Geostatistics

Deterministic and Probabilistic Modeling

The problem motivating the early pioneers of geostatistics, and still the most common goal of contemporary geostatistical analyses, can be stated in the simplest terms as the prediction of a variable (e.g., the concentration of an ore grade) at locations where it has not been sampled (e.g., across a potentially minable deposit). Such predictions require, at least implicitly, the use of a model describing in some way how the phenomenon of interest behaves at these unsampled locations. Various conceptual approaches exist for the formulation of such a model, and a useful categorization is between deterministic and probabilistic models. In a deterministic model, each unknown value is predicted as a single value with no associated prediction error. Such models are best employed when the physical mechanisms that govern the variable of interest are well understood and established physical equations exist that allow calculation of the unknown value with negligible or no error. In fields such as environmental, epidemiological, and public health sciences, however, the systems of interest are often of such complexity and scale that they retain an inherent unpredictability, even when many of the constituent processes are understood in detail, meaning that a deterministic model may be neither feasible nor appropriate. Probabilistic models represent an alternative paradigm to deterministic approaches by considering the underlying mechanism that generates observed data and, by extension, that determines values at unsampled locations, as a random process. Although the mechanism in question is rarely, if ever, entirely random, the adoption of a probabilistic model provides a framework that can prove extremely useful in both predicting unsampled values and assessing the uncertainty of those predictions. Instead of predicting a single value for each unsampled location with assumed zero error, probabilistic models allow the prediction of a set of possible values with corresponding probabilities of occurrence. Unlike deterministic models, probabilistic models do not necessarily require knowledge of the physical process that generated sample data. Rather, most of the information used is derived from the data.

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