Regression Theory for Categorical Time Series

Categorical---or qualitative---time series data with random time-dependent covariates are frequently encountered in diverse applications as the list of examples shows. As with "ordinary'' time series, the data analyst is faced with the same problems of modeling, estimation, model checking, diagnostics and prediction. The present work shows that these questions can be attacked by means of regression theory for categorical time series whose foundation is based on generalized linear models and partial likelihood inference. A variety of models are provided to illustrate the selection of the link function and recent large sample results are reviewed. The theory is developed without resorting to the Markov assumption and to the notion of stationarity. Moreover, regression methods for categorical time series allow for parsimonious modeling and incorporation of random time-dependent covariates as opposed to other procedures. In particular, nominal and ordinal time series are analyzed and compared empirically to Markov chains and mixture transition distribution models

Prediction and Classification of Non-stationary Categorical Time Series

Partial likelihood analysis of a general regression model for the analysis of non-stationary categorical time series is presented, taking into account stochastic time dependent covariates. The model links the probabilities of each category to a covariate process through a vector of time invariant parameters. Under mild regularity conditions, we establish good asymptotic properties of the estimator by appealing to martingale theory. Certain diagnostic tools are presented for checking the adequacy of the fit

A stochastic approximation algorithm for the adaptive control of time series following generalized linear models

A recursive estimation method for time series models following generalized linear models is developed in two ways. The estimation procedure, suitably modified, gives rise to a stochastic approximation scheme. We use the modified estima‐tion procedure to illustrate a connection between control theory and generalized linear models by employing a logistic regression model

Partial likelihood inference for time series following generalized linear models

The present article offers a certain unifying approach to time series regression modelling by combining partial likelihood (PL) inference and generalized linear models. An advantage gained by resorting to PL is that the joint distribution of the response and the covariates is left unspecified, and furthermore, PL allows for temporal or sequential conditional inference with respect to a filtration generated by all that is known to the observer at the time of observation. Two real data examples illustrate the methodology

Predicting precipitation level

We present a generalized logistic regression model for the statistical analysis of multicategorical time series. The model is suitably parameterized and partial likelihood inference is proposed for estimation of the unknown parameters. A goodness of fit statistic is derived to judge the quality of fit. The analysis is applied to data from the Tropical Ocean and Global Atmosphere/Coupled Ocean‐Atmosphere Response Experiment

Statistical comparison of algorithms

A "reference" algorithm or instrument and its various "distortions" are considered, where the distortions carry some valid information about the reference. The objective is to combine data from the reference and the distortions together in some manner in order to extract information from both the reference, as well as the distortions, and produce improved inference about the true reference algorithm. This is illustrated in terms of m precipitation radars and semiparametric estimation of the reference distribution and the distortion parameters