Bayesian Analysis of Time Series

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 SeriesIntroductionBayesian AnalysisFundamentals of Time Series AnalysisBasic Random ModelsTime Series and RegressionTime Series and StationarityTime Series and Spectral AnalysisDynamic Linear ModelThe Shift Point ProblemResiduals and Diagnostic TestsReferences 2. Bayesian AnalysisIntroductionBayes’ TheoremPrior InformationThe Binomial DistributionThe Normal DistributionPosterior InformationThe Binomial DistributionThe Normal DistributionThe Poisson DistributionInferenceIntroductionEstimationTesting HypothesesPredictive InferenceIntroductionThe Binomial PopulationForecasting from a Normal PopulationChecking Model AssumptionsIntroductionForecasting from an Exponential, but Assuming a Normal PopulationA Poisson PopulationThe Wiener ProcessTesting the Multinomial AssumptionComputingIntroductionMonte Carlo Markov ChainsIntroductionThe Metropolis AlgorithmGibbs SamplingThe Common Mean of Normal PopulationsAn ExampleComments and ConclusionsExercisesReferences 3. Preliminary Considerations for Time SeriesTime SeriesAirline Passenger BookingsSunspot DataLos Angeles Annual RainfallGraphical TechniquesPlot of Air Passenger BookingsSunspot DataGraph of Los Angeles Rainfall DataTrends, Seasonality, and TrajectoriesDecompositionDecompose Air Passenger BookingsAverage Monthly Temperatures for Debuque, IowaGraph of Los Angeles Rainfall DataMean, Variance, Correlation and General Sample Characteristic of a Time SeriesOther Fundamental ConsiderationsSummary and ConclusionsExercisesReferences 4. Basic Random ModelsIntroductionWhite NoiseA Random WalkAnother ExampleGoodness of FitPredictive DistributionsComments and ConclusionsExercisesReferences 5. Time Series and RegressionIntroductionLinear ModelsLinear Regression with Seasonal Effects and Autoregressive ModelsBayesian Inference for a Non-Linear Trend in Time SeriesNonlinear Trend with Seasonal EffectsRegression with AR(2) ErrorsSimple Linear Regression ModelNonlinear Regression with Seasonal EffectsComments and ConclusionsExercisesReferences 6. Time Series and StationarityMoving Average ModelsRegression Models with Moving Average ErrorsRegression Model with MA Errors and Seasonal EffectsAutoregressive Moving Average ModelsAnother Approach for the Bayesian analysis of MA ProcessesSecond Order Moving Average ProcessQuadratic Regression With MA(2) ResidualsRegression Model With MA(2) Errors and Seasonal EffectsForecasting with Moving Average ProcessesAnother ExampleTesting HypothesesForecasting with a Moving Average Time SeriesExercisesReferences 7. Time Series and Spectral AnalysisIntroductionThe FundamentalsUnit of Measurement of Frequency The SpectrumExamplesBayesian Spectral Analysis of Autoregressive Moving Average SeriesMA(1) ProcessMA(2) SeriesThe AR(1) Time SeriesAR(2)ARMA(1,1) Time SeriesSunspot CycleComments and ConclusionsExercisesReferences 8. Dynamic Linear ModelsIntroductionDiscrete Time Linear Dynamic SystemsEstimation of the StatesFilteringSmoothingPredictionThe Control problemExampleThe Kalman FilterThe Control ProblemAdaptive EstimationAn Example of Adaptive EstimationTesting HypothesesSummaryExercisesReferences 9. The Shift Point Problem in Time SeriesIntroductionA Shifting Normal SequenceStructural Change in an Autoregressive Time Series One Shift in a MA(1) Time SeriesChanging Models in EconometricsRegression Model with Autocorrelated ErrorsAnother Example of Structural ChangeTesting HypothesesAnalyzing Threshold Autoregression with the Bayesian ApproachA Numerical Example of Threshold AutoregressionComments and ConclusionsExercisesReferences 10. Residuals and Diagnostic TestsIntroductionDiagnostic Checks for Autoregressive ModelsResiduals for Model of Color DataResiduals and Diagnostic Checks for Regression Models with AR(1) ErrorsDiagnostic Tests for Regression Models with Moving Average Time SeriesComments and ConclusionsExercisesReferences

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.