Hidden Markov Models for Time Series: An Introduction Using R

by Zucchini; Walter

• ISBN13: 9781584885733
• ISBN10: 1584885734
• Format: Hardcover
• Copyright: 2009-04-28
• Publisher: Chapman & Hall/

Summary

This book illustrates the wonderful flexibility of HMMs as general-purpose models for time series data. It presents an accessible overview of HMMs for analyzing time series data, from continuous-valued, circular, and multivariate series to binary data, bounded and unbounded counts, and categorical observations. It explores a variety of applications in animal behavior, finance, epidemiology, climatology, and sociology. The authors discuss how to employ the freely available computing environment R to carry out computations for parameter estimation, model selection and checking, decoding, and forecasting. They provide all of the data sets analyzed in the text online.

Table of Contents

Preface	p. xvii
Notation and abbreviations	p. xxi
Model structure, properties and methods	p. 1
Preliminaries: mixtures and Markov chains	p. 3
Introduction	p. 3
Independent mixture models	p. 6
Definition and properties	p. 6
Parameter estimation	p. 9
Unbounded likelihood in mixtures	p. 10
Examples of fitted mixture models	p. 11
Markov chains	p. 15
Definitions and example	p. 16
Stationary distributions	p. 18
Reversibility	p. 19
Autocorrelation function	p. 19
Estimating transition probabilities	p. 20
Higher-order Markov chains	p. 22
Exercises	p. 24
Hidden Markov models: definition and properties	p. 29
A simple hidden Markov model	p. 29
The basics	p. 30
Definition and notation	p. 30
Marginal distributions	p. 32
Moments	p. 34
The likelihood	p. 35
The likelihood of a two-state Bernoulli-HMM	p. 35
The likelihood in general	p. 37
The likelihood when data are missing at random	p. 39
The likelihood when observations are interval-censored	p. 40
Exercises	p. 41
Estimation by direct maximization of the likelihood	p. 45
Introduction	p. 45
Scaling the likelihood computation	p. 46
Maximization subject to constraints	p. 47
Reparametrization to avoid constraints	p. 47
Embedding in a continuous-time Markov chain	p. 49
Other problems	p. 49
Multiple maxima in the likelihood	p. 49
Starting values for the iterations	p. 50
Unbounded likelihood	p. 50
Example: earthquakes	p. 50
Standard errors and confidence intervals	p. 53
Standard errors via the Hessian	p. 53
Bootstrap standard erros and confidence intervals	p. 55
Example: parametric bootstrap	p. 55
Exercises	p. 57
Estimation by the EM algorithm	p. 59
Forward and backward probabilities	p. 59
Forward probabilities	p. 60
Backward probabilities	p. 61
Properties of forward and backward probabilities	p. 62
The EM algorithm	p. 63
EM in general	p. 63
EM for HMMs	p. 64
M step for Poisson-and normal-HMMs	p. 66
Starting from a specified state	p. 67
EM for the case in which the Markov chain is stationary	p. 67
Examples of EM applied to Poisson-HMMs	p. 68
Earthquakes	p. 68
Foetal movement counts	p. 70
Discussion	p. 72
Exercises	p. 73
Forecasting, decoding and state prediction	p. 75
Conditional distributions	p. 76
Forecast distributions	p. 77
Decoding	p. 80
State probabilities and local decoding	p. 80
Global decoding	p. 82
State prediction	p. 86
Exercises	p. 87
Model selection and checking	p. 89
Model selection by AIC and BIC	p. 89
Model checking with pseudo-residuals	p. 92
Introducing pseudo-residuals	p. 93
Ordinary pseudo-residuals	p. 96
Forecast pseudo-residuals	p. 97
Examples	p. 98
Ordinary pseudo-residuals for the earthquakes	p. 98
Dependent ordinary pseudo-residuals	p. 98
Discussion	p. 100
Exercises	p. 101
Bayesian inference for Poisson-HMMs	p. 103
Applying the Gibbs sampler to Poisson-HMMs	p. 103
Generating sample paths of the Markov chain	p. 105
Decomposing observed counts	p. 106
Updating the parameters	p. 106
Bayesian estimation of the number of states	p. 106
Use of the integrated likelihood	p. 107
Model selection by parallel sampling	p. 108
Example: earthquakes	p. 108
Discussion	p. 110
Exercises	p. 112
Extensions of the basic hidden Markov model	p. 115
Introduction	p. 115
HMMs with general univariate state-dependent distribution	p. 116
HMMs based on a second-order Markov chain	p. 118
HMMs for multivariate series	p. 119
Series of multinomial-like observations	p. 119
A model for categorical series	p. 121
Other multivariate models	p. 122
Series that depend on covariates	p. 125
Covariates in the state-dependent distributions	p. 125
Covariates in the transition probabilities	p. 126
Models with additional dependencies	p. 128
Exercises	p. 129
Applications	p. 133
Epileptic seizures	p. 135
Introduction	p. 135
Models fitted	p. 135
Model checking by pseudo-residuals	p. 138
Exercises	p. 140
Eruptions of the Old Faithful geyser	p. 141
Introduction	p. 141
Binary time series of short and long eruptions	p. 141
Markov chain models	p. 142
Hidden Markov models	p. 144
Comparison of models	p. 147
Forecast distributions	p. 148
Normal-HMMs for durations and waiting times	p. 149
Bivariate model for durations and waiting times	p. 152
Exercises	p. 153
Drosophila speed and change of direction	p. 155
Introduction	p. 155
Von Mises distributions	p. 156
Von Mises-HMMs for the two subjects	p. 157
Circular autocorrelation functions	p. 158
Bivariate model	p. 161
Exercises	p. 165
Wind direction at Koeberg	p. 167
Introduction	p. 167
Wind direction classified into 16 categories	p. 167
Three HMMs for hourly averages of wind direction	p. 167
Model comparisons and other possible models	p. 170
Conclusion	p. 173
Wind direction as a circular variable	p. 174
Daily at hour 24: von Mises-HMMs	p. 174
Modelling hourly change of direction	p. 176
Transition probabilities varying with lagged speed	p. 176
Concentration parameter varying with lagged speed	p. 177
Exercises	p. 180
Models for financial series	p. 181
Thinly traded shares	p. 181
Univariate models	p. 181
Multivariate models	p. 183
Discussion	p. 185
Multivariate HMM for returns on four shares	p. 186
Stochastic volatility models	p. 190
Stochastic volatility models without leverage	p. 190
Application: FTSE 100 returns	p. 192
Stochastic volatility models with leverage	p. 193
Application: TOPIX returns	p. 195
Discussion	p. 197
Births at Edendale Hospital	p. 199
Introduction	p. 199
Models for the proportion Caesarean	p. 199
Models for the total number of deliveries	p. 205
Conclusion	p. 208
Homicides and suicides in Cape Town	p. 209
Introduction	p. 209
Firearm homicides as a proportion of all homicides, suicides and legal intervention homicides	p. 209
The number of firearm homicides	p. 211
Firearm homicide and suicide proportions	p. 213
Proportion in each of the five categories	p. 217
Animal behaviour model with feedback	p. 219
Introduction	p. 219
The model	p. 220
Likelihood evaluation	p. 222
The likelihood as a multiple sum	p. 223
Recursive evaluation	p. 223
Parameter estimation by maximum likelihood	p. 224
Model checking	p. 224
Inferring the underlying state	p. 225
Models for a heterogeneous group of subjects	p. 226
Models assuming some parameters to be constant across subjects	p. 226
Mixed models	p. 227
Inclusion of covariates	p. 227
Other modifications of extensions	p. 228
Increasing the number of states	p. 228
Changing the nature of the state-dependent distribution	p. 228
Application to caterpillar feeding behaviour	p. 229
Date description and preliminary analysis	p. 229
Parameter estimates and model checking	p. 229
Runlength distributions	p. 233
Joint models for seven subjects	p. 235
Discussion	p. 236
Examples of R code	p. 239
Stationary Poisson-HMM, numerical maximization	p. 239
Transform natural parameters to working	p. 240
Transform working parameters to natural	p. 240
Log-likelihood of a stationary Poisson-HMM	p. 240
ML estimation of a stationary Poisson-HMM	p. 241
More on Poisson-HMMs, including EM	p. 242
Generate a realization of a Poisson-HMM	p. 242
Forward and backward probabilities	p. 242
EM estimation of a Poisson-HMM	p. 243
Viterbi algorithm	p. 244
Conditional state probabilities	p. 244
Local decoding	p. 245
State prediction	p. 245
Forecast distributions	p. 246
Conditional distribution of one observation given the rest	p. 246
Ordinary pseudo-residuals	p. 247
Bivariate normal state-dependent distributions	p. 248
Transform natural parameters to working	p. 248
Transform working parameters to natural	p. 249
Discrete log-likelihood	p. 249
MLEs of the parameters	p. 250
Categorical HMM, constrained optimization	p. 250
Log-likelihood	p. 251
MLEs of the parameters	p. 252
Some proofs	p. 253
Factorization needed for forward probabilities	p. 253
Two results for backward probabilites	p. 255
Conditional independence of Xt1 and $$	p. 256
References	p. 257
Author index	p. 267
Subject index	p. 271
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