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

✍ Scribed by Junichiro Hagiwara

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 350
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

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.  

✦ Table of Contents

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