Flexible models for nonstationary dependence: Methodology and examples

There are many situations when modelling environmental phenomena for which it is not appropriate to assume a stationary dependence structure. citesampson1992 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. citesampson1992 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 citebornn2012, 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 citesampson1992, and then on an example related to risk analysis, which culminates in producing simulations of extreme rainfall for part of Colorado, US.