Essays on Bayesian Modeling and Analysis of Time Series of Counts
Open AccessIn this dissertation, we propose new models and efficient algorithms for time series of counts. The first essay presents a new class of integer-valued autoregressive processes by combining integer autoregression and innovations that assume a state-space evolution. We term our new model Latent Factor Integer Autoregressive Process of Order One and further introduce its multivariate extension via a random environment for modeling component correlations. The probabilistic structure of the model features an exact Gibbs sampler and a Particle Learning algorithm for inference. Using data on the weekly number of visits to grocery stores, we show that our model outperforms competing models in out-of-sample one-step-ahead forecasts. The second essay proposes two models for zero-inflated count series and presents their Bayesian analysis. The first model augments the Latent Factor Integer Autoregressive Process from the first essay with zero-inflated Poisson innovations that allow the model to handle sparse count series, and we devise an analytic Gibbs sampler for inference. The second model considers a multivariate first-order integer-valued autoregressive process with zero-inflated Poisson innovations, where the probability that the innovations come from a Dirac distribution at zero varies according to a state space dynamic. The time-varying mixing probability for the innovations equips the model with extra flexibility to handle count series whose degree of sparsity changes over time. We name the model Dynamic Zero-inflated Integer Autoregressive Process of Order One and introduce its multivariate generalization. For inference, we present an analytic Gibbs sampler and an Approximate Particle Learning algorithm for online updates. Using actual data, we show that the Dynamic Zero-inflated Integer Autoregressive process substantially outperforms the univariate integer autoregressive process with standard zero-inflated Poisson innovations, and the multivariate generalization brings in additional performance lift by leveraging information from correlated series.
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