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Advancement: Mixture Transition Distribution Models for Non-Gaussian Temporal and Spatial Processes

Speaker Name: 
Xiaotian Zheng
Speaker Title: 
Ph.D. Student
Start Time: 
Friday, May 22, 2020 - 9:00am
End Time: 
Friday, May 22, 2020 - 10:00am

Abstract: Mixture transition distribution (MTD) time series models build high-order dependence through a weighted combination of first-order transition densities for each one of a specified number of lags. We will present a framework to construct stationary MTD models that extend beyond linear, Gaussian dynamics. We study conditions for first-order strict stationarity which allow for different constructions with either continuous or discrete families for the first-order transition densities, and with general forms for the resulting conditional expectations. Moreover, we develop conditions for weak stationarity which are substantially easier to implement than existing conditions that rely on solutions of equations involving complex functions of all model parameters. We will then extend the univariate stationary MTD to the setting of multivariate time series, with emphasis on modeling marked point processes. In particular, we propose a copula multiva! riate MTD model for the process of intervals between successive event times and for their corresponding marks, resulting in a new class of marked point processes that evolve dependent on historical events. The key of the proposed model is a nonparametric prior based on a kernel copula process. A copula MTD marked point process with a marginal distribution that arises from this prior implies a dynamic interdependence between the inter-arrival process and the mark process. The mixture formulation of the MTD will be propagated to the conditional intensity function with local coefficients. This allows for non-standard intensity shapes, thus supporting flexible inference for marked point processes, upon which first- and second-order properties of the process will be explored. Finally, we will generalize the MTD model for non-Gaussian spatial data. Equipped with a directed acyclic graph, we develop a class of nearest neighbor spatial processes extended from a finite-dimensional distribution based on the MTD model. This proce! ss will be useful in the analysis of continuous spatial-referenced responses that exhibit skewness or kurtosis, and in spatial classification problems where the response is categorical. Space-dependent mixture weights will be developed to improve prediction that relies on the nearest neighbors. From the computational point of view, we will investigate the extent to which the proposed model can provide general, scalable inference, and its practical utility as a spatially varying prior embedded in a hierarchical regression model. Throughout all the proposed models, inference and prediction will be developed under the Bayesian framework with particular emphasis on flexible, structured priors for the mixture weights.

Event Type: 
Adancement/Defense
Advisor: 
Athanasios Kottas and Bruno Sansó
Graduate Program: 
Statistical Science