ABSTRACT

In this chapter we take up the problem of multivariate spatial modeling. Spatial data is often multivariate in the sense that multiple (i.e. more than one) outcomes are measured at each spatial unit. As in the univariate case, the spatial units can be referenced by points or by areal units. Examples of multivariate point-referenced data abound in the sciences. For example, at a particular environmental monitoring station, levels of several pollutants would typically be measured (e.g., ozone, nitric oxide, carbon monoxide, PM2.5, etc.). In atmospheric modeling, at a given site we may observe surface temperature, precipitation, and wind speed. In examining commercial real estate markets, for an individual property we may observe both selling price and total rental income. In forestry, investigators seek to produce spatially explicit predictions of multiple forest attributes (e.g., abundance and basal area) using a multi-source forest inventory approach. In each of these illustrations, we anticipate both dependence between measurements at a particular location, and association between measurements across locations. Multivariate areal data are conspicuous in public health where each county or administrative unit supplies counts or rates for a number of diseases. Again, we expect dependence between diseases within each county as well as across counties.