Decomposing data sets into skewness modes

TL;DR

This paper introduces a method to decompose data sets into modes that maximize skewness under variance constraints, using nonlinear equations and gradient flows, with applications to atmospheric data.

Contribution

It develops a novel approach combining nonlinear equations and gradient flows to identify skewness-maximizing modes, extending to flatness modes and demonstrated on atmospheric data.

Findings

Maximal-skewness modes correspond to localized atmospheric flows

The method effectively decomposes data into modes with specific statistical properties

Extensions to maximal-flatness modes are possible

Abstract

We derive the nonlinear equations satisfied by the coefficients of linear combinations that maximize their skewness when their variance is constrained to take a specific value. In order to numerically solve these nonlinear equations we develop a gradient-type flow that preserves the constraint. In combination with the Karhunen-Lo\`eve decomposition this leads to a set of orthogonal modes with maximal skewness. For illustration purposes we apply these techniques to atmospheric data; in this case the maximal-skewness modes correspond to strongly localized atmospheric flows. We show how these ideas can be extended, for example to maximal-flatness modes.