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

2026-01-16

AI Summary

This study addresses the numerical instability and high computational cost commonly encountered in maximum likelihood estimation of the smoothness parameter in Matérn covariance functions for large-scale spatial data. The authors propose a novel approach that integrates the Fisher scoring algorithm with a backtracking line search strategy and employs a series approximation of modified Bessel functions, implemented within the ExaGeoStat high-performance computing framework. This method substantially enhances convergence speed and numerical stability, overcoming the efficiency limitations of traditional derivative-free optimization techniques such as BOBYQA and Nelder-Mead. Experimental results on both synthetic datasets and real-world soil moisture measurements from the Mississippi River basin demonstrate that the proposed approach achieves comparable estimation accuracy while significantly reducing computational overhead.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Matérn covariance
• smoothness parameter
• maximum likelihood estimation
• numerical instability
• large spatial datasets

Innovation

• Fisher scoring
• Matérn covariance
• smoothness parameter estimation
• numerical stability
• high-performance computing