Analysis of nonlinear modes of variation for functional data

TL;DR

This paper introduces a new variance analysis method for functional data that captures nonlinear modes of variation, providing more interpretable insights than traditional PCA, with applications in biological data analysis.

Contribution

The paper develops a novel variance decomposition technique for functional data that accounts for nonlinear modes using differential geometry, enhancing interpretability.

Findings

Method successfully applied to biological data

Allows comparison of genetic tradeoffs

Quantifies effects of selection on evolution

Abstract

A set of curves or images of similar shape is an increasingly common functional data set collected in the sciences. Principal Component Analysis (PCA) is the most widely used technique to decompose variation in functional data. However, the linear modes of variation found by PCA are not always interpretable by the experimenters. In addition, the modes of variation of interest to the experimenter are not always linear. We present in this paper a new analysis of variance for Functional Data. Our method was motivated by decomposing the variation in the data into predetermined and interpretable directions (i.e. modes) of interest. Since some of these modes could be nonlinear, we develop a new defined ratio of sums of squares which takes into account the curvature of the space of variation. We discuss, in the general case, consistency of our estimates of variation, using mathematical tools from differential geometry and shape statistics. We successfully applied our method to a motivating example of biological data. This decomposition allows biologists to compare the prevalence of different genetic tradeoffs in a population and to quantify the effect of selection on evolution.