Predicting a space-time process from aggregate data exemplified by the animation of mumps disease.

摘要

The current understanding of the world comes from the space-time observations of various phenomena (which are, of course, explained by models of varying complexity, e.g., Kepler's laws, Newton's laws, Murphy's laws, etc.). The phenomena may be expressed as a function (possibly a finite or infinite-dimensional vector function) over the space-time. The observations of that function are taken at some points in space-time and then the function is determined given those observations. In many cases the observations can not be physically taken at the points in space-time, but rather are gathered as averages over small (or not-so-small) regions.;In order to estimate (predict) the value of this function (model) one should know the dependence structure of the underlying stochastic process. The existence and form of spatial-temporal dependencies is also a potentially important question. A method is proposed to estimate the covariance function from the integrals of a stationary stochastic process. The method poses the problem as a set of integral equations which are then solved via least squares. To solve the equations efficiently in the case of an isotropic covariance function in two dimensions a closed form expression for the kernel functions (the functions that are convolved with the covariance function in the integral equations) is obtained.;Two approaches to predict a space-time process given its integrals are discussed: a simple-to-implement kernel type method and a statistically-motivated best linear unbiased predictor (BLUP or kriging) method. The latter method requires the knowledge of the trend and the covariance function of the process.;The ways to present the predicted surface in space-time as an animation are discussed. This type of visualization is a natural way to present a space-time process because it has both space and time components. Finally we apply the above methods to produce animated maps of mumps in the United States.;This thesis is concerned about ways to determine the underlying function (model) when the observations are integrals or averages over some irregularly shaped regions in space-time (or just in space). Those types of regions are most common in applications where the data is gathered for administrative, political, geographic, or agricultural regions.