Table of Contents
MODEL STRUCTURE, PROPERTIES, AND METHODS
Mixture Distributions and Markov Chains
Introduction
Independent mixture models
Markov chains
Hidden Markov Models: Definition and Properties
A simple hidden Markov model
The basics
The likelihood
Estimation by Direct Maximization of the Likelihood
Introduction
Scaling the likelihood computation
Maximization subject to constraints
Other problems
Example: earthquakes
Standard errors and confidence intervals
Example: parametric bootstrap
Estimation by the EM Algorithm
Forward and backward probabilities
The EM algorithm
Examples of EM applied to Poisson HMMs
Discussion
Forecasting, Decoding, and State Prediction
Conditional distributions
Forecast distributions
Decoding
State prediction
Model Selection and Checking
Model selection by AIC and BIC
Model checking with pseudo-residuals
Examples
Discussion
Bayesian Inference for Poisson HMMs
Applying the Gibbs sampler to Poisson HMMs
Bayesian estimation of the number of states
Example: earthquakes
Discussion
Extensions of the Basic Hidden Markov Model
Introduction
HMMs with general univariate state-dependent distribution
HMMs based on a second-order Markov chain
HMMs for multivariate series
Series which depend on covariates
Models with additional dependencies
APPLICATIONS
Epileptic Seizures
Introduction
Models fitted
Model checking by pseudo-residuals
Eruptions of the Old Faithful Geyser
Introduction
Binary time series of short and long eruptions
Normal HMMs for durations and waiting times
Bivariate model for durations and waiting times
Drosophila Speed and Change of Direction
Introduction
Von Mises distributions
Von Mises HMMs for the two subjects
Circular autocorrelation functions
Bivariate model
Wind Direction at Koeberg
Introduction
Wind direction as classified into 16 categories
Wind direction as a circular variable
Models for Financial Series
Thinly traded shares
Multivariate HMM for returns on four shares
Stochastic volatility models
Births at Edendale Hospital
Introduction
Models for the proportion Caesarean
Models for the total number of deliveries
Conclusion
Cape Town Homicides and Suicides
Introduction
Firearm homicides as a proportion of all homicides, suicides, and legal intervention homicides
The number of firearm homicides
Firearm homicide and suicide proportions
Proportion in each of the five categories
Animal-Behavior Model with Feedback
Introduction
The model
Likelihood evaluation
Parameter estimation by maximum likelihood
Model checking
Inferring the underlying state
Models for a heterogeneous group of subjects
Other modifications or extensions
Application to caterpillar feeding behavior
Discussion
Appendix A: Examples of R code
Stationary Poisson HMM, numerical maximization
More on Poisson HMMs, including EM
Bivariate normal state-dependent distributions
Categorical HMM, constrained optimization
Appendix B: Some Proofs
Factorization needed for forward probabilities
Two results for backward probabilities
Conditional independence of Xt1 and XTt+1
References
Author Index
Subject Index
Exercises appear at the end of most chapters.
