Smoothness Priors Analysis of Time Series

by Genshiro Kitagawa, Will Gersch

Overview

Smoothness Priors Analysis of Time Series addresses some of the problems of modeling stationary and nonstationary time series primarily from a Bayesian shastic regression "smoothness priors" state space point of view. Prior distributions on model coefficients are parametrized by hyperparameters. Maximizing the likelihood of a small number of hyperparameters permits the robust modeling of a time series with relatively complex structure and a very large number of implicitly inferred parameters. The critical statistical ideas in smoothness priors are the likelihood of the Bayesian model and the use of likelihood as a measure of the goodness of fit of the model. The emphasis is on a general state space approach in which the recursive conditional distributions for prediction, filtering, and smoothing are realized using a variety of nonstandard methods including numerical integration, a Gaussian mixture distribution-two filter smoothing formula, and a Monte Carlo "particle-path tracing" method in which the distributions are approximated by many realizations. The methods are applicable for modeling time series with complex structures.

Product Details

ISBN-13:	9780387948195
Publisher:	Springer New York
Publication date:	08/09/1996
Series:	Lecture Notes in Statistics , #116
Edition description:	1996
Pages:	280
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

1 Introduction.- 1.1 Background.- 1.2 What is in the Book.- 1.3 Time Series Examples.- 2 Modeling Concepts and Methods.- 2.1 Akaike’s AIC: Evaluating Parametric Models.- 2.2 Least Squares Regression by Householder Transformation.- 2.3 Maximum Likelihood Estimation and an Optimization Algorithm.- 2.4 State Space Methods.- 3 The Smoothness Priors Concept.- 3.1 Introduction.- 3.2 Background, History and Related Work.- 3.3 Smoothness Priors Bayesian Modeling.- 4 Scalar Least Squares Modeling.- 4.1 Estimating a Trend.- 4.2 The Long AR Model.- 4.3 Transfer Function Estimation.- 5 Linear Gaussian State Space Modeling.- 5.1 Introduction.- 5.2 Standard State Space Modeling.- 5.3 Some State Space Models.- 5.4 Modeling With Missing Observations.- 5.5 Unequally Spaced Observations.- 5.6 An Information Square-Root Filter/Smoother.- 6 Contents General State Space Modeling.- 6.1 Introduction.- 6.2 The General State Space Model.- 6.3 Numerical Synthesis of the Algorithms.- 6.4 The Gaussian Sum-Two Filter Formula Approximation.- 6.5 A Monte Carlo Filtering and Smoothing Method.- 6.6 A Derivation of the Kalman filter.- 7 Applications of Linear Gaussian State Space Modeling.- 7.1 AR Time Series Modeling.- 7.2 Kullback-Leibler Computations.- 7.3 Smoothing Unequally Spaced Data.- 7.4 A Signal Extraction Problem.- 8 Modeling Trends.- 8.1 State Space Trend Models.- 8.2 State Space Estimation of Smooth Trend.- 8.3 Multiple Time Series Modeling: The Common Trend Plus Individual Component AR Model.- 8.4 Modeling Trends with Discontinuities.- 9 Seasonal Adjustment.- 9.1 Introduction.- 9.2 A State Space Seasonal Adjustment Model.- 9.3 Smooth Seasonal Adjustment Examples.- 9.4 Non-Gaussian Seasonal Adjustment.- 9.5 Modeling Outliers.- 9.6 Legends.- 10 Estimation of Time Varying Variance.- 10.1Introduction and Background.- 10.2 Modeling Time-Varying Variance.- 10.3 The Seismic Data.- 10.4 Smoothing the Periodogram.- 10.5 The Maximum Daily Temperature Data.- 11 Modeling Scalar Nonstationary Covariance Time Series.- 11.1 Introduction.- 11.2 A Time Varying AR Coefficient Model.- 11.3 A State Space Model.- 11.4 PARCOR Time Varying AR Modeling.- 11.5 Examples.- 12 Modeling Multivariate Nonstationary Covariance Time Series.- 12.1 Introduction.- 12.2 The Instantaneous Response-Orthogonal Innovations Model.- 12.3 State Space Modeling.- 12.4 Time Varying PARCOR VAR Modeling.- 12.5 Examples.- 13 Modeling Inhomogeneous Discrete Processes.- 13.1 Nonstationary Discrete Process.- 13.2 Nonstationary Binary Processes.- 13.3 Nonstationary Poisson Process.- 14 Quasi-Periodic Process Modeling.- 14.1 The Quasi-periodic Model.- 14.2 The Wolfer Sunspot Data.- 14.3 The Canadian Lynx Data.- 14.4 Other Examples.- 14.5 Predictive Properties of Quasi-periodic Process Modeling.- 15 Nonlinear Smoothing.- 15.1 Introduction.- 15.2 State Estimation.- 15.3 A One Dimensional Problem.- 15.4 A Two Dimensional Problem.- 16 Other Applications.- 16.1 A Large Scale Decomposition Problem.- 16.2 Markov State Classification.- 16.3 SPVAR Modeling for Spectrum Estimation.- References.- Author Index.