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Predicting the spatiotemporal variability of terrestrial vegetation is important for many environmental applications. In this paper, we propose a predictive model for the space-time dynamics of vegetation indexes (VIs) obtained from remote-sensing data through a two-step approach. In the first step, we quantify the deterministic component of the vegetation variations with an additive model that includes seasonal components, interannual trends, and jump discontinuities. The second step captures the dynamics of random residuals (anomalies) with a predictive multidimensional model accounting for the autoregressive (AR) characteristics and spatial correlations of the vegetation fields. The overall goal of the model is to provide short lead-time predictions (from two weeks to one month ahead) of vegetation for management purposes, including irrigation applications in croplands and fire-hazard assessment in natural forests. The proposed model is tested using MODIS-based normalized difference VI (NDVI) and enhanced VI (EVI) data over the period of March 2000 to December 2006 over Italy. Model performance is quantified for five major land covers, including agricultural areas and natural forests. Finally, the analysis is carried out at different levels of spatial aggregation (1-8 km) to investigate the efficacy of the model at different spatial scales. The results show that a simple additive deterministic model explains well the temporal vegetation variability, while the spatial-AR contributions generally improve the forecast accuracy, particularly for croplands. Coherent biophysical patterns are also detected through the proposed predictive model.