Detecting spatial patterns with the cumulant function. Part I: The theory

TL;DR

This paper introduces a new method based on the cumulant function to detect spatial patterns in non-Gaussian climate data, overcoming PCA limitations by incorporating higher order moments.

Contribution

It proposes a simple, fast algorithm that maximizes the cumulant function to identify meaningful spatial patterns in non-Gaussian datasets, extending beyond PCA's second-order statistics.

Findings

Algorithm effectively identifies spatial patterns in non-Gaussian data.

Maximizes spread over probability density tails in projected data.

Applicable to various multivariate random vectors.

Abstract

In climate studies, detecting spatial patterns that largely deviate from the sample mean still remains a statistical challenge. Although a Principal Component Analysis (PCA), or equivalently a Empirical Orthogonal Functions (EOF) decomposition, is often applied on this purpose, it can only provide meaningful results if the underlying multivariate distribution is Gaussian. Indeed, PCA is based on optimizing second order moments quantities and the covariance matrix can only capture the full dependence structure for multivariate Gaussian vectors. Whenever the application at hand can not satisfy this normality hypothesis (e.g. precipitation data), alternatives and/or improvements to PCA have to be developed and studied. To go beyond this second order statistics constraint that limits the applicability of the PCA, we take advantage of the cumulant function that can produce higher order moments information. This cumulant function, well-known in the statistical literature, allows us to propose a new, simple and fast procedure to identify spatial patterns for non-Gaussian data. Our algorithm consists in maximizing the cumulant function. To illustrate our approach, its implementation for which explicit computations are obtained is performed on three family of of multivariate random vectors. In addition, we show that our algorithm corresponds to selecting the directions along which projected data display the largest spread over the marginal probability density tails.