Spatial prediction in an arbitrary location, based on a spatial set of observations, is usually performed by Kriging, being the best linear unbiased predictor (BLUP) in a least-square sense. In order to predict a continuous surface over a spatial domain a grid representation is most often used. Kriging predictions and prediction variances are computed in the nodes of a grid covering the spatial domain, and the continuous surface is assessed from this grid representation. A precise representation usually requires the number of grid nodes to be considerably larger than the number of observations. For a Gaussian random field model the Kriging predictor coinsides with the conditional expectation of the spatial variable given the observation set. An alternative expression for this conditional expectation provides a spatial predictor on functional form which does not rely on a spatial grid discretization. This functional predictor, called the Kernel predictor, is identical to the asymptotic grid infill limit of the Kriging-based grid representation, and the computational demand is primarily dependent on the number of observations - not the dimension of the spatial reference domain nor any grid discretization. We explore the potential of this Kernel predictor with associated prediction variances. The predictor is valid for Gaussian random fields with any eligible spatial correlation function, and large computational savings can be obtained by using a finite-range spatial correlation function. For studies with a huge set of observations, localized predictors must be used, and the computational advantage relative to Kriging predictors can be very large. Moreover, model parameter inference based on a huge observation set can be efficiently made. The methodology is demonstrated in a couple of examples.

翻译：基于空间观测集对任意位置进行空间预测通常采用克里金法，该方法在最小二乘意义下是最佳线性无偏预测器（BLUP）。为预测空间域上的连续曲面，网格表示法最为常用。在覆盖空间域的网格节点上计算克里金预测值和预测方差，并通过该网格表示评估连续曲面。精确表示通常要求网格节点数远大于观测数。对于高斯随机场模型，克里金预测器等同于给定观测集条件下空间变量的条件期望。该条件期望的另一种表达式提供了不依赖空间网格离散化的函数形式空间预测器。这种被称为核预测器的函数型预测器，与基于克里金网格表示的渐进网格填充极限相同，其计算需求主要取决于观测数量，而非空间参考域维度或任何网格离散化方式。本文探讨了该核预测器及其关联预测方差的潜在应用。该预测器适用于任意合格空间相关函数的高斯随机场，采用有限程空间相关函数可大幅降低计算量。对于含大规模观测集的研究，必须使用局部化预测器，其相较于克里金预测器的计算优势极为显著。此外，基于大规模观测集的模型参数推断也能高效实现。本文通过若干实例验证了该方法的有效性。