Rate-Preserving Shrinking-Support Gaussian Process Prediction

Abstract

Gaussian process prediction is a central tool in spatial statistics, but standard implementations require dense matrix operations that become prohibitive for large datasets. We propose a scale-adjusted compactly supported working correlation for Gaussian process prediction, using the generalized Wendland family with a support radius $\phi_n$ that is allowed to decrease with the sample size $n$. Under fixed-domain asymptotics with quasi-uniform designs, a fixed support radius does not yield asymptotic sparsity, whereas a shrinking support radius can make the covariance matrix sparse. We show that $\phi_n$ and the regularization parameter can be jointly chosen so that the resulting predictor preserves the optimal integrated mean squared prediction error rate while reducing the number of nonzero covariance entries and the associated sparse matrix-vector multiplication cost. We also establish an analogous rate-preserving sparsification result for kernel ridge regression. Simulations and an ERA5 temperature application show that the proposed method achieves competitive prediction accuracy while retaining the computational efficiency provided by sparse linear algebra.