Statistical Methodology 9 (2012) 115–116

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Statistical Methodology

journal homepage: www.elsevier.com/locate/stamet

Editorial for the special issue on spatial statistics

The modeling and analysis of spatially referenced data sets have received much attention over thelast decade, especially with the emergence of highly efficient Geographical Information Systems (GIS)databases. Spatially-referenced data sets and their analysis using GIS arise in diverse areas of scientificinvestigations. As spatially-referenced data sets become more prevalent in scientific applications,the desire for full statistical inference and accurate assessment of uncertainty become increasinglyimportant. Statistical theory andmethods play a crucial role in themodeling and analysis of such databy developing sound tools to deliver better prediction and estimation, which help in enhancing ourunderstanding of the processes being studied.

This special issue presents a collection of twelve original research articles focusing upon differentaspects ofmethodological advancements in spatial statistics. They subsumedifferent aspects of spatialstatistics including diverse methods for spatial and space–time data, spatial experimental design andstrategies for comparing and selecting models.

Given the burgeoning interest in space–timemodeling, it is not surprising that the current volumefeatures multiple articles in that domain. For example, Zhuang and Cressie present a spatiotemporalmodel within a Bayesian hierarchical framework using a dynamic Markov random field to analyze awell-known Sudden Infant Death Syndrome (SIDS) data from North Carolina. Dynamic linear modelsalso feature prominently in the article by Sahu and Bakar who compare them with hierarchicalautoregressive models. They point out theoretical differences between these approaches in terms ofthe induced spatial correlation structures as well as in terms of the performance of these models inanalyzing ozone concentration levels. Yet another novel contribution on space–timemodeling comesfromDebarsy, Ertur and Lesage, where the application domain is spatial econometrics. They elucidateon the interpretability of parameters in dynamic space–time panel data models and illustrate theirapproach using an interesting data set concerning the demand for cigarettes over a thirty yearperiod.

The volume also features three articles primarily focusing upon spatial design. The article byVer Hoef considers experimental designs for spatially autocorrelated data. He demonstrates whyclassical randomization strategiesmaynot always lead to optimal experimental designswhen the datacomes from spatial processes. Using computer-intensive genetic algorithms, Ver Hoef obtains optimaldesigns for spatial data and offers insight into robustness issues. A different approach is taken by Ruiz-Cardenas, Ferreira and Schmidt who propose an evolutionary Markov chain Monte Carlo frameworkfor optimal design of large-scalemonitoring networks. Theyworkwithin a Bayesian decision theoreticframework and achieve optimality with respect to expected utility functions. The offer a veryeffective demonstration of their method to optimally redesign a network of monitoring stations forspatiotemporal ground level ozone in the eastern United States. A third contribution on spatial designcomes from Spoeck and Hussain. They feel that the objective functions often used in spatial samplingdesigns are often unnecessarily complicated and, instead, propose applying deterministic algorithmsto mathematically tractable representations of the averaged kriging variance. A special feature of this

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article is the use of external drift variables, such as rainfall, humidity and elevation, to arrive at optimalmonitoring networks.

The remaining articles in the issue present novel spatial methodologies for some more specific,but challenging, problems. The analysis of physical processes on a global scale requires betterunderstanding of the properties of spatial processes on the sphere. Hitczenko and Stein undertaketheoretical investigations for a such a class of anisotropic processes that are invariant to shiftsin longitude. Using analytically tractable forms for the spherical harmonic representation for thecovariance functions of these processes, they are able to derive local properties of the processesincluding conditions when these models will not render consistent parameter estimation and alsotheir ability to capture local isotropy. The article by Song and Oliveira undertake some theoreticalinvestigations into formal Bayesian model selection in spatial lattice models. Such models includeconditional autoregressive (CAR) and simultaneous autoregressive (SAR)models that enjoy enormousapplicability in spatial analysis. Indeed, formal model comparison approaches will constitute animportant ingredient in the Bayesian toolbox. Their approach uses a selection criterion based uponthe posteriormodel probabilities computed by using default priors for themodel parameters. Anotherinteresting article comes from Zimmerman and Fang, who address the very relevant problem ofincomplete geocoding. They consider inference for relative risk in spatial epidemiology and point outthe adverse impact of incomplete geocoding for relative risk estimation. They propose coarsened-datamethods using both nonparametric and parametric procedures and demonstrate their methods usingsimulations as well as real example of childhood asthma cases in an Iowa county.

The remaining entries offer methodological advancements in the domain of Bayesian hierarchicalmodeling. Ghosh, Gelfand and Molhave propose a Bayesian hierarchical approach to assigninguncertainty to spatial surfaces obtained from deterministic spatial interpolation. They propose a datafusion model and specifically consider two settings: one where the obtained surface is presumed tobe continuous and the other where it is discontinuous. The article by Reich and Fuentes addressesthe common exercise of estimating spatial correlation function in a spatial process. Adopting anonparametric Bayesian approach, they use spectral methods and the Dirichlet process to propose aflexible class of priors and apply their models to the analysis of air pollution data from California. Last,but not the least, Sanso and Lemos present very interesting work on parsimonious representationsof non-homogeneous spatial random fields using linear combinations of basis functions obtainedfrom spatially-varying and compactly supported kernels. Their approach is related to low-rank anddimension reduction methods that are useful for estimating spatial processes from very large datasets. They discuss similarities and differences between their approach and other methods such asGaussian predictive processes.

In summary, we believe that this special issue serves an assortment of stimulating and originalresearch articles that represent current methodological trends in spatial statistics and that will helpgenerate further methodological advancements in a variety of application domains.

Sudipto BanerjeeDipak K. Dey