The application of geostatistical and machine learning methods based on Gaussian processes to big space–time data is beset by the requirement for storing and numerically inverting large and dense covariance matrices. Computationally efficient representations of space–time correlations can be constructed using local models of conditional dependence which can reduce the computational load. We formulate a stochastic local interaction model for regular and scattered space–time data that incorporates interactions within controlled space–time neighborhoods. The strength of the interaction and the size of the neighborhood are defined by means of kernel functions and adaptive local bandwidths. Compactly supported kernels lead to finite-size local neighborhoods and consequently to sparse precision matrices that admit explicit expression. Hence, the stochastic local interaction model’s requirements for storage are modest and the costly covariance matrix inversion is not needed. We also derive a semi-explicit prediction equation and express the conditional variance of the prediction in terms of the diagonal of the precision matrix. For data on regular space–time lattices, the stochastic local interaction model is equivalent to a Gaussian Markov Random Field.

基于高斯过程的地统计学和机器学习方法在大时空数据上的应用受到存储和数值反转大而密集的协方差矩阵的需求所困扰。可以使用条件依赖的局部模型来构建时空相关性的计算有效表示，这可以减少计算量。我们为规则和分散的时空数据制定了一个随机的局部相互作用模型，该模型将受控时空邻域内的相互作用纳入其中。交互作用的强度和邻域的大小是通过内核函数和自适应本地带宽定义的。紧密支持的内核导致局部邻域的大小有限，因此导致稀疏的精度矩阵无法接受显式表达式。因此，随机局部交互模型对存储的要求不高，不需要昂贵的协方差矩阵求逆。我们还导出了一个半显式预测方程，并根据精度矩阵的对角线来表示预测的条件方差。对于规则时空格上的数据，随机局部相互作用模型等效于高斯马尔可夫随机场。