A multiresolution tree-structured spatial linear model

Multiresolution spatial models are able to capture complex dependence correlation in spatial data and are excellent alternatives to the traditional random field models for mapping spatial processes. Because of the multiresolution structures, spatial process prediction can be obtained by direct and fast computation algorithms. However, the existing multiresolution models usually assume a simple constant mean structure, which may not be suitable in practice. In this article, we focus on a multiresolution tree-structured spatial model and extend the model to incorporate a linear regression mean. We explore the properties of the multiresolution tree-structured spatial linear model in depth and estimate the parameters in the linear regression mean and the spatial-dependence structure simultaneously. An expectation-maximization algorithm is adopted to obtain the maximum likelihood estimates of the model parameters and the corresponding information matrix. Given the estimated parameters, a one-pass change-of-resolution Kalman filter algorithm is implemented to obtain the best linear unbiased predictor of the true underlying spatial process. For illustration, the methodology is applied to optimally map crop yield in a Wisconsin field, after accounting for the field conditions by a linear regression.