Statistical Analysis of Categorical Time Series through Sparse Markov Models

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Analysis of a sequence of categorical values is best done by a model that captures the statistical properties of the sequence, while being simple enough so that statistical analysis is feasible. In many cases, the analysis of such data has been simplified by the sequence having a short memory or Markov property in the sense that conditional probabilities depend on only the very recent past. Whereas Markov models are applied extensively, typically low-order models are fit because the number of estimated conditional probabilities grows geometrically as the number of past observations used for conditioning increases. Another drawback involves the lack of model flexibility, as the number of possible Markov models is limited. Sparse Markov models (SMM) help with these two problems, thus allowing better model fits. While theoretical results for Markov models are prevalent, those for SMMs are relatively rare, only being considered in the last decade. Thus, there is tremendous potential for the furthering of knowledge related to theory and applications of SMMs to analyze categorical time series. This is the fundamental aim of this project. Sparse Markov models allow the fitting of a parsimonious model that is also flexible enough so that a better trade-off is obtained between bias that arises from conditioning contexts that are shorter than truth and variance from having many parameters to estimate, thus improving inference. This project studies theoretical properties and applications of sparse Markov models; the special case of variable length Markov chains (VLMCs) to the analysis of categorical time series is considered. Related objectives are (i) To develop theory for prediction of data from SMMs, central limit theorems, model fitting through regularized regression, and comparisons of asymptotic and finite-sample properties of derived and current methods; (ii) To develop a new model called hidden sparse Markov models (HSMMs); (iii) To extend methods for efficient computation of distributions of pattern statistics in Markovian sequences to both sparse and hidden sparse Markov models; (iv) To apply VLMCs/SMMs and HSMMs to improve the analysis and inference of various categorical time series.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.