Flexible models for nonstationary dependence: Methodology and examples
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There are many situations when modelling environmental phenomena for which it is not appropriate to assume a stationary dependence structure. <cit.> proposed an approach to allowing nonstationarity in dependence based on a deformed space: coordinates from original geographic "G" space are mapped to a new dispersion "D" space in which stationary dependence is a reasonable assumption. <cit.> achieve this with two deformation functions, which are chosen as thin plate splines, each representing how one of the two coordinates in D-space relates to the original G-space coordinates. This works extends the deformation approach, and the dimension expansion approach of <cit.>, to a regression-based framework in which all dimensions in D-space are treated as "smooths" as found, for example, in generalized additive models. The framework offers an intuitive and user-friendly approach to specifying D-space, allows different levels of smoothing for dimensions in D-space, and allows objective inference for all model parameters. Furthermore, a numerical approach is proposed to avoid non-bijective deformations, should they occur, which applies to any deformation. The proposed framework is demonstrated on the solar radiation data studied in <cit.>, and then on an example related to risk analysis, which culminates in producing simulations of extreme rainfall for part of Colorado, US.
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