Multi-Level Restricted Maximum Likelihood Covariance Estimation and Kriging for Large Non-Gridded Spatial Datasets

TL;DR

This paper introduces a multi-level restricted maximum likelihood method for efficient covariance estimation and kriging in large, irregular, non-gridded spatial datasets, overcoming computational challenges of traditional approaches.

Contribution

The paper presents a novel multi-level contrast approach that decouples covariance parameter estimation from deterministic components, enabling scalable analysis of large spatial datasets.

Findings

Successfully applied to datasets with up to 512,000 observations

Achieved fast decay in multi-level covariance matrices for efficient computation

Handled highly irregular observation placements and numerical instability

Abstract

We develop a multi-level restricted Gaussian maximum likelihood method for estimating the covariance function parameters and computing the best unbiased predictor. Our approach produces a new set of multi-level contrasts where the deterministic parameters of the model are filtered out thus enabling the estimation of the covariance parameters to be decoupled from the deterministic component. Moreover, the multi-level covariance matrix of the contrasts exhibit fast decay that is dependent on the smoothness of the covariance function. Due to the fast decay of the multi-level covariance matrix coefficients only a small set is computed with a level dependent criterion. We demonstrate our approach on problems of up to 512,000 observations with a Matern covariance function and highly irregular placements of the observations. In addition, these problems are numerically unstable and hard to solve with traditional methods.