Hidden Markov Models for Time Series An Introduction Using R

About this book

Implements all methods in R
Hidden Markov Models for Time Series applies hidden Markov models (HMMs) to a wide range of time series types, from continuous-valued, circular, and multivariate series to binary data, bounded and unbounded counts, and categorical observations. It also discusses how to employ the freely available computing environment R to carry out computations for parameter estimation, model selection and checking, decoding, and forecasting.

Illustrates the methodology in action
After presenting the simple Poisson HMM, Hidden Markov Models for Time Series covers estimation, forecasting, decoding, prediction, model selection, and Bayesian inference. Through examples and applications, the authors describe how to extend and generalize the basic model so it can be applied in a rich variety of situations. They also provide R code for some of the examples, enabling the use of the codes in similar applications.

Effectively interpret data using HMMs
Hidden Markov Models for Time Series illustrates the wonderful flexibility of HMMs as general-purpose models for time series data. It provides a broad understanding of the models and their uses.

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 X^t₁ and X^T_t+1

References

Author Index

Subject Index

Exercises appear at the end of most chapters.

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