Space-time covariance models on networks

The second-order, small-scale dependence structure of a stochastic process deffined in the space-time domain is key to prediction (or kriging). When the underlying stochastic process is spatio-temporally dependent, adopting an adequate space-time model instead of the identically independently distributed assumption of the error term would substantially reduce the prediction error. While great efforts have been dedicated to developing models for cases in which the spatial domain is either a finite-dimensional Euclidean space or a unit sphere, counterpart developments on a generalized linear network are practically non-existent. Merely replacing the Euclidean distance in a standard geostatistical model with the shortest distance along the network might lead to an invalid covariance function on the network. To fill this gap, I develop a broad range of parametric, non-separable space-time covariance models on generalized linear networks and then an important subgroup — Euclidean trees — by the space embedding technique in concert with the generalized Gneiting class of models and 1-symmetric characteristic functions in the literature, and the scale mixture approach. I give examples from each class of models and investigate the geometric features of these covariance functions near the origin and at infinity. I also show the linkage between different classes of space-time covariance models on Euclidean trees. I illustrate the use of models constructed by different methodologies on a daily stream temperature data set and compare model predictive performance by cross validation. The generalized Gneiting class of models introduced in the thesis can also be applied to non-dendritic networks, e.g. traffic networks. Though I emphasize the decisive role such space-time covariance functions have in space-time geostatistical models, they allow direct extension to space-time log Gaussian process on generalized linear networks.