Analysis of Data of Different Spatial Support: A Multivariate Process Approach

Inherent to a spatial variable is the unit of support at which it is measured. In many studies, variables are observed at different support. For example, disease rates might be measured at an aggregated level while temperature is usually measured at specific points. It is still an interesting problem to study the relationship of variables having different support. However, it may be a different problem to statistically model the relationship of variables of different support, particularly when the supports do not have a hierarchical structure. Currently, cokriging, the use of one or more spatial variables to predict another variable, is applied to variables of the same support. In this work, I extend cokriging for use with variables of different support by constructing a nonparametric cross-covariance matrix. This method is flexible as it applies to any marginal spatial model and is suited to large datasets because it uses latent variables which can assist with dimension reduction. The proposed nonparametric method is demonstrated with two correlated variables which are measured at different spatial units. In addition, the method is implemented using two algorithms; one which yields an optimized matrix (Wang, 2011) and the other which produces an approximately optimized matrix but is computationally more efficient (Hu 2013). The results show that the method is appropriate for predicting data of different support and that it outperforms some competing methods with respect to predictive performance. Furthermore, as expected, the approximately optimized matrix does not perform as well as the alternative algorithm, but it performs better than the comparative methods