Bayesian Spanning Tree Models for Complex Spatial Data

Abstract

In many applications, spatial data often display heterogeneous dependence patterns and may be subject to irregular geographic constraints. In light of these challenges, this dissertation develops several novel Bayesian methodologies for modeling non-trivial spatial data.

The first part of this dissertation develops a Bayesian partition prior model for a finite number of spatial locations using random spanning trees (RSTs) of a spatial graph, which guarantees contiguity in clustering and allows to detect clusters with arbitrary shapes and sizes. We embed this model within a hierarchical modeling framework to estimate spatially clustered coeÿcients and their uncertainty measures in a regression model. We prove posterior concentration results and design an eÿcient Markov chain Monte Carlo algorithm.

In the second part, we propose a new class of locally stationary stochastic processes, where local spatially contiguous partitions are modeled by a soft partition process via predictive RSTs for flexible cluster shapes. This valid nonstationary process model allows to knit together local models such that both parameter estimation and prediction can be performed under a coherent framework, and to capture both abrupt changes and smoothness in a spatial random field. We study the posterior concentration theories for this Bayesian process model.

Finally, we consider Bayesian ensemble models for nonparametric regression on complex constrained domains. We first propose a Bayesian additive regression model using RST man-ifold partition models as weak learners, which are capable of capturing any irregularly shaped spatially contiguous partitions while respecting intrinsic geometries and domain boundary constraints. For applications that also involve possibly high dimensional features without known multivariate structures, we further develop a Bayesian additive multivariate decision trees model that combines univariate split rules and novel multivariate split rules in each weak learner. The proposed multivariate split rules are built upon predictive spanning tree bipartition models on reference knots, which are capable of achieving flexible nonlinear decision boundaries on manifold feature spaces while reducing computations.