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Hidden Markov Models for Time Series_ An Introduction Using R.pdf
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Contents
Preface xxi
Preface to first edition xxiii
Notation and abbreviations xxvii
I Model structure, properties and methods 1
1 Preliminaries: mixtures and Markov chains 3
1.1 Introduction 3
1.2 Independent mixture models 6
1.2.1 Definition and properties 6
1.2.2 Parameter estimation 9
1.2.3 Unbounded likelihood in mixtures 11
1.2.4 Examples of fitted mixture models 12
1.3 Markov chains 14
1.3.1 Definitions and example 14
1.3.2 Stationary distributions 17
1.3.3 Autocorrelation function 18
1.3.4 Estimating transition probabilities 19
1.3.5 Higher-order Markov chains 20
Exercises 23
2 Hidden Markov models: definition and properties 29
2.1 A simple hidden Markov model 29
2.2 The basics 30
2.2.1 Definition and notation 30
2.2.2 Marginal distributions 32
2.2.3 Moments 33
2.3 The likelihood 34
2.3.1 The likelihood of a two-state Bernoulli–HMM 35
2.3.2 The likelihood in general 36
2.3.3 HMMs are not Markov processes 39
2.3.4 The likelihood when data are missing 40
xixii CONTENTS
2.3.5 The likelihood when observations are interval-
censored 41
Exercises 41
3 Estimation by direct maximization of the likelihood 47
3.1 Introduction 47
3.2 Scaling the likelihood computation 48
3.3 Maximization of the likelihood subject to constraints 50
3.3.1 Reparametrization to avoid constraints 50
3.3.2 Embedding in a continuous-time Markov chain 52
3.4 Other problems 53
3.4.1 Multiple maxima in the likelihood 53
3.4.2 Starting values for the iterations 53
3.4.3 Unbounded likelihood 53
3.5 Example: earthquakes 54
3.6 Standard errors and confidence intervals 56
3.6.1 Standard errors via the Hessian 56
3.6.2 Bootstrap standard errors and confidence
intervals 58
3.7 Example: the parametric bootstrap applied to the
three-state model for the earthquakes data 59
Exercises 60
4 Estimation by the EM algorithm 65
4.1 Forward and backward probabilities 65
4.1.1 Forward probabilities 66
4.1.2 Backward probabilities 67
4.1.3 Properties of forward and backward probabil-
ities 68
4.2 The EM algorithm 69
4.2.1 EM in general 70
4.2.2 EM for HMMs 70
4.2.3 M step for Poisson– and normal–HMMs 72
4.2.4 Starting from a specified state 73
4.2.5 EM for the case in which the Markov chain is
stationary 73
4.3 Examples of EM applied to Poisson–HMMs 74
4.3.1 Earthquakes 74
4.3.2 Foetal movement counts 76
4.4 Discussion 77
Exercises 78
5 Forecasting, decoding and state prediction 81
5.1 Introduction 81CONTENTS xiii
5.2 Conditional distributions 82
5.3 Forecast distributions 83
5.4 Decoding 85
5.4.1 State probabilities and local decoding 86
5.4.2 Global decoding 88
5.5 State prediction 92
5.6 HMMs for classification 93
Exercises 94
6 Model selection and checking 97
6.1 Model selection by AIC and BIC 97
6.2 Model checking with pseudo-residuals 101
6.2.1 Introducing pseudo-residuals 101
6.2.2 Ordinary pseudo-residuals 105
6.2.3 Forecast pseudo-residuals 105
6.3 Examples 106
6.3.1 Ordinary pseudo-residuals for the earthquakes 106
6.3.2 Dependent ordinary pseudo-residuals 108
6.4 Discussion 109
Exercises 109
7 Bayesian inference for Poisson–hidden Markov models 111
7.1 Applying the Gibbs sampler to Poisson–HMMs 111
7.1.1 Introduction and outline 111
7.1.2 Generating sample paths of the Markov chain 113
7.1.3 Decomposing the observed counts into regime
contributions 114
7.1.4 Updating the parameters 114
7.2 Bayesian estimation of the number of states 114
7.2.1 Use of the integrated likelihood 115
7.2.2 Model selection by parallel sampling 116
7.3 Example: earthquakes 116
7.4 Discussion 119
Exercises 120
8 R packages 123
8.1 The package depmixS4 123
8.1.1 Model formulation and estimation 123
8.1.2 Decoding 124
8.2 The package HiddenMarkov 124
8.2.1 Model formulation and estimation 124
8.2.2 Decoding 126
8.2.3 Residuals 126
8.3 The package msm 126xiv CONTENTS
8.3.1 Model formulation and estimation 126
8.3.2 Decoding 128
8.4 The package R2OpenBUGS 128
8.5 Discussion 129
II Extensions 131
9 HMMs with general state-dependent distribution 133
9.1 Introduction 133
9.2 General univariate state-dependent distribution 133
9.2.1 HMMs for unbounded counts 133
9.2.2 HMMs for binary data 134
9.2.3 HMMs for bounded counts 134
9.2.4 HMMs for continuous-valued series 135
9.2.5 HMMs for proportions 135
9.2.6 HMMs for circular-valued series 136
9.3 Multinomial and categorical HMMs 136
9.3.1 Multinomial–HMM 136
9.3.2 HMMs for categorical data 137
9.3.3 HMMs for compositional data 138
9.4 General multivariate state-dependent distribution 138
9.4.1 Longitudinal conditional independence 138
9.4.2 Contemporaneous conditional independence 140
9.4.3 Further remarks on multivariate HMMs 141
Exercises 142
10 Covariates and other extra dependencies 145
10.1 Introduction 145
10.2 HMMs with covariates 145
10.2.1 Covariates in the state-dependent distributions 146
10.2.2 Covariates in the transition probabilities 147
10.3 HMMs based on a second-order Markov chain 148
10.4 HMMs with other additional dependencies 150
Exercises 152
11 Continuous-valued state processes 155
11.1 Introduction 155
11.2 Models with continuous-valued state process 156
11.2.1 Numerical integration of the likelihood 157
11.2.2 Evaluation of the approximate likelihood via
forward recursion 158
11.2.3 Parameter estimation and related issues 160
11.3 Fitting an SSM to the earthquake data 160
11.4 Discussion 162CONTENTS xv
12 Hidden semi-Markov models and their representation
as HMMs 165
12.1 Introduction 165
12.2 Semi-Markov processes, hidden semi-Markov models
and approximating HMMs 165
12.3 Examples of HSMMs represented as HMMs 167
12.3.1 A simple two-state Poisson–HSMM 167
12.3.2 Example of HSMM with three states 169
12.3.3 A two-state HSMM with general dwell-time
distribution in one state 171
12.4 General HSMM 173
12.5 R code 176
12.6 Some examples of dwell-time distributions 178
12.6.1 Geometric distribution 178
12.6.2 Shifted Poisson distribution 178
12.6.3 Shifted negative binomial distribution 179
12.6.4 Shifted binomial distribution 180
12.6.5 A distribution with unstructured start and
geometric tail 180
12.7 Fitting HSMMs via the HMM representation 181
12.8 Example: earthquakes 182
12.9 Discussion 184
Exercises 184
13 HMMs for longitudinal data 187
13.1 Introduction 187
13.2 Models that assume some parameters to be constant
across component series 189
13.3 Models with random effects 190
13.3.1 HMMs with continuous-valued random effects 191
13.3.2 HMMs with discrete-valued random effects 193
13.4 Discussion 195
Exercises 196
III Applications 197
14 Introduction to applications 199
15 Epileptic seizures 201
15.1 Introduction 201
15.2 Models fitted 201
15.3 Model checking by pseudo-residuals 204
Exercises 206xvi CONTENTS
16 Daily rainfall occurrence 207
16.1 Introduction 207
16.2 Models fitted 207
17 Eruptions of the Old Faithful geyser 213
17.1 Introduction 213
17.2 The data 213
17.3 The binary time series of short and long eruptions 214
17.3.1 Markov chain models 214
17.3.2 Hidden Markov models 216
17.3.3 Comparison of models 219
17.3.4 Forecast distributions 219
17.4 Univariate normal–HMMs for durations and waiting
times 220
17.5 Bivariate normal–HMM for durations and waiting times 223
Exercises 224
18 HMMs for animal movement 227
18.1 Introduction 227
18.2 Directional data 228
18.2.1 Directional means 228
18.2.2 The von Mises distribution 228
18.3 HMMs for movement data 229
18.3.1 Movement data 229
18.3.2 HMMs as multi-state random walks 230
18.4 A basic HMM for Drosophila movement 232
18.5 HMMs and HSMMs for bison movement 235
18.6 Mixed HMMs for woodpecker movement 238
Exercises 242
19 Wind direction at Koeberg 245
19.1 Introduction 245
19.2 Wind direction classified into 16 categories 245
19.2.1 Three HMMs for hourly averages of wind
direction 245
19.2.2 Model comparisons and other possible models 248
19.3 Wind direction as a circular variable 251
19.3.1 Daily at hour 24: von Mises–HMMs 251
19.3.2 Modelling hourly change of direction 253
19.3.3 Transition probabilities varying with lagged
speed 253
19.3.4 Concentration parameter varying with lagged
speed 254
Exercises 257CONTENTS xvii
20 Models for financial series 259
20.1 Financial series I: A multivariate normal–HMM for
returns on four shares 259
20.2 Financial series II: Discrete state-space stochastic
volatility models 262
20.2.1 Stochastic volatility models without leverage 263
20.2.2 Application: FTSE 100 returns 265
20.2.3 Stochastic volatility models with leverage 265
20.2.4 Application: TOPIX returns 268
20.2.5 Non-standard stochastic volatility models 270
20.2.6 A model with a mixture AR(1) volatility
process 271
20.2.7 Application: S&P 500 returns 272
Exercises 273
21 Births at Edendale Hospital 275
21.1 Introduction 275
21.2 Models for the proportion Caesarean 275
21.3 Models for the total number of deliveries 282
21.4 Conclusion 285
22 Homicides and suicides in Cape Town, 1986–1991 287
22.1 Introduction 287
22.2 Firearm homicides as a proportion of all homicides,
suicides and legal intervention homicides 287
22.3 The number of firearm homicides 289
22.4 Firearm homicides as a proportion of all homicides, and
firearm suicides as a proportion of all suicides 291
22.5 Proportion in each of the five categories 295
23 A model for animal behaviour which incorporates feed-
back 297
23.1 Introduction 297
23.2 The model 298
23.3 Likelihood evaluation 300
23.3.1 The likelihood as a multiple sum 301
23.3.2 Recursive evaluation 301
23.4 Parameter estimation by maximum likelihood 302
23.5 Model checking 302
23.6 Inferring the underlying state 303
23.7 Models for a heterogeneous group of subjects 304
23.7.1 Models assuming some parameters to be
constant across subjects 304
23.7.2 Mixed models 305xviii CONTENTS
23.7.3 Inclusion of covariates 306
23.8 Other modifications or extensions 306
23.8.1 Increasing the number of states 306
23.8.2 Changing the nature of the state-dependent
distribution 306
23.9 Application to caterpillar feeding behaviour 307
23.9.1 Data description and preliminary analysis 307
23.9.2 Parameter estimates and model checking 307
23.9.3 Runlength distributions 311
23.9.4 Joint models for seven subjects 313
23.10 Discussion 314
24 Estimating the survival rates of Soay sheep from mark–
recapture–recovery data 317
24.1 Introduction 317
24.2 MRR data without use of covariates 318
24.3 MRR data involving individual-specific time-varying
continuous-valued covariates 321
24.4 Application to Soay sheep data 324
24.5 Conclusion 328
A Examples of R code 331
A.1 The functions 331
A.1.1 Transforming natural parameters to working 332
A.1.2 Transforming working parameters to natural 332
A.1.3 Computing minus the log-likelihood from the
working parameters 332
A.1.4 Computing the MLEs, given starting values for
the natural parameters 333
A.1.5 Generating a sample 333
A.1.6 Global decoding by the Viterbi algorithm 334
A.1.7 Computing log(forward probabilities) 334
A.1.8 Computing log(backward probabilities) 334
A.1.9 Conditional probabilities 335
A.1.10 Pseudo-residuals 336
A.1.11 State probabilities 336
A.1.12 State prediction 336
A.1.13 Local decoding 337
A.1.14 Forecast probabilities 337
A.2 Examples of code using the above functions 338
A.2.1 Fitting Poisson–HMMs to the earthquakes
series 338
A.2.2 Forecast probabilities 339CONTENTS xix
B Some proofs 341
B.1 A factorization needed for the forward probabilities 341
B.2 Two results needed for the backward probabilities 342
B.3 Conditional independence of X
References 345
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