Partition-Based Nonstationary Covariance Estimation Using the Stochastic Score Approximation

We introduce computational methods that allow for effective estimation of a flexible nonstationary spatial model when the field size is too large to compute the multivariate normal likelihood directly. In this method, the field is defined as a weighted spatially varying linear combination of a globally stationary process and locally stationary processes. Often in such a model, the difficulty in its practical use is in the definition of the boundaries for the local processes, and therefore, we describe one such selection procedure that generally captures complex nonstationary relationships. We generalize the use of a stochastic approximation to the score equations in this nonstationary case and provide tools for evaluating the approximate score in O(n log ⁡n) operations and <i>O</i>(<i>n</i>) storage for data on a subset of a grid. We perform various simulations to explore the effectiveness and speed of the proposed methods and conclude by predicting average daily temperature. Supplementary materials for this article are available online.

本文提出一种计算方法，可在空间场规模过大而无法直接求解多元正态似然的场景下，有效估计灵活非平稳空间模型。该方法将空间场定义为全局平稳过程（globally stationary process）与局部平稳过程（locally stationary processes）的加权空间变系数线性组合。此类模型在实际应用中往往面临局部过程边界定义的难题，为此本文提出一类通用的边界选择方法，可有效捕捉复杂的非平稳空间关联关系。本文将随机逼近（stochastic approximation）推广至该非平稳场景下的得分方程（score equations）求解，并针对网格子集上的空间数据，提供了可在O(n log n)次运算与O(n)级存储开销下完成近似得分评估的工具集。我们通过多组模拟实验探究了所提方法的有效性与运算效率，并最终以日均气温预测任务作为应用案例收尾。本文配套补充材料可在线获取。