A Generalized Unified Skew-Normal Process with Neural Bayes Inference

TL;DR

This paper introduces a flexible spatial process model called GSUN that captures non-Gaussian behaviors in spatial data, and develops a neural Bayes inference method using deep attention networks, demonstrating improved accuracy and stability.

Contribution

The paper proposes a generalized unified skew-normal spatial process and a neural Bayes inference method with deep attention networks, enhancing modeling of non-Gaussian spatial data.

Findings

GSUN is a valid spatial process with appropriate correlation decay.

Neural Bayes estimators outperform CNNs in stability and accuracy.

GSUN differs from Gaussian and Tukey g-and-h processes as shown by PIT analysis.

Abstract

In recent decades, statisticians have been increasingly encountering spatial data that exhibit non-Gaussian behaviors such as asymmetry and heavy-tailedness. As a result, the assumptions of symmetry and fixed tail weight in Gaussian processes have become restrictive and may fail to capture the intrinsic properties of the data. To address the limitations of the Gaussian models, a variety of skewed models has been proposed, of which the popularity has grown rapidly. These skewed models introduce parameters that govern skewness and tail weight. Among various proposals in the literature, unified skewed distributions, such as the Unified Skew-Normal (SUN), have received considerable attention. In this work, we revisit a more concise and intepretable re-parameterization of the SUN distribution and apply the distribution to random fields by constructing a generalized unified skew-normal (GSUN) spatial process. We demonstrate that the GSUN is a valid spatial process by showing its vanishing correlation in large distances and provide the corresponding spatial interpolation method. In addition, we develop an inference mechanism for the GSUN process using the concept of neural Bayes estimators with deep graphical attention networks (GATs) and encoder transformer. We show the superiority of our proposed estimator over the conventional CNN-based architectures regarding stability and accuracy by means of a simulation study and application to Pb-contaminated soil data. Furthermore, we show that the GSUN process is different from the conventional Gaussian processes and Tukey g-and-h processes, through the probability integral transform (PIT).