Nonparametric Estimation of Spatial Covariance Function using Mixtures of Gaussian Kernels

Music City Center 

Estimating the covariance function of a spatial process is important for model estimation and spatial prediction. Many spatial models, such as Gaussian Processes, rely on covariance functions to define their structure. However, parametric estimation can suffer from model misspecification leading to biased predictions if the chosen covariance structure is incorrect. In this work, we study a nonparametric approach to estimate the covariance function of an isotropic stationary process in R^d. We focus on a class of covariance functions that are valid in all dimensions d>=1, which includes popular parametric kernels such as Matern kernel. Leveraging the fact that such covariance functions can be represented as infinite mixtures of scaled Gaussian kernels, we propose two estimation methods: least squares and nonparametric maximum likelihood estimation for estimating the mixing measure of scaled Gaussian kernels. We also develop computationally efficient methods to solve the optimizations using non-negative least squares and fisher-scoring updates. Finally, we evaluate our proposed methods through simulations and real data, comparing them against parametric and nonparametric approaches.

Stationary isotropic processes

Spatial covariance function

Nonparametric estimation

Gaussian mixtures

Fast computation 

Section on Nonparametric Statistics