On Estimation and Prediction for Multivariate Multiresolution Tree-structured Spatial Linear Models

Multiresolution tree-structured models are attractive when dealing with large amounts of spatial data in environmental sciences. With the multiresolution tree structure, a change-of-resolution Kalman filter algorithm has been devised to predict spatial processes in a computationally efficient manner (see, e.g., Huang and Cressie (1997) and Huang, Cressie and Gabrosek (2002)). In this article, we extend the multiresolution tree-structured model to account for multiple response variables. Despite the increased model complexity, we derive the theoretical properties of statistical inference and develop direct and fast algorithms for computation. For spatial process prediction, we develop a general theory of optimal projection and generalize the existing change-of-resolution Kalman filter to accommodate singularity. For model parameter estimation, we consider a factorization of the likelihood function to ensure computational efficiency. Moreover, under a fairly mild condition, we derive the distributional properties of both maximum likelihood estimates and restricted maximum likelihood estimates. We evaluate the theory and methods developed here by a simulation study.