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In many branches of science relevant observations are taken sequentially over time. *Bayesian Analysis of Time Series* discusses how to use models that explain the probabilistic characteristics of these time series and then utilizes the Bayesian approach to make inferences about their parameters. This is done by taking the prior information and via Bayes theorem implementing Bayesian inferences of estimation, testing hypotheses, and prediction. The methods are demonstrated using both R and WinBUGS. The R package is primarily used to generate observations from a given time series model, while the WinBUGS packages allows one to perform a posterior analysis that provides a way to determine the characteristic of the posterior distribution of the unknown parameters.

Features

- Presents a comprehensive introduction to the Bayesian analysis of time series.
- Gives many examples over a wide variety of fields including biology, agriculture, business, economics, sociology, and astronomy.
- Contains numerous exercises at the end of each chapter many of which use R and WinBUGS.
- Can be used in graduate courses in statistics and biostatistics, but is also appropriate for researchers, practitioners and consulting statisticians.

About the author

Lyle D. Broemeling, Ph.D., is Director of Broemeling and Associates Inc., and is a consulting biostatistician. He has been involved with academic health science centers for about 20 years and has taught and been a consultant at the University of Texas Medical Branch in Galveston, The University of Texas MD Anderson Cancer Center and the University of Texas School of Public Health. His main interest is in developing Bayesian methods for use in medical and biological problems and in authoring textbooks in statistics. His previous books for Chapman & Hall/CRC include *Bayesian Biostatistics and Diagnostic Medicine*, and *Bayesian Methods for Agreement.*

**Table of Contents**

**1. Introduction to the Bayesian Analysis of Time Series**

Introduction

Bayesian Analysis

Fundamentals of Time Series Analysis

Basic Random Models

Time Series and Regression

Time Series and Stationarity

Time Series and Spectral Analysis

Dynamic Linear Model

The Shift Point Problem

Residuals and Diagnostic Tests

References

**2. Bayesian Analysis**

Introduction

Bayes’ Theorem

Prior Information

The Binomial Distribution

The Normal Distribution

Posterior Information

The Binomial Distribution

The Normal Distribution

The Poisson Distribution

Inference

Introduction

Estimation

Testing Hypotheses

Predictive Inference

Introduction

The Binomial Population

Forecasting from a Normal Population

Checking Model Assumptions

Introduction

Forecasting from an Exponential, but Assuming a Normal Population

A Poisson Population

The Wiener Process

Testing the Multinomial Assumption

Computing

Introduction

Monte Carlo Markov Chains

Introduction

The Metropolis Algorithm

Gibbs Sampling

The Common Mean of Normal Populations

An Example

Comments and Conclusions

Exercises

References

**3. Preliminary Considerations for Time Series**

Time Series

Airline Passenger Bookings

Sunspot Data

Los Angeles Annual Rainfall

Graphical Techniques

Plot of Air Passenger Bookings

Sunspot Data

Graph of Los Angeles Rainfall Data

Trends, Seasonality, and Trajectories

Decomposition

Decompose Air Passenger Bookings

Average Monthly Temperatures for Debuque, Iowa

Graph of Los Angeles Rainfall Data

Mean, Variance, Correlation and General Sample Characteristic of a Time Series

Other Fundamental Considerations

Summary and Conclusions

Exercises

References

**4. Basic Random Models**

Introduction

White Noise

A Random Walk

Another Example

Goodness of Fit

Predictive Distributions

Comments and Conclusions

Exercises

References

**5. Time Series and Regression**

Introduction

Linear Models

Linear Regression with Seasonal Effects and Autoregressive Models

Bayesian Inference for a Non-Linear Trend in Time Series

Nonlinear Trend with Seasonal Effects

Regression with AR(2) Errors

Simple Linear Regression Model

Nonlinear Regression with Seasonal Effects

Comments and Conclusions

Exercises

References

**6. Time Series and Stationarity**

Moving Average Models

Regression Models with Moving Average Errors

Regression Model with MA Errors and Seasonal Effects

Autoregressive Moving Average Models

Another Approach for the Bayesian analysis of MA Processes

Second Order Moving Average Process

Quadratic Regression With MA(2) Residuals

Regression Model With MA(2) Errors and Seasonal Effects

Forecasting with Moving Average Processes

Another Example

Testing Hypotheses

Forecasting with a Moving Average Time Series

Exercises

References

**7. Time Series and Spectral Analysis**

Introduction

The Fundamentals

Unit of Measurement of Frequency

The Spectrum

Examples

Bayesian Spectral Analysis of Autoregressive Moving Average Series

MA(1) Process

MA(2) Series

The AR(1) Time Series

AR(2)

ARMA(1,1) Time Series

Sunspot Cycle

Comments and Conclusions

Exercises

References

**8. Dynamic Linear Models**

Introduction

Discrete Time Linear Dynamic Systems

Estimation of the States

Filtering

Smoothing

Prediction

The Control problem

Example

The Kalman Filter

The Control Problem

Adaptive Estimation

An Example of Adaptive Estimation

Testing Hypotheses

Summary

Exercises

References

**9. The Shift Point Problem in Time Series**

Introduction

A Shifting Normal Sequence

Structural Change in an Autoregressive Time Series

One Shift in a MA(1) Time Series

Changing Models in Econometrics

Regression Model with Autocorrelated Errors

Another Example of Structural Change

Testing Hypotheses

Analyzing Threshold Autoregression with the Bayesian Approach

A Numerical Example of Threshold Autoregression

Comments and Conclusions

Exercises

References

**10. Residuals and Diagnostic Tests**

Introduction

Diagnostic Checks for Autoregressive Models

Residuals for Model of Color Data

Residuals and Diagnostic Checks for Regression Models with AR(1) Errors

Diagnostic Tests for Regression Models with Moving Average Time Series

Comments and Conclusions

Exercises

References

In many branches of science relevant observations are taken sequentially over time. *Bayesian Analysis of Time Series* discusses how to use models that explain the probabilistic characteristics of these time series and then utilizes the Bayesian approach to make inferences about their parameters. This is done by taking the prior information and via Bayes theorem implementing Bayesian inferences of estimation, testing hypotheses, and prediction. The methods are demonstrated using both R and WinBUGS. The R package is primarily used to generate observations from a given time series model, while the WinBUGS packages allows one to perform a posterior analysis that provides a way to determine the characteristic of the posterior distribution of the unknown parameters.

Features

- Presents a comprehensive introduction to the Bayesian analysis of time series.
- Gives many examples over a wide variety of fields including biology, agriculture, business, economics, sociology, and astronomy.
- Contains numerous exercises at the end of each chapter many of which use R and WinBUGS.
- Can be used in graduate courses in statistics and biostatistics, but is also appropriate for researchers, practitioners and consulting statisticians.

About the author

Lyle D. Broemeling, Ph.D., is Director of Broemeling and Associates Inc., and is a consulting biostatistician. He has been involved with academic health science centers for about 20 years and has taught and been a consultant at the University of Texas Medical Branch in Galveston, The University of Texas MD Anderson Cancer Center and the University of Texas School of Public Health. His main interest is in developing Bayesian methods for use in medical and biological problems and in authoring textbooks in statistics. His previous books for Chapman & Hall/CRC include *Bayesian Biostatistics and Diagnostic Medicine*, and *Bayesian Methods for Agreement.*

**Table of Contents**

**1. Introduction to the Bayesian Analysis of Time Series**

Introduction

Bayesian Analysis

Fundamentals of Time Series Analysis

Basic Random Models

Time Series and Regression

Time Series and Stationarity

Time Series and Spectral Analysis

Dynamic Linear Model

The Shift Point Problem

Residuals and Diagnostic Tests

References

**2. Bayesian Analysis**

Introduction

Bayes’ Theorem

Prior Information

The Binomial Distribution

The Normal Distribution

Posterior Information

The Binomial Distribution

The Normal Distribution

The Poisson Distribution

Inference

Introduction

Estimation

Testing Hypotheses

Predictive Inference

Introduction

The Binomial Population

Forecasting from a Normal Population

Checking Model Assumptions

Introduction

Forecasting from an Exponential, but Assuming a Normal Population

A Poisson Population

The Wiener Process

Testing the Multinomial Assumption

Computing

Introduction

Monte Carlo Markov Chains

Introduction

The Metropolis Algorithm

Gibbs Sampling

The Common Mean of Normal Populations

An Example

Comments and Conclusions

Exercises

References

**3. Preliminary Considerations for Time Series**

Time Series

Airline Passenger Bookings

Sunspot Data

Los Angeles Annual Rainfall

Graphical Techniques

Plot of Air Passenger Bookings

Sunspot Data

Graph of Los Angeles Rainfall Data

Trends, Seasonality, and Trajectories

Decomposition

Decompose Air Passenger Bookings

Average Monthly Temperatures for Debuque, Iowa

Graph of Los Angeles Rainfall Data

Mean, Variance, Correlation and General Sample Characteristic of a Time Series

Other Fundamental Considerations

Summary and Conclusions

Exercises

References

**4. Basic Random Models**

Introduction

White Noise

A Random Walk

Another Example

Goodness of Fit

Predictive Distributions

Comments and Conclusions

Exercises

References

**5. Time Series and Regression**

Introduction

Linear Models

Linear Regression with Seasonal Effects and Autoregressive Models

Bayesian Inference for a Non-Linear Trend in Time Series

Nonlinear Trend with Seasonal Effects

Regression with AR(2) Errors

Simple Linear Regression Model

Nonlinear Regression with Seasonal Effects

Comments and Conclusions

Exercises

References

**6. Time Series and Stationarity**

Moving Average Models

Regression Models with Moving Average Errors

Regression Model with MA Errors and Seasonal Effects

Autoregressive Moving Average Models

Another Approach for the Bayesian analysis of MA Processes

Second Order Moving Average Process

Quadratic Regression With MA(2) Residuals

Regression Model With MA(2) Errors and Seasonal Effects

Forecasting with Moving Average Processes

Another Example

Testing Hypotheses

Forecasting with a Moving Average Time Series

Exercises

References

**7. Time Series and Spectral Analysis**

Introduction

The Fundamentals

Unit of Measurement of Frequency

The Spectrum

Examples

Bayesian Spectral Analysis of Autoregressive Moving Average Series

MA(1) Process

MA(2) Series

The AR(1) Time Series

AR(2)

ARMA(1,1) Time Series

Sunspot Cycle

Comments and Conclusions

Exercises

References

**8. Dynamic Linear Models**

Introduction

Discrete Time Linear Dynamic Systems

Estimation of the States

Filtering

Smoothing

Prediction

The Control problem

Example

The Kalman Filter

The Control Problem

Adaptive Estimation

An Example of Adaptive Estimation

Testing Hypotheses

Summary

Exercises

References

**9. The Shift Point Problem in Time Series**

Introduction

A Shifting Normal Sequence

Structural Change in an Autoregressive Time Series

One Shift in a MA(1) Time Series

Changing Models in Econometrics

Regression Model with Autocorrelated Errors

Another Example of Structural Change

Testing Hypotheses

Analyzing Threshold Autoregression with the Bayesian Approach

A Numerical Example of Threshold Autoregression

Comments and Conclusions

Exercises

References

**10. Residuals and Diagnostic Tests**

Introduction

Diagnostic Checks for Autoregressive Models

Residuals for Model of Color Data

Residuals and Diagnostic Checks for Regression Models with AR(1) Errors

Diagnostic Tests for Regression Models with Moving Average Time Series

Comments and Conclusions

Exercises

References

In many branches of science relevant observations are taken sequentially over time. *Bayesian Analysis of Time Series* discusses how to use models that explain the probabilistic characteristics of these time series and then utilizes the Bayesian approach to make inferences about their parameters. This is done by taking the prior information and via Bayes theorem implementing Bayesian inferences of estimation, testing hypotheses, and prediction. The methods are demonstrated using both R and WinBUGS. The R package is primarily used to generate observations from a given time series model, while the WinBUGS packages allows one to perform a posterior analysis that provides a way to determine the characteristic of the posterior distribution of the unknown parameters.

Features

- Presents a comprehensive introduction to the Bayesian analysis of time series.
- Gives many examples over a wide variety of fields including biology, agriculture, business, economics, sociology, and astronomy.
- Contains numerous exercises at the end of each chapter many of which use R and WinBUGS.
- Can be used in graduate courses in statistics and biostatistics, but is also appropriate for researchers, practitioners and consulting statisticians.

About the author

Lyle D. Broemeling, Ph.D., is Director of Broemeling and Associates Inc., and is a consulting biostatistician. He has been involved with academic health science centers for about 20 years and has taught and been a consultant at the University of Texas Medical Branch in Galveston, The University of Texas MD Anderson Cancer Center and the University of Texas School of Public Health. His main interest is in developing Bayesian methods for use in medical and biological problems and in authoring textbooks in statistics. His previous books for Chapman & Hall/CRC include *Bayesian Biostatistics and Diagnostic Medicine*, and *Bayesian Methods for Agreement.*

**Table of Contents**

**1. Introduction to the Bayesian Analysis of Time Series**

Introduction

Bayesian Analysis

Fundamentals of Time Series Analysis

Basic Random Models

Time Series and Regression

Time Series and Stationarity

Time Series and Spectral Analysis

Dynamic Linear Model

The Shift Point Problem

Residuals and Diagnostic Tests

References

**2. Bayesian Analysis**

Introduction

Bayes’ Theorem

Prior Information

The Binomial Distribution

The Normal Distribution

Posterior Information

The Binomial Distribution

The Normal Distribution

The Poisson Distribution

Inference

Introduction

Estimation

Testing Hypotheses

Predictive Inference

Introduction

The Binomial Population

Forecasting from a Normal Population

Checking Model Assumptions

Introduction

Forecasting from an Exponential, but Assuming a Normal Population

A Poisson Population

The Wiener Process

Testing the Multinomial Assumption

Computing

Introduction

Monte Carlo Markov Chains

Introduction

The Metropolis Algorithm

Gibbs Sampling

The Common Mean of Normal Populations

An Example

Comments and Conclusions

Exercises

References

**3. Preliminary Considerations for Time Series**

Time Series

Airline Passenger Bookings

Sunspot Data

Los Angeles Annual Rainfall

Graphical Techniques

Plot of Air Passenger Bookings

Sunspot Data

Graph of Los Angeles Rainfall Data

Trends, Seasonality, and Trajectories

Decomposition

Decompose Air Passenger Bookings

Average Monthly Temperatures for Debuque, Iowa

Graph of Los Angeles Rainfall Data

Mean, Variance, Correlation and General Sample Characteristic of a Time Series

Other Fundamental Considerations

Summary and Conclusions

Exercises

References

**4. Basic Random Models**

Introduction

White Noise

A Random Walk

Another Example

Goodness of Fit

Predictive Distributions

Comments and Conclusions

Exercises

References

**5. Time Series and Regression**

Introduction

Linear Models

Linear Regression with Seasonal Effects and Autoregressive Models

Bayesian Inference for a Non-Linear Trend in Time Series

Nonlinear Trend with Seasonal Effects

Regression with AR(2) Errors

Simple Linear Regression Model

Nonlinear Regression with Seasonal Effects

Comments and Conclusions

Exercises

References

**6. Time Series and Stationarity**

Moving Average Models

Regression Models with Moving Average Errors

Regression Model with MA Errors and Seasonal Effects

Autoregressive Moving Average Models

Another Approach for the Bayesian analysis of MA Processes

Second Order Moving Average Process

Quadratic Regression With MA(2) Residuals

Regression Model With MA(2) Errors and Seasonal Effects

Forecasting with Moving Average Processes

Another Example

Testing Hypotheses

Forecasting with a Moving Average Time Series

Exercises

References

**7. Time Series and Spectral Analysis**

Introduction

The Fundamentals

Unit of Measurement of Frequency

The Spectrum

Examples

Bayesian Spectral Analysis of Autoregressive Moving Average Series

MA(1) Process

MA(2) Series

The AR(1) Time Series

AR(2)

ARMA(1,1) Time Series

Sunspot Cycle

Comments and Conclusions

Exercises

References

**8. Dynamic Linear Models**

Introduction

Discrete Time Linear Dynamic Systems

Estimation of the States

Filtering

Smoothing

Prediction

The Control problem

Example

The Kalman Filter

The Control Problem

Adaptive Estimation

An Example of Adaptive Estimation

Testing Hypotheses

Summary

Exercises

References

**9. The Shift Point Problem in Time Series**

Introduction

A Shifting Normal Sequence

Structural Change in an Autoregressive Time Series

One Shift in a MA(1) Time Series

Changing Models in Econometrics

Regression Model with Autocorrelated Errors

Another Example of Structural Change

Testing Hypotheses

Analyzing Threshold Autoregression with the Bayesian Approach

A Numerical Example of Threshold Autoregression

Comments and Conclusions

Exercises

References

**10. Residuals and Diagnostic Tests**

Introduction

Diagnostic Checks for Autoregressive Models

Residuals for Model of Color Data

Residuals and Diagnostic Checks for Regression Models with AR(1) Errors

Diagnostic Tests for Regression Models with Moving Average Time Series

Comments and Conclusions

Exercises

References

*Bayesian Analysis of Time Series* discusses how to use models that explain the probabilistic characteristics of these time series and then utilizes the Bayesian approach to make inferences about their parameters. This is done by taking the prior information and via Bayes theorem implementing Bayesian inferences of estimation, testing hypotheses, and prediction. The methods are demonstrated using both R and WinBUGS. The R package is primarily used to generate observations from a given time series model, while the WinBUGS packages allows one to perform a posterior analysis that provides a way to determine the characteristic of the posterior distribution of the unknown parameters.

Features

- Presents a comprehensive introduction to the Bayesian analysis of time series.
- Contains numerous exercises at the end of each chapter many of which use R and WinBUGS.

About the author

*Bayesian Biostatistics and Diagnostic Medicine*, and *Bayesian Methods for Agreement.*

**Table of Contents**

**1. Introduction to the Bayesian Analysis of Time Series**

Introduction

Bayesian Analysis

Fundamentals of Time Series Analysis

Basic Random Models

Time Series and Regression

Time Series and Stationarity

Time Series and Spectral Analysis

Dynamic Linear Model

The Shift Point Problem

Residuals and Diagnostic Tests

References

**2. Bayesian Analysis**

Introduction

Bayes’ Theorem

Prior Information

The Binomial Distribution

The Normal Distribution

Posterior Information

The Binomial Distribution

The Normal Distribution

The Poisson Distribution

Inference

Introduction

Estimation

Testing Hypotheses

Predictive Inference

Introduction

The Binomial Population

Forecasting from a Normal Population

Checking Model Assumptions

Introduction

Forecasting from an Exponential, but Assuming a Normal Population

A Poisson Population

The Wiener Process

Testing the Multinomial Assumption

Computing

Introduction

Monte Carlo Markov Chains

Introduction

The Metropolis Algorithm

Gibbs Sampling

The Common Mean of Normal Populations

An Example

Comments and Conclusions

Exercises

References

**3. Preliminary Considerations for Time Series**

Time Series

Airline Passenger Bookings

Sunspot Data

Los Angeles Annual Rainfall

Graphical Techniques

Plot of Air Passenger Bookings

Sunspot Data

Graph of Los Angeles Rainfall Data

Trends, Seasonality, and Trajectories

Decomposition

Decompose Air Passenger Bookings

Average Monthly Temperatures for Debuque, Iowa

Graph of Los Angeles Rainfall Data

Mean, Variance, Correlation and General Sample Characteristic of a Time Series

Other Fundamental Considerations

Summary and Conclusions

Exercises

References

**4. Basic Random Models**

Introduction

White Noise

A Random Walk

Another Example

Goodness of Fit

Predictive Distributions

Comments and Conclusions

Exercises

References

**5. Time Series and Regression**

Introduction

Linear Models

Linear Regression with Seasonal Effects and Autoregressive Models

Bayesian Inference for a Non-Linear Trend in Time Series

Nonlinear Trend with Seasonal Effects

Regression with AR(2) Errors

Simple Linear Regression Model

Nonlinear Regression with Seasonal Effects

Comments and Conclusions

Exercises

References

**6. Time Series and Stationarity**

Moving Average Models

Regression Models with Moving Average Errors

Regression Model with MA Errors and Seasonal Effects

Autoregressive Moving Average Models

Another Approach for the Bayesian analysis of MA Processes

Second Order Moving Average Process

Quadratic Regression With MA(2) Residuals

Regression Model With MA(2) Errors and Seasonal Effects

Forecasting with Moving Average Processes

Another Example

Testing Hypotheses

Forecasting with a Moving Average Time Series

Exercises

References

**7. Time Series and Spectral Analysis**

Introduction

The Fundamentals

Unit of Measurement of Frequency

The Spectrum

Examples

Bayesian Spectral Analysis of Autoregressive Moving Average Series

MA(1) Process

MA(2) Series

The AR(1) Time Series

AR(2)

ARMA(1,1) Time Series

Sunspot Cycle

Comments and Conclusions

Exercises

References

**8. Dynamic Linear Models**

Introduction

Discrete Time Linear Dynamic Systems

Estimation of the States

Filtering

Smoothing

Prediction

The Control problem

Example

The Kalman Filter

The Control Problem

Adaptive Estimation

An Example of Adaptive Estimation

Testing Hypotheses

Summary

Exercises

References

**9. The Shift Point Problem in Time Series**

Introduction

A Shifting Normal Sequence

Structural Change in an Autoregressive Time Series

One Shift in a MA(1) Time Series

Changing Models in Econometrics

Regression Model with Autocorrelated Errors

Another Example of Structural Change

Testing Hypotheses

Analyzing Threshold Autoregression with the Bayesian Approach

A Numerical Example of Threshold Autoregression

Comments and Conclusions

Exercises

References

**10. Residuals and Diagnostic Tests**

Introduction

Diagnostic Checks for Autoregressive Models

Residuals for Model of Color Data

Residuals and Diagnostic Checks for Regression Models with AR(1) Errors

Diagnostic Tests for Regression Models with Moving Average Time Series

Comments and Conclusions

Exercises

References

*Bayesian Analysis of Time Series* discusses how to use models that explain the probabilistic characteristics of these time series and then utilizes the Bayesian approach to make inferences about their parameters. This is done by taking the prior information and via Bayes theorem implementing Bayesian inferences of estimation, testing hypotheses, and prediction. The methods are demonstrated using both R and WinBUGS. The R package is primarily used to generate observations from a given time series model, while the WinBUGS packages allows one to perform a posterior analysis that provides a way to determine the characteristic of the posterior distribution of the unknown parameters.

Features

- Presents a comprehensive introduction to the Bayesian analysis of time series.
- Contains numerous exercises at the end of each chapter many of which use R and WinBUGS.

About the author

*Bayesian Biostatistics and Diagnostic Medicine*, and *Bayesian Methods for Agreement.*

**Table of Contents**

**1. Introduction to the Bayesian Analysis of Time Series**

Introduction

Bayesian Analysis

Fundamentals of Time Series Analysis

Basic Random Models

Time Series and Regression

Time Series and Stationarity

Time Series and Spectral Analysis

Dynamic Linear Model

The Shift Point Problem

Residuals and Diagnostic Tests

References

**2. Bayesian Analysis**

Introduction

Bayes’ Theorem

Prior Information

The Binomial Distribution

The Normal Distribution

Posterior Information

The Binomial Distribution

The Normal Distribution

The Poisson Distribution

Inference

Introduction

Estimation

Testing Hypotheses

Predictive Inference

Introduction

The Binomial Population

Forecasting from a Normal Population

Checking Model Assumptions

Introduction

Forecasting from an Exponential, but Assuming a Normal Population

A Poisson Population

The Wiener Process

Testing the Multinomial Assumption

Computing

Introduction

Monte Carlo Markov Chains

Introduction

The Metropolis Algorithm

Gibbs Sampling

The Common Mean of Normal Populations

An Example

Comments and Conclusions

Exercises

References

**3. Preliminary Considerations for Time Series**

Time Series

Airline Passenger Bookings

Sunspot Data

Los Angeles Annual Rainfall

Graphical Techniques

Plot of Air Passenger Bookings

Sunspot Data

Graph of Los Angeles Rainfall Data

Trends, Seasonality, and Trajectories

Decomposition

Decompose Air Passenger Bookings

Average Monthly Temperatures for Debuque, Iowa

Graph of Los Angeles Rainfall Data

Mean, Variance, Correlation and General Sample Characteristic of a Time Series

Other Fundamental Considerations

Summary and Conclusions

Exercises

References

**4. Basic Random Models**

Introduction

White Noise

A Random Walk

Another Example

Goodness of Fit

Predictive Distributions

Comments and Conclusions

Exercises

References

**5. Time Series and Regression**

Introduction

Linear Models

Linear Regression with Seasonal Effects and Autoregressive Models

Bayesian Inference for a Non-Linear Trend in Time Series

Nonlinear Trend with Seasonal Effects

Regression with AR(2) Errors

Simple Linear Regression Model

Nonlinear Regression with Seasonal Effects

Comments and Conclusions

Exercises

References

**6. Time Series and Stationarity**

Moving Average Models

Regression Models with Moving Average Errors

Regression Model with MA Errors and Seasonal Effects

Autoregressive Moving Average Models

Another Approach for the Bayesian analysis of MA Processes

Second Order Moving Average Process

Quadratic Regression With MA(2) Residuals

Regression Model With MA(2) Errors and Seasonal Effects

Forecasting with Moving Average Processes

Another Example

Testing Hypotheses

Forecasting with a Moving Average Time Series

Exercises

References

**7. Time Series and Spectral Analysis**

Introduction

The Fundamentals

Unit of Measurement of Frequency

The Spectrum

Examples

Bayesian Spectral Analysis of Autoregressive Moving Average Series

MA(1) Process

MA(2) Series

The AR(1) Time Series

AR(2)

ARMA(1,1) Time Series

Sunspot Cycle

Comments and Conclusions

Exercises

References

**8. Dynamic Linear Models**

Introduction

Discrete Time Linear Dynamic Systems

Estimation of the States

Filtering

Smoothing

Prediction

The Control problem

Example

The Kalman Filter

The Control Problem

Adaptive Estimation

An Example of Adaptive Estimation

Testing Hypotheses

Summary

Exercises

References

**9. The Shift Point Problem in Time Series**

Introduction

A Shifting Normal Sequence

Structural Change in an Autoregressive Time Series

One Shift in a MA(1) Time Series

Changing Models in Econometrics

Regression Model with Autocorrelated Errors

Another Example of Structural Change

Testing Hypotheses

Analyzing Threshold Autoregression with the Bayesian Approach

A Numerical Example of Threshold Autoregression

Comments and Conclusions

Exercises

References

**10. Residuals and Diagnostic Tests**

Introduction

Diagnostic Checks for Autoregressive Models

Residuals for Model of Color Data

Residuals and Diagnostic Checks for Regression Models with AR(1) Errors

Diagnostic Tests for Regression Models with Moving Average Time Series

Comments and Conclusions

Exercises

References

*Bayesian Analysis of Time Series* discusses how to use models that explain the probabilistic characteristics of these time series and then utilizes the Bayesian approach to make inferences about their parameters. This is done by taking the prior information and via Bayes theorem implementing Bayesian inferences of estimation, testing hypotheses, and prediction. The methods are demonstrated using both R and WinBUGS. The R package is primarily used to generate observations from a given time series model, while the WinBUGS packages allows one to perform a posterior analysis that provides a way to determine the characteristic of the posterior distribution of the unknown parameters.

Features

- Presents a comprehensive introduction to the Bayesian analysis of time series.
- Contains numerous exercises at the end of each chapter many of which use R and WinBUGS.

About the author

*Bayesian Biostatistics and Diagnostic Medicine*, and *Bayesian Methods for Agreement.*

**Table of Contents**

**1. Introduction to the Bayesian Analysis of Time Series**

Introduction

Bayesian Analysis

Fundamentals of Time Series Analysis

Basic Random Models

Time Series and Regression

Time Series and Stationarity

Time Series and Spectral Analysis

Dynamic Linear Model

The Shift Point Problem

Residuals and Diagnostic Tests

References

**2. Bayesian Analysis**

Introduction

Bayes’ Theorem

Prior Information

The Binomial Distribution

The Normal Distribution

Posterior Information

The Binomial Distribution

The Normal Distribution

The Poisson Distribution

Inference

Introduction

Estimation

Testing Hypotheses

Predictive Inference

Introduction

The Binomial Population

Forecasting from a Normal Population

Checking Model Assumptions

Introduction

Forecasting from an Exponential, but Assuming a Normal Population

A Poisson Population

The Wiener Process

Testing the Multinomial Assumption

Computing

Introduction

Monte Carlo Markov Chains

Introduction

The Metropolis Algorithm

Gibbs Sampling

The Common Mean of Normal Populations

An Example

Comments and Conclusions

Exercises

References

**3. Preliminary Considerations for Time Series**

Time Series

Airline Passenger Bookings

Sunspot Data

Los Angeles Annual Rainfall

Graphical Techniques

Plot of Air Passenger Bookings

Sunspot Data

Graph of Los Angeles Rainfall Data

Trends, Seasonality, and Trajectories

Decomposition

Decompose Air Passenger Bookings

Average Monthly Temperatures for Debuque, Iowa

Graph of Los Angeles Rainfall Data

Mean, Variance, Correlation and General Sample Characteristic of a Time Series

Other Fundamental Considerations

Summary and Conclusions

Exercises

References

**4. Basic Random Models**

Introduction

White Noise

A Random Walk

Another Example

Goodness of Fit

Predictive Distributions

Comments and Conclusions

Exercises

References

**5. Time Series and Regression**

Introduction

Linear Models

Linear Regression with Seasonal Effects and Autoregressive Models

Bayesian Inference for a Non-Linear Trend in Time Series

Nonlinear Trend with Seasonal Effects

Regression with AR(2) Errors

Simple Linear Regression Model

Nonlinear Regression with Seasonal Effects

Comments and Conclusions

Exercises

References

**6. Time Series and Stationarity**

Moving Average Models

Regression Models with Moving Average Errors

Regression Model with MA Errors and Seasonal Effects

Autoregressive Moving Average Models

Another Approach for the Bayesian analysis of MA Processes

Second Order Moving Average Process

Quadratic Regression With MA(2) Residuals

Regression Model With MA(2) Errors and Seasonal Effects

Forecasting with Moving Average Processes

Another Example

Testing Hypotheses

Forecasting with a Moving Average Time Series

Exercises

References

**7. Time Series and Spectral Analysis**

Introduction

The Fundamentals

Unit of Measurement of Frequency

The Spectrum

Examples

Bayesian Spectral Analysis of Autoregressive Moving Average Series

MA(1) Process

MA(2) Series

The AR(1) Time Series

AR(2)

ARMA(1,1) Time Series

Sunspot Cycle

Comments and Conclusions

Exercises

References

**8. Dynamic Linear Models**

Introduction

Discrete Time Linear Dynamic Systems

Estimation of the States

Filtering

Smoothing

Prediction

The Control problem

Example

The Kalman Filter

The Control Problem

Adaptive Estimation

An Example of Adaptive Estimation

Testing Hypotheses

Summary

Exercises

References

**9. The Shift Point Problem in Time Series**

Introduction

A Shifting Normal Sequence

Structural Change in an Autoregressive Time Series

One Shift in a MA(1) Time Series

Changing Models in Econometrics

Regression Model with Autocorrelated Errors

Another Example of Structural Change

Testing Hypotheses

Analyzing Threshold Autoregression with the Bayesian Approach

A Numerical Example of Threshold Autoregression

Comments and Conclusions

Exercises

References

**10. Residuals and Diagnostic Tests**

Introduction

Diagnostic Checks for Autoregressive Models

Residuals for Model of Color Data

Residuals and Diagnostic Checks for Regression Models with AR(1) Errors

Diagnostic Tests for Regression Models with Moving Average Time Series

Comments and Conclusions

Exercises

References