Estimating Dependencies from Spatial Averages

Abstract

Modeling of space-time functions can be done using observations in the form of averages of the function over a set of irregularly shaped regions in space-time. Such observations are most common in applications where the data are gathered for administrative, political, geographic, or agricultural regions. The value of such functions can be predicted by first estimating the dependence structure of the underlying stochastic process. Our proposed method for estimating the covariance function from the integrals of a stationary isotropic stochastic process poses the problem as a set of integral equations. To test this proposal we applied it to epidemiological data on the incidence rates of three diseases in the United States between 1980 and 1994. Spatial correlations obtained in this way reasonably described the mechanism by which those diseases spread. We therefore conclude that it is possible to reliably estimate covariance functions from aggregate observations. The estimate of the covariance functions provides valuable insights into the nature of the space-time process—in the epidemiological data it described a possible mechanism by which the diseases spread.