Flexible models for nonstationary dependence: Methodology and examples

TL;DR

This paper introduces a flexible, regression-based framework for modeling nonstationary dependence in environmental data using deformation functions, allowing for intuitive specification, variable smoothing, and objective inference, demonstrated on solar radiation and rainfall data.

Contribution

It extends deformation and dimension expansion methods into a regression framework with smooth functions, enabling flexible, user-friendly modeling of nonstationary dependence with objective parameter inference.

Findings

Successfully modeled nonstationary dependence in solar radiation data.

Produced realistic simulations of extreme rainfall in Colorado.

Demonstrated the framework's flexibility and effectiveness in real-world examples.

Abstract

There are many situations when modelling environmental phenomena for which it is not appropriate to assume a stationary dependence structure. \cite{sampson1992} 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. \cite{sampson1992} 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 \cite{bornn2012}, 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 \cite{sampson1992}, and then on an example related to risk analysis, which culminates in producing simulations of extreme rainfall for part of Colorado, US.