Kriging is a fundamental tool for spatial prediction, but its computational complexity of $O(N^3)$ becomes prohibitive for large datasets. While local kriging using $K$-nearest neighbors addresses this issue, the selection of $K$ typically relies on ad-hoc criteria that fail to account for spatial correlation structure. We propose a penalized kriging framework that incorporates LASSO-type penalties directly into the kriging equations to achieve automatic, data-driven neighbor selection. We further extend this to adaptive LASSO, using data-driven penalty weights that account for the spatial correlation structure. Our method determines which observations contribute non-zero weights through $\ell_1$ regularization, with the penalty parameter selected via a novel criterion based on effective sample size that balances prediction accuracy against information redundancy. Numerical experiments demonstrate that penalized kriging automatically adapts neighborhood structure to the underlying spatial correlation, selecting fewer neighbors for smoother processes and more for highly variable fields, while maintaining prediction accuracy comparable to global kriging at substantially reduced computational cost.

翻译：克里金法是空间预测的基本工具，但其$O(N^3)$的计算复杂度对于大规模数据集而言难以承受。虽然使用$K$近邻的局部克里金法解决了这一问题，但$K$的选择通常依赖于临时性准则，未能考虑空间相关结构。我们提出了一个惩罚性克里金框架，将LASSO型惩罚项直接纳入克里金方程组，以实现自动化的数据驱动邻域选择。我们进一步将其扩展至自适应LASSO，采用能反映空间相关结构的数据驱动惩罚权重。该方法通过$\ell_1$正则化确定哪些观测值具有非零权重，其惩罚参数通过基于有效样本量的新准则进行选择，该准则在预测精度与信息冗余之间取得平衡。数值实验表明，惩罚性克里金法能根据底层空间相关性自动调整邻域结构：对平滑过程选择较少邻域点，对高变异性场选择较多邻域点，同时以显著降低的计算成本保持与全局克里金法相当的预测精度。