Hidden Markov and Other Models for Discrete-valued Time Series

Transcription

1 Hidden Markov and Other Models for Discrete-valued Time Series Iain L. MacDonald University of Cape Town South Africa and Walter Zucchini University of Gottingen Germany CHAPMAN & HALL London Weinheim l New York Tokyo Melbourne Madras

2 Contents Preface xiii PART ONE Survey of models 1 1 A survey of models for discrete-valued time series Introduction: the need for discrete-valued time series models Markov chains Saturated Markov chains A nonhomogeneous Markov chain model for binary time series Higher-order Markov chains The DARMA models of Jacobs and Lewis Models based on thinning Models with geometric marginal Models with negative binomial marginal Models with Poisson marginal Models with binomial marginal Results not based on any explicit distributional assumption The bivariate geometric models of Block, Langberg and Staffer Moving average models with bivariate geometric distribution Autoregressive and autoregressive moving average models with bivariate geometric distribution Markov regression models Parameter-driven models State-space models 45

3 x CONTENTS 1.10 Miscellaneous models Discussion 49 PART TWO Hidden Markov models 53 2 The basic models Introduction Some theoretical aspects of hidden Markov models in speech processing Hidden Markov time series models: definition and notation Correlation properties The autocorrelation function of a Poissonhidden Markov model The autocorrelation function of a binomialhidden Markov model The partial autocorrelation function Evaluation of the likelihood function Distributional properties Marginal, joint and conditional distributions of the observations The Markov chain conditioned on the observations Runlength distributions for binary hidden Markov models Parameter estimation Computing maximum likelihood estimates Asymptotic properties of maximum likelihood estimators Use of the parametric bootstrap Identification of outliers Reversibility Discussion Extensions and modifications Introduction Models based on a second-order Markov chain Multinomial-hidden Markov models The likelihood Marginal properties and cross-correlations "A model for categorical time series Multivariate models 121

4 CONTENTS xi The likelihood function for multivariate models Cross-correlations of models assuming contemporaneous conditional independence Cross-correlations of models not assuming contemporaneous conditional independence Multivariate models with time lags Multivariate models in which some variables are discrete and others continuous Models with state-dependent probabilities depending on covariates Models in which the Markov chain is homogeneous but not assumed stationary Models in which the Markov chain is nonhomogeneous Joint models for the numbers of trials and the numbers of successes in those trials Discussion Applications Introduction The durations of successive eruptions of the Old Faithful geyser Markov chain models Hidden Markov models Comparison of models Forecast distributions Epileptic seizure counts Births at Edendale hospital Models for the proportion Caesarean Models for the total number of deliveries Conclusion Locomotory behaviour of Locusta migratoria Multivariate models ' Univariate models Conclusion Wind direction at Koeberg Three hidden Markov models for hourly averages of wind direction Model comparisons and other possible models Conclusion Evapotranspiration 177

5 xii CONTENTS 4.8 Thinly traded shares on the Johannesburg Stock Exchange Univariate models Multivariate models Discussion Daily rainfall at Durban Homicides and suicides, Cape Town, Models for firearm homicides as a proportion of all homicides, suicides and legal intervention homicides Models for the number of firearm homicides Firearm homicides as a proportion of all homicides, and firearm suicides as a proportion of all suicides Models for the proportions in each of the five categories of death Conclusion 201 Appendices 203 A Proofs of results used in the derivation of the Baum-Welch algorithm 203 B Data. 207 References 217 Author index 227 Subject index 231