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.

翻译：采用Matérn协方差函数的高斯随机场因其在捕捉空间场不同特征方面的灵活性，已成为建模空间过程的强大框架。然而，平滑度参数的最大似然估计具有挑战性，这源于基于高斯随机场完整协方差矩阵评估似然函数时存在的数值不稳定性。此外，对于大规模空间数据集，最大似然估计在计算上仍然难以实现。为应对这一挑战，我们提出了Fisher-BT方法，该方法将Fisher评分算法与回溯线搜索策略相结合，并采用修正贝塞尔函数的级数近似。该方法利用ExaGeoStat高性能计算框架，实现了对空间数据集的高效最大似然估计。与无导数优化方法相比，我们提出的方法不仅减少了迭代次数并加速了收敛，还提高了平滑度参数估计的数值稳定性。通过模拟实验以及使用覆盖密西西比河流域的土壤湿度数据集进行的实际数据分析，我们证明所提出的Fisher-BT方法在达到与现有方法相当精度的同时，在计算效率和数值稳定性方面显著优于BOBYQA和Nelder-Mead等无导数算法。