Introduction to Time Series Modeling with Applications in R
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Book Description
Praise for the first edition:
[This book] reflects the extensive experience and significant contributions of the author to non-linear and non-Gaussian modeling. … [It] is a valuable book, especially with its broad and accessible introduction of models in the state-space framework.
–Statistics in Medicine
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.
–MAA Reviews
Introduction to Time Series Modeling with Applications in R, Second Edition covers numerous stationary and nonstationary time series models and tools for estimating and utilizing them. The goal of this book is to enable readers to build their own models to understand, predict and master time series. The second edition makes it possible for readers to reproduce examples in this book by using the freely available R package TSSS to perform computations for their own real-world time series problems.
This book employs the state-space model as a generic tool for time series modeling and presents the Kalman filter, the non-Gaussian filter and the particle filter as convenient tools for recursive estimation for state-space models. Further, it also takes a unified approach based on the entropy maximization principle and employs various methods of parameter estimation and model selection, including the least squares method, the maximum likelihood method, recursive estimation for state-space models and model selection by AIC.
Along with the standard stationary time series models, such as the AR and ARMA models, the book also introduces nonstationary time series models such as the locally stationary AR model, the trend model, the seasonal adjustment model, the time-varying coefficient AR model and nonlinear non-Gaussian state-space models.
About the Author:
Genshiro Kitagawa is a project professor at the University of Tokyo, the former Director-General of the Institute of Statistical Mathematics, and the former President of the Research Organization of Information and Systems.
Table of Contents
1 Introduction and Preparatory Analysis
1.1 Time Series Data
1.2 Classification of Time Series
1.3 Objectives of Time Series Analysis
1.4 Pre-processing of Time Series
1.4.1 Transformation of variables
1.4.2 Differencing
1.4.3 Month-to-month basis and year-over-year
1.4.4 Moving average
1.5 Organization of This Book
2 The Covariance Function
2.1 The Distribution of Time Series and Stationarity
2.2 The Autocovariance Function of Stationary Time Series
2.3 Estimation of the Autocovariance Function
2.4 Multivariate Time Series and Scatterplots
2.5 Cross-covariance Function and Cross-correlation Function
3 The Power Spectrum and the Periodogram
3.1 The Power Spectrum
3.2 The Periodogram
3.3 Averaging and Smoothing of the Periodogram
3.4 Computational Method of Periodogram
3.5 Computation of the Periodogram by Fast Fourier Transform
4 Statistical Modeling
4.1 Probability Distributions and Statistical Models
4.2 K-L Information and Entropy Maximization Principle
4.3 Estimation of the K-L Information and the Log-likelihood
4.4 Estimation of Parameters by the Maximum Likelihood Method
4.5 AIC (Akaike Information Criterion)
4.5.1 Evaluation of C1
4.5.2 Evaluation of C3
4.5.3 Evaluation of C2
4.5.4 Evaluation of C and AIC
4.6 Transformation of Data
5 The Least Squares Method
5.1 Regression Models and the Least Squares Method
5.2 Householder Transformation Method
5.3 Selection of Order by AIC
5.4 Addition of Data and Successive Householder Reduction
5.5 Variable Selection by AIC
6 Analysis of Time Series Using ARMA Models
6.1 ARMA Model
6.2 The Impulse Response Function
6.3 The Autocovariance Function
6.4 The Relation Between AR Coefficients and PARCOR 98
6.5 The Power Spectrum of the ARMA Process 98
6.6 The Characteristic Equation 102
6.7 The Multivariate AR Model 106
7 Estimation of an AR Model
7.1 Fitting an AR Mode
7.2 Yule-Walker Method and Levinson’s Algorithm
7.3 Estimation of an AR Model by the Least Squares Method
7.4 Estimation of an AR Model by the PARCOR Method
7.5 Large Sample Distribution of the Estimates
7.6 Estimation of Multivariate AR Model by Yule-Walker Method
7.7 Estimation of Multivariate AR Model by Least Squares Method
8 The Locally Stationary AR Model
8.1 Locally Stationary AR Model
8.2 Automatic Partitioning of the Time Interval
8.3 Precise Estimation of the Change Point
8.4 Posterior Probability of the Change Point
9 Analysis of Time Series with a State-Space Model
9.1 The State-Space Model
9.2 State Estimation via the Kalman Filter
9.3 Smoothing Algorithms
9.4 Long-term Prediction of the State
9.5 Prediction of Time Series
9.6 Likelihood Computation and Parameter Estimation for Time Series Models
9.7 Interpolation of Missing Observations
10 Estimation of the ARMA Model
10.1 State-Space Representation of the ARMA Model
10.2 Initial State Distribution for an AR Model
10.3 Initial State Distribution of an ARMA Model
10.4 Maximum Likelihood Estimates of an ARMA Model
10.5 Initial Estimates of Parameters
11 Estimation of Trends
11.1 The Polynomial Trend Model
11.2 Trend Component Model – Model for Gradual Changes
11.3 Trend Model
12 The Seasonal Adjustment Model
12.1 Seasonal Component Model
12.2 Standard Seasonal Adjustment Model
12.3 Decomposition Including an AR Component
12.4 Decomposition Including a Trading-day Effect
13 Time-Varying Coefficient AR Model
13.1 Time-varying Variance Model
13.2 Time-varying Coefficient AR Model
13.3 Estimation of the Time-varying Spectrum
13.4 The Assumption on System Noise for the Time-varying Coefficient AR Model
13.5 Abrupt Changes of Coefficients
14.1 Necessity of Non-Gaussian Models
14.2 Non-Gaussian State-Space Models and State Estimation
14.3 Numerical Computation of the State Estimation Formula
14.4 Non-Gaussian Trend Model
14.5 A Time-varying Variance Model
14.6 Further Applications of Non-Gaussian State-Space Model
14.6.1 Processing of the outliers by a mixture of Gaussian distributions
14.6.2 A nonstationary discrete process
14.6.3 A direct method of estimating the time-varying variance
14.6.4 Nonlinear state-spece models
15 Particle Filter
15.1 The Nonlinear Non-Gaussian State-Space Model and Approximations of Distributions
15.2 Particle Filter
15.2.1 One-step-ahead prediction
15.2.2 Filtering
15.2.3 Algorithm for the particle filter
15.2.4 Likelihood of a model
15.2.5 On the re-sampling method
15.2.6 Numerical examples
15.3 Particle Smoothing Method
15.4 Nonlinear Smoothing
16 Simulation
16.1 Generation of Uniform Random Numbers
16.2 Generation of White Noise
16.2.1 χ2 distribution
16.2.2 Cauchy distribution
16.2.3 Arbitrary distribution
16.3 Simulation of ARMA models
16.4 Simulation Using a State-Space Model
16.5 Simulation with Non-Gaussian Model
A Algorithms forNonlinearOptimizationBibliography
Author(s)
Biography
Genshiro Kitagawa is a project professor at the University of Tokyo, the former Director-General of the Institute of Statistical Mathematics, and the former President of the Research Organization of Information and Systems.