Hidden markov models for time series : an introduction using R

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010226352	QA280 Z83 2009	Open Access Book	Book	Searching... Unknown
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Summary

Reveals How HMMs Can Be Used as General-Purpose Time Series Models

Implements all methods in R
Hidden Markov Models for Time Series: An Introduction Using Rapplies 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, the book 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
This book 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.

Preface	p. xvii
Notation and abbreviations	p. xxi
Part 1 Model structure, properties and methods	p. 1
1 Preliminaries: mixtures and Markov chains	p. 3
1.1 Introduction	p. 3
1.2 Independent mixture models	p. 6
1.2.1 Definition and properties	p. 6
1.2.2 Parameter estimation	p. 9
1.2.3 Unbounded likelihood in mixtures	p. 10
1.2.4 Examples of fitted mixture models	p. 11
1.3 Markov chains	p. 15
1.3.1 Definitions and example	p. 16
1.3.2 Stationary distributions	p. 18
1.3.3 Reversibility	p. 19
1.3.4 Autocorrelation function	p. 19
1.3.5 Estimating transition probabilities	p. 20
1.3.6 Higher-order Markov chains	p. 22
Exercises	p. 24
2 Hidden Markov models: definition and properties	p. 29
2.1 A simple hidden Markov model	p. 29
2.2 The basics	p. 30
2.2.1 Definition and notation	p. 30
2.2.2 Marginal distributions	p. 32
2.2.3 Moments	p. 34
2.3 The likelihood	p. 35
2.3.1 The likelihood of a two-state Bernoulli-HMM	p. 35
2.3.2 The likelihood in general	p. 37
2.3.3 The likelihood when data are missing at random	p. 39
2.3.4 The likelihood when observations are interval-censored	p. 40
Exercises	p. 41
3 Estimation by direct maximization of the likelihood	p. 45
3.1 Introduction	p. 45
3.2 Scaling the likelihood computation	p. 46
3.3 Maximization subject to constraints	p. 47
3.3.1 Reparametrization to avoid constraints	p. 47
3.3.2 Embedding in a continuous-time Markov chain	p. 49
3.4 Other problems	p. 49
3.4.1 Multiple maxima in the likelihood	p. 49
3.4.2 Starting values for the iterations	p. 50
3.4.3 Unbounded likelihood	p. 50
3.5 Example: earthquakes	p. 50
3.6 Standard errors and confidence intervals	p. 53
3.6.1 Standard errors via the Hessian	p. 53
3.6.2 Bootstrap standard erros and confidence intervals	p. 55
3.7 Example: parametric bootstrap	p. 55
Exercises	p. 57
4 Estimation by the EM algorithm	p. 59
4.1 Forward and backward probabilities	p. 59
4.1.1 Forward probabilities	p. 60
4.1.2 Backward probabilities	p. 61
4.1.3 Properties of forward and backward probabilities	p. 62
4.2 The EM algorithm	p. 63
4.2.1 EM in general	p. 63
4.2.2 EM for HMMs	p. 64
4.2.3 M step for Poisson-and normal-HMMs	p. 66
4.2.4 Starting from a specified state	p. 67
4.2.5 EM for the case in which the Markov chain is stationary	p. 67
4.3 Examples of EM applied to Poisson-HMMs	p. 68
4.3.1 Earthquakes	p. 68
4.3.2 Foetal movement counts	p. 70
4.4 Discussion	p. 72
Exercises	p. 73
5 Forecasting, decoding and state prediction	p. 75
5.1 Conditional distributions	p. 76
5.2 Forecast distributions	p. 77
5.3 Decoding	p. 80
5.3.1 State probabilities and local decoding	p. 80
5.3.2 Global decoding	p. 82
5.4 State prediction	p. 86
Exercises	p. 87
6 Model selection and checking	p. 89
6.1 Model selection by AIC and BIC	p. 89
6.2 Model checking with pseudo-residuals	p. 92
6.2.1 Introducing pseudo-residuals	p. 93
6.2.2 Ordinary pseudo-residuals	p. 96
6.2.3 Forecast pseudo-residuals	p. 97
6.3 Examples	p. 98
6.3.1 Ordinary pseudo-residuals for the earthquakes	p. 98
6.3.2 Dependent ordinary pseudo-residuals	p. 98
6.4 Discussion	p. 100
Exercises	p. 101
7 Bayesian inference for Poisson-HMMs	p. 103
7.1 Applying the Gibbs sampler to Poisson-HMMs	p. 103
7.1.1 Generating sample paths of the Markov chain	p. 105
7.1.2 Decomposing observed counts	p. 106
7.1.3 Updating the parameters	p. 106
7.2 Bayesian estimation of the number of states	p. 106
7.2.1 Use of the integrated likelihood	p. 107
7.2.2 Model selection by parallel sampling	p. 108
7.3 Example: earthquakes	p. 108
7.4 Discussion	p. 110
Exercises	p. 112
8 Extensions of the basic hidden Markov model	p. 115
8.1 Introduction	p. 115
8.2 HMMs with general univariate state-dependent distribution	p. 116
8.3 HMMs based on a second-order Markov chain	p. 118
8.4 HMMs for multivariate series	p. 119
8.4.1 Series of multinomial-like observations	p. 119
8.4.2 A model for categorical series	p. 121
8.4.3 Other multivariate models	p. 122
8.5 Series that depend on covariates	p. 125
8.5.1 Covariates in the state-dependent distributions	p. 125
8.5.2 Covariates in the transition probabilities	p. 126
8.6 Models with additional dependencies	p. 128
Exercises	p. 129
Part 2 Applications	p. 133
9 Epileptic seizures	p. 135
9.1 Introduction	p. 135
9.2 Models fitted	p. 135
9.3 Model checking by pseudo-residuals	p. 138
Exercises	p. 140
10 Eruptions of the Old Faithful geyser	p. 141
10.1 Introduction	p. 141
10.2 Binary time series of short and long eruptions	p. 141
10.2.1 Markov chain models	p. 142
10.2.2 Hidden Markov models	p. 144
10.2.3 Comparison of models	p. 147
10.2.4 Forecast distributions	p. 148
10.3 Normal-HMMs for durations and waiting times	p. 149
10.4 Bivariate model for durations and waiting times	p. 152
Exercises	p. 153
11 Drosophila speed and change of direction	p. 155
11.1 Introduction	p. 155
11.2 Von Mises distributions	p. 156
11.3 Von Mises-HMMs for the two subjects	p. 157
11.4 Circular autocorrelation functions	p. 158
11.5 Bivariate model	p. 161
Exercises	p. 165
12 Wind direction at Koeberg	p. 167
12.1 Introduction	p. 167
12.2 Wind direction classified into 16 categories	p. 167
12.2.1 Three HMMs for hourly averages of wind direction	p. 167
12.2.2 Model comparisons and other possible models	p. 170
12.2.3 Conclusion	p. 173
12.3 Wind direction as a circular variable	p. 174
12.3.1 Daily at hour 24: von Mises-HMMs	p. 174
12.3.2 Modelling hourly change of direction	p. 176
12.3.3 Transition probabilities varying with lagged speed	p. 176
12.3.4 Concentration parameter varying with lagged speed	p. 177
Exercises	p. 180
13 Models for financial series	p. 181
13.1 Thinly traded shares	p. 181
13.1.1 Univariate models	p. 181
13.1.2 Multivariate models	p. 183
13.1.3 Discussion	p. 185
13.2 Multivariate HMM for returns on four shares	p. 186
13.3 Stochastic volatility models	p. 190
13.3.1 Stochastic volatility models without leverage	p. 190
13.3.2 Application: FTSE 100 returns	p. 192
13.3.3 Stochastic volatility models with leverage	p. 193
13.3.4 Application: TOPIX returns	p. 195
13.3.5 Discussion	p. 197
14 Births at Edendale Hospital	p. 199
14.1 Introduction	p. 199
14.2 Models for the proportion Caesarean	p. 199
14.3 Models for the total number of deliveries	p. 205
14.4 Conclusion	p. 208
15 Homicides and suicides in Cape Town	p. 209
15.1 Introduction	p. 209
15.2 Firearm homicides as a proportion of all homicides, suicides and legal intervention homicides	p. 209
15.3 The number of firearm homicides	p. 211
15.4 Firearm homicide and suicide proportions	p. 213
15.5 Proportion in each of the five categories	p. 217
16 Animal behaviour model with feedback	p. 219
16.1 Introduction	p. 219
16.2 The model	p. 220
16.3 Likelihood evaluation	p. 222
16.3.1 The likelihood as a multiple sum	p. 223
16.3.2 Recursive evaluation	p. 223
16.4 Parameter estimation by maximum likelihood	p. 224
16.5 Model checking	p. 224
16.6 Inferring the underlying state	p. 225
16.7 Models for a heterogeneous group of subjects	p. 226
16.7.1 Models assuming some parameters to be constant across subjects	p. 226
16.7.2 Mixed models	p. 227
16.7.3 Inclusion of covariates	p. 227
16.8 Other modifications of extensions	p. 228
16.8.1 Increasing the number of states	p. 228
16.8.2 Changing the nature of the state-dependent distribution	p. 228
16.9 Application to caterpillar feeding behaviour	p. 229
16.9.1 Date description and preliminary analysis	p. 229
16.9.2 Parameter estimates and model checking	p. 229
16.9.3 Runlength distributions	p. 233
16.9.4 Joint models for seven subjects	p. 235
16.10 Discussion	p. 236
A Examples of R code	p. 239
A.1 Stationary Poisson-HMM, numerical maximization	p. 239
A.1.1 Transform natural parameters to working	p. 240
A.1.2 Transform working parameters to natural	p. 240
A.1.3 Log-likelihood of a stationary Poisson-HMM	p. 240
A.1.4 ML estimation of a stationary Poisson-HMM	p. 241
A.2 More on Poisson-HMMs, including EM	p. 242
A.2.1 Generate a realization of a Poisson-HMM	p. 242
A.2.2 Forward and backward probabilities	p. 242
A.2.3 EM estimation of a Poisson-HMM	p. 243
A.2.4 Viterbi algorithm	p. 244
A.2.5 Conditional state probabilities	p. 244
A.2.6 Local decoding	p. 245
A.2.7 State prediction	p. 245
A.2.8 Forecast distributions	p. 246
A.2.9 Conditional distribution of one observation given the rest	p. 246
A.2.10 Ordinary pseudo-residuals	p. 247
A.3 Bivariate normal state-dependent distributions	p. 248
A.3.1 Transform natural parameters to working	p. 248
A.3.2 Transform working parameters to natural	p. 249
A.3.3 Discrete log-likelihood	p. 249
A.3.4 MLEs of the parameters	p. 250
A.4 Categorical HMM, constrained optimization	p. 250
A.4.1 Log-likelihood	p. 251
A.4.2 MLEs of the parameters	p. 252
B Some proofs	p. 253
B.1 Factorization needed for forward probabilities	p. 253
B.2 Two results for backward probabilites	p. 255
B.3 Conditional independence of Xt1 and $$	p. 256
References	p. 257
Author index	p. 267
Subject index	p. 271

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