Statistical analysis of stochastic processes in time

J. K. Lindsey0521837413, 9780521837415, 9780511217319

This introduction to ways of modelling a wide variety of phenomena that occur over time is accessible to anyone with a basic knowledge of statistical ideas. J.K. Lindsey concentrates on tractable models involving simple processes for which explicit probability models, hence likelihood functions, can be specified. (These models are the most useful in statistical applications modelling empirical data.) Examples are drawn from physical, biological and social sciences, to show how the book’s underlying ideas can be applied, and data sets and R code are supplied for them. Author resource page: http://popgen.unimaas.nl/~jlindsey/books.html

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Table of contents :
Cover……Page 1
Half-title……Page 2
Series-title……Page 3
Title……Page 4
Copyright……Page 5
Contents……Page 6
Preface……Page 10
Notation and symbols……Page 14
Part I Basic principles……Page 16
1.1 Definition……Page 18
1.1.1 Time……Page 19
1.1.2 State space……Page 20
1.1.3 Randomness……Page 21
Multivariate distributions……Page 22
Ergodicity……Page 23
1.1.5 Replications……Page 24
1.2.1 Constructing multivariate distributions……Page 25
1.2.2 Markov processes……Page 26
1.2.4 Serial dependence……Page 27
1.2.5 Birth processes……Page 28
1.3.1 Preliminary questions……Page 29
1.3.2 Inference……Page 30
Exercises……Page 31
2.1 Descriptive statistics……Page 33
2.1.2 Graphics……Page 34
2.2 Linear regression……Page 36
2.2.2 Fitting regression lines……Page 37
Likelihood……Page 38
Multiple regression……Page 39
Interactions……Page 40
Baseline constraint……Page 41
Mean constraint……Page 42
Interactions……Page 43
Gamma distribution……Page 44
Weibull distribution……Page 47
Other distributions……Page 48
Logistic growth curve……Page 49
Exercises……Page 50
Part II Categorical state space……Page 52
Interevent times……Page 54
Absorbing states……Page 55
3.1.2 Incomplete data……Page 56
Stopping rules……Page 57
Time alignment……Page 58
3.1.3 Survivor and intensity functions……Page 60
3.1.4 Likelihood function……Page 61
3.2 Right censoring……Page 62
Proportional hazards……Page 63
Accelerated failure times……Page 64
3.2.2 Intensity and survivor functions……Page 65
3.3.1 Probability models……Page 68
3.4.1 Probability models……Page 72
3.5.1 Durations and counts of events……Page 75
3.6.2 Overdispersion……Page 79
Exercises……Page 81
4 Recurrent events……Page 86
Basic concepts……Page 87
Some simple special cases……Page 88
Modelling intensities……Page 89
Exponential distribution……Page 91
Nonhomogeneous Poisson processes……Page 92
4.1.3 Departures from randomness……Page 94
Stationarity……Page 95
Recurrence times……Page 96
Types of failure……Page 97
4.2.1 Detecting trends……Page 98
Cumulative events and counts of events……Page 99
4.2.2 Detecting time dependence……Page 100
4.2.3 Kaplan–Meier curves……Page 102
4.3 Counts of recurrent events……Page 103
4.3.1 Poisson regression……Page 104
4.3.2 Over- and underdispersion……Page 105
4.4.1 Renewal processes……Page 106
Further reading……Page 108
Exercises……Page 109
5 Discrete-time Markov chains……Page 116
Marginal and conditional probabilities……Page 117
Reversibility……Page 118
Aggregation……Page 119
Duration in a state……Page 120
Recurrence of a state……Page 121
General case……Page 122
5.2 Binary point processes……Page 123
5.2.1 Transition matrices……Page 124
5.2.2 Logistic regression……Page 126
Binary time series……Page 127
5.2.3 Log linear models……Page 128
5.3 Checking the assumptions……Page 129
Global homogeneity……Page 130
Local homogeneity……Page 132
5.3.2 Order……Page 133
5.4.1 Reversibility and equilibrium……Page 134
5.4.2 Random walks……Page 136
5.4.3 Mover–stayer model……Page 137
Exercises……Page 139
6.1.1 Diagrams……Page 148
6.1.3 Continuous-time Markov chains……Page 149
Differences with respect to discrete time……Page 151
6.1.4 Semi-Markov processes……Page 152
6.2.1 Erratic and permanent missingness……Page 153
6.2.2 Trends in missingness……Page 156
6.3.1 Constant intensities within states……Page 157
6.3.3 Intensities depending on covariates……Page 159
Further reading……Page 160
Exercises……Page 161
7 Dynamic models……Page 166
7.1.1 Single count responses……Page 167
7.1.2 Binary count responses……Page 169
7.1.3 Overdispersion……Page 172
7.1.4 Changing variability over time……Page 174
7.2.1 Theory……Page 176
Hidden state space……Page 177
Fitting a model……Page 178
Discretized hidden Poisson process……Page 179
7.2.2 Clustered point process……Page 180
7.3.1 Theory……Page 182
Gamma mixture……Page 184
Updating the parameters……Page 186
7.3.2 Frailty……Page 190
7.3.3 Longitudinal dependence……Page 192
7.4 Overdispersed series of counts……Page 193
7.4.1 Theory……Page 194
Exercises……Page 195
8.1.1 Birth or contagion……Page 198
8.1.2 Learning models……Page 201
8.1.3 Overdispersion……Page 203
8.2.1 Theory……Page 206
Conditional exponential family……Page 207
Exponential dispersion family……Page 208
Compound Poisson processes……Page 210
8.3.2 Markov chains with marks……Page 211
8.4 Doubly stochastic processes……Page 213
Gamma mixture of Weibull distributions……Page 214
8.5.1 Theory……Page 217
8.5.2 Hidden Markov model……Page 220
Exercises……Page 221
Part III Continuous state space……Page 226
9.1.1 Graphics……Page 228
Autocorrelation function……Page 229
Partial autocorrelation function……Page 230
9.2.1 AR(1)……Page 231
Covariance matrix……Page 232
Autocorrelation functions……Page 233
9.2.2 Transformations……Page 235
9.2.3 Random walks……Page 238
9.2.4 Heteroscedasticity……Page 239
9.3.1 Periodograms……Page 241
Estimation……Page 242
Autoregression……Page 243
Exponential model……Page 244
Exercises……Page 245
10.1.1 Theory……Page 248
Measurement error……Page 249
Likelihood function……Page 250
10.2.1 Theory……Page 253
10.2.2 Modelling velocity……Page 254
10.3 Heavy-tailed distributions……Page 255
Definition……Page 256
Estimation……Page 257
10.3.2 Other heavy-tailed distributions……Page 261
Normal distribution……Page 264
10.4.2 Biological variability……Page 265
Exercises……Page 268
11.1.1 Theory……Page 270
Autoregression models……Page 271
Model specification……Page 272
11.1.2 Continuous-time autoregression……Page 273
11.2 Hidden Markov models……Page 274
11.3 Overdispersed responses……Page 277
Further reading……Page 279
Exercises……Page 280
12.1 Characteristics……Page 283
12.2.1 Exponential growth……Page 284
Distributional assumptions……Page 285
12.2.2 Monomolecular growth……Page 287
12.3.1 Logistic growth……Page 290
12.3.2 Gompertz growth……Page 292
12.4 Richards growth curve……Page 293
Further reading……Page 294
Exercises……Page 295
13.1 Theory……Page 300
13.1.1 First-order kinetics……Page 301
13.1.2 Open, first-order, one-compartment model……Page 302
13.2.1 Random walk……Page 304
13.2.2 Gamma distribution……Page 306
13.3.1 Models for proportions……Page 308
Further reading……Page 310
Exercises……Page 311
14.1.1 Mixture models……Page 318
Choice of random parameters……Page 319
Other mixtures……Page 320
14.2.1 Collections of time series……Page 321
14.2.2 Cross-over trials……Page 323
14.3 Normal random coefficients……Page 325
14.3.1 Random coefficients in time……Page 326
14.4 Gamma random effects……Page 327
Further reading……Page 328
Exercises……Page 329
References……Page 332
Author index……Page 342
Subject index……Page 345