Time Series Analysis for the State-Space Model with R/Stan

Description

This book provides a comprehensive and concrete illustration of time series analysis focusing on the state-space model, which has recently attracted increasing attention in a broad range of fields. The major feature of the book lies in its consistent Bayesian treatment regarding whole combinations of batch and sequential solutions for linear Gaussian and general state-space models: MCMC and Kalman/particle filter. The reader is given insight on flexible modeling in modern time series analysis. The main topics of the book deal with the state-space model, covering extensively, from introductory and exploratory methods to the latest advanced topics such as real-time structural change detection. Additionally, a practical exercise using R/Stan based on real data promotes understanding and enhances the reader’s analytical capability.   Preface Contents 1 Introduction 1.1 What Is Time Series Analysis? 1.2 Two Approaches in Time Series Analysis 1.3 Use of R 1.3.1 Library in R and External Software 1.3.2 Code and Data in This Book 1.4 Notation in This Book References 2 Fundamentals of Probability and Statistics 2.1 Probability 2.2 Mean and Variance 2.3 Normal Distribution 2.4 Relation Among Multiple Random Variables 2.5 Stochastic Process 2.6 Covariance and Correlation 2.7 Stationary and Nonstationary Processes 2.8 Maximum Likelihood Estimation and Bayesian Estimation References 3 Fundamentals of Handling Time Series Data with R 3.1 Object for Handling Time Series 3.2 Handling of Time Information Reference 4 Quick Tour of Time Series Analysis 4.1 Confirmation of the Purpose and Data Collection 4.2 Preliminary Examination of Data 4.2.1 Plot with Horizontal Axis as Time 4.2.2 Histogram and Five-Number Summary 4.2.3 Autocorrelation Coefficient 4.2.4 Frequency Spectrum 4.3 Model Definition 4.4 Specification of Parameter Values 4.5 Execution of Filtering, Prediction, and Smoothing 4.6 Diagnostic Checking for the Results 4.7 Guideline When Applying the State-Space Model References 5 State-Space Model 5.1 Stochastic Model 5.2 Definition of State-Space Model 5.2.1 Representation by Graphical Model 5.2.2 Representation by Probability Distribution 5.2.3 Representation by Equation 5.2.4 Joint Distribution of State-Space Model 5.3 Features of State-Space Model 5.4 Classification of State-Space Models References 6 State Estimation in the State-Space Model 6.1 State Estimation Through the Posterior Distribution 6.2 How to Obtain the State Sequentially 6.2.1 A Simple Example 6.2.2 Conceptual Diagram of Recursion 6.2.3 Formulation of Filtering Distribution 6.2.4 Formulation of Predictive Distribution 6.2.5 Formulation of the Smoothing Distribution 6.3 Likelihood and Model Selection in the State-Space Model 6.4 Treatment of Parameters in the State-Space Model 6.4.1 When Parameters are Not Regarded as Random Variables 6.4.2 When Parameters are Regarded as Random Variables References 7 Batch Solution for Linear Gaussian State-Space Model 7.1 Wiener Filter 7.1.1 Wiener Smoothing 7.2 Example: AR(1) Model Case References 8 Sequential Solution for Linear Gaussian State-Space Model 8.1 Kalman Filter 8.1.1 Kalman Filtering 8.1.2 Kalman Prediction 8.1.3 Kalman Smoothing 8.2 Example: Local-level Model Case 8.2.1 Confirmation of the Purpose and Data Collection 8.2.2 Preliminary Examination of Data 8.2.3 Model Definition 8.2.4 Specification of Parameter Values 8.2.5 Execution of Filtering, Prediction, and Smoothing 8.2.6 Diagnostic Checking for the Results References 9 Introduction and Analysis Examples of a Well-Known Component Model in the Linear Gaussian State-Space Model 9.1 Combination of Individual Models 9.2 Local-Level Model 9.2.1 Example: Artificial Local-Level Model 9.3 Local-Trend Model 9.4 Seasonal Model 9.4.1 Approach from the Time Domain 9.4.2 Approach from the Frequency Domain 9.4.3 Example: CO2 Concentration in the Atmosphere 9.5 ARMA Model 9.5.1 Example: Japanese Beer Production 9.6 Regression Model 9.6.1 Example: Nintendo's Stock Price 9.6.2 Example: Flow Data of the Nile (Considering the Rapid Decrease in 1899) 9.6.3 Example: Family Food Expenditure (Considering Effects Depending on the Days of the Week) 9.7 Supplement to Modeling References 10 Batch Solution for General State-Space Model 10.1 MCMC 10.1.1 MCMC Fundamentals 10.1.2 Notes on Using the MCMC Method 10.2 State Estimation with MCMC 10.3 Use of Library 10.3.1 Various Libraries 10.3.2 Example: Artificial Local-Level Model 10.4 Estimation Example in General State-Space Model 10.5 Technique for Improving Estimation Accuracy 10.5.1 Case in Which the Linear Gaussian State-Space Model is Partially Applicable 10.5.2 Example: Artificial Local-Level Model 10.5.3 Example: Monthly Totals of Car Drivers in the UK Killed or Injured References 11 Sequential Solution for General State-Space Model 11.1 Particle Filter 11.1.1 Particle Filtering 11.1.2 Particle Prediction 11.1.3 Particle Smoothing 11.2 State Estimation with Particle Filter 11.2.1 Example: Artificial Local-Level Model 11.2.2 Attention to Numerical Computation 11.3 Use of Library 11.4 Estimation Example in General State-Space Model 11.4.1 Example: A Well-Known Nonlinear Benchmark Model 11.4.2 Application of a Particle Filter 11.5 Technique for Improving Estimation Accuracy 11.5.1 Auxiliary Particle Filter 11.5.2 Case in Which the Linear Gaussian State-Space Model is Partially Applicable References 12 Example of Applied Analysis in General State-Space Model 12.1 Consideration of Structural Change 12.2 Approach Using a Kalman Filter (Known Change Point) 12.2.1 Time-Invariant Model Studied thus Far 12.2.2 Utilizing Prior Information in the Linear Gaussian State-Space Model 12.2.3 Numerical Result 12.3 Approach Using MCMC (Unknown Change Point) 12.3.1 Time-Invariant Model Studied thus Far 12.3.2 Use of a Horseshoe Distribution in the General State-Space Model 12.3.3 Numerical Result 12.4 Approach Using a Particle Filter (Unknown Change Point) 12.4.1 Time-Invariant Model Studied thus Far 12.4.2 Use of a Horseshoe Distribution in the General State-Space Model 12.4.3 Numerical Results 12.5 Real-Time Detection for an Unknown Change Point References Appendix A Library in R and External Software A.1 dlm A.2 Stan A.3 pomp A.4 NIMBLE Appendix B Library dlm B.1 Handling of Model B.2 Setting of Time-Varying Model B.3 Square Root Algorithm B.4 Functions Primarily Used in This Book B.4.1 dlmFilter() B.4.2 dlmForecast() B.4.3 dlmSmooth() B.4.4 dlmBSample() B.4.5 dlmSvd2var() B.4.6 dlmLL() B.4.7 dlmMLE() B.4.8 dlmModPoly() B.4.9 dlmModSeas() B.4.10 dlmModTrig() B.4.11 dlmModARMA() B.4.12 dlmModReg() B.4.13 ARtransPars() B.4.14 weighted.quantile() Appendix C Supplement on Conditional Independence in the State-Space Model C.1 Derivation of Eq.(5.3摥映數爠eflinkeq:spsYuukouspsBunrisps15.35) Appendix D Symbol Assignment in the Linear Gaussian State-Space Model Appendix E Algorithm Derivation E.1 Wiener Filter E.1.1 Auxiliary Information for Derivation E.1.1.1 Frequency Domain Representation E.1.1.2 Linear Time-Invariant System E.1.2 Wiener Smoothing E.1.3 Derivation of Eq.(7.6摥映數爠eflinkeq:ARsps1spsspsnispsTaisuruspsuinaspsfirutasps7.67) E.2 Kalman Filter E.2.1 Auxiliary Information for Derivation E.2.1.1 Matrix Inversion Lemmas E.2.1.2 Bayesian Estimation for Linear Gaussian Regression Model E.2.2 Kalman Filtering E.2.2.1 One-Step-Ahead Predictive Distribution E.2.2.2 One-Step-Ahead Predictive Likelihood E.2.2.3 Filtering Distribution E.2.3 Kalman Prediction E.2.4 Kalman Smoothing E.3 Solution Using MCMC E.3.1 FFBS E.4 Particle Filter E.4.1 Particle Filtering E.4.2 Particle Prediction E.4.3 Particle Smoothing E.4.3.1 Kitagawa Algorithm E.4.3.2 FFBSi Algorithm Appendix F Execution of Particle Filtering with Library F.1 Example: Artificial Local-Level Model F.1.1 pomp F.1.2 NIMBLE Index