Fisher Scoring for Exact Matérn Covariance Estimation through Stable Smoothness Optimization

TL;DR

The paper tackles the challenge of estimating the Matérn covariance smoothness parameter $\nu$ in Gaussian random fields via exact maximum likelihood, which is numerically unstable and computationally demanding for large data. It introduces Fisher-BackTracking (Fisher-BT), a derivative-based optimization that merges Fisher scoring with backtracking line search and a Nelder–Mead fallback, augmented by a stable series-based derivative for $\partial_\nu\ell$ and implemented on the ExaGeoStat HPC framework for scalability. Empirical results show Fisher-BT achieves competitive accuracy with substantially fewer log-likelihood evaluations and improved numerical stability across a range of $\nu$, outperforming derivative-free methods in most settings. The method extends exact MLE capability to very large spatial datasets and offers a practical benchmark for exact Matérn estimation, with potential extensions to other covariance models and hybrid approximation techniques.

Abstract

Gaussian Random Fields (GRFs) with Matérn 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.

Figures (10)

• Figure 1: Computational time, the number of iterations, and the number of calling the derivative functions for different MLE algorithms for moderate true values of $\nu$.
• Figure 2: Computational time, the number of iterations, and the number of calls to the derivative functions for different MLE algorithms for small and large true values of $\nu$.
• Figure 3: Boxplots of the computational time saving between the proposed method and two ExaGeoStat methods.
• Figure 4: Boxplots of the computational time saving between the proposed method and two ExaGeoStat methods with respect to $n$.
• Figure 5: Boxplots of estimates from different MLE algorithms for moderate true values of $\nu$.