Bayesian Statistics in Spatial Analysis

Bayesian maximum entropy (BME) is a group of spatial analysis techniques that are used to study natural systems (physical, biological, health, social, financial, and cultural) and their attributes. Methodologically, spatial BME analysis is a synthesis of stochastics theory with epistematics principles. Stochastics theory involves random fields (spatial and spatiotemporal; ordinary and generalized). Epistematics introduces a fusion of evolutionary concepts and methods from brain and behavioral sciences (e.g., intentionality, adaptation, teleology of reason, complementarity, and prescriptivity) in real-world problem solving. Consequently, instead of a dry presentation of spatial analysis within a hermetically sealed mathematics discourse, the BME approach focuses on the basic inquiry process that investigates the problem's conceptual background and knowledge status, and presents it within a general methodological context that accounts for objective reality-human [Page 188]agent interactions and introduces quantitative tools of mathematics in an environment of realistic uncertainty.

BME analysis and techniques have certain important advantages, both theoretical and computational, compared with mainstream spatial statistics methods (including statistical regression, standard Bayesian inference, Markovian analysis, Kalman filters, Gaussian process, and geostatistical kriging techniques). The BME techniques (ordinary and generalized) incorporate various kinds of knowledge bases (core and site specific) in a rigorous and unified framework rather than in an ad hoc and artificial manner (which is the case, for example, with many statistical regression methods). The BME core knowledge base may include physical laws, scientific models, theoretical equations, social structures, logical rules, and reasoning principles; they assume spatial coordinate systems that accommodate Euclidean and non-Euclidean metrics; and the site-specific knowledge base includes hard and soft data, secondary information, empirical relationships, fuzzy sets, and engineering charts. BME techniques can study systems and attributes with heterogeneous space-time dependence patterns; they make no restrictive assumptions concerning estimator linearity and probabilistic normality (nonlinear estimators and non-Gaussian laws are automatically incorporated rather than been restricted by the linear/linearized estimators of mainstream spatial statistics); and they can be used in the spatiotemporal domain too. Also, BME analysis can consider uncertain yet valuable information at the estimation (prediction) points themselves, when available; it provides a complete predictive probability density function at every spatial point rather than just an attribute estimate (in this way, more than one possibility is available at each spatial point, as far as estimation is concerned); and it incorporates higherorder spatial moments (e.g., effect of skewness on spatial analysis).

BME techniques have been used in a wide range of scientific disciplines, including medical geography, epidemiology, earth and atmospheric sciences, biology, human exposure, global health, ecology, and environmental engineering. In these applications, BME analysis has been shown to rigorously and effectively process multisourced uncertainty sources (conceptual, technical, onto-logic, and epistemic); to account for measurable and categorical variables; and to work in the case of multiple attributes (vector or co-BME) and different space-time supports (functional BME). Noticeably, the BME techniques are particularly suitable for multidisciplinary real-world projects. A number of computer software packages exist for the implementation of spatial and spatiotemporal BME statistics concepts and techniques in practice. One such package is the SEKS-GUI software library (Spatiotemporal Epistematics Knowledge Synthesis & Graphical User Interface; see http://homepage.ntu.edu.tw/~hlyu/software/SEKSGUI/SEKSHome.html). A comprehensive user's manual is available at the same Web site, which addresses a broad audience ranging from the novice spatiotemporal modeling user to the field expert.

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