Hierarchical Spatial Models
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Hierarchical Spatial Models A LI A RAB1 , M EVIN B. H OOTEN 2, C HRISTOPHER K. W IKLE 1 1 Department of Statistics, University of Missouri-Columbia, Columbia, MO, USA 2 Department of Mathematics and Statistics, Utah State University, Logan, UT, USA
Definition A hierarchical spatial model is the product of conditional distributions for data conditioned on a spatial process and parameters, the spatial process conditioned on the parameters defining the spatial dependencies between process locations, and the parameters themselves. Historical Background Scientists across a wide range of disciplines have long recognized the importance of spatial dependencies in their data and the underlying process of interest. Initially due to computational limitations, they dealt with such dependencies by randomization and blocking rather than the explicit characterization of the dependencies in their models. Early developments in spatial modeling started in the 1950’s and 1960’s motivated by problems in mining engineering and meteorology [11], followed by the introduction of Markov random fields [2]. The application of hierarchical spatial and spatio-temporal models have become increasingly popular since the advancements of computational techniques, such as MCMC methods, in the later years of the 20th century. Scientific Fundamentals Methods for spatial and spatio-temporal modeling are becoming increasingly important in the environmental sciences and other sciences where data arise from processes in spatial settings. Unfortunately, the application of traditional covariance-based spatial statistical models is either inappropriate or computationally inefficient in many problems. Moreover, conventional methods are often incapable of allowing the researcher to quantify uncertainties corresponding to the model parameters since the parameter space of most complex spatial and spatio-temporal models is very large.
Synonyms Hierarchical dynamic spatio-temporal models; Geostatistical models; Hierarchies; Autoregressive models; Process model
Hierarchical Models A main goal in the rigorous characterization of natural phenomena is the estimation and prediction of processes as
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Hierarchical Spatial Models
well as the parameters governing processes. Thus a flexible framework capable of accommodating complex relationships between data and process models while incorporating various sources of uncertainty is necessary. Traditional likelihood based approaches to modeling have allowed for scientifically meaningful data structures, though, in complicated situations with heavily parameterized models and limited or missing data, estimation
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