Statistical Methods & Applications (2003) 12:61-78 DOI: 10.1007/s 10260-002-0046-7 SMA (~) Spfinger-Verlag 2003 Dynamic models for space-time prediction via Karhunen-Lo ve expansion Lara Fontanella, Luigi Ippoliti Dipartimento di Metodi Quantitativi e Teoria Economica, Universit~ degli Studi "G. d'Annunzio", Viale Pindaro, 42, 65127 Pescara, Italy (e-mail: {lfontan,ippoliti} @dmqte.unich.it) Abstract. The paper is concerned with the spatio-temporal prediction of space- time processes. By combining the state-space model with the kriging predictor and Karhunen-Lobve Expansion, we present a parsimonious space-time model which is spatially descriptive and temporally dynamic. We consider the difficulties of apply- ing principal component analysis of stochastic processes observed on an irregular network. Using the Voronoi tessellation we make adjustments to the Fredholm in- tegral equation to avoid distorted loading patterns and derive an "adjusted" kriging spatial predictor. This allows for the specification of a space-time model which achieves dimension reduction in the analysis of large spatial and spatio-temporal data sets. As a practical example, the model is applied to study the evolution of the Nitrogen Dioxide (NO2) measurements recorded in the Milan district. Key words: Kalman filter, ARIMA models, Karhunen-Lobve expansion, Dynamic linear model, Kriging 1. Introduction All data have location in time and space. Sometimes it is necessary to take these lo- cations explicitly into account in the analysis. Spatio-temporal models have gained widespread popularity in recent years. One reason for this is an abundance of im- portant and challenging new applications arising in the environmental and health sciences. Some space-time models are based on the assumption of temporal station- arity. In this context STARMA (Pfeifer and Deutsch 1980), STARMAX (Stoffer 1986) and STARMAG (Di Giacinto 1994; Terzi 1995; Ippoliti 2001) models add spatial covariance matrices to standard vector ARIMA (Liitkepohl 1993) models. More recently, models for spatio-temporal data have been constructed by combin- ing dynamic linear models (DLMs), or state space models, with variogram-based models from spatial statistics. Many of the authors have considered a Bayesian