Sparse Precision Matrix Selection for Fitting Gaussian Random Field Models to Large Data Sets

摘要

Iterative methods for fitting a Gaussian Random Field (GRF) model to spatialdata via maximum likelihood (ML) require $\mathcal{O}(n^3)$ floating pointoperations per iteration, where $n$ denotes the number of data locations. Forlarge data sets, the $\mathcal{O}(n^3)$ complexity per iteration together withthe non-convexity of the ML problem render traditional ML methods inefficientfor GRF fitting. The problem is even more aggravated for anisotropic GRFs wherethe number of covariance function parameters increases with the process domaindimension. In this paper, we propose a new two-step GRF estimation procedurewhen the process is second-order stationary. First, a \emph{convex} likelihoodproblem regularized with a weighted $\ell_1$-norm, utilizing the availabledistance information between observation locations, is solved to fit a sparse\emph{{precision} (inverse covariance) matrix to the observed data using theAlternating Direction Method of Multipliers. Second, the parameters of the GRFspatial covariance function are estimated by solving a least squares problem.Theoretical error bounds for the proposed estimator are provided; moreover,convergence of the estimator is shown as the number of samples per locationincreases. The proposed method is numerically compared with state-of-the-artmethods for big $n$. Data segmentation schemes are implemented to handle largedata sets.