Fréchet Kriging for Metric Space-Valued Spatial Data

Abstract

Spatial prediction of non-Euclidean data, such as probability distributions, symmetric positive definite matrices, and networks, arises in many modern applications. Such data are naturally modeled as elements of general metric spaces, where classical kriging methods that rely on vector space structure are no longer directly applicable. We introduce Fréchet kriging, a general framework for spatial prediction of metric space-valued data. Fréchet kriging replaces the weighted average in classical ordinary kriging with a weighted Fréchet mean, while the weights are determined by characterizing the spatial dependence structure through a generalized variogram. This construction allows the proposed method to preserve the intrinsic geometry of the response space without requiring an embedding into a linear space. We establish theoretical properties of the proposed predictor, including a surrogate optimality property and consistency under variogram misspecification for a fixed working variogram under suitable regularity conditions. Simulation studies for univariate distributions and symmetric positive definite matrices demonstrate favorable predictive performance compared with existing approaches. We further illustrate the practical utility of the proposed framework through applications to exceedance probabilities of daily average temperatures, joint probability densities of daily minimum and maximum temperatures, and inter-industry monetary flow networks.