New systems for modeling, estimating, and predicting a multivariate spatio–temporal process

Abstract Two new systems for modeling, estimating, and predicting a multivariate spatio–temporal process are given. In the first system, a cylinder containing a fraction of the multivariate data with axis along the temporal dimension is centered at a spatio–temporal prediction location. After fitting a linear or nonlinear within‐cylinder trend function using minimum distance, an optimal prediction of the observable multivariate process is computed at the cylinder's center. The local nature of this system makes it suited to the analysis of large data sets often associated with ecosystem analyses. The second system consists of a global model of both the spatio–temporal trend and autocovariance function of the residual process. Each of these is formed by the weighted sum of component global models with weights computed from spatio–temporal kernels centered at each component's most characteristic spatio–temporal location. Minimum distance is again used to estimate all parameters. Predictions are then computed from this fitted global model. Spatio–temporal autocovariance within and between residual processes is modeled with parametric functions and need not be separable in space and time. The fast Fourier transform is used to approximately verify Hermitian positive semi‐definiteness of the covariogram matrix. Asymmetry of the cross‐covariogram in the temporal dimension is modeled as the sum of parametric even and odd functions. These functions have interpretable parameters. Long temporal memory is also modeled with an interpretable parametric function—obviating the need for fractional differencing across time. These covariance structure models are believed to be new. The use of a combined John–Draper and Box–Cox transformation allows continuous, count, categorical–ordinal, and categorical–nominal process data to be fitted within the same modeling, estimation, and prediction system, thus allowing mixed continuous–discrete multivariate predictions to be computed that take into account correlations between all constituent processes. Numerical examples consist of, first, the local system being used to redesign a spatial monitoring network with simulated data, and then both systems being applied to several different real data sets: multivariate spatio–temporal data (prediction of sulfate deposition in the conterminous U.S. with and without nitrate observations), multivariate temporal‐only data (prediction of mink–muskrat pelt counts), and univariate temporal‐only data with long memory (prediction of 5000 years of annual Lake Saki mud thickness). Copyright © 2002 John Wiley & Sons, Ltd.