The de Wijs process (also known as the Gaussian free field in statistical physics) is a fundamental spatial process that arises as the scaling limit of lattice based Gaussian Markov random fields and generalizes Brownian motion in two-dimensions. However, at present, there is a wide gap between the theory of Gaussian free field (including the subsequent theory of random fields) in statistical physics and modern probability, and the current practice of spatial statistics via lattice based Gaussian Markov random fields. Thus, there is great need to bridge this gap to develop a principled framework for statistics and inference of spatial models and to pursue novel computations that make such inferences feasible. This project will consider formulating appropriate functionals of the de Wijs process to construct useful random fields and novel matrix-free computations via conjugate gradient and other methods, and will focus on developing new areas of scientific applications. The proposed research will also shed new light on and allow deeper understanding of theoretical and computational issues discussed by many researchers in spatial statistics in the past decades. Novel matrix-free computations will provide further impetus to study parametric bootstrap methods and multi-scale modeling, and to construct a new class of non-Gaussian random fields. The project will contribute to obtaining enhanced scientific understanding in studies of environmental bioassays, arsenic contamination of groundwater and distributions of galaxies.
Advances in the field of spatial statistics are important because new statistical methods can be applied to a wide range of scientific questions in fields such as astronomy, agriculture, biomedical imaging, computer vision, climate and environmental studies, epidemiology and geology. The de Wijs process is one fundamental spatial process that generalizes Brownian motion from time to space. Using the de Wijs process as a fundamental building block, this project will develop novel mathematics and derive fast, efficient and large-scale statistical computations so that various scientific questions can be answered in a practical way. This will lead to new developments for the analysis of continuum spatial data and spatial point patterns, and will allow us to obtain enhanced scientific understanding in studies of environmental bioassays, arsenic contamination of groundwater and distributions of galaxies. The statistics and the computations that will be developed in this project will also be particularly relevant for various research problems that arise in environmental or global change, and in health studies. Finally, this project will integrate research and educational activities through the development of new graduate and undergraduate courses and will also provide valuable training and learning opportunities for students at graduate and undergraduate levels.
