Adaptive Estimation and Prediction of Univariate Vehicular Traffic Condition Series

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ABSTRACT

GUO, JIANHUA. Adaptive Estimation and Prediction of Univariate Vehicular Traffic Condition Series. (Under the direction of Dr. Billy M. Williams)

Aimed at providing the anticipatory ability for the proactive traffic control systems, a

new adaptive online short-term univariate traffic condition forecasting method is presented in

this dissertation by assimilating knowledge from previous research. Using 15-minute traffic

flow series as a representative, this methodology is based on the hypothesis that the first two

conditional moments of univariate traffic flow series can be modeled as a

SARIMA+GARCH structure, based on which an online forecasting system can be developed

using seasonal exponential smoothing and Kalman filter. Supplementary components,

including missing value imputation and outlier detection, can be incorporated into the system

to meet the requirements of real traffic data collection situations.

The development of the system follows two steps. In step (1), the SARIMA model is

separated into a seasonal component handling traffic flow level due to historical traffic

information, and a short-term component handling local variation after the traffic level due to

historical information is subtracted. The seasonal component is processed using seasonal

exponential smoothing by recognizing the theoretical equivalence between ARIMA model

and exponential smoothing; the local variation is processed using Kalman filter by

constructing a state space model. Afterwards, GARCH model is processed using Kalman

filter based on the recognition that GARCH has an equivalent representation as ARMA in the

sense of squared series. In step (2), missing values are replaced with one-step-ahead

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Outliers, indicating extraordinary patterns, are detected based on intervention analysis and

likelihood ratio test. Outliers are proved to be assimilated into the system through an

investigation showing outliers do have significant influences on the forecasting system.

Additionally, the square root transformation is applied in the system.

Using real traffic operation data from four regions, the research hypothesis is validated

and the proposed methodology is implemented and proved to bear desirable attributes,

including competencies, adaptability, computational efficiency, robustness, and

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ADAPTIVE ESTIMATION AND PREDICTION OF UNIVARIATE VEHICULAR TRAFFIC CONDITION SERIES

by

JIANHUA GUO

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

CIVIL ENGINEERING

Raleigh

2005

APPROVED BY:

(Dr. Nagui M. Rouphail)

(Dr. Joseph E. Hummer)

(Dr. Peter Bloomfield)

Chair of Advisory Committee

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BIOGRAPHY

Jianhua Guo was born in eastern China on May 5, 1976. He started his career in civil

engineering from 1993 when he was admitted into Southeast University in China. He

received a B.S. and a M.S. in 1997 and 1999, respectively. After receiving a Ph.D. at North

Carolina State University, he would like to pursue a research career in transportation

engineering. His research interests mainly focused on applications under the broad umbrella

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ACKNOWLEDGEMENTS

I would like to express my sincerest gratitude for having the opportunity to complete this

dissertation. In the trials and tribulations of this process, it would never be possible without

the support, guidance, encouragement, and confidence in me from my advisor Dr. Billy M.

Williams. I aspire someday to reach a modicum of his personal and professional stature.

I want to thank Dr. Peter Bloomfield, Dr. Nagui M. Rouphail, and Dr. Joseph E. Hummer

for severing on my advisory committee. I am indebted to Dr. Bloomfield for the suggestions

and discussions on the statistical analyses in this dissertation. I admire his keen insight and

broad knowledge in statistics. I am grateful to Dr. Rouphail for his help in my course work

and the constructive suggestions on this dissertation that improve its quality greatly. My

special thank goes to Dr. Hummer who stands on my committee for the entire community of

traffic engineers. His challenges and comments benefit this dissertation and will benefit my

future research career as well.

My thanks also go to Shashank Shekhar for discussing and sharing ideas on traffic

forecasting; to Angshuman Guin, through whose work I benefit a lot; to Ting Yi and Jisun

Lee for working together.

I thank the generous sharing of the archived traffic operations data to the third party

researchers as the author by the Minnesota Department of Transportation through the

University of Minnesota at Duluth, the Washington State Department of Transportation

through the Intelligent Transportation Systems Research Program at the University of

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through the Center of Advanced Transportation Technology Laboratory at the University of

Maryland. I also thank the United Kingdoms Highways Agency for providing the traffic data

used in this research. The author takes all the responsibility on the analyses and views

presented in this dissertation.

Finally, I would like to take this opportunity to express my thanks to my family for

always being with me. Their unconditional and endless love and insurmountable faith in me

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LIST OF TABLES

Table 3.1: Overview of Short-Term Traffic Forecasting Methods 63

Table 4.1: Transformation Descriptions 72

Table 4.2: Changing Variance Testing Strategies 77

Table 4.3: Prediction Confidence Interval Summary Statistics 91

Table 4.4: Groups for Prediction Confidence Interval Evaluation by Traffic Level 92

Table 4.5: Groups for Prediction Confidence Interval Evaluation by Time of Day 93

Table 5.1: Station Description for 1996 UK Data 96

Table 5.2: Station Description for 2002 UK Data 96

Table 5.3: Station Description for WS Data 99

Table 5.4: Station Description for 2004 MD Data 99

Table 5.5: Station Description for 2000 and 2004 MN Data 102

Table 5.6: Threshold Test for Microscopic Data Validation 103

Table 5.7: Traffic Condition Based on Occupancy Group 106

Table 5.8: Overview of Selected Traffic Flow Data 107

Table 5.9: Traffic Condition Illustration Using Traffic Flow Levels 108

Table 5.10: Traffic Condition Illustration Using Occupancy Data 109

Table 6.1: Transformations for Stations 118

Table 6.2: Results of Heteroscedasticity Testing by Month 121

Table 6.3: Results of Heteroscedasticity Testing by Week 122

Table 6.4: Results of Heteroscedasticity Testing by Day 123

Table 6.5: Repeated t-test for Batch Mode Forecasting Performance 126

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Table 6. 7: Results of Prediction CI Width to Flow Ratio by Traffic Level Comparison 129

Table 6.8: Results of Prediction CI Kickoff Percentage by Time of Day Comparison 131

Table 6.9: Results of Prediction CI Width to Flow Ratio by Time of Day Comparison 132

Table 6.10: Transformations for Best Forecasting Performance Measures 134

Table 6.11: Data Selected for Results Illustration 135

Table 6.12: Parameters and Initial Conditions for Basic System 143

Table 6.13: Repeated t-test for Basic System Forecasting Performance 145

Table 6.14: Comparison Results of Basic System Prediction Confidence Interval Kickoff

Percentage by Traffic Level 148

Table 6.15: Comparison for Basic System Prediction Confidence Interval Width to Flow Ratio

by Traffic Level 148

Table 6.16: Comparison Results of Basic System Prediction Confidence Interval Kickoff

Percentage by Time of Day 150

Table 6.17: Comparison Results of Basic System Prediction Confidence Interval Width to

Flow Ratio by Time of Day 151

Table 6.18: Transformations for Best Forecasting Performance Measures of Basic System 153

Table 6.19: Forecasting Performance Comparison Results 158

Table 6.20: Comparison Results of Prediction CI Kickoff Percentage by Traffic Level 160

Table 6.21: Comparison Results of Prediction CI Width to Flow Ratio by Traffic Level 160

Table 6.22: Comparison Results of Prediction CI Kickoff Percentage by Time of Day 161

Table 6.23: Comparison Results of Prediction CI Width to Flow Ratio by Time of Day 162

Table 6.24: Outliers by Hour of Day for Choice A 165

Table 6.25: Level of Outlier Clustering for Choice A 166

Table 6.26: Outliers by Hour of Day for Choice B 167

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Table 6.28: Correspondence between System Structure Changes and Outliers 173

Table 6.29: Repeated t-test for System Robustness 175

Table 6.30: Transferability Investigation of Full System 177

Table A.1: Batch Mode SARIMA Estimation Results 202

Table A.2: GARCH Identification 203

Table B.1: Batch Mode MAE 204

Table B.2: Batch Mode MAPE 205

Table B.3: Batch Mode RMSE 206

Table B.4: Batch Mode RMSEP 207

Table B.5: Batch Mode LPI 208

Table B.6: Batch Mode Prediction CI Kickoff Percentage by Traffic Level 209

Table B.7: Batch Mode Prediction CI Width to Flow Ratio by Traffic Level 212

Table B.8: Batch Mode Prediction CI Kickoff Percentage by Time of Day 215

Table B.9: Batch Mode Prediction CI Width to Flow Ratio by Time of Day 221

Table C.1: Basic System MAE 227

Table C.2: Basic System MAPE 228

Table C.3: Basic System RMSE 229

Table C.4: Basic System RMSEP 230

Table C.5: Basic System LPI 231

Table C.6: Basic System Prediction CI Kickoff Percentage by Traffic Level 232

Table C.7: Basic System Prediction CI Width to Flow Ratio by Traffic Level 235

Table C.8: Basic System Prediction CI Kickoff Percentage by Time of Day 238

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Table D.1: Full System Forecasting MOE (SQR Transformation) 250

Table D.2: Full System Prediction CI Kickoff Percentage by Traffic Level (SQR

Transformation) 251

Table D.3: Full System Prediction CI Kickoff Percentage by Time of Day (SQR

Transformation) 252

Table E.1: Forecasting Performance of Seasonal Exponential Smoothing with Random Walk

(Smoothing parameter = 0.15) 253

Table E.2: Forecasting Performance of AKF (SQR Transformation) 254

Table E.3: AKF Prediction CI Kickoff Percentage by Traffic Level 255

Table E.4: AKF Prediction CI Width to Flow Ratio by Traffic Level 256

Table E.5: AKF Prediction CI Kickoff Percentage by Time of Day (SQR Transformation) 257

Table E.6: AKF Prediction CI Width to Flow Ratio by Time of Day (SQR Transformation) 258

Table F.1: Batch Mode Forecasting Performances in the Presence of Missing Values and

Outliers 259

Table F.2: Basic System Forecasting Performances (SQR Transformation) in the Presence of

Missing Values and Outliers 260

Table G.1: Forgetting Factor Index 261

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LIST OF FIGURES

Figure 2.1: Typical Layout of an Instrumented Freeway Section 8

Figure 2.2: Forecasting Problem Illustration in the Time-Space Diagram 8

Figure 2.3: Traffic State Propagation at 15-minute Interval 13

Figure 4.1: Online Traffic Forecasting System Structure 69

Figure 5.1: Overview of the Motorway Network in the UK 95

Figure 5.2: Detector Configuration for Washington State, US 98

Figure 5.3: Detector Map for Maryland Data 100

Figure 5.4: Detector Map for Minnesota Data 101

Figure 6.1: Results Illustration of lsh (up) and sqr (down) for UK 4762a, Oct 02, 1996 137

Figure 6.2: Results Illustration of lsh (up) and sqr (down) for UK 6951a, Feb 1, 2002 138

Figure 6.3: Results Illustration of lsh (up) and sqr (down) for MD 9b, Jan 16, 2004 139

Figure 6.4: Results Illustration of lsh (up) and sqr (down) for MN 882, Jul 29, 2000 140

Figure 6.5: Results Illustration of lsh (up) and sqr (down) for MN 882, Mar 30, 2004 141

Figure 6.6: Results Illustration of lsh (up) and sqr (down) for WS ES_130D_MS_Stn, Sep 9,

2004 142

Figure 6.7: Outlier Detection Illustration for Choice A (up) and Choice B (down) 171

Figure 6.8: Full System Illustration for UK 4762a, Oct 02, 1996 181

Figure 6.9: Full System Illustration for UK 6951a, Feb 1, 2002 181

Figure 6.10: Full System Illustration for MD 9b, Jan 16, 2004 182

Figure 6.11: Full System Illustration for MN 882, Jul 29, 2000 182

Figure 6.12: Full System Illustration for MN 882, Mar 30, 2004 183

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Figure A.1: Sample ACF for UK 4762b, 1996 199

Figure A.2: Sample ACF for UK 4680b, 2002 199

Figure A.3: Sample ACF for MD 2a, 2004 200

Figure A.4: Sample ACF for MN 60, 2000 200

Figure A.5: Sample ACF for MN 737, 2004 201

Figure A.6: Sample ACF for WS ES_130D_MN_Stn, 2004 201

Figure G.1: Sensitivity of MAE to Forgetting Factor for Short-Term Kalman Filter 262

Figure G.2: Sensitivity of MAPE to Forgetting Factor for Short-Term Kalman Filter 262

Figure G.3: Sensitivity of RMSE to Forgetting Factor for Short-Term Kalman Filter 263

Figure G.4: Sensitivity of RMSEP to Forgetting Factor for Short-Term Kalman Filter 263

Figure G.5: Sensitivity of LPI to Forgetting Factor for Short-Term Kalman Filter 264

Figure G.6: Sensitivity of Prediction Confidence Interval Kickoff Percentage for All Traffic

Level to Forgetting Factor for GARCH filter 264

Figure G.7: Sensitivity of MAE to Seasonal Exponential Smoothing Parameter 265

Figure G.8: Sensitivity of MAPE to Seasonal Exponential Smoothing Parameter 265

Figure G.9: Sensitivity of RMSE to Seasonal Exponential Smoothing Parameter 266

Figure G.10: Sensitivity of RMSEP to Seasonal Exponential Smoothing Parameter 266

Figure G.11: Sensitivity of Prediction Confidence Interval Kickoff Percentage for All Traffic

Levels to Seasonal Exponential Smoothing Parameter 267

Figure H.1: Results of UK 4762a, Oct 02, 1996 (lambda = 1) 268

Figure H.2: Prediction CI from Different Lambda for UK 4762a, Oct 02, 1996 268

Figure H.3: Results of UK 6951a, Feb 01, 2002 (lambda = 1) 269

Figure H.4: Prediction CI from Different Lambda for UK 6951a, Feb 01, 2002 269

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Figure I.2: Sensitivity of MAPE to AKF Algorithm Memory 270

Figure I.3: Sensitivity of RMSE to AKF Algorithm Memory 271

Figure I.4: Sensitivity of RMSEP to AKF Algorithm Memory 271

Figure I.5: Sensitivity of LPI to AKF Algorithm Memory 272

Figure I.6: Sensitivity of Prediction Confidence Interval Kickoff Percentage for All Traffic

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LIST OF ABBREVIATIONS

ACF AutoCorrelation Function

AIC Akaike Information Criterion

AKF Adaptive Kalman Filter

AO Additive Outlier

AR AutoRegressive

ARCH AutoRegressive Conditional Heteroscedasticity

ARIMA AutoRegressive Integrated Moving Average

ARMA AutoRegressive Moving Average

ATIS Advanced Traveler Information System

ATMS Advanced Traffic Management System

CI Confidence Interval

CLS Conditional Least Square

CT Cell Transmission

DA Data Augmentation

DOT Department Of Transportation

DSPM Discrete State Propagation Model

EM Expectation Maximization

ERLS Extended Recursive Least Square

FHWA Federal HighWay Administration

FWRBFNN Fuzzy Wavelet Radial Basis Function Neural Network

GARCH Generalized AutoRegressive Conditional Heteroscedasticity

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HCM Highway Capacity Manual

HOV High Occupancy Vehicle

HW Holt-Winter

IO Innovational Outlier

IMA Integrated Moving Average

ITS Intelligent Transportation Systems

K-NN K-Nearest Neighbor

KARIMA Kohonen maps with AutoRegressive Integrated Moving Average

KF Kalman Filter

LCT Lagged Cell Transmission

LMS Least Mean Square

LPI Leeds Performance Index

LS Level Shift

LSD Least Significance Difference

LWR Lighthill – Whitham – Richards

MAE Mean Absolute Error

MAPE Mean Absolute Percentage Error

MIDAS Motorway Incident Detection and Automatic Signaling

ML Maximum Likelihood

MLE Maximum Likelihood Estimation

PACF Partial AutoCorrelation Function

RLS Recursive Least Square

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RMSEP Root Mean Square Error Proportional

SACF Sample AutoCorrelation Function

SARIMA Seasonal Autoregressive Integrated Moving Average

SBC Schwarz Bayesian Criterion

SCAT Sydney Coordinated Adaptive Traffic system

SCOOT Split Cycle Offset Optimization Technique system

STARIMA Space-Time AutoRegressive Integrated Moving Average

TC Temporary Change

TDAD Traffic Data And Distribution

TMC Traffic Management Center

ULS Unconditional Least Square

UTCS Urban Traffic Control System

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CHAPTER 1

BACKGROUND, MOTIVATION, AND ORGANIZATION

1.1

B

ACKGROUND

Congestion and its side effects such as environmental pollution, safety degradation,

mobility deterioration, etc. have been great concerns for both the public and the

transportation engineers in developed countries. Congestion, defined as traffic demand

exceeding traffic capacity, occurs both in urban arterials and freeway systems. Of particular

interest are the freeway systems within metropolitan areas where congestion often occurs in

peak hours when the traffic demand is more compact and intense. A well-known

phenomenon of congestion is traffic break down, or stop-and-go traffic, which, ironically,

reduces the capacity when it is most urgently needed and causes a large amount of delay.

Due to the limitation in constructing new road in already crowded metropolitan areas,

freeway management has been an important alternative to improve freeway operations.

According to the Freeway Management and Operations Handbook (FHWA 2003), freeway

management is defined as “the implementation of policies, strategies and technologies to

improve freeway performance” with its over-riding objectives to “minimize congestion

(and its side effects), improve safety, enhance overall mobility, and provide support to other

agencies during emergencies”. Major freeway management techniques identified in the

handbook include lane management, ramp management, high occupancy vehicle (HOV)

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ramp metering has been regarded as “the most direct and efficient way to control and

upgrade freeway traffic” (Parageorgiou et al. 2003).

Proactive traffic control systems call for an ability to anticipate the traffic condition in

the near future, yielding so called short-term traffic forecasting problem. Freeway

management systems can be proactive or reactive according to the traffic condition based

upon which the control strategies are developed. If the real time data indicating current

traffic condition are utilized, the resultant control system is at most reactive;

correspondingly, if forecasts indicating the near future traffic condition are utilized, the

resultant control system is proactive. Obviously, proactive traffic control actions have a

better chance of success if the future traffic conditions are accurately predicted. Thus, to

make the most from the freeway management systems, predictive ability has been

identified as one of the major challenges for advanced traffic management system (ATMS)

and advanced traveler information system (ATIS), as is also pointed out in the Ten Year ITS

program plan (ITS America 2002). Specifically, for ramp metering control, current

strategies in practice are at most reactive, using either historical traffic information

(pre-timed control) or real-time traffic measurements (local or coordinated

traffic-responsive control) (Chu et al. 2004). From a proactive point of view, a

comprehensive optimal control concept was discussed in (Parageorgiou and Kotsalos, 2002)

and (Parageorgiou et al. 2003), requiring an explicit consideration of “demand predictions

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algorithm has also been recognized in (Jacobson et al. 1989).

Recent advancement in the analysis of freeway traffic operation data also provides a

firm support for pursuing accurate prediction capability as a means to improve the

effectiveness of freeway control systems. A two regime, discontinuous model of freeway

traffic flow as is argued in (Payne 1984) has been implicitly recognized in freeway control

systems, especially for ramp metering algorithms. In this two regime model, freeway traffic

condition is described as either being in a non-congestion state, a congestion state, or a

connecting transient state. When congestion occurs, the traffic state will switch from

non-congestion to congestion through a short transient state. Should the transient state be

detected or anticipated, control measures can be applied and be expected to prevent or

delay the occurrence of the congestion. The same argument is also presented in the

Freeway Management and Operations Handbook (FHWA 2003).

Two aspects are inherently implied in a forecasting problem, i.e., the accuracy of the

forecast and its credibility, indicated by the prediction confidence interval. A plausible

forecasting system is expected to provide the answers to both of the aspects. The accuracy

of the forecast enables the control system relying on that to work appropriately while the

credibility will shed light on the uncertainty of the adopted control strategy. Actually in

current customer-oriented traffic system, the knowledge of the uncertainty is an essential

attribute of mobility. Unfortunately, so far the research on forecast confidence interval is

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Various control methods in a freeway control system call for a corresponding traffic

forecasting framework. Intelligent Transportation Systems (ITS), the application of

advanced technology in transportation and an important enabler for freeway management

systems, helps to maintain an information supply chain in a freeway management system.

In this information supply chain, different end users have different requirements for the

forecasting algorithm. For example, commuters desire a forecast of travel time, rather than

a forecast of flow rate; however, the forecast of travel time might not be readily applicable

to a proactive ramp metering algorithm. In addition to commuters, other end users might be

traffic engineers, freeway control algorithms, or other decision making procedures. Thus, in

order to address the user specific requirements, clarification of the short-term traffic

forecasting attributes will be the first step of developing a forecasting strategy.

1.2

M

OTIVATION

This research aimed to develop a robust online forecasting strategy based on previous

research in order to provide the anticipatory ability required by a proactive freeway

management and control system. This strategy can forecast traffic condition in the near

future and generate its confidence interval based on real time freeway operation data.

This strategy can be implemented at traffic management centers (TMC) to facilitate the

operation of the freeway systems in the US or other countries, such as the motorway

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freeway control strategies, such as ramp metering algorithms, to provide traffic condition

forecasts for the purpose of a better control. Compared to previous work, this effort

explicitly separates the forecasting strategy development into two major steps: the batch

mode investigation of traffic dynamics and the online implementation of the identified

batch mode traffic dynamics model.

1.3

T

HESIS

O

RGANIZATION

This research intends to develop an online traffic condition forecasting system. In all,

seven chapters including this introductory chapter are presented and the organization of the

dissertation is summarized below.

Chapter 2 discusses the research problem, including the general short-term traffic

forecasting problem definition, attributes of “ideal” forecasting methods, and the

specifications of the proposed forecasting system.

Chapter 3 presents a literature review of previous research on short-term traffic

forecasting, including the classification of the forecasting methods and the discussion of the

methodological highlights of each method. Afterwards, robust forecasting is reviewed. A

comparison table summarizes the review.

Chapter 4 describes the methodology for developing the online forecasting system,

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Chapter 5 provides a description of the real traffic condition data utilized in this work.

Chapter 6 performs the development and testing of the online forecasting system,

including experimental design, traffic dynamics investigation, online system

implementation and testing.

Chapter 7 summarizes the research findings and presents the recommendations for

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CHAPTER 2

PROBLEM STATEMENT

This chapter presents the problem to be addressed in this study. Firstly, the general

short-term traffic forecasting problem and the associated attributes are defined and

discussed. Secondly, the characteristics of “ideal” traffic condition forecasting algorithms

are discussed for the purpose of algorithm evaluation. Finally, the specifications of the

forecasting algorithm proposed in this work are articulated in compliance with the previous

discussions on the general forecasting algorithm characteristics.

2.1

G

ENERAL

S

HORT

-T

ERM

T

RAFFIC

F

ORECASTING

P

ROBLEM

D

EFINITION

Inductive loop detectors are the most widely applied traffic monitoring devices used in

freeway management systems. A typical three lane freeway section instrumented with

inductive loop detectors is demonstrated in Figure 2.1. In this layout, detectors are

distributed spatially across each lane and the three adjacent detectors at a specific location

form a detection station, e.g., station i. For each time interval indexed by time stamp t,

each loop detector produces a set of traffic data indicating traffic conditions for this time

interval, typically including flow rate, occupancy, and speed if a speed trap with two

closely-spaced detectors is used. Data from three detectors form the station data,

corresponding to the aggregation of a rectangle to the left of line AB in Figure 2.2. An

important point worth mentioning is that the size of these rectangles is closely related to the

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historical traffic condition database

{

Ω

( )

t , t =1,2,3,...

}

. Clearly, the historical traffic

condition database is spatial and temporal in nature.

Inductive Loop Detector Traffic Direction

i-1 i i+1

l_i-1 l_i

Figure 2.1: Typical Layout of an Instrumented Freeway Section

i i+1

t-1 t t+1

Distance

Time i-1

t-2

A

B Vehicle

Trajectories

t-3

?

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The general traffic condition prediction can be stated as: Given a historical traffic

condition database up to current time interval indexed by time stamp t , i.e.,

( )

{

Ωt , t =1,2,3,...,t

}

, find the traffic condition set

{

Ω

( )

t ,t =t+1,t+2,...,t+k

}

, where k is

the prediction horizon with k =1 for the one-step-ahead prediction. In Figure 2.2, the historical traffic condition database is represented as the aggregation of traffic condition

within each rectangle to the left of the line AB and the forecasts at different locations are

represented as the aggregation of traffic condition within the rectangles with question

marks that are to the right of the line AB.

In the short-term traffic forecasting problem, although by-lane traffic condition data

are available and by-lane forecasting is feasible, all the previous work (to the best

knowledge of the author) and this work included intend to forecast traffic condition at a

station. Note that site is also used in the literature indicating a station as defined previously.

In this dissertation, “site” and “station” are used interchangeably hereafter. One point need

to point out is that in order for a fair comparison between locations with different number

of lanes, the traffic operation data for each station is normalized by the number of lanes. In

the following analyses, unless specifically pointed out, the traffic data utilized are after the

normalization.

There are many specifications which are necessary to characterize a traffic forecasting

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(1) Prediction Variable

According to the number of prediction variables employed, a forecasting algorithm can

be univariate or multivariate. Univariate processing can also be termed as point forecasting,

which make use of only one traffic variable at a single station. For example, traffic flow

rate is typically used for ramp metering. Although less information is utilized, univariate

processing is still promising in that relative simple forecasting methods with satisfying

accuracy can be formulated, especially for forecasting at longer time intervals where traffic

state propagation can be captured by using traffic data from a single station. Compared

with univariate processing, more information, for example, traffic data from surrounding

sites and/or additional traffic variables, is applied in multivariate forecasting methods. A

recent multivariate application was developed in Guin (2004), in which 20-second traffic

flow rates are forecasted using the upstream and downstream station data for the purpose of

automatic incident detection. The selection of the prediction variable is closely related to

the forecasting horizon, which is discussed in the subsequent section. For forecasting at a

higher time scale, the calibration of the relationship between these traffic variables is a

complicated task, although multivariate processing seems appealing. Other concerns such

as time varying cross-correlation are also to be considered for multivariate processing.

(2) Time Scale

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algorithm can be microscopic or macroscopic. The typical time scales for microscopic

processing are 20-second, 30-second, or 1-minute. Microscopic processing is necessary for

time-critical applications such as incident detection. The most widely used time scale for

macroscopic processing is 15-minute conforming to the convention of Highway Capacity

Manual (HCM). Other intervals such as 5-minute, 10-minute are also presented in the

literature. Macroscopic processing relies on the aggregated traffic condition data, and

compared with microscopic processing, macroscopic processing can provide the traffic

condition prediction into a further future.

The selection of traffic forecasting variable and its aggregation time scale has a close

relationship with traffic state propagation. Figure 2.3 illustrates a hypothetical freeway

section with adjacent detection stations i and i+1 at a spacing of 0.5 mile. Assume the traffic on this section has an almost stationary state with a space mean speed 60 mile per

hour. So within 15 minutes, a traffic state can propagate to 60 4=15 miles downstream

further. For station i, the traffic condition from t to t+15minutescan be represented by the aggregation of vehicles crossing the zone ABEF. Similarly, for station i+1, the traffic condition from t to t+15mincan be represented by the aggregation of vehicles crossing the zone CDGH. Using assumed traffic stream speed and detection station spacing, the

overlay between these two zones, i.e., zone CDEF, has approximately a length of

5 . 14 5 . 0

15− = miles; in other words, around

(

14.515.5

)

×100%=93.5%of the 15-minute

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15-minute univariate flow rate at a single station as the forecasting variable in this work.

For this nearly stationary traffic, these condition zones for time interval from t to

minutes 15

+

t will propagate to zones for time interval from t+15minutes to

minutes 30

+

t , i.e., ABEF to FIKM, CDGH to HJLN, and overlap CDEF to HJKM. For real traffic, the propagation of traffic condition is governed by the traffic dynamics.

Therefore, the problem of short-term traffic forecasting is to propagate current traffic

condition to forecasting horizon according to pre-identified traffic dynamics. Intuitively,

this pre-identified traffic dynamics model will in a sense dominate the performance of the

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i i+1

t t+15min

t+30min Distance

Time 14.5 mile

0.5 mile

15 mile Vehicle

Trajectories

A B

C D

E

F G

H

I J K L

M N

Figure 2.3: Traffic State Propagation at 15-minute Interval

It can be deduced that as the aggregation interval decreases, the traffic overlap between

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station data into the forecasting strategy in order to improve the forecast. This is often the

case for microscopic level processing. The most recent development to this end is the

discrete state propagation model developed in Guin (2004) based on a revision of Lagged

Cell Transmission Model (Daganzo 1999).

(3) Algorithm Memory

A forecasting algorithm can be implemented as a batch mode process or a recursive

process according to the size of algorithm memory. In batch mode processing, a finite set of

historical traffic data is retained in the forecasting system. With the incoming traffic stream,

the historical database is updated and the forecast is produced based on the entire historical

database, causing a heavy computational burden. To ensure the forecasting quality, the

historical database should be routinely maintained. Compared with batch mode processing,

recursive processing has a smaller algorithm memory and the forecast is produced based on

the predefined recursion equation when new traffic data are available. Due to the reduced

algorithm memory, recursive processing has a reduced amount of computation and thus a

higher computational performance. Batch mode processing would seem to promise more

accurate forecasts; however, it is not suitable for online traffic applications where traffic

data are collected in stream. Therefore, recursive processing has a greater potential in time

critical applications as is often the case in Intelligent Transportation Systems. For example,

the timely detection of an incident will greatly reduce its negative effects on traffic

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(4) Adaptability

A short-term traffic forecasting method can be either adaptive or non-adaptive

depending on the way the algorithm responds to the incoming traffic data. Traffic is

dynamic in nature; thus, the traffic condition data stream arising from the traffic is dynamic.

In adaptive processing, the incoming traffic data is first used to update the forecasting

system; then the forecast is produced based on the updated forecasting system. Adaptive

processing is promising in that it provides a mechanism for learning the evolution of the

traffic process. Compared with adaptive processing, non-adaptive processing cannot adapt

to the incoming of traffic stream and has a high chance of deviating from the real traffic

condition state over time (Ahmed 1989). For example, most heuristic forecasting methods

are non-adaptive forecasting methods. Under these circumstances, a restart of the

forecasting algorithm might be needed to reflect the current traffic conditions. The

adaptability is further discussed in the succeeding section.

2.2

I

DEAL

S

HORT

-T

ERM

T

RAFFIC

F

ORECASTING

M

ETHODS

Although different forecasting algorithms within the forecasting framework have

different evaluation criteria, theoretically, “ideal” traffic forecasting algorithms share some

common characteristics which are discussed below.

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Competencies for a short-term traffic forecasting algorithm include at least two aspects:

the generation of satisfactory forecasts to meet the specific requirements and the provision

of the credibility of the forecast. Actually these two aspects constitute the basic essential

requirements for any forecasting structure. Failure in providing of either one will degrade

the usefulness of the forecasting algorithm.

(2) Implementation Efficiency or Transferability

Implementation efficiency means that a short-term traffic forecasting system should be

easily implemented across various locations, or a forecasting algorithm would be “plug and

play” in nature. This characteristic describes the transferability of a forecasting method. In

this regard, a forecasting algorithm should first be easily implemented into existing traffic

condition surveillance and control systems without requiring much revision of the current

system. Secondly, the new component should work without massive manual calibration to

capture the location specific attributes through, for example, a test run within a short period

of time. To summarize, the involvement of the location specific parameters and the way

these parameters are calibrated or identified are two of the most important factors affecting

the forecasting system transferability.

(3) Computation Efficiency

In a real freeway operation situation, traffic condition data will be continuously

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control. Under this situation, a forecasting algorithm, as a preprocessor before succeeding

control strategies, is expected to generate the forecast with the least possible computational

burden in order to guarantee a fast response. Therefore, the forecasting algorithm in a

recursive form with limited algorithm memory is suitable to produce the forecast only

using the current traffic data with all the previous information encapsulated in the recursion

equation.

(4) Adaptability

A forecasting algorithm is expected to be self-tuning to the incoming data stream.

Obviously, traffic is far from stationary. For example, traffic condition data may have

pronounced patterns such as a weekly pattern and/or a daily pattern, which perturb the level

of traffic during different times of the day and different days of the week. Additionally,

traffic also suffers from exogenous disturbances, such as weather changes, holidays, special

events, etc., all of which call for a learning scheme in the forecasting algorithm to adapt to

these temporary changes of traffic condition.

The adaptability of the forecasting system can embrace two different strategies, i.e., (1)

updating the system structure indicated by the system parameters and/or the historical

databases, and (2) switching the forecast generation equation based on changes in traffic

dynamics. The first strategy is clear in that the forecasting system state indicated by system

(39)

when the forecasting system has a set of forecast equations. A typical example is the

discrete state propagation model in Guin (2004) where a set of forecasting equations were

established for different traffic regimes and then selected according to traffic dynamics

changes.

(5) Robustness

Robustness is an important factor in enabling normal operation of a forecasting system.

Due to the somewhat unavoidable errors in the traffic condition data collection system,

suspect data and erroneous data are common in traffic condition databases. A robust

forecasting strategy should only respond to the “good” data with “bad” data handled or

ignored. Missing values and outliers are two of the most widely recognized difficulties in

developing a forecasting strategy. Missing values have been regarded as a common

attribute of ITS traffic operations data due to the continuous nature of the traffic monitoring

equipment (Turner 2001). Missing value imputation has been studied extensively and

heuristic approaches and statistical approaches have been developed. Outliers, observations

that are significantly different from other observations, are generated by an unknown

process which is different from the process under study. A most common source of outliers

might be traffic incidents that perturb the traffic dynamics temporarily. So far, there is

limited research on outlier detection on traffic condition data forecasting except for

Williams (1999). Actions should be taken for a robust forecasting strategy to detect and

(40)

Adaptability and robustness are closely related in that both attributes require the

forecasting system to make a response when something unusual happens. Obviously, the

characters of the unusual events, or exogenous disturbances, are important for a proper

differentiation of adaptability and robustness. One important attribute of relevance is

whether or not the unusual event changes the travel pattern. For example, a special event

might temporarily change the travel pattern for days. In this case, a forecasting algorithm

might need to capture the pattern change and make forecasts that are consistent with the

changed pattern. The forecasting algorithm will adapt back to normal again when the event

is over. However, for a traffic incident, supposing this incident information is not available

to the public, the travel pattern will remain the same and traffic will return to normal when

the incident is cleared. In this case, the forecasting algorithm might need to detect the

disturbance and ignore it. One rough condition for testing the travel pattern change might

be the duration of the effect of the exogenous events. There is also an urgent need for a

forecasting framework, within which the proper responses of a forecasting algorithm for

various traffic control and management scenarios need be clearly defined. In other words,

the forecasting system must be tailored to its real application purposes.

2.3

S

PECIFICATIONS OF

P

ROPOSED

A

LGORITHM IN

T

HIS

S

TUDY

The traffic condition forecasting strategy to be developed in this research has the

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(1) Univariate Traffic Flow Rate at 15-minute Level

This forecasting strategy uses 15-minute traffic flow rate at a single station as the

forecasting variable. The forecasted value will be the 15-minute traffic flow rate at the

same station for the next 15-minute. This choice is supported by the freeway traffic state

propagation demonstration illustrated in Figure 2.3.

It is important to emphasize that the 15-minute interval selected for this study is just a

representative of a spectrum of aggregation intervals. As discussed previously, the decrease

in the time scale calls for the incorporation of temporal and spatial information, yielding a

cutoff time interval, above which there is a spectrum of intervals that the proposed

univariate forecasting method can be applied to. Therefore, the proposed methodology is

expected to work for similar aggregation intervals, say 20-minute and/or 10-minute, within

this spectrum, although no detailed investigation will be performed in this study to

determine the cutoff interval. In addition, this strategy can also be extended to model other

traffic condition variables such as occupancy and/or speed.

(2) Sound Traffic Dynamics Investigation

The development of the forecasting strategy will follow two stages. Firstly, the

dynamics of traffic condition series (15-minute flow rate series in this study) will be

investigated in batch mode to capture the inner regularity embedded in the data. The first

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(conditional variance) of the series will be modeled. Secondly, the batch mode modeling

results will be converted into an online version using the techniques of seasonal

exponential smoothing and Kalman filter theory.

(3) Joint Production of Forecasts and Forecast Confidence Intervals

The forecasts and the associated forecast confidence intervals are jointly produced in

this online forecasting system. The forecasts are produced using the first order conditional

moment model and the forecast confidence intervals are produced using the second order

conditional moment model. The first order conditional moment is modeled as the Seasonal

Integrated Autoregressive Moving Average (SARIMA) process, which is implemented

using the technique of seasonal exponential smoothing and Kalman filter. The modeling of

the first conditional moment has been conducted in the literature and Williams (1999) is the

most up-to-date work where a sound theoretical justification of the SARIMA model was

firstly recognized based on the Wold Decomposition Theorem. The second order

conditional moment is modeled as the Generalized Autoregressive Conditional

Heteroscedasticity (GARCH) process, which is implemented by a Kalman filter using the

squared forecasting error as inputs.

(4) Transferability across Locations

This forecasting strategy is expected to possess an ability of learning the

(43)

locations without massive manual calibration. In this study, the system transferability will

be demonstrated using real traffic data from various regions.

(5) Recursive Processing with Reduced Computational Burden

This forecasting strategy is based on the techniques of seasonal exponential smoothing

and Kalman filter, both of which are recursive in nature. Therefore, the forecasting

algorithm has limited algorithm memory and the computational burden is greatly reduced

since it only responds to incoming traffic condition data with previous traffic information

encapsulated in the current system state.

(6) Adaptability

This online forecasting strategy employs Kalman filter, which itself can adapt to

changing traffic dynamics. Efforts to apply an adaptive Kalman filter will be conducted and

tested to fine-tune the Kalman filter. The system adaptability will be investigated using real

traffic operations data.

(7) Robustness

This forecasting strategy has the ability to handle missing values and detect outliers to

conduct robust forecasting. Unlike the outlier detection in Williams (1999), where the

constant process variance was implicitly assumed, time varying conditional variance is

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occurrence of missing values and the response of the system to outliers will be investigated.

This forecasting strategy can be implemented at a TMC, and the forecasting results can

be directly fed into the development of traffic control measures. The end users of this

strategy are professional traffic engineers who can respond to the forecast of traffic flow

rate and intermittent freeway control systems that make use of the forecast in the control

measurement development. Furthermore, the quantified uncertainty indicated by the time

varying prediction confidence intervals can be explored in succeeding research for the

purpose of enhancing the mobility of transportation systems.

2.4

S

UMMARY

Different uses of the traffic condition forecast call for different treatments of the

forecasting system. This research is to establish an online short-term univariate traffic

condition prediction system incorporating online missing value imputation and outlier

detection. The algorithm uses 15-minute traffic flow rates as a representative time interval

and is intended to provide the proactive ability for advanced traffic control. The prediction

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CHAPTER 3

LITERATURE REVIEW

In this chapter, a review of the short-term traffic forecasting methods is presented to

reveal the connection and improvement of the proposed method over previous efforts.

Firstly, the short-term traffic forecasting methods are classified into seven categories.

Detailed discussions on the methodological highlights and the applications of each method

are provided in the subsequent section. A concise review on the traffic condition data

validation which is closely related to robust forecasting is presented afterwards. The

chapter concludes with a comparison table summarizing the forecasting methods review.

3.1

C

LASSIFICATION OF

F

ORECASTING

M

ETHODS

A variety of short-term traffic forecasting methods have been developed since the start

of the research several decades ago. These methods can be generally classified into seven

categories discussed below.

3.1.1 Heuristic Methods

The efforts until early 1980s were on the heuristic methods, or ad hoc methods,

including random walk, historical average, informed historical average, and UTCS

predictors. The historical traffic pattern and current traffic condition are two important

components in constructing these methods. The heuristic methods share a common

(46)

arbitrary utilization of traffic data, where personnel knowledge and engineering judgment

are crucial for a successful deployment. Moreover, none of these methods has the ability of

adapting to the changing traffic condition (Stephanedes et al. 1981).

3.1.2 Linear Statistical Methods

Researches in 1980s and 1990s saw an adoption of the linear statistical theory. The

most widely tested method is the univariate Box-Jenkins time series method in time domain.

This method assumes a linear combination of past observations and forecasting errors can

capture the traffic dynamics, thereby satisfactory forecasts can be constructed. The method

essence is to capture the linear correlation structure embedded in the traffic condition series.

Correspondingly, in frequency domain, based on the inherent assumption of a stable daily

traffic pattern, the spectral representation method using the orthogonal series

decomposition also falls into this category. Multivariate time series models belong to this

category, too, in that although complicated in form, it is still the linear structure of the

multivariate traffic condition series to be modeled. Three multivariate time series methods

were developed, i.e., vector ARIMA (VARIMA) model, space-time ARIMA (STARIMA)

model, and ARIMAX model.

Exponential smoothing also falls into this category due to its equivalent reformulation

to ARIMA model (McKenzie 1984). In exponential smoothing, the traffic condition data

(47)

exponential smoothing generally works well in practice (Chatfield et al. 1988).

All these methods are constructed based on prescribed parametric models, recognizing

the linear traffic structure. However, should no online estimation procedure be incorporated,

these methods cannot provide the ability of adaptation.

3.1.3 Nonlinear Methods

Based on the recognition of the seemingly nonlinear/chaotic nature of the traffic,

nonlinear methods have been investigated extensively since 1990s with an expectation that

an appropriately designed nonlinear structure can simulate the traffic dynamics and

generate a good forecast. The neural network method from artificial intelligence and the

non-parametric method from nonlinear statistical theory are two major approaches. The

neural network model applies interconnected and layered neurons to model the nonlinearity

in traffic data and a variety of neural network models have been devised and tested. The

non-parametric method embraces a set of techniques for curve estimation without making

strong assumptions on the true curve. K-nearest neighbor (K-NN), kernel smoothing, and

local linear regression are three methods that have been applied to traffic forecasting. In

these methods, localized sets of traffic condition data are selected and utilized to generate

the forecasts. It is worthwhile to note that the bandwidth or span denoting the size of the set

is of central role in these methods (Altman 1992).

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on which the ability of adaptation is induced. The learning strategy generally requires either

a historical traffic database or a localized set with arbitrary size, both of which might

impose certain negative effects. For example, better forecasting accuracy and learning

efficiency requires a sizable historical database, which will increase the computational

burden.

Recent researches conducted on the traffic state transition using the loop detector data

undermined the strength of the nonlinear approach. It was shown that the traffic dynamics

can be best modeled using the simple first order continuum theory and complicated traffic

phenomena, even the most mysterious one as traffic hysterics, can be well explained.

Daganzo (2002) presented a summary of these studies. These researches suggested a simple

linear intrinsic regularity in the traffic dynamics; thus linear theory based methods are

expected to yield better results than those based on nonlinear theory.

3.1.4 Hybrid Methods

Hybrid methods include the combination of methods and the combination of forecasts.

The combination of methods is to construct a new forecasting method by combining two or

more forecasting methods. Efforts were made to combine the linear methods and nonlinear

methods where a nonlinear method is applied to construct clusters of traffic condition data

and a linear method is applied to each cluster to produce the forecasts. Typical methods are

(49)

disadvantage of destroying the possible inner mechanism across time in the traffic

condition series through clustering operation. For example, the data close to the edge of

each cluster might not be sharply different from each other while the forecasting strategies

are totally different. The combination of the forecasts, termed by prediction combination in

this study, is to produce a forecast by combining the forecasts of the same quantity from

two or more individual forecasting methods based on the premise that the combination of

the forecasts is almost always better than selecting a single “best” forecast.

Intuitively, the characters of a hybrid method are determined partly by the characters of

involved individual forecasting method. For example, if a neural network is applied, the

hybrid method might inherit a disadvantage of heavy computational burden, while the

advantage of adaptation might also be transferred to the hybrid method as well. In addition,

the hybrid methods generally have complicated structures, restricting the cross location

implementation.

3.1.5 Traffic Flow Theory Based Methods

Traffic flow theory that describes the law of traffic dynamics provides a natural

forecasting method. So far the field of traffic flow theory includes the simple first order

continuum theory (Lighthill and Whitham 1955, Richard 1956) and the higher order

continuum theory (Whitham 1974 and Payne 1979). Due to the recent researches on traffic

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order theory (Daganzo 2002). Discretized versions of the first order continuum model, e.g.

Cell Transmission Model (Daganzo 1994, 1995) and Lagged Cell Transmission Model

(Daganzo 1999), are good candidates for constructing forecasting methods. Discrete state

propagation model (DSPM) by Guin (2004) was developed based on a revision of the LCT

model for the purpose of automatic incident detection.

Traffic flow theory based method is natural to incorporate multiple traffic variables

into the forecasting system. However, as is discussed in the traffic state propagation, the

forecasting horizon is generally at low level for this approach.

3.1.6 Adaptive Forecasting Methods

The non-stationary traffic dynamics calls for an adaptive forecasting strategy and the

awareness of bringing adaptability into traffic forecasting system can be traced back to the

early stages of the research (Stephanedes et al. 1981).

Adaptive filters adopted from other disciplines, mainly digital signal processing and

dynamic system control, are the most widely applied adaptive forecasting methods. The

adopted adaptive filters include recursive least square (RLS), Kalman filter, generalized

linear model (GLM), and least mean square (LMS) filters. These methods are promising in

imparting a self-adjusting ability into the forecasting system; however, contrary to the

discussion in Stephanedes et al. (1981) that forecasting method should be built upon solid

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autoregressive models for the traffic dynamics based on personal judgment or preference.

3.1.7 Other Methods

This category includes traffic forecasting methods which cannot be clearly classified

into previous categories for the purpose of completeness. A typical method is Gaussian

Maximum Likelihood Formulation in Lin (2001) and Lin et al. (2002) based on the

investigation of the traffic level and the traffic level increment.

3.2

M

ETHODOLOGICAL

H

IGHLIGHTS OF

F

ORECASTING

M

ETHODS

This section describes the methodological highlights of the traffic forecasting methods.

Applications are presented and references provided for interested readers.

3.2.1 Random Walk

Random walk model is the simplest forecasting method which assumes traffic

condition for the next time interval is the same as that in the previous one, i.e., Vˆ_t₊₁=V_t, where Vˆ_t₊₁ is the forecast for time interval t+1 and V_t is the traffic condition for time interval t. No historical traffic pattern is utilized herein. Without parameters in the

forecasting system, random walk model is easily to be implemented across locations and is

implicitly applied in any system that responds to the current traffic conditions, such as the

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3.2.2 Historical Average

Compared with the random walk model that ignores the historical pattern, historical

average takes the forecast as an average of all the observations for the same time interval of

the day. Obviously, totally ignoring current traffic condition, it cannot respond to the time

varying traffic condition and is not appropriate for dynamic traffic control. Additionally,

although historical pattern can be extracted by this method, the exponential smoothing is a

more powerful tool in capturing the historical pattern by assigning an exponential

weighting scheme to historical traffic observations. Simple in form, the historical average

method is easy to be implemented as is the random walk model.

3.2.3 Informed Historical Average

Informed historical average aims to combine the historical pattern and the current

traffic condition in order to improve the forecasting performance, and it is also easy to be

implemented as no parameters are involved in the forecasting system. As is pointed out in

Williams (1999), such a combination is still “classified as simple if it requires no fitting or

fine-tuning to site specific characteristics”.

An informed historical average method was proposed in the ALI-SCOUT system for

travel time forecasting (Kaysi et al. 1993). This method firstly computes the ratio of the

historical travel time on a specific link to the current travel time; then the ratio is used

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remedy the random walk assumption, Kutsopoulos and Xu (1993) suggested the use of the

information discounting, a strategy to adjust or penalize the current link travel time

according to the amount of the value and the standard error of the historical travel time on

the same link.

3.2.4 UTCS Predictor

Urban Traffic Control System was developed for the testing of advanced network

control methods and strategies. Three generations of predictors have been developed. In the

first generation UTCS, the forecast relied heavily on the historical data; the second

generation predictor, i.e., UTCS-2, utilized both historical data and current data; and the

third generation predictor, i.e., UTCS-3, generated prediction according to merely current

data. Though having complicated forms, these methods are still heuristic because of the

arbitrary treatment of the traffic data and the lack of rigorous justification. A

comprehensive comparison conducted in Stephanedes et al. (1981) showed that UTCS-2

had the best performance in the sense of mean square error and mean absolute error when

compared with random walk, historical average, and UTCS-3. Since there are some

parameters involved in the forecasting system, the parameter calibration required for

appropriate deployments will restrict the transferability.

3.2.5 Univariate Box-Jenkins Method

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of adaptability and rigorous justification), attention was paid to the investigation of traffic

dynamics in order to find a better method with sound theoretical foundation. The

recognition of traffic process to be a point process leaded to the adoption of the univariate

time series modeling. Box-Jenkins approach (Box and Jenkins 1994) is the most widely

applied time series modeling method. Since this method is closely related to the proposed

forecasting method in this work, a brief description of the method is presented below. See

(Fuller 1996) for a detailed discussion.

A time series defined as

{

X_t :t∈T

}

can be strictly stationary or weakly stationary. Given F_X

( )

⋅ as the probability distribution function and lag h, a time series is strictly stationary if

(

n

)

t h t h tn h

(

n

)

n t t

t X X t t t X X X t t t

X x x x F x x x

F , ,..., ₁, ₂,...,

2 1 2

1 2

1, ,..., = + , + ,..., + (3-1)

for all possible sets of t₁,t₂,...,t_n andt₁+h,t₂ +h,...,t_n +h. This definition states that the joint probability of any given subset of a strictly stationary time series is independent of the

lag.

A time series is weakly stationary if (1) the expected value of X_t is constant for all t; and (2) the covariance matrix of

(

X_t₁,X_t₂,...,X_t_n

)

is the same as that of

(

Xt1+h,Xt2+h,...,Xtn+h

)

for all finite sets

(

t1,t2,...,tn

)

and h. The definition of the weakly

stationarity only restricts the first and second moment of the time series, and hence weakly

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Theorem, a weakly stationary time series Y_t defined on T =

{

0,±1,±2,...

}

can be represented as

t t

t X Z

Y = + , (3-2)

where Z_t is a deterministic component and X_t can be modeled as an autoregressive moving average process defined as φ(B)X_t =θ(B)ε_t which is a subset of SARIMA process defined in the following paragraph.

For a non-stationary time series{X_t}, the Box-Jenkins approach assumes the nonstationarity can be handled by proper differencing, thus the