Smoothness Priors Analysis of Time Series: An Introduction

Smoothness Priors Analysis of Time Series addresses some of the problems of modeling stationary and nonstationary time series primarily from a Bayesian stochastic 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.

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.