On the Theoretical Guarantees for Parameter Estimation of Gaussian Random Field Models: A Sparse Precision Matrix Approach
This addresses the computational bottleneck in spatial statistics for researchers and practitioners, offering a more efficient approach with theoretical guarantees, though it is incremental as it builds on existing sparse methods.
The paper tackles the computationally expensive and nonconvex problem of parameter estimation for Gaussian Random Field models, especially in high dimensions, by proposing a two-stage method that first estimates a sparse precision matrix and then fits covariance parameters, with theoretical error bounds provided.
Iterative methods for fitting a Gaussian Random Field (GRF) model via maximum likelihood (ML) estimation requires solving a nonconvex optimization problem. The problem is aggravated for anisotropic GRFs where the number of covariance function parameters increases with the dimension. Even evaluation of the likelihood function requires $O(n^3)$ floating point operations, where $n$ denotes the number of data locations. In this paper, we propose a new two-stage procedure to estimate the parameters of second-order stationary GRFs. First, a convex likelihood problem regularized with a weighted $\ell_1$-norm, utilizing the available distance information between observation locations, is solved to fit a sparse precision (inverse covariance) matrix to the observed data. Second, the parameters of the covariance function are estimated by solving a least squares problem. Theoretical error bounds for the solutions of stage I and II problems are provided, and their tightness are investigated.