Attaching uncertainty to deterministic spatial interpolations
Deterministic spatial interpolation algorithms such as the natural neighbor interpolation (NNI) or the Cressman interpolation schemes are widely used to interpolate environmental features. In particular, the former have been applied to digital elevation models (DEM's), the latter to weather data and pollutant exposure. However, they are unsatisfying in that they fail to provide any uncertainty assessment. Such schemes are not model-based; rather, they provide a set of rules, usually geometrically motivated, by which point-level data is interpolated to a grid. We distinguish this setting from the case where the deterministic model is essentially a mapping from inputs to outputs in which case a joint model can be formulated to assign uncertainty. In our setting we have no inputs, only an interpolated surface at some spatial resolution.We propose a general approach to handle the non model-based setting. In fact, the approach can be used to assign uncertainty to any supplied surface regardless of how it was created. We first formulate a useful notion of uncertainty and then show, with additional external validation data, that we can attach uncertainty using a convenient version of a data fusion model. We also clarify the distinction between this setting and the more usual case where we are trying to build an explanatory model to explain an environmental surface.We discuss two settings for such interpolation, one where the surface is presumed to be continuous such as elevation or temperature and the other where the surface would be discontinuous such as with precipitation where, at any location, there would be a point mass in the distribution at 0. We work within a hierarchical Bayesian framework and illustrate with a DEM within the Cape Floristic Region of South Africa. © 2011 Elsevier B.V..
Ghosh, S; Gelfand, AE; Mølhave, T
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