Regression Models for Time Series Analysis. Benjamin Kedem, Konstantinos Fokianos Copyright 0 2002 John Wiley & Sons, Inc. ISBN. 0-471-36355-3 Index Adaptive rejection sampling, 233 Adjacent categories logits, 98 Airline passenger study, seasonal time series, 278-281 Akaike s information criterion (AIC) applications: binary time series, 73-74, 79 categorical time series, 115, 117 generalized linear model (GLM), 23, 25, 35 mixture transition distribution (MTD) model, 293 Alternative approachesimodels: autoregressive conditionally heteroscedastic (ARCH) models, 199-20 1 discrete autoregressive moving average (DARMA) models, 189-190 hidden Markov models, 194-1 97 integer autoregressive and moving average models, 175-1 89 longitudinal data, mixed models for, 205-208 mixture transition distribution (MTD) model, 19&194 sinusoidal regression model, 201-205 variable mixture models, 197-1 99 Alternative modeling, categorical time series, 125 Anscombe residuals, 26 ARCH( 1 ) model, 199-200 Asymptotic distribution: categorical time series, 107-108, 112, 115 generalized linear model (GLM), 16-24, 27 quasi-partial likelihood, 28 Asymptotic relative efficiency (ARE), binary time series, 6465 Asymptotic theory: categorical time series, 130-137 count time series, 156 generalized linear model (GLM), 1&20 Autocorrelation estimation, 294295 Autocorrelation function (acf): binary time series, 60-62, 75 categorical time series, 94-96, 10&101, 118 count time series, 143, 145, 147-148, 150, 152 hidden Markov models, 196 integer moving average (MMA) models, 186 stationary processes, 287-288 Autocovariance function: integer autoregressive models, 184 stationary processes, 286, 289, 292 Autoregressive moving average (ARMA): count time series, 141 Gaussian, 257-258 linear state space models, 2 17 prediction and, 249 327
~~ 328 INDEX Autoregressive moving average (ARMA) (continued) process, 294 Autoregressive process, binary time series, 53 Bayesian forecasting, 2 15 Bayesian inference, 232 Bayesian information criterion (BIC) applications: binary time series, 73-75,79 categorical time series, 115, 117 generalized linear model (GLM), 23,25, 35-36,39,41 mixture transition distribution (MTD) model, 293 Bayesian mixture models, 237 Bayesian spatial prediction: auxiliary Gaussian process, 258,260 likelihood, 260-262 model parameters, prior and posterior, 262-263 transformations, normalizing, 265 Z, Prediction, 263-264 Bayes theorem, 82,223, 225, 230,237-238, 262 Bernoulli data, 10 Bernoulli random variables, integer autoregressive models, 178-1 80 Best linear unbiased predictor (BLUP), 252 Binary time series: characteristics of, generally, 9-10 goodness of fit, 65-69 inference, for logistic regression, 59-65 link functions, 50-56 partial likelihood estimation, 56-59 real data examples, 70-80 regression models, 49-50 state space model, 228 Binomial data, 10 Binomial thinning, 178, 188 Bivariate exponential autoregressive moving average models, 190 Box-Cox power transformation: Bayesian spatial prediction, 265, 267 generalized linear model (GLM), 7 Box-Pierce portmanteau statistic, 27 Branching process: with immigration, 173, 175-178 size-dependent, 154 BTG algorithm: defined, 264-265 kriging, comparison with, 274 seasonal time series, 278-281 spatial rainfall prediction, 267-273 time series prediction, 274-277 BTG program, 265 Buffon s needle experiment, 23 1 Bum-in period, 233 Canonical links/parameters: count time series, Poisson model, 142, 148, 156 generalized linear model (GLM), 5, 8 partial likelihood inference, 13 Categorical time series: alternative modeling, 125 asymptotic theory, 130-137 characteristics of, 89-90 data examples, 11 6124 goodness of fit, 110-1 15 link functions, 92-101 longitudinal data, 125 modeling, 90-92 partial likelihood estimation, 101-1 10 spectral analysis, 125 Central limit theorem, 3, 18, 130-131 Chi-square: approximation, 24 asymptotic, 207 distribution,68, 110-111, 113, 117, 158-159 Neyman-modified statistic, 113 random variable, 2 I, 23 Cholesky square root, 130, 134 Complementary log-log (cloglog), 10, 54, 73-74 Composite hypothesis, 20 Computation, generalized linear model example, 33 Computer software: btg program, 265-267 GMTD, 192 MTD, 192 SAS, 33 S-PLUS, 33,203,223 state space models, 223 Conditional covariance matrices:
INDEX 329 binary time series, 60 categorical time series, 105, 130 Conditional density: count time series, 140-141 doubly truncated Poisson, 141 Gaussian, 219-220 generalized linear model (GLM), 9 Conditional distribution: Gaussian, 225 INAR(2), 184 simulation-based state space models, 235 variable mixture models, 198-199 Conditional expectation: categorical time series, 114 count time series, 140 Conditional Gaussian marginals, 8 Conditional inference, 1,4, 12, 206 Conditional information matrix: binary time series, 58 categorical time series, 10&107, 109 generalized linear model (GLM), 14-15, 17 Conditional likelihood, 3, 108 Conditional linear autoregressive process (CLAR(I)), 188-189 Conditional maximum likelihood estimates, integer autoregressive models, 184 Conditional mean, count time series, 142 Conditional probability: binary time series, 5 1 success, 49 Conditional variance: count time series, 157 working, 31 Confidence bands, 123 Conjugate analysis, nonlinear and non- Gaussian state space models, 228-230 Continuation ratio model, 98 Contraction mapping (CM): algorithm, 203-205 method, 202-203 Contraction parameter, 203 Correlation function, 286 Count time series: asymptotic theory, 156 characteristics of, generally, 139 data examples, 159-1 67 doubly truncated Poisson model, 148-152 goodness of fit, 159 hypothesis testing, 158 inference, 154-1 58 link functions, 142-152 modeling, 140-142 over-dispersion, 3&3 1 partial likelihood estimation, 154-156 Poisson model, 142-147, 154-156 prediction intervals, 157 Zeger-Qaqish model, 153-154, 157-1 58 Covariance function: in spatial prediction, 25 1 stationary processes 286 Covariance matrix: generalized linear model (GLM), 20 linear Gaussian state space models, 219, 222 mixed models, 207 Covariate(s), generally: in generalized linear model (GLM), 1 matrix, categorical time series, 107 process, defined, 5 random time dependent, 3,90 vector, see Covariate vector Covariate vector: binary time series, 50, 56 count time series, 153 in generalized linear model (GLM), 6,28 Covariogram, 251-252 Cox s proportional hazards, 54 Cramtr-Wold device, 130, 132, 136 Cross-correlation, categorical time series, 96, 101 Cross-validation studies, 267-268,273, 275,277-278,280 Cumulative conditional information matrix, 12 Cumulative distribution function (cdf): binary time series, 5 I, 54 categorical time series, 97-98 generalized linear model (GLM), 9 Cumulative odds, categorical time series, 92-93 Data examples, see specific fypes of regression models Degrees of freedom: binary time series, 68, 71
INDEX Degrees of freedom (continued) categorical time series, 110, 112-1 13, 117 count time series, 158-159 generalized linear model (GLM), 35 mixed models, 207 Delta method, 59, 108 Densityidensities: conditional, 3,9, 14C141, 219-220 probability, 7, 56, 231-232,265,286, 289 Dependence: binary time series, 53 Markov, 2-3 nonhomogeneous Markovian, 7 in partial likelihood, 2, 10 Deviance: binary time series, 65 count time series, 159-160, 164 generalized linear model (GLM), 23-25 partial likelihood inference, 14 Deviance residuals, 25-26 Diagnostics: categorical time series, 115 generalized linear model (GLM), 23-28 Discrete autoregressive moving average (DARMA), 125, 189-190 Dispersion parameter: defined, 6 estimation of, 14 DNA sequence data analysis: categorical time series example, 1llsl19 mixture transition distribution model, example of, 194 Double chain Markov model, 191 Doubly truncated Poisson model, count time series, 141, 148-152 Dynamic generalized linear models (DGLM): characteristics of, 227-228 conjugate analysis, 228-229 defined, 227 formulation of, 228 linear Bayes estimation, 228-230 posterior mode estimation, 23@-231 EM algorithm, 222-223 Empirical likelihood, 4 Ergodic binary time series, 5 1 Estimation, generally: autocorrelation, 294-295 GARMA models, 7 linear Gaussian state space models, 218-223 Existence, maximum partial likehood estimators, 17 Exponential autoregressive moving average (EARMA), 188 Exponential autoregressive process of order 1 (EAR(I), 188 Exponential correlation, 251, 267,271, 275 Exponential dispersion model, 7 Exponential families, nonlinear and non- Gaussian state space models, 228-230 Exponential moving average (EMA( I)), 188 Extensions: autoregressive conditionally heterscedastic (ARCH) models, 20&201 integer autogressive and moving average models, 188-189 Fast Fourier transform (FFT), 202,204 Filtered data, 34-35 Fi I tering: Kalman, 2 18-2 19 Monte Carlo method, 227 nonlinear and non-gaussian state space models, 225-227,23 1 Finite dimensional distribution, 286 First-order Markov chains, 90 First-order Taylor expansion, 27 Fisher scoring, 12, 14-15, 155 Fixed effects, in mixed models, 205 FORTRAN 77,265 Fourier frequencies, 202, 204 Full likelihood, binary time series, 64-65 Galton-Watson process, 173, 175 Gamma, generally: defined, 5 distribution, 188 Gaussian ARMA, 184 Gaussian distribution, 227,23 1 Gaussian process, 286 Gaussian random fields, 253-255
INDEX 33 1 Gaussian-sum filter, 227 Gaussian white noise, 27-28, 36 Generalized ARCH (GARCH) models, 2ock201 Generalized Autoregressive Moving Average (GARMA), 6 Generalized estimating equations (GEE) method, 31-33 Generalized linear models (GLM): applications of, generally, 1 asympotic theory, 16-20 data examples, 33-42 defined, 5 diagnostics, 23-28 hypotheses testing, 2&23 partial likelihood, 14, 10-16 quasi-partial likelihood, 28-33 random component, 5-6 systematic component, 5-8 time series and, 4-1 0 Generalized Linear Autoregressive Moving Average (GLARMA), 6 Geometric autoregressive moving average models, 190 Gibbs sampling algorithm, 232-234,237 Goodness of fit: binary time series, 65-69, 7 I categorical time series, 110-1 I5 count time series, 159 Herglotz, G., 289 Hidden Markov models, 125, 194108 Hyperparameters: linear Gaussian state space models, 22 1 nonlinear and non-gaussian state space models, 225 simulation-based state space models, 235 Hypotheses testing: categorical time series, 109-1 10 count time series, 158 generalized linear model (GLM), 20-23 Importance, sequence sampling (SIS), 238-240 INARb), 184185,188 Independence, generally: binary time series, 52 of irrelevant alternatives, 93 Inference: Bayesian, 232 conditional, 1, 12,206 count time series, 154-158 likelihood, 240 logistic regression, binary time series, 59-65 Markov Chain Monte Carlo for state space models, 233-237 mixture, 206 nonparametric, 4 partial likelihood, see Partial likelihood Infinite moving average, 293 INMA(q), 185-1 86, 188 Integer autoregressive models, regression analysis of, 185 Integer autoregressive models of order I (INAR( I)): characteristics of, 178-1 83 estimation for, 183-1 84 Integer autoregressive models of order p (INAR(2)), 184185 Integer autoregressive moving average (INARMA), 188 Integer moving average (TNMA) models, 185-188 Integrated GARCH (IGARCH) models, 201 Interpolation, kriging, 257, 259 Intervals: categorical time series, 93 prediction, 20 Inverse Gamma distribution, 234 Inverse link function, categorical time series, 92 Isotropic correlation function, 25 1 Iterative reweighted least squares, 15-1 6 Kalman, R. E., 214 Kalman filter/filtering: defined 2 14 in linear Gaussian state space model estimation, see Kalman filtering, linear Gaussian state space model estimation in space-time data, 241 simulation-based state space models, 236 Kalman filtering, linear Gaussian state space model estimation: characteristics of, 2 18-2 19
332 INDEX Kalman filtering, linear Gaussian state space model estimation (continued) structural model, 222-223 Kalman prediction, 2 18 Kalman smoothing, linear Gaussian state space model estimation, 2 19-22 1 Kriging: BTG algorithm compared with, 274 trans-gaussian, 258,260, 274-275 variance, 256 Large sample theory, categorical time series, 107-108 Left truncated Poisson distribution, 141 Linear Bayes estimation, nonlinear and non- Gaussian state space models, 228-230 Linear Gaussian state space models: characteristics of, 2 15 estimation, generally, 221-223 estimation, by Kalman filtering and smoothing, 218-22 1 examples of, 216217 Linear mixed-effects model, 207-208 Linear predictor, 6 Linear regression: classical, 5, 8, 265 ordinary, 1 Link functions: binary time series, 5&56 categorical time series, 92-1 01 count time series, 142-1 52 generalized linear model (GLM), 6, 8 Log-likelihood: estimation, mixture transition distribution (MTD) model, 192 linear Gaussian state space models, 22 1 Log-linear model: count time series, 142-144, 146, 153, 156 generalized linear model (GLM), 7 Log-partial likelihood: binary time series, 7 1 categorical time series, 102, 105 count time series, 155 generalized linear model (GLM), 10, 12, 19,21-22,40 ratio test, 40 Logistic autoregression model, binary time series, 53-54,6&62 Logistic model, 9 Logistic regression: binary time series, 52-53 generalized linear model (GLM), 9,39, 41,53 Logi t( s): adjacent, 98 defined, 53 multinomial, 92-96, 108-109 Longitudinal datdstudies: categorical time series, 125 generalized estimating equations (GEE), 31 linear mixed models for, 205-208 simulation-based state space models, 24 1 Marginal partial likelihood, 4 Markov assumption, categorical time series, 108 Markov chain(s): branching processes, 175 binary time series, 73 categorical time series, 89-90, 110, 125 generalized linear model (GLM), 41 hidden, 194-197 homogeneous, 194 mixture transition distribution (MTD) model, 190-194 Markov Chain Monte Carlo (MCMC) techniques: applications, generally, 221, 225 defined, 232 Gibbs sampling algorithm, 232-234, 237 inference, for state space models, 233-237 Metropolis-Hastings algorithm, 233 Monte Carlo simulation techniques, 231-232 Markov dependence, 2 Markov process, 7, 182,225 Markov s inequality, 132, 135 Martingale(s): binary time series, 57, 67--68 categorical time series, 114 Central Limit Theorem, 67, 13&13 1 difference, 176 generalized linear model (GLM), 3, 18, 26,29
INDEX 333 Matern correlation, 25 I, 267,271,273-275, 278 Maximum distribution function, categorical time series, 98 Maximum likelihood estimation (MLE): asymptotic properties of, 3 autoregressive conditionally heterscedastic (ARCH) models, 200 hidden Markov models, 197 WAR( 1) process, 183-1 84 linear Gaussian state space models, 223 mixed models, 206 simulation-based state space models, 240 Maximum partial likelihood estimator (MPLE): binary time series, 57, 59, 63-65 categorical time series, 105, 107, 110, 114, 118, 130, 135-136 count time series, 156 generalized linear model (GLM), 4, 12, 16-19,24,29 Mean response model, categorical time series, 98 Mean square error (MSE), 72 Mean square prediction error (MSE), 274 Metropolis-Hastings algorithm, 233 Minimal distribution function, categorical time series, 98 Minimum eigenvalue, categorical time series, 131 Mixing distribution, 206 Mixture inference, 206 Mixture models, see Bayesian mixture models; Variable mixture models Mixture transition distribution (MTD) model: characteristics of, 190-19 1 data examples, 192-1 94 estimation in, 192 Model(s), generally: adequacy, binary time series, 68 building, 33-34 selection, 25 Modeling: categorical time series, 90-92 count time series, 140-142 Modifications, autoregressive moving average models, 188-1 89 Monte Carlo, generally: algorithm, Bayesian spatial prediction, 264 Markov Chain techniques, 22 1 Moving average: discrete autoregressive models, 189-1 90 generalized autoregressive (GARMA), 6 generalized linear autoregressive (GLARM), 6 generalized linear autoregressive moving average (GLARMA), 6 integer models, 185-188 Multi-matrix mixture transition distribution (MTDg) model, 191, 192 Multinomial distribution, integer autoregressive models, 182, 184 Multinomial logit, categorical time series: defined, 92-93 inference for, 108-1 09 nominal time series model, 93-94 with periodic component, 9&96 Multinomial thinning, 178, 183 Multiplicative error model, count time series, 153 Multiplicative model, count time series, 153 Multivariate, generally: functions, 102 generalized linear model, 92, 130 normal distribution, 219,251,287 state space models, 24 1 NASA, Apollo space program, 214 Natural parameter, in generalized linear model (GLM), 5-7 Neural network prediction models, 249-250 Newton-Raphson procedure, 12, 14-1 5 Nominal categorical time series, 93-96, 1 16 Non-Gaussian fields, 256 Non-Gaussian processes, 29CL29 1 Nonhomogeneous Markovian dependence,7 Nonlinear filters, 249 Nonlinear least squares, quasi-partial likelihood, 30 Nonlinear transformation, 258 Non-Markovian processes, 4 Nonnegative definite, 89 Nonparametric prediction, binary time series, 5 1 Nonstationary binary time series, 52
334 INDEX Nonstationary response, 49 Normal distribution: binary time series, 61 categorical time series, 115 generalized linear model (GLM), 8, 18, 29 multivariate, 219,251, 287 simulation-based state space models, 236 stationary processes, 286-287 Null hypothesis, 21-23,25, 71 Observation equation, 213,228 Observed information matrix, 58 Old Faithful eruptions: binary time series example, 72-76 mixture transition distribution model, example of, 192-193 One-step prediction errorshnnovations, 22 1 Ordinal categorical time series, 93, 97--101, 120-121 Oscillation: binary time series, 50 count time series, 163 Over-dispersion: in count data analysis, 3&3 1 count time series, 158 quasi-partial score, 29 Parametric hazard models, 54 Parsimonious modeling, 90 Partial likelihood: binary time series, 5659, 61 characteristics of, 1-3 count time series, Poisson model, 154-156 defined, 3 4 estimation, see Partial likelihood estimation inference, 10-1 6 quasi-, 28-33 ratio test, see Partial likelihood ratio test variable mixture models, 198 Partial likelihood estimation: binary time series, 5C59 categorical time series, 101-1 10 count time series, 15&156 generalized linear model (GLM), 33 Partial likelihood ratio test: categorical time series, 110, 112-1 13 count time series, 158 Partial score: categorical time series, 105-106, 110, 130 count time series, 158 generalized linear model, 1 I, 14,28 Pearson goodness of fit statistic, 66, 110, 112-1 13, 159, 164 Pearson residuals: autocorrelation of, 2C27 binary time series, 80 categorical time series, 115, 118, 123-124 count time series, 159-1 60, 165-1 66 generalized linear model (GLM), 25&27, 30,35,41 defined, 25 quasi-partial likelihood, 30 Periodic component: binary time series, 6&61 categorical time series, 94-96 count time series, 143-145, 148 Periodogram: binary time series, 79-80 categorical time series, 123-1 24 count time series, 166 sinusoidal regression model, 201-202 Permutation sampling, 221 Poisson distribution: integer autoregressive models, 180, 186 generalized linear model (GLM), 5,9,28 integer moving average (TNMA) models, 186 Poisson-gamma, 28 Poisson GARMA, 6 Poisson hidden Markov model, 195-196 Poisson INAR( I), 182 Poisson model, count time series: characteristics of, 141-147 doubly truncated 141-142, 148-152 partial likelihood estimation, 154-1 56 Poisson regression, 19-20,26,30-3 I, 34, 141 Positive Poisson distribution, 141 Posterior distributions, Bayesian spatial predictions, 262-263 Posterior mode estimation, nonlinear and non-gaussian state space models, 23&23 1
INDEX 335 Power divergence: categorical time series, 111-1 13, 118, 123 generalized linear model (GLM), 40,42 Power transformations, Bayesian spatial prediction, 265 Prediction: Bayesian spatial, 250, 258-267 BTG algorithm, applications of, 267-281 implications of, generally, 249-250 intervals, see Prediction intervals kriging algorithm, 252,256258 nonlinear and non-gaussian state space models, 227,231 ordinary kriging, 252, 256-258 stationary random fields, elements of, 25 1-252 Prediction intervals: binary time series, 59, 77 categorical time series, 108 count time series, 157 generalized linear model (GLM), 20 Predictive density: Bayesian spatial prediction, 260,264, 273,275,28 1 BTG algorithm, 264-265 simulation-based state space models, 240 Prior distributions, Bayesian spatial predictions, 262-263 Prior weight, 6 Probability density function (pdf) binary time series, 56 generalized linear model (GLM), 8 Monte Carlo simulation techniques, 23 1-232 Probability distribution, 265, 286, 289 Probit regression: binary time series, 55, 73 generalized linear model (GLM), 10,41 Proportional odds, categorical time series, 97,99-100, 121, 123 Pseudo-likelihoods, 4 Pulse function. I63 Quasi-partial likelihood: characteristics of, 28-29 generalized estimating equations (GEE), 31-33 nonlinear least squares example, 30 over-dispersion in count data, 30-3 1 Quasi-partial score, 29,3 1 Quasi-score, 157 Rainfall prediction, binary time series example, 70-72 Random walk method, 234 Rational quadratic correlation, 252, 275 Raw residuals: categorical time series, 1 I5 count time series, 165 generalized linear model (GLM), 25 Real-valued stationary time series, 294 Rejection sampling algorithm, 233 Residual matrix, 58 Residuals, see specific types of residuals analysis, binary time series, 74-75 categorical time series, 114 count time series, 159 generalized linear model (GLM), 25-28 Response classification: binary time series, 65-69 categorical time series, 111-1 12 Response residuals, 25 Response series, 4-5 Rice s formula. 290 Sample autocorrelation (sample ACF), 294-295 Sampling: adaptive rejection, 233 asymptotic theory, 17-20 permutation, 22 1 rejection algorithm, 233 sequential importance (SIS), 238-240 sequential Monte Carlo methods, 237-240 Scaled deviance: categorical time series, 110, 1 15 generalized linear model (GLM), 23-24 Score statistic, 21-22 Seasonal time series, BTG algorithm, 278-28 1 Second-order generalized estimating equations (GEE2), 33 Second-order stationary, 287 Self-exciting threshold autoregressive (SETAR) model, 198-199
INDEX Sequential importance sampling (SIS), 238-240 Sequential Monte Carlo sampling methods, 237-240 Simulation-based state space models: characteristics of, 231-232 likelihood inference, 240 longitudinal data, 241 Markov Chain Monte Carlo (MCMC), 232-237 Monte Carlo, generally, 23 1-232 sequential Monte Carlo sampling methods, 237-240 Sinusoidal component: binary time series, 62 categorical time series, 99 count time series, 143, 148 Sinusoidal regression models, 201-205 Size-dependent branching, 154 Sleep prediction: binary time series example, 79-80 categorical time series example, 121-1 24 Slow convergence, simulation-based state space models, 236 Slutsky s theorem, 136 Smoothed data, 34 Smoothing, generally: density, simulation-based state space models, 235 Kalman, 219-22 1 nonlinear and non-gaussian state space models, 225-226,23 1 Soccer forecasting: categorical time series example, 119-1 2 1 mixture transition distribution model, example of, 193-1 94 Spatial data analysis, 4 Spatial MTD model, 191 Spatial rainfall, prediction with BTG algorithm, 267-273 Spectral coherence, 34 Spectral density, stationary processes, 292, 294 Spectral distribution function, 289,291 Spectrum, zero-crossing rate and, 29&29 1 Spherical correlation, 25 I, 275 S-PLUS, 33,203,223 Square integrable zero-mean martingale, 133 Squared Pearson residuals, 115, 118 State space models: characteristics of, 213 defined, 2 13 historical perspective, 214-2 15 Kalman filtering, space-time data, 241 linear, 214 linear Gaussian, 215-223 non-gaussian, 223-23 1 nonlinear, 223-23 1 simulation-based methods, 231-241 State space representation, linear state space models, 216-2 17 Stationarity, 285-288 Stationary AR( 1) process, 291-293 Stationary AR(p) process, 293-294 Stationary binary time series, 51-52 Stationary Gaussian random field, 274 Stationary Gaussian time series, 295 Stationary in the wide sense, 287 Stationary processes, elements of: complex-valuated, 288-295 stationarity, 285-288 Stock price prediction, binary time series example, 76-78 Strict stationarity, 287 Structural time series: Kalman filtering, 222 linear state space models, 2 16 Systematic component, in generalized linear model (GLM), 5-8 System equation, 213 Taylor expansion, 27,68, 134135, 137 Testing data, binary time series, 68 Test statistics, 23, 30, 110 Thinning operation, integer autoregressive models, 178-180 Time series: prediction, BTG algorithm, 274-277 seasonal, BTG algorithm, 278-281 Tourist arrivals, count time series example, 139-1 40, 163-167 Training data, binary time series, 68 Trans-Gaussian kriging, 258,260, 274-275 Transition, generally: density, 240 probabilities, 5, 113-114, 194
INDEX 337 Two step response model, categorical time series, 98 Unconditional covariance matrix, categorical time series, 130 Unconditional information matrix: categorical time series, 106, 131 count time series, 155 generalized linear model (GLM), 12, 17 Unemployed women, Bayesian spatial prediction, 274277 Uniqueness, maximum partial likelihood estimators, 17-1 9 Univariate zero-mean martingale, 130 Variable length Markov chains (VLMC), 125 Variable mixture models: overview, 197-198 partial likelihood inference, 198 threshold models, 198-199 Variance components, in mixed models, 207 Variance function, 8 Vector INAR(I), 182-183 Volterra-type expansion models, 249-250 Wald statistic, 21, 23, 110, 158 Wald test, 23, 165 Wavelet methods, 124 Weakly stationary, 287 Weight, 6 Weighted least squares: estimation, 15 generalized linear model (GLM), 15-16 integer autoregressive and moving average models, 176, 185 White noise, 27-28, 75,214 Wide sense stationary, 290 Working covariance matrix, 32-33 Working residuals: binary time series, 72 count time series, 159 generalized linear model (GLM), 25-26, 35 Working variance, quasi-partial likelihood, 28 Yule-Walker equation, 19 I, 293 Z,, Prediction of, 263-264 Zeger-Qaqish model, count time series, 153-154, 157-158 Zero-crossing rate, 290-29 1 Zero-mean martingale, 68, 114 Zero mean square integrable martingale, 3, 130, 136