Stochastic modeling and analysis: an overview

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  • Date: 14 Mar 2013

Perceptions regarding the meaning of the term space-time stochastic modeling (STSM) are not as uniform as one might think. Depending on the application, one refers to stochastic models as environmental, genetic, epidemiologic, medical, and so on. Common factor in all these cases is that STSM is concerned with mathematically rigorous and scientifically meaningful representation, explanation, and prediction of natural phenomena (physical, life, social) in conditions of multisourced uncertainty and space-time variation. In situ uncertainty may be due to investigator's epistemic situation and/or ontic aspects of the phenomenon (measurement errors, erratic observation fluctuations, natural heterogeneities, insufficient understanding). Practicing STSM relies on the interplay between critical thinking, internally consistent methodology, and consciousness of the phenomenon's space-time domain. To achieve this goal, STSM relies on the powerful blending of a formal component (mathematical structure, logical process, and theoretical representation) with an interpretive component (real world applications of the formal component, including the physical meaning of mathematical structure, observation methods, and connections to other empirical phenomena).

Formal STSM deals with a wide range of topics, including random fields, stochastic differential equations, probability theory, space-time geometries, rules of logic, and multiobjective optimization theories. The challenge of applying STSM techniques in human exposure is often not in the formal component, but in the validity of the interpretive component that goes beyond mathematics into the realms of scientific knowledge and empirical observation. Interpretation issues are relevant when one needs to establish correspondence relationships between human exposure and the formal mathematics which describe it, to measure and test the formal structure, or to justify methodological steps. Fruitful interaction of formal and interpretive investigations lies at the heart of STSM's success in environmental health sciences. The STSM paradigm is founded on natural laws and phenomenological representations, rather than merely on formal techniques of pattern fitting. This remarkable STSM feature enhances its scientific content and makes it a central force in the study of such diverse phenomena as pollutant transport in heterogeneous media, toxicokinetics of biologic burden in the human body, multistage representation of carcinogenesis, gene–environment interactions, cellular and biochemical processes, patterns of population exposure in space-time, and maps of spatiotemporal health risk indicators [1-7] (Christakos, George 'Space-Time Stochastic Modeling in Human Exposure', Encyclopedia of Environmetrics, 2013).

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