Fisher Scoring for Exact Mat\'ern Covariance Estimation through Stable Smoothness Optimization

TL;DR

This paper introduces Fisher-BT, a new method combining Fisher scoring and backtracking to efficiently and stably estimate the smoothness parameter of Matérn covariance functions in Gaussian Random Fields, especially for large datasets.

Contribution

The paper presents Fisher-BT, a novel Fisher scoring-based approach with backtracking and series approximation, improving stability and efficiency over existing methods for Matérn covariance estimation.

Findings

Fisher-BT reduces iteration count and accelerates convergence.

It improves numerical stability in smoothness parameter estimation.

Outperforms derivative-free methods in large spatial datasets.

Abstract

Gaussian Random Fields (GRFs) with Mat\'ern covariance functions have emerged as a powerful framework for modeling spatial processes due to their flexibility in capturing different features of the spatial field. However, the smoothness parameter is challenging to estimate using maximum likelihood estimation (MLE), which involves evaluating the likelihood based on the full covariance matrix of the GRF, due to numerical instability. Moreover, MLE remains computationally prohibitive for large spatial datasets. To address this challenge, we propose the Fisher-BackTracking (Fisher-BT) method, which integrates the Fisher scoring algorithm with a backtracking line search strategy and adopts a series approximation for the modified Bessel function. This method enables an efficient MLE estimation for spatial datasets using the ExaGeoStat high-performance computing framework. Our proposed method not only reduces the number of iterations and accelerates convergence compared to derivative-free optimization methods but also improves the numerical stability of the smoothness parameter estimation. Through simulations and real-data analysis using a soil moisture dataset covering the Mississippi River Basin, we show that the proposed Fisher-BT method achieves accuracy comparable to existing approaches while significantly outperforming derivative-free algorithms such as BOBYQA and Nelder-Mead in terms of computational efficiency and numerical stability.