Hidden Markov Models for Time Series: An Introduction Using R

Title......Page 5
Contents......Page 8
Preface......Page 15
Notation......Page 18
Abbreviations......Page 19
PART ONE: Model structure, properties and methods......Page 20
1.1 Introduction......Page 21
1.2.1 Definition and properties......Page 24
1.2.2 Parameter estimation......Page 27
1.2.3 Unbounded likelihood in mixtures......Page 28
Mixtures of Poisson distributions......Page 29
Mixtures of normal distributions......Page 31
1.3 Markov chains......Page 33
1.3.1 Definitions and example......Page 34
1.3.2 Stationary distributions......Page 36
1.3.4 Autocorrelation function......Page 37
1.3.5 Estimating transition probabilities......Page 38
1.3.6 Higher-order Markov chains......Page 40
Exercises......Page 42
2.1 A simple hidden Markov model......Page 47
2.2.1 Definition and notation......Page 48
Univariate distributions......Page 50
Bivariate distributions......Page 51
2.2.3 Moments......Page 52
2.3.1 The likelihood of a two-state Bernoulli–HMM......Page 53
2.3.2 The likelihood in general......Page 55
2.3.3 The likelihood when data are missing at random......Page 57
2.3.4 The likelihood when observations are interval-censored......Page 58
Exercises......Page 59
3.1 Introduction......Page 63
3.2 Scaling the likelihood computation......Page 64
3.3.1 Reparametrization to avoid constraints......Page 65
3.4.1 Multiple maxima in the likelihood......Page 67
3.5 Example: earthquakes......Page 68
3.6.1 Standard errors via the Hessian......Page 71
Recursive computation of the Hessian......Page 72
3.7 Example: the parametric bootstrap applied to the three-state model for the earthquakes data......Page 73
Exercises......Page 75
4.1 Forward and backward probabilities......Page 77
4.1.1 Forward probabilities......Page 78
4.1.2 Backward probabilities......Page 79
4.1.3 Properties of forward and backward probabilities......Page 80
4.2.1 EM in general......Page 81
4.2.2 EM for HMMs......Page 82
4.2.3 M step for Poisson– and normal–HMMs......Page 84
4.2.5 EM for the case in which the Markov chain is stationary......Page 85
4.3.1 Earthquakes......Page 86
4.3.2 Foetal movement counts......Page 88
4.4 Discussion......Page 90
Exercises......Page 91
C05 Forecasting, decoding and state prediction......Page 93
5.1 Conditional distributions......Page 94
5.2 Forecast distributions......Page 95
5.3.1 State probabilities and local decoding......Page 98
5.3.2 Global decoding......Page 100
5.4 State prediction......Page 104
Exercises......Page 105
6.1 Model selection by AIC and BIC......Page 106
6.2 Model checking with pseudo-residuals......Page 109
6.2.1 Introducing pseudo-residuals......Page 110
6.2.2 Ordinary pseudo-residuals......Page 113
6.2.3 Forecast pseudo-residuals......Page 114
6.3.2 Dependent ordinary pseudo-residuals......Page 115
6.4 Discussion......Page 117
Exercises......Page 118
7.1 Applying the Gibbs sampler to Poisson–HMMs......Page 120
7.1.1 Generating sample paths of the Markov chain......Page 122
7.2 Bayesian estimation of the number of states......Page 123
7.2.1 Use of the integrated likelihood......Page 124
7.3 Example: earthquakes......Page 125
7.4 Discussion......Page 127
Exercises......Page 129
8.1 Introduction......Page 131
8.2 HMMs with general univariate state-dependent distribution......Page 132
8.3 HMMs based on a second-order Markov chain......Page 134
8.4.1 Series of multinomial-like observations......Page 135
8.4.2 A model for categorical series......Page 137
8.4.3 Other multivariate models......Page 138
8.5.1 Covariates in the state-dependent distributions......Page 141
8.5.2 Covariates in the transition probabilities......Page 142
8.6 Models with additional dependencies......Page 144
Exercises......Page 145
PART TWO: Applications......Page 149
9.2 Models fitted......Page 150
9.3 Model checking by pseudo-residuals......Page 153
Exercises......Page 155
10.2 The binary time series of short and long eruptions......Page 156
10.2.1 Markov chain models......Page 157
10.2.2 Hidden Markov models......Page 159
10.2.3 Comparison of models......Page 162
10.2.4 Forecast distributions......Page 163
10.3 Univariate normal–HMMs for durations and waiting times......Page 164
10.4 Bivariate normal–HMM for durations and waiting times......Page 167
Exercises......Page 168
11.1 Introduction......Page 170
11.2 Von Mises distributions......Page 171
11.3 Von Mises–HMMs for the two subjects......Page 172
11.4 Circular autocorrelation functions......Page 173
11.5 A bivariate series with one component linear and one circular: Drosophila speed and change of direction......Page 176
Exercises......Page 180
12.2.1 Three HMMs for hourly averages of wind direction......Page 182
12.2.2 Model comparisons and other possible models......Page 185
12.2.3 Conclusion......Page 188
12.3.1 Daily at hour 24: von Mises–HMMs......Page 189
12.3.3 Transition probabilities varying with lagged speed......Page 191
12.3.4 Concentration parameter varying with lagged speed......Page 192
Exercises......Page 195
13.1.1 Univariate models......Page 196
13.1.2 Multivariate models......Page 198
13.1.3 Discussion......Page 200
13.2 Financial series II: a multivariate normal–HMM for returns on four shares......Page 201
13.3.1 Stochastic volatility models without leverage......Page 205
13.3.2 Application: FTSE 100 returns......Page 207
13.3.3 Stochastic volatility models with leverage......Page 208
13.3.4 Application: TOPIX returns......Page 210
13.3.5 Discussion......Page 212
14.2 Models for the proportion Caesarean......Page 213
14.3 Models for the total number of deliveries......Page 219
14.4 Conclusion......Page 222
15.2 Firearm homicides as a proportion of all homicides, suicides and legal intervention homicides......Page 223
15.3 The number of firearm homicides......Page 225
15.4 Firearm homicides as a proportion of all homicides, and firearm suicides as a proportion of all suicides......Page 227
15.5 Proportion in each of the five categories......Page 231
16.1 Introduction......Page 233
16.2 The model......Page 234
16.3 Likelihood evaluation......Page 236
16.3.2 Recursive evaluation......Page 237
16.5 Model checking......Page 238
16.6 Inferring the underlying state......Page 239
16.7.1 Models assuming some parameters to be constant across subjects......Page 240
16.7.3 Inclusion of covariates......Page 241
16.8.2 Changing the nature of the state-dependent distribution......Page 242
16.9.2 Parameter estimates and model checking......Page 243
16.9.3 Runlength distributions......Page 247
16.9.4 Joint models for seven subjects......Page 249
16.10 Discussion......Page 250
References......Page 252
A.1 Fit a stationary Poisson–HMM by direct numerical maximization......Page 261
A.1.3 Log-likelihood of a stationary Poisson–HMM......Page 262
A.1.4 ML estimation of a stationary Poisson–HMM......Page 263
A.2.2 Forward and backward probabilities......Page 264
A.2.3 EM estimation of a Poisson–HMM......Page 265
A.2.5 Conditional state probabilities......Page 266
A.2.7 State prediction......Page 267
A.2.9 Conditional distribution of one observation given the rest......Page 268
A.2.10 Ordinary pseudo-residuals......Page 269
A.3.1 Transform natural parameters to working......Page 270
A.3.3 Discrete log-likelihood......Page 271
A.4 Fitting a categorical HMM by constrained optimization......Page 272
A.4.1 Log-likelihood......Page 273
A.4.2 MLEs of the parameters......Page 274
B.1 A factorization needed for the forward probabilities......Page 275
B.2 Two results needed for the backward probabilities......Page 277
B.3 Conditional independence of Xt1 and…......Page 278