A Nonstationary Soft Partitioned Gaussian Process Model via Random Spanning Trees

There has been a long-standing challenge in developing locally stationary Gaussian process models concerning how to obtain flexible partitions and make predictions near boundaries. In this work, we develop a new class of locally stationary stochastic processes, where local partitions are modeled by a soft partition process via predictive random spanning trees that leads to highly flexible spatially contiguous subregion shapes. This valid nonstationary process model knits together local models such that both parameter estimation and prediction can be performed under a unified and coherent framework, and it captures both discontinuities/abrupt changes and local smoothness in a spatial random field. We propose a theoretical framework to study the Bayesian posterior concentration concerning the behavior of this Bayesian nonstationary process model. The performance of the proposed model is illustrated with simulation studies and real data analysis of precipitation rates over the contiguous United States.