Introduction to Time Series Modeling: 1st Edition (Hardback) book cover

Introduction to Time Series Modeling

1st Edition

By Genshiro Kitagawa

Chapman and Hall/CRC

296 pages | 80 B/W Illus.

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pub: 2010-04-21
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Description

In time series modeling, the behavior of a certain phenomenon is expressed in relation to the past values of itself and other covariates. Since many important phenomena in statistical analysis are actually time series and the identification of conditional distribution of the phenomenon is an essential part of the statistical modeling, it is very important and useful to learn fundamental methods of time series modeling. Illustrating how to build models for time series using basic methods, Introduction to Time Series Modeling covers numerous time series models and the various tools for handling them.

The book employs the state-space model as a generic tool for time series modeling and presents convenient recursive filtering and smoothing methods, including the Kalman filter, the non-Gaussian filter, and the sequential Monte Carlo filter, for the state-space models. Taking a unified approach to model evaluation based on the entropy maximization principle advocated by Dr. Akaike, the author derives various methods of parameter estimation, such as the least squares method, the maximum likelihood method, recursive estimation for state-space models, and model selection by the Akaike information criterion (AIC). Along with simulation methods, he also covers standard stationary time series models, such as AR and ARMA models, as well as nonstationary time series models, including the locally stationary AR model, the trend model, the seasonal adjustment model, and the time-varying coefficient AR model.

With a focus on the description, modeling, prediction, and signal extraction of times series, this book provides basic tools for analyzing time series that arise in real-world problems. It encourages readers to build models for their own real-life problems.

Reviews

My first reaction before opening this book was if there is a market for yet another book on the subject. However, my skepticism disappeared fast once I started reading the book. … The content of the book is well chosen … I would strongly recommend this book for readers who want to have a first glance to the notion of time series analysis and modelling. It is also a valuable book for teaching a first course on time series modelling both for graduate and/or undergraduate students.

Journal of Time Series Analysis, Volume 32, May 2011

This book provides an introduction to time series analysis with emphasis on the state space approach. It reflects the extensive experience and significant contributions of the author to non-linear and non-Gaussian modeling. … The material from Chapter 8 on is worth reading by anybody wishing to extend their range of time series tools. … This is a valuable book, especially with its broad and accessible introduction of models in the state space framework.

Statistics in Medicine, 2011, 30

… What distinguishes this book from comparable introductory texts is the use of state space modeling. Along with this come a number of valuable tools for recursive filtering and smoothing including the Kalman filter, as well as non-Gaussian and sequential Monte Carlo filters. … a useful reference for the application of state space modeling to time series.

MAA Reviews, October 2010

Table of Contents

Introduction and Preparatory Analysis

Time Series Data

Classification of Time Series

Objectives of Time Series Analysis

Preprocessing of Time Series

Organization of This Book

The Covariance Function

The Distribution of Time Series and Stationarity

The Autocovariance Function of Stationary Time Series

Estimation of the Autocovariance Function

Multivariate Time Series and Scatterplots

Cross-Covariance Function and Cross-Correlation Function

The Power Spectrum and the Periodogram

The Power Spectrum

The Periodogram

Averaging and Smoothing of the Periodogram

Computational Method of Periodogram

Computation of the Periodogram by Fast Fourier Transform

Statistical Modeling

Probability Distributions and Statistical Models

K-L Information and the Entropy Maximization Principle

Estimation of the K-L Information and Log-Likelihood

Estimation of Parameters by the Maximum Likelihood Method

Akaike Information Criterion (AIC)

Transformation of Data

The Least Squares Method

Regression Models and the Least Squares Method

Householder Transformation Method

Selection of Order by AIC

Addition of Data and Successive Householder Reduction

Variable Selection by AIC

Analysis of Time Series Using ARMA Models

ARMA Model

The Impulse Response Function

The Autocovariance Function

The Relation between AR Coefficients and the PARCOR

The Power Spectrum of the ARMA Process

The Characteristic Equation

The Multivariate AR Model

Estimation of an AR Model

Fitting an AR Model

Yule–Walker Method and Levinson’s Algorithm

Estimation of an AR Model by the Least Squares Method

Estimation of an AR Model by the PARCOR Method

Large Sample Distribution of the Estimates

Yule–Walker Method for MAR Model

Least Squares Method for MAR Model

The Locally Stationary AR Model

Locally Stationary AR Model

Automatic Partitioning of the Time Interval

Precise Estimation of a Change Point

Analysis of Time Series with a State-Space Model

The State-Space Model

State Estimation via the Kalman Filter

Smoothing Algorithms

Increasing Horizon Prediction of the State

Prediction of Time Series

Likelihood Computation and Parameter Estimation for a Time Series Model

Interpolation of Missing Observations

Estimation of the ARMA Model

State-Space Representation of the ARMA Model

Initial State of an ARMA Model

Maximum Likelihood Estimate of an ARMA Model

Initial Estimates of Parameters

Estimation of Trends

The Polynomial Trend Model

Trend Component Model—Model for Probabilistic Structural Changes

Trend Model

The Seasonal Adjustment Model

Seasonal Component Model

Standard Seasonal Adjustment Model

Decomposition Including an AR Component

Decomposition Including a Trading-Day Effect

Time-Varying Coefficient AR Model

Time-Varying Variance Model

Time-Varying Coefficient AR Model

Estimation of the Time-Varying Spectrum

The Assumption on System Noise for the Time-Varying Coefficient AR Model

Abrupt Changes of Coefficients

Non-Gaussian State-Space Model

Necessity of Non-Gaussian Models

Non-Gaussian State-Space Models and State Estimation

Numerical Computation of the State Estimation Formula

Non-Gaussian Trend Model

A Time-Varying Variance Model

Applications of Non-Gaussian State-Space Model

The Sequential Monte Carlo Filter

The Nonlinear Non-Gaussian State-Space Model and Approximations of Distributions

Monte Carlo Filter

Monte Carlo Smoothing Method

Nonlinear Smoothing

Simulation

Generation of Uniform Random Numbers

Generation of Gaussian White Noise

Simulation Using a State-Space Model

Simulation with Non-Gaussian Model

Appendix A: Algorithms for Nonlinear Optimization

Appendix B: Derivation of Levinson’s Algorithm

Appendix C: Derivation of the Kalman Filter and Smoother Algorithms

Appendix D: Algorithm for the Monte Carlo Filter

Bibliography

About the Author

Genshiro Kitagawa is the Director-General of the Institute of Statistical Mathematics in Tokyo, Japan.

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

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Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General