This work develops a multivariate extension of the Fixed Rank Kriging (FRK) framework for spatial prediction in settings where multiple spatial processes may provide complementary information. The goal is to preserve the computational efficiency, the ability to operate without assuming stationarity over the domain, and the spatial support flexibility of FRK, while incorporating cross-process dependence. To this end, we employ a multiresolution coregionalization structure for the latent spatial effects, in which spatial basis functions are combined with Gaussian Markov Random Field coefficients. An estimation procedure based on the expectation-maximization algorithm is developed, designed to exploit the multiresolution latent structure. Through simulation studies, we examine when the proposed joint modeling is beneficial. We consider cases in which one process is observed more sparsely or is entirely unobserved in a subregion and find that the multivariate formulation is able to borrow information from the more densely observed process, producing coherent and accurate predictions even where direct observations are limited or absent. Finally, the model is applied to the analysis of PM10 concentrations in Northern Italy, illustrating its applicability in a real environmental context.

翻译：本文发展了固定秩克里金（FRK）框架的多元扩展，用于多个空间过程可能提供互补信息的空间预测场景。其目标是在保留FRK计算效率、无需假设域内平稳性的操作能力及空间支撑灵活性的同时，融入跨过程依赖性。为实现此目标，我们对潜在空间效应采用多分辨率协同区域化结构，其中空间基函数与高斯马尔可夫随机场系数相结合。基于期望最大化算法，我们开发了一种估计程序，该程序旨在利用多分辨率潜在结构。通过模拟研究，我们考察了何时联合建模是有益的。我们考虑了其中一个过程在子区域内观测更稀疏或完全未观测到的情况，发现多元公式能够从更密集观测的过程中借用信息，即使在直接观测有限或缺失的区域也能产生连贯且准确的预测。最后，该模型被应用于意大利北部PM10浓度分析，展示了其在真实环境背景中的适用性。