Index
A
ABM, see Agent based model
Absolutely regular, 88
ACF, see Autocorrelation function
Adaptive Gaussian quadrature (AGQ), 68–71
Agent Based Model (ABM), 349–364
AGQ, see Adaptive Gaussian quadrature
AIC, see Akaike Information Criterion
Akaike Information Criterion (AIC), 205, 267,
278–281, 321
ARFIMA, see Autoregressive fractionally
integrated moving-average
ARMA, see Autoregressive Moving Average
Asymptotics, 89, 94, 223, 226, 228, 232, 237, 240
Autocorrelation function (ACF), 6–7, 125, 191,
208, 377, 414
component residual, 198–199
hidden Markov model, 273, 279
INARMA model, 149
INAR model, 32, 192, 208
latent process, 62, 132
Markov models for counts, 35, 37, 41
multivariate counts, 414
Pearson residuals, 13, 196, 213
renewal based models, 103–104, 107, 112
residuals, 12, 196–198
state space model (SSM), 125–127
Autocovariance, 73, 101–103, 111–115, 126–128,
137, 142, 149, 312, 448, 451, 455
Autologistic regression model, 367–369, 371,
373–384
Autoregressive, 7, 22, 34, 58, 65, 78–80, 82–83,
97, 112, 122, 152, 176, 181, 183, 190,
196, 208, 211, 213, 220–222, 225, 228,
236, 241, 315, 320, 336–337, 389, 395,
449, 455
binary, 219, 226, 230–231
conditional double Poisson, 420
conditional Poisson (ACP), 12, 31, 146
exponential, 9
nonlinear, 78–80
Poisson, 78, 219–220, 227–229, 232–233, 246
smooth transition, 9
threshold, 97, 337
Autoregressive fractionally integrated
moving-average (ARFIMA), 455–456
Autoregressive Moving Average models, 36, 44,
54, 448
B
Backward probabilities, 275–276
Baum-Welch algorithm, 275
Bayes factor, 246
Bayesian count time series models, 245–263
long memory, 454–456
Bayesian inference, 172, 246, 263, 284, 336,
382, 440
autologistic regression, 374–375, 378
HMDM, 434–435, 440
MDFM, 425–429, 437–440
Bayesian Information Criterion (BIC), 176,
205–207, 267, 278–281
Bernoulli, 33–34, 56–57, 73, 93, 103–105, 123, 133,
327–328, 331–333, 352, 368, 382,
411–412, 429, 442, 448
Beta, 199, 247, 249, 253, 437, 443
Beta-binomial, 34, 39, 42, 167, 190
Binary, 225–227, 268, 311, 313, 315, 317–319, 321,
323, 352–353, 363, 365, 447, 451, 456
autoregressive (BAR), 226, 230–231
data, 59, 332, 353, 367–371, 383
moving average (BINMA), 417
Pareto lifetimes, 454
Binary segmentation, 224, 233
Binomial, 52, 54, 57–61, 73, 79, 98, 103–104, 118,
124, 128, 167, 183, 190, 209, 268, 352,
383, 388–389, 397, 411, 416, 448, 453
Bootstrap, 11, 15, 21, 88–89, 129, 133, 137,
191–192, 194, 209, 215, 373, 378
Branching process, 34, 396
Brownian bridge, 223, 230, 239
C
Calibration, 16–19, 189, 355–356, 360, 362, 430
Canonical parameter, 80, 123–124
Categorical, 4, 287
time series, 98, 270, 287–288, 291–292, 298
Central Limit Theorem, 56, 83, 127, 129, 137, 236,
239, 383
Change point, 219–224, 226–229, 231, 236, 395
459
460
Coherence, 311, 313–317, 319–321
Conditional distributions, 16–18, 43, 74, 87,
122–123, 125, 142, 145–147, 159,
169–170, 198, 205, 247, 251–252, 255,
270, 277, 329, 332, 351–352, 371–373,
377, 392, 436, 440–442, 455
Conditional independence, 130, 269–270,
329–331
Conditional predictive ordinate, surveillance
(SCPO), 398–400
Convolution-closed and innitely divisible
(CCID) distributions, 39, 44
Conway-Maxwell Poisson (CMP), 328, 330,
332–333, 337
Copula, 29–32, 41–44, 115–116, 409–411
Correlation
auto, 6, 12, 62, 103, 107, 149
cross, 65, 115
Kendall’s tau, 409
Coupling, 4, 87, 89, 94, 373
Covariate, 7, 12, 21, 23, 29–31, 38–46, 52–54,
56–57, 61, 118, 124, 126, 129, 138, 166,
175, 226, 246–250, 255, 259–260, 262,
271–274, 284, 298–300, 317–338,
358–359, 371, 376, 378, 383, 395, 412,
415, 419–420, 429, 436
Cumulative sum (CUSUM), 235–236
D
Decoding, 277–279, 282–283
Deviance information criterion (DIC), 176,
390, 431
DGLM, see Dynamic Generalized Linear Model
Diagnostics, 4, 12, 191, 193–199, 201–202, 209,
215–217, 246, 285, 398
Discrete autoregressive moving-average
(DARMA), 101, 448
D-vines, 43
Dynamical, 327, 334–336, 338, 346, 349, 354,
364–365
Dynamic Generalized Linear Model (DGLM),
167, 173, 175, 333–335, 426, 435
Dynamic model, 126, 167, 173, 333–335, 349, 426,
428, 431–432
E
e-chain, 85–86
EE, see Estimating equation
EF, see Estimating function
EM, see Expectation maximization
Index
Empirical orthogonal function (EOF), 335,
339–342, 362
Emulator, 349, 355–357, 360, 362–364
Ergodicity, 8–9, 22, 57, 87–92, 225–227, 229, 242
geometric, 90–92, 94, 96–97
Estimating equation (EE), 146–148, 152–154,
383, 448
Estimating function (EF), 147, 152, 155,
157–160, 222
combined, 146, 148, 155, 157, 159
linear, 146, 153, 156
Expectation maximization (EM) algorithm, 267,
274–277, 281, 284, 378, 382, 417
Monte Carlo, 129, 379
Exponential family, 52–53, 57, 73, 80–81,
127, 131–132, 147, 166, 328, 331, 376,
383, 426
canonical link, 69
one-parameter, 93, 123
Exponentially weighted moving-average
(EWMA), 235–236, 398
F
Feller
chain, 226
property, 84–87, 90, 94
Filtering, 121, 141, 146, 161–162, 248–249, 254,
256–257, 263, 306–307, 320–321, 431
Kalman, 132, 135, 154, 170, 390, 418–419, 426
particle, 168, 170–172, 176, 180, 263
Forecasting, 17, 31, 45, 74, 126, 141–142, 189–190,
200, 217, 233, 246, 255, 263, 277, 279,
285, 311, 328, 337, 339, 344, 355
Forward Filtering Backward Smoothing (FFBS),
246, 252, 255, 257, 431, 436
Forward probabilities, 272, 275–276
Fourier transform, 294, 300, 314
Fractional differencing, 450
G
Gamma, 5, 179, 199, 247–249, 252–255, 276, 419
function, 161, 223
prior, 390
process, 245
Gegenbauer exponential model, 456
Generalized autoregressive conditionally
heteroscedastic (GARCH), 5, 9, 79, 82,
150, 219–220
Generalized autoregressive score (GAS), 146,
152, 159–161
Index 461
Generalized linear autoregressive moving
average (GLARMA), 51–74, 425
asymptotic properties, 56–57
binary response, 53–54, 58–59
package, 52–54, 56, 58, 61–65
Generalized linear mixed model (GLMM), 63,
130, 327, 329, 331, 333, 335, 337, 339,
341, 343, 349, 364, 383, 397
Generalized Linear Model (GLM), 7, 9, 20, 22,
54–55, 126–128, 138–139, 166–167, 225,
327, 426
Geometric
decay, 103
distribution, 98, 102, 108–112, 123, 333,
417, 452
drift, 84
moment, 103
moment contracting, 83, 91, 93, 103
rate, 88, 92, 117, 137
Gibbs sampler, 140, 168–169, 180, 246, 251, 253,
255, 373, 375–376, 426, 431, 436–437
GLARMA
see Generalized Linear Autoregressive
Moving Average
GLM, see Generalized Linear Model
Goodness of t test, 4, 12–15, 19, 98, 202
H
Hájek-Rényi-type inequalities, 237, 239
Hammersley-Clifford Theorem, 368–369, 377
Hessian, 72, 95, 118, 129, 131, 133, 228
Heteroscedasticity, 32, 35, 37, 40, 45, 79, 219
Hidden Markov Model (HMM), 146, 267–284,
396–397, 429
multivariate count time series, 421
Hierarchical model, 167, 172–173, 334, 350,
373, 389
Bayesian, 125, 327–330, 337, 339, 343–344,
357, 363, 373–374, 383, 390, 397, 401
dynamic generalized linear mixed model,
327–344
dynamic linear model (HDLM), 426, 430
dynamic spatio-temporal, 349–353
multivariate dynamic model, 432
HMM, see Hidden Markov Model
Hyperparameters, 166, 173, 179, 181, 183, 331,
339, 352–357, 431, 435
Importance sampling, 126–127, 129, 132–135,
140–141, 144, 170, 246, 374, 418
approximate (AIS), 134, 139
numerically accelerated (NAIS), 135
Innite divisibility, 30, 39
Information matrix
conditional, 10
Fisher, 374, 379
Godambe, 147
test, 199–200, 205
Integer Autoregression (INAR), 32, 34, 38,
190–217, 411–414
bivariate (BINAR), 415–417
multivariate (MINAR), 412–415
multivariate generalized (MGINAR), 412
Poisson (PINAR), 190–212
Integer autoregressive moving-average
(INARMA), 36, 40, 101, 149, 412, 448
Integer moving-average (INMA), 36, 40
Bivariate (BINMA), 417
Integer-valued Generalized Autoregressive
Conditionally Heteroscedastic
(INGARCH), 79, 146, 150–151, 161,
419–420
Integrated Nested Laplace Approximation
(INLA), 168, 172–176, 180–184, 328,
383, 393, 440
Interaction, 311, 315–316, 318–323, 336, 350, 352,
354–355, 389–390, 397
Interval censored, 272
Intervention, 21–22, 64, 72, 98, 220, 397
Invariance principle, 237, 239–242
Inverse Gaussian distribution, 201
Inverse probability transform, 43
Ising model, 369
K
Kalman recursions, 121
Kullback-Leibler divergence, 42
L
Laplace approximation, 68–73, 127, 129,
131–133, 140, 174
Lasso, 383
Latent
factor model, 418
Gaussian model, 172–174
process, 21, 23, 125–128, 131–133, 138–140,
269, 284, 328, 330, 337–338, 340,
419, 434
state, 166–167, 175
Least squares, 228, 415, 417
conditional (CLS), 32, 37, 44, 157, 417
I
462
Lifetime, 101–104, 107, 109, 112, 114–117,
450–452, 454
Likelihood function
complete data, 275–276
composite, 44, 119, 127, 135–139,
416–417, 421
GLARMA, 54–55
HMM, 271–273
penalized quasi (PQL), 127, 139
Poisson, 9, 11, 184
quasi, 16, 22, 117, 119, 139
Likelihood ratio, 219, 375, 379
statistic, 64, 72, 220, 222
test, 55–56, 66, 71–72, 133, 220, 319
Link function, 7, 53, 123–124, 128, 166–167, 170,
176, 183, 225, 331–332, 383, 415
canonical link, 53, 69, 123–124, 131, 225
Logistic regression, 167, 311, 317–323, 332,
373, 378
Log-linear models, 7, 167
Long memory (Long range dependence), 103,
105, 107–108, 117, 161, 447–456
M
Markov
chain, 43, 78–81, 83–86, 88, 90, 96, 116, 190,
226, 268–271, 273, 275–278
ergodic chain, 88
latent, 268–270, 278, 284
model, 29–33, 35, 41, 44–45, 47, 97, 252–253
order, 32–34, 39–46, 78, 97, 331, 338
process, 89
structure, 267–271, 370
Markov chain Monte Carlo (MCMC), 12, 141,
165, 169–170, 176–182, 246, 251–253,
255, 339, 352, 360, 372–375, 389–390,
393, 431, 440
Markov Random Field (MRF), 349–352, 368–370
Gaussian (GMRF), 173
Markov switching, 269
Martingale, 200, 318
Martingale difference, 53, 55–56, 94, 147–149,
152–155, 159, 161
Maximum likelihood estimates
asymptotic properties, 57, 64, 129
EM algorithm, 275–277
GLARMA, 64, 67
HMM, 274–276
pseudo, 372–373, 378, 382–383
quasi, 16, 22, 127–128
Index
Metropolis-Hastings (MH) sampling, 129, 140,
169–172, 246, 251, 374, 426, 431,
436–437
random walk, 443
Metropolis sampling
adaptive random walk, 170–172, 176, 180
Missing data, 74, 272, 337, 393, 437
Mixing, 87–89, 92, 135, 137, 169, 225–226, 228,
237, 241, 409
strongly, 87–88, 137
Mixture models
Poisson-Gamma, 425
Model selection, 160
Akaike Information Criterion, 59, 65,
205–207, 209, 278–281, 321
HMM, 278–279
pseudo residuals, 278–279
Model validation, 189, 207, 215, 217, 379
Moments, 146, 148–152, 161, 273, 300, 329, 447
Monte Carlo, 127, 129, 133–134, 139–141, 192,
230, 252
Moving-average (MA), 36, 44, 412, 449, 455
Multinomial, 34, 171, 245, 270, 292, 332, 394, 442
Multivariate counts, 22, 31, 113, 246, 253–255,
258, 268, 270, 407–408, 411, 432
Multivariate Dynamic Finite Mixture Model,
426, 436
N
Negative binomial, 5–11, 13, 15–16, 18, 29, 47,
52–54, 73, 93–94, 98, 123–124, 127–128,
150–151, 167, 190, 248–250, 254–255,
269, 276, 278, 330, 417, 426, 452
Negpotential function, 368–369
Newton-Raphson, 54–55, 64, 69, 129–131, 379
Monte Carlo, 129, 139
Nonlinear model, 11, 18, 20, 80, 94, 98, 153, 167
general quadratic (GQN), 335
Nonstationary, 147, 227, 257
O
Observation driven models, 7–8, 52–53, 145,
245, 421
Observation equation, 21, 166–167
Overdispersion, 30, 32, 35, 37–38, 44–45, 150,
189, 199, 207–209, 268–269, 273, 280
P
Panel data, 246
Parameter driven models, 22–23, 78, 123, 145,
246, 269
463 Index
Pareto distribution, 451–452, 454
Pareto lifetimes, 107
Partial sum process, 226
Periodicity, see Seasonality
Periodogram, 289, 294–295, 300, 302, 315,
321–322; see also Spectral density
Perturbation method, 90–91, 94–96
Poisson, 29–47, 123–125, 128, 136, 167, 190, 209,
230, 245–249, 255, 257–258, 263,
268–273, 276, 278, 280, 283, 327–328,
330–333, 337–338, 340–344, 346–347,
388–389, 395, 397–398, 402, 426,
451–452, 455,
conditional, 31, 455
generalized, 29–47, 206, 208–214
marginal, 105
model, 180, 246–251
multivariate, 107, 432
probabilities, 107
process, 5, 77, 79–81, 87, 96–97, 247
Poisson-gamma, 425
Posterior distribution, 168–170, 250–259, 375
Prediction, 16–17, 29, 59, 117–118, 121, 145–146,
202, 204–205, 249, 277, 327, 339–342,
356, 372, 374–376, 382–383, 390,
396–397, 415–416, 426, 431, 437
density, 141, 171
error, 117
root mean square prediction error (RMSPE),
45–46
Principal components, 287, 291, 293,
305–306, 362
Prior distribution, 166–168, 175, 248–256
Probability integral transform (PIT), 16–18, 59,
201–203
histogram, 17–18
Q
Qualitative time series, 98, 270, 287–288,
291–292, 298
Quantile-quantile (QQ) plot, 201, 279, 281–282
Quasi-binomial, 40, 42, 190
R
Random coefcient thinning, 33, 40
Random effects, 62, 184
Random iterative approach, 81–83
Random walk, 450
Recursions, 57, 153
Regression time series model, 124, 142
Renewal Process and series, 448, 450–453
ARMA based approaches, 119
binary sequence, 102–103
bivariate process, 113–114
count process, 111, 117–118
long memory, 451
probabilities, 102
process, 101–102
theorem, 102
Resampling
see also Bootstrap, 171, 192–195, 203
Residual, 12–14, 196, 397
component, 197–199
Pearson, 12, 53, 65, 196–197, 206, 209,
215, 217
pseudo, 277, 279
quantile, 278–282
score-type, 53
Residual analysis, 195–199, 246, 323
Residual coherence, see Coherence
R-INLA, 174–175
Robust estimation, 22
S
Score
binary model, 226–227
statistic, 160–161, 221–223, 238, 241
test, 21, 160
Score function, 152–153, 159
quasi, 147
Scoring rules, 18–20, 204–207
Fisher, 55
Seasonality, 45, 118, 166, 183–184, 197–198, 200,
205–206, 213, 260, 262, 271, 456
Self-generalizable distribution, 35, 37
Sequential Monte Carlo (SMC), 170–180
Serial correlation, 56, 73
Shannon Entropy, 204
Sharpness, 18
Single block inference, 167–168
Site philopatry, 339–340, 344
Smoothing, 170, 252, 263
Spatial
lattice, 367
neighborhood, 399
random effects, 334
Spatio-Temporal, 349–353, 355, 357, 359, 361,
363–365, 367–384
dynamical (DSTM), 334–336
Spectral density, 191, 291–292, 299–300, 305,
307, 312
Spectral density matrix, 294, 305
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