Hidden Markov Models for Time Series An Introduction Using R Chapman amp Hall CRC Monographs on Statistics amp Applied Probability
目录
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 10
1.2.4 Examples of fitted mixture models 11
1.3 Markov chains 15
1.3.1 Definitions and example 16
1.3.2 Stationary distributions 18
1.3.3 Reversibility 19
1.3.4 Autocorrelation function 19
1.3.5 Estimating transition probabilities 20
1.3.6 Higher-order Markov chains 22
Exercises 24
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 34
2.3 The likelihood 35
2.3.1 The likelihood of a two-state Bernoulli–HMM 35
2.3.2 The likelihood in general 37
2.3.3 The likelihood when data are missing at
random 39
ix
x CONTENTS
2.3.4 The likelihood when observations are intervalcensored
40
Exercises 41
3 Estimation by direct maximization of the likelihood 45
3.1 Introduction 45
3.2 Scaling the likelihood computation 46
3.3 Maximization subject to constraints 47
3.3.1 Reparametrization to avoid constraints 47
3.3.2 Embedding in a continuous-time Markov chain 49
3.4 Other problems 49
3.4.1 Multiple maxima in the likelihood 49
3.4.2 Starting values for the iterations 50
3.4.3 Unbounded likelihood 50
3.5 Example: earthquakes 50
3.6 Standard errors and confidence intervals 53
3.6.1 Standard errors via the Hessian 53
3.6.2 Bootstrap standard errors and confidence
intervals 55
3.7 Example: parametric bootstrap 55
Exercises 57
4 Estimation by the EM algorithm 59
4.1 Forward and backward probabilities 59
4.1.1 Forward probabilities 60
4.1.2 Backward probabilities 61
4.1.3 Properties of forward and backward probabilities
62
4.2 The EM algorithm 63
4.2.1 EM in general 63
4.2.2 EM for HMMs 64
4.2.3 M step for Poisson– and normal–HMMs 66
4.2.4 Starting from a specified state 67
4.2.5 EM for the case in which the Markov chain is
stationary 67
4.3 Examples of EM applied to Poisson–HMMs 68
4.3.1 Earthquakes 68
4.3.2 Foetal movement counts 70
4.4 Discussion 72
Exercises 73
5 Forecasting, decoding and state prediction 75
5.1 Conditional distributions 76
CONTENTS xi
5.2 Forecast distributions 77
5.3 Decoding 80
5.3.1 State probabilities and local decoding 80
5.3.2 Global decoding 82
5.4 State prediction 86
Exercises 87
6 Model selection and checking 89
6.1 Model selection by AIC and BIC 89
6.2 Model checking with pseudo-residuals 92
6.2.1 Introducing pseudo-residuals 93
6.2.2 Ordinary pseudo-residuals 96
6.2.3 Forecast pseudo-residuals 97
6.3 Examples 98
6.3.1 Ordinary pseudo-residuals for the earthquakes 98
6.3.2 Dependent ordinary pseudo-residuals 98
6.4 Discussion 100
Exercises 101
7 Bayesian inference for Poisson–HMMs 103
7.1 Applying the Gibbs sampler to Poisson–HMMs 103
7.1.1 Generating sample paths of the Markov chain 105
7.1.2 Decomposing observed counts 106
7.1.3 Updating the parameters 106
7.2 Bayesian estimation of the number of states 106
7.2.1 Use of the integrated likelihood 107
7.2.2 Model selection by parallel sampling 108
7.3 Example: earthquakes 108
7.4 Discussion 110
Exercises 112
8 Extensions of the basic hidden Markov model 115
8.1 Introduction 115
8.2 HMMs with general univariate state-dependent distribution
116
8.3 HMMs based on a second-order Markov chain 118
8.4 HMMs for multivariate series 119
8.4.1 Series of multinomial-like observations 119
8.4.2 A model for categorical series 121
8.4.3 Other multivariate models 122
8.5 Series that depend on covariates 125
8.5.1 Covariates in the state-dependent distributions 125
8.5.2 Covariates in the transition probabilities 126
xii CONTENTS
8.6 Models with additional dependencies 128
Exercises 129
PART TWO Applications 133
9 Epileptic seizures 135
9.1 Introduction 135
9.2 Models fitted 135
9.3 Model checking by pseudo-residuals 138
Exercises 140
10 Eruptions of the Old Faithful geyser 141
10.1 Introduction 141
10.2 Binary time series of short and long eruptions 141
10.2.1 Markov chain models 142
10.2.2 Hidden Markov models 144
10.2.3 Comparison of models 147
10.2.4 Forecast distributions 148
10.3 Normal–HMMs for durations and waiting times 149
10.4 Bivariate model for durations and waiting times 152
Exercises 153
11 Drosophila speed and change of direction 155
11.1 Introduction 155
11.2 Von Mises distributions 156
11.3 Von Mises–HMMs for the two subjects 157
11.4 Circular autocorrelation functions 158
11.5 Bivariate model 161
Exercises 165
12 Wind direction at Koeberg 167
12.1 Introduction 167
12.2 Wind direction classified into 16 categories 167
12.2.1 Three HMMs for hourly averages of wind
direction 167
12.2.2 Model comparisons and other possible models 170
12.2.3 Conclusion 173
12.3 Wind direction as a circular variable 174
12.3.1 Daily at hour 24: von Mises–HMMs 174
12.3.2 Modelling hourly change of direction 176
12.3.3 Transition probabilities varying with lagged
speed 176
CONTENTS xiii
12.3.4 Concentration parameter varying with lagged
speed 177
Exercises 180
13 Models for financial series 181
13.1 Thinly traded shares 181
13.1.1 Univariate models 181
13.1.2 Multivariate models 183
13.1.3 Discussion 185
13.2 Multivariate HMM for returns on four shares 186
13.3 Stochastic volatility models 190
13.3.1 Stochastic volatility models without leverage 190
13.3.2 Application: FTSE 100 returns 192
13.3.3 Stochastic volatility models with leverage 193
13.3.4 Application: TOPIX returns 195
13.3.5 Discussion 197
14 Births at Edendale Hospital 199
14.1 Introduction 199
14.2 Models for the proportion Caesarean 199
14.3 Models for the total number of deliveries 205
14.4 Conclusion 208
15 Homicides and suicides in Cape Town 209
15.1 Introduction 209
15.2 Firearm homicides as a proportion of all homicides,
suicides and legal intervention homicides 209
15.3 The number of firearm homicides 211
15.4 Firearm homicide and suicide proportions 213
15.5 Proportion in each of the five categories 217
16 Animal behaviour model with feedback 219
16.1 Introduction 219
16.2 The model 220
16.3 Likelihood evaluation 222
16.3.1 The likelihood as a multiple sum 223
16.3.2 Recursive evaluation 223
16.4 Parameter estimation by maximum likelihood 224
16.5 Model checking 224
16.6 Inferring the underlying state 225
16.7 Models for a heterogeneous group of subjects 226
16.7.1 Models assuming some parameters to be
constant across subjects 226
xiv CONTENTS
16.7.2 Mixed models 227
16.7.3 Inclusion of covariates 227
16.8 Other modifications or extensions 228
16.8.1 Increasing the number of states 228
16.8.2 Changing the nature of the state-dependent
distribution 228
16.9 Application to caterpillar feeding behaviour 229
16.9.1 Data description and preliminary analysis 229
16.9.2 Parameter estimates and model checking 229
16.9.3 Runlength distributions 233
16.9.4 Joint models for seven subjects 235
16.10 Discussion 236
A Examples of R code 239
A.1 Stationary Poisson–HMM, numerical maximization 239
A.1.1 Transform natural parameters to working 240
A.1.2 Transform working parameters to natural 240
A.1.3 Log-likelihood of a stationary Poisson–HMM 240
A.1.4 ML estimation of a stationary Poisson–HMM 241
A.2 More on Poisson–HMMs, including EM 242
A.2.1 Generate a realization of a Poisson–HMM 242
A.2.2 Forward and backward probabilities 242
A.2.3 EM estimation of a Poisson–HMM 243
A.2.4 Viterbi algorithm 244
A.2.5 Conditional state probabilities 244
A.2.6 Local decoding 245
A.2.7 State prediction 245
A.2.8 Forecast distributions 246
A.2.9 Conditional distribution of one observation
given the rest 246
A.2.10 Ordinary pseudo-residuals 247
A.3 Bivariate normal state-dependent distributions 248
A.3.1 Transform natural parameters to working 248
A.3.2 Transform working parameters to natural 249
A.3.3 Discrete log-likelihood 249
A.3.4 MLEs of the parameters 250
A.4 Categorical HMM, constrained optimization 250
A.4.1 Log-likelihood 251
A.4.2 MLEs of the parameters 252
B Some proofs 253
B.1 Factorization needed for forward probabilities 253
B.2 Two results for backward probabilities 255
CONTENTS xv
B.3 Conditional independence of Xt
1 and XT t+1 256
References 257
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发表于 2013-2-15 08:47:22