0.31 0.28

8B 0.56 0.49 0.38 0.54 0.46 0.38 0.38 0.38 0.33 0.41 0.41 0.31

9A 0.38 0.32 0.26 0.34 0.30 0.24 0.24 0.20 0.20 0.42 0.36 0.28

9B 0.52 0.52 0.52 0.54 0.54 0.54 0.50 0.50 0.50 0.54 0.54 0.54

10A 0.69 0.67 0.67 0.64 0.64 0.64 0.60 0.58 0.58 0.50 0.50 0.50

10B 0.85 0.85 0.83 0.79 0.79 0.77 0.77 0.77 0.75 0.62 0.62 0.62

11A 0.38 0.26 0.12 0.32 0.20 0.08 0.26 0.18 0.06 0.30 0.22 0.12

11B 0.74 0.70 0.60 0.62 0.62 0.58 0.58 0.58 0.56 0.52 0.52 0.48

12A 0.16 0.16 0.14 0.16 0.16 0.14 0.18 0.18 0.14 0.18 0.18 0.14

12B 0.74 0.72 0.70 0.70 0.68 0.65 0.65 0.63 0.61 0.47 0.47 0.42

13A 0.42 0.42 0.35 0.32 0.32 0.25 0.32 0.32 0.29 0.58 0.58 0.54

13B 0.63 0.63 0.63 0.63 0.63 0.63 0.58 0.58 0.58 0.91 0.91 0.91

132


BL(3,0,1,1)

MODEL


2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5 2.5 3.0 3.5

1A 0.30 0.30 0.20 0.34 0.32 0.22 0.24 0.22 0.14 0.28 0.26 0.16

1B 0.60 0.60 0.58 0.66 0.66 0.62 0.56 0.56 0.50 0.60 0.60 0.56

2A 0.38 0.38 0.13 0.50 0.38 0.13 0.38 0.38 0.13 0.26 0.26 0.13

2B 0.69 0.69 0.69 0.69 0.69 0.69 0.54 0.54 0.54 0.27 0.27 0.26

3A 0.24 0.24 0.24 0.30 0.30 0.30 0.36 0.36 0.36 0.30 0.30 0.30

3B 0.72 0.72 0.68 0.68 0.68 0.68 0.60 0.60 0.60 0.56 0.56 0.48

4A 0.41 0.32 0.29 0.35 0.27 0.25 0.32 0.27 0.25 0.38 0.32 0.29

4B 0.74 0.74 0.69 0.69 0.69 0.64 0.84 0.84 0.75 0.75 0.75 0.69

5A 0.38 0.38 0.13 0.50 0.50 0.26 0.38 0.38 0.13 0.13 0.13 0.13

5B 0.73 0.73 0.73 0.82 0.82 0.82 0.73 0.73 0.73 0.73 0.73 0.73

6A 0.09 0.09 0.09 0.18 0.18 0.18 0.27 0.27 0.27 0.09 0.09 0.09

6B 0.55 0.55 0.44 0.66 0.66 0.55 0.55 0.55 0.44 0.44 0.44 0.33

7A 0.34 0.28 0.08 0.38 0.28 0.10 0.34 0.24 0.10 0.28 0.18 0.08

7B 0.70 0.66 0.60 0.68 0.64 0.56 0.56 0.52 0.48 0.68 0.64 0.58

8A 0.44 0.42 0.21 0.37 0.35 0.21 0.39 0.39 0.24 0.29 0.26 0.13

8B 0.78 0.76 0.76 0.80 0.78 0.78 0.73 0.72 0.72 0.51 0.51 0.49

9A 0.40 0.34 0.10 0.46 0.38 0.10 0.40 0.34 0.08 0.30 0.24 0.10

9B 0.86 0.86 0.84 0.80 0.80 0.78 0.76 0.76 0.74 0.64 0.64 0.62

10A 0.33 0.28 0.21 0.33 0.28 0.21 0.31 0.29 0.22 0.27 0.21 0.15

10B 0.73 0.73 0.70 0.84 0.84 0.81 0.73 0.73 0.73 0.52 0.52 0.46

11A 0.44 0.36 0.16 0.42 0.34 0.18 0.34 0.30 0.16 0.40 0.36 0.16

11B 0.90 0.90 0.86 0.92 0.92 0.84 0.82 0.82 0.74 0.86 0.86 0.82

12A 0.26 0.23 0.07 0.20 0.17 0.04 0.23 0.23 0.07 0.14 0.14 0.04

12B 0.47 0.47 0.39 0.47 0.47 0.42 0.50 0.50 0.39 0.22 0.22 0.20

13A 0.34 0.34 0.32 0.43 0.43 0.39 0.39 0.39 0.34 0.13 0.10 0.10

13B 0.56 0.56 0.56 0.62 0.62 0.62 0.56 0.56 0.56 0.18 0.18 0.18

133


6.18, respectively. From Table 6.17, the results show that, with critical value 2.5 and

5=ω , proportion of performance of the outlier detection procedure performed well for

all procedures. Overall, the proportion of correctly detecting AO for almost all models

and procedures were greater than 55% except for MB procedure. MB procedure gave the

lowest proportion of correctly detecting AO. The proportion detecting AO improves

when larger value of ω was considered. The results in IO case were almost similar to the

AO case, but the proportion of correctly detecting AO is higher than that of the IO case.

MB procedure still gave the lowest proportion of correct detection of the outlier.

In summary, the performance of the detection procedure is good. The

performance of outlier detection procedure is better when larger ω is used.

6.4 SUMMARY

In this chapter, the performance of test criteria and outlier detection procedure

were studied through simulation works. Results show that the performance of the

procedure for detecting outliers were better when larger ω is used. In all cases, the

performances of test criteria of AO and IO individually were generally better than the

outlier detection procedure. In general, the procedure performed well.

134

CHAPTER SEVEN

ANALYSIS OF DATA

In this chapter, analysis on rainfall data and air quality data will be carried out.

The analysis will compare the modeling results of linear and BL(p,0,1,1) models, where

p=1,2,3 . It will be shown that BL(p,0,1,1) model can be alternative to the ARIMA

(p,d,q) model. The data are then used to illustrate the outlier detection procedure.

7.1 KAMPUNG ARING MONTHLY RAINFALL DATA

The first data set was the rainfall data collected from Kampung Aring weather

station, Kelantan, Malaysia, for the period of August 1995 till July 2002. The plot of

monthly average in millimeter was given in Figure 7.1. It can be observed that the data

was generally stationary in mean and variance except at time point 41 and 77, where

rainfalls were heavy.

Figure 7.1 Plot of the Kampung Aring rainfall data

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

N u m b e r o f m o n th s

020

040

060

080

0

Rai

nfal

l (m

m)

T im e p o in t 4 1D e c e m b e r 1 9 9 8

T im e p o in t 7 7D e c e m b e r 2 0 0 1

135

The possibility of fitting nonlinear model on the rainfall data was investigated.

When nonlinearity test were applied, the p-values of Keenan’ test and F-test were 0.0293

and 0.2777 respectively. The Keenan’s test strongly suggested that the data was

nonlinear. Hence the BL(p,0,1,1) models were considered for the data for p=1,2,3. Table

7.1 presents the diagnostic result when ARIMA(p,d,q) and BL(p,0,1,1) models were

fitted to the rainfall data for p=1,2,3. From the diagnostic stage, the AIC, BIC and SBIC

for bilinear models were slightly smaller than that of ARIMA model. BL(1,0,1,1) models

could be considered for modelling the data.

When the detection was applied to the data, an additive outlier (AO) was detected

at time point 41 for all cases except model-based (MB) case which detected an

innovational outlier (IO) but at the same time point. Results are given in Table 7.2. Time

points 41 corresponded to December 1998.

Table 7.1 Summary of diagnostic results for the Kampung Aring rainfall data

MODEL AIC BIC SBIC 2ˆ eσ ARIMA(1,0,0) 1093.682 862.1617 860.1617 27717.71 ARIMA(1,2,1) 1101.897 866.9463 865.9463 28456.64 ARIMA(2,0,0) 1090.959 862.8695 859.8695 24829.55 ARIMA(2,1,0) 1094.26 862.7398 860.7398 25372.13 ARIMA(3,0,0) 1091.384 866.7252 862.7252 23888.32 ARIMA(3,3,3) 1104.124 876.035 873.035 27862.31

BL(1,0,1,1) 1073.714 842.1941 840.1941 60116 BL(2,0,1,1) 1076.387 848.2977 845.2977 59290.51 BL(3,0,1,1) 1080.142 855.4838 851.4838 59293.49

Table 7.2 The test statistic value of outlier detection procedure on the Kampung Aring

rainfall data

MODEL STANDARD TM MAD MB BL(1,0,1,1)AO 5.0270 (41) 7.4124(41) 6.0252(41) 3.7953(41) BL(1,0,1,1)IO 4.6702 (41) 6.6282 (41) 4.7345(41) 5.6913(41)

136

7.2 KUALA LUMPUR AIR QUALITY DATA

The second data set that we considered for this study was the Kuala Lumpur air

quality data on particulate matter (PM) for the period of January 1997 till January 1998.

Figure 7.2 gives plot of daily average concentration of PM. It can be seen that, there was

sudden increase in average concentration of PM on 23rd January 1997 corresponding to

time point 54.

0 1 0 0 2 0 0 3 0 0 4 0 0

4060

8010

012

014

016

0

N u m b e r o f d a ys

Con

cent

ratio

n of

Par

ticul

ate

Mat

ter (

ug/m

3)

T im e p o in t 5 4(2 3 rd J a n u a ry 1 9 9 7 )

Figure 7.2 Plot of the Kuala Lumpur air quality data

When nonlinearity test were applied, the p-values of Keenan’ test and F-test were

0.0943 and 0.2383 respectively. The Keenan’s test strongly suggested that the data was

nonlinear. Table 7.3 gives the summary of diagnostic result based on ARIMA(p,d,q) and

BL(p,0,1,1) models, p=1,2,3. It can be seen that, in general, BL(3,0,1,1) models fit the

data better than ARIMA(p,d,q) as the values of AIC, BIC and SBIC were also reduced.

When the detection was applied to the data, an additive outlier (AO) was detected

at time point 54 for the standard and MAD cases. Otherwise, an innovational outlier (IO)

137

is detected for the trimmed mean (TM) and model-based (MB) cases but still at the same

time point. Results were given in Table 7.4. Time points 54 correspond to 23rd January

1997.

Table 7.3 Summary of diagnostic results for the Kuala Lumpur air quality data

MODEL AIC BIC SBIC 2ˆ eσ ARIMA(1,0,0) 3191.708 2072.89 2071.89 184.3693 ARIMA(1,3,1) 3499.352 2380.534 2379.534 400.9434 ARIMA(2,0,0) 3186.69 2072.854 2070.854 181.1309 ARIMA(2,2,2) 3171.343 2057.507 2055.507 174.2454 ARIMA(3,0,0) 3174.738 2065.883 2062.883 174.8602 ARIMA(3,2,3) 3188.516 2079.661 2076.661 181.0511

BL(1,0,1,1) 3176.78 2057.962 2056.962 177.5485 BL(2,0,1,1) 3174.383 2060.546 2058.546 175.588 BL(3,0,1,1) 3161.28 2052.424 2049.424 169.0172

Table 7.4 The test statistic value of outlier detection procedure on the Kuala Lumpur air

quality data

MODEL STANDARD TM MAD MB BL(3,0,1,1)AO 8.1026(54) 10.1829(54) 8.7401(54) 6.4530(54) BL(3,0,1,1)IO 8.0841(54) 10.3948(54) 8.7236(54) 7.3521(54)

7.3 SUMMARY

In this chapter, analysis of data was carried out on rainfall data and air quality

data. For Kampung Aring rainfall data and Kuala Lumpur air quality data, the Keenan’s

test and F-test suggest that the data were nonlinear. The results of data analysis suggest

that bilinear model can be alternative model for data sets which are classified as nonlinear

by at least one nonlinearity test.

138

The proposed outlier detection procedure was applied on two data sets. The

procedure successfully detected change point in Kampung Aring rainfall data as an AO

of type of outlier. On the other hand, one time points of high recorded in Kuala Lumpur

air quality data was correctly detected as AO and IO depending on the case that we used.

These results suggest that the performance of the suggested outlier detection procedure

was very desirable. The procedure was able to detect different type of outliers according

to the pattern of the outliers.

139

CHAPTER EIGHT

CONCLUSION AND FURTHER RESEARCH

This chapter presents the results, contributions of the study and areas of further

research.

8.1 SUMMARY OF THE STUDY

The study proposed an outlier detection procedure for the BL(p,0,1,1) models,

where 3,2,1=p . In this process, a time series was first fitted by the models using the

Box-Jenkins approach. In the estimation stage, the parameter estimates for the model

were found using the nonlinear least squares method.

In the model identification process, the BL(p,0,1,1) models, with 3,2,1=p were

considered. The selected model may not necessarily be the best fitted bilinear model for

the data since the choice of the highest order considered was (3,0,1,1). However, it is

shown that the selected models can be the candidate for modeling the monthly rainfall

measured at Kampung Aring weather station, Kelantan, Malaysia, (see section 7.1) and

monthly concentration of particulate matter in Kuala Lumpur air quality index (see

section 7.2).

Two types of outliers generally found in time series data were considered. They

are additive outlier (AO) and innovational outlier (IO). The characteristics of these

outliers in the BL(p,0,1,1) models, with 3,2,1=p were studied and shown to be similar

140

to the linear case. For a given ω , the effects of introducing outliers on the values of

residuals were found to be dependent on the magnitude of parameters a and b.

The statistics for measuring the outlier effects, ω , were derived using the least

squares method. Due to the complexity of the statistics, bootstrapping was used to find

the variance of the statistics. Based on the bootstrap samples, three different formulae

were used to calculate the variance, namely the standard formula, trimmed mean (TM)

and median absolute deviance (MAD). The appropriate test criteria and test statistics to

identify the occurrence of outliers were found by standardizing the observed ω giving

three different bootstrap-based procedures. These procedures were then compared to the

model-based (MB) procedure.

The outlier detection procedure for the BL(p,0,1,1) models, with 3,2,1=p was

proposed. The procedure determined the outlier types that occur at a particular time point

t by comparing the values of the test statistics for both types of outlier. Simulation studies

have shown that, in general, the procedure works well in detecting the outliers. For both

cases, the proportions of correct detection depend on large values of ω ; when the value

of ω is larger the proportion of correct detection increase. . In all cases, the performances

of test criteria of AO and IO individually are generally better than the outlier detection

procedure.

The detection procedure was applied on the rainfall data collected at Kampung

Aring, Kelantan, Malaysia. An additive outlier (AO) was detected at time point 41 for all

cases except model-based case which detected an innovational outlier (IO) but at the

same time point. Time points 41 correspond to December 1998 in this case. Meanwhile,

when the detection was applied to the Kuala Lumpur air quality data, an additive outlier

141

(AO) was detected at time point 54 for the standard and MAD cases. Otherwise, an

innovational outlier (IO) is detected for the TM and MB cases but still at the same time

point. Time points 54 correspond to 23rd January 1997.

8.2 CONTRIBUTIONS

The study was focused on the detection of outliers in the BL(p,0,1,1) models for

3,2,1=p . The contributions of the study are as follows:

a) Zaharim et al. [2006] formulated of outlier effect on observation and residual for

BL(1,1,1,1) processes. Further, statistics to measure the effect of AO, IO, TC

and LC were derived. In this study, we extended the scope of work to general

BL(p,q,r,s) processes for AO and IO types of outlier.

b) Zaharim et al. [2006] studied the performance of the outlier detection procedure

for BL(1,1,1,1) models via simulation. In this study, we performed similar

approach to study the performance of the proposed procedure for the BL(p,0,1,1)

models, with 3,2,1=p for AO and IO.

c) Battaglia and Orfei [2005] had used model-based procedure for identifying AO

and IO for general nonlinear time series models. In this study we proposed three

bootstrap-based procedures for similar proposes. We showed that the bootstrap-

based procedures performed better than the model-based procedure for

BL(p,0,1,1) models through simulation study.

d) This study has shown that bilinear model can be an alternative choice to linear

model when applied to local rainfall data and air quality index data

142

8.3 FURTHER RESEARCH

In this study, we considered two types of outliers; the additive outlier (AO) and

the innovational outlier (IO). The study can be extended using other types of outliers,

namely, the level change (LC) and the temporary change (TC). The work can also be

extended for general bilinear models.

The outlier detection procedure developed in this study is intended to detect a

single outlier in a single iteration. A natural extension of this procedure should include a

procedure for adjustment of the data to take into account the presence of more than one

outlier. The process of detecting an outlier using the proposed procedure can be iterated

until the data is free from outliers. The iterative procedure for ARMA models has been

described in Chen and Liu [1993a]. An extension to include other types of outliers based

on this approach can be further explored.

143

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APPENDIX

KAMPUNG ARING MONTHLY RAINFALL DATA

Year Month Rainfall (mm)



1995 8 273 1998 8 118 2001 8 208 9 357.5 9 309 9 266 10 243 10 296 10 321.5 11 144.5 11 227 11 339.5 12 316.5 12 859 12 792

1996 1 92.5 1999 1 287 2001 1 89 2 87 2 124.5 2 51.5 3 46 3 161.5 3 103 4 139.5 4 301 4 58.5 5 150 5 185 5 212.5 6 202.5 6 235.5 6 163 7 200 7 126 7 107.5 8 189.5 8 372.3 9 289 9 110.3 10 264.5 10 189.6 11 180.5 11 202.9 12 88 12 389

1997 1 0 2000 1 247.5 2 165 2 166.5 3 146.5 3 263.5 4 392.5 4 196.5 5 83.5 5 147.5 6 164.5 6 61.5 7 53 7 181 8 153 8 319.5 9 85 9 106.5 10 91 10 221.5 11 234.5 11 374.5 12 347 12 222

1998 1 215 2001 1 289.5 2 21 2 117 3 10 3 246.5 4 12.5 4 63.5 5 79.5 5 104 6 359 6 276.5 7 109 7 229

161

KUALA LUMPUR DAILY AIR QUALITY DATA ON PARTICULATE MATTER (PM)

Month

of Year

Day PM 3/ mug

Month of

Year

Day PM 3/ mug

Monthof

Year

Day PM 3/ mug

December of

1997

1 87

9 81 17 81 2 96 10 66 18 70 3 74 11 71 19 61 4 90 12 83 20 75 5 81 13 84 21 83 6 88 14 83 22 73 7 112 15 81 23 83 8 102 16 85 24 70 9 77 17 68 25 68 10 81 18 60 26 88 11 102 19 62 27 79 12 91 20 70 28 82 13 87 21 68 March

of 1998

1 64 14 77 22 77 2 79 15 71 23 164 3 89 16 92 24 97 4 73 17 99 25 75 5 94 18 92 26 64 6 85 19 81 27 50 7 90 20 78 28 53 8 77 21 84 29 52 9 85 22 83 30 40 10 69 23 71 31 40 11 88 24 70 February

of 1998

1 38 12 78 25 68 2 43 13 123 26 73 3 50 14 83 27 92 4 60 15 76 28 85 5 64 16 82 29 81 6 69 17 91 30 69 7 69 18 86 31 85 8 61 19 103

January of

1998

1 75 9 51 20 99 2 87 10 50 21 97 3 68 11 56 22 118 4 70 12 80 23 87 5 83 13 75 24 108 6 86 14 66 25 87 7 71 15 68 26 83 8 75 16 76 27 73

162

CONTINUE: KUALA LUMPUR DAILY AIR QUALITY DATA ON PARTICULATE MATTER (PM)

Month

of Year

Day PM 3/ mug

Month of

Year

Day PM 3/ mug

Month of

Year

Day PM 3/ mug

28 86

6 78 14 80 29 83 7 70 15 93 30 79 8 76 16 65 31 82 9 87 17 49

April of

1998

1 73 10 73 18 49 2 76 11 52 19 62 3 62 12 60 20 45 4 65 13 78 21 66 5 59 14 82 22 86 6 79 15 98 23 84 7 91 16 67 24 64 8 91 17 52 25 50 9 108 18 81 26 78 10 92 19 97 27 63 11 98 20 81 28 72 12 78 21 88 29 87 13 84 22 101 30 101 14 93 23 80 July

of 1998

1 76 15 92 24 63 2 53 16 104 25 75 3 66 17 101 26 100 4 78 18 109 27 93 5 97 19 102 28 81 6 54 20 111 29 81 7 78 21 123 30 86 8 94 22 71 31 66 9 65 23 111 June

of 1998

1 83 10 77 24 123 2 72 11 86 25 105 3 73 12 76 26 88 4 75 13 69 27 111 5 80 14 78 28 101 6 62 15 77 29 93 7 56 16 88 30 95 8 58 17 84

May of

1998

1 83 9 74 18 69 2 99 10 64 19 62 3 116 11 81 20 49 4 105 12 80 21 60 5 79 13 82 22 63

163


Month

of Year

Day PM 3/ mug

Month of

Year

Day PM 3/ mug

Month of

Year

Day PM 3/ mug

23 76 31 34 9 66 24 68 September

of 1998

1 46 10 83 25 60 2 57 11 66 26 40 3 70 12 57 27 63 4 68 13 59 28 64 5 65 14 39 29 56 6 66 15 34 30 84 7 61 16 52 31 78 8 62 17 40

August of

1998

1 62 9 67 18 32 2 63 10 63 19 34 3 76 11 72 20 45 4 63 12 73 21 36 5 61 13 52 22 30 6 86 14 55 23 43 7 59 15 71 24 54 8 50 16 57 25 52 9 42 17 62 26 43 10 63 18 69 27 54 11 66 19 64 28 52 12 76 20 49 29 46 13 66 21 64 30 59 14 63 22 68 31 57 15 63 23 61 November

of 1998

1 46 16 42 24 74 2 41 17 49 25 56 3 46 18 59 26 55 4 44 19 45 27 59 5 55 20 59 28 68 6 46 21 54 29 70 7 37 22 53 30 62 8 50 23 63 October

of 1998

1 62 9 51 24 50 2 92 10 54 25 60 3 75 11 54 26 82 4 71 12 41 27 74 5 41 13 43 28 64 6 36 14 49 29 53 7 62 15 42 30 51 8 44 16 30

164


Month

of Year

Day PM 3/ mug

Month of

Year

Day PM 3/ mug

17 42 26 56 18 58 27 60 19 71 28 60 20 53 29 47 21 41 30 54 22 45 31 51 23 46 24 41 25 34 26 39 27 57 28 73 29 74 30 66

December of

1998

1 44 2 47 3 45 4 78 5 50 6 63 7 62 8 68 9 74 10 59 11 50 12 77 13 65 14 71 15 89 16 79 17 70 18 38 19 55 20 39 21 47 22 54 23 59 24 60 25 50

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