具有稀疏精度矩阵的时空插值随机局部相互作用模型
Stochastic Local Interaction Model with Sparse Precision Matrix for Space-Time Interpolation
摘要：
基于高斯过程的地统计和机器学习方法在大时空数据中的应用受到存储和数值反演大而密集的协方差阵的要求的困扰。利用条件依赖的局部模型可以构造时空相关性的计算高效表示，这可以减少计算量。我们制定了一个随机的局部相互作用模型，定期和分散的时空数据，包括控制的时空邻域内的相互作用。通过核函数和自适应局部带宽来定义相互作用的强度和邻域的大小。紧支持的内核导致有限大小的局部邻域，从而稀疏精度矩阵，承认明确的表达。因此，随机局部相互作用模型对存储的要求是适度的，并且不需要昂贵的协方差阵求逆。我们还推导出一个半显式的预测方程，并表示预测的条件方差的精度矩阵的对角线。对于正则空间-时间格点上的数据，随机局部相互作用模型等价于高斯马尔可夫随机场。
英文摘要：
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.
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