Tangent phylogenetic PCA

TL;DR

Tangent p-PCA extends phylogenetic PCA to data on Riemannian manifolds, enabling dimension reduction of complex shape data while accounting for evolutionary relationships.

Contribution

It introduces a novel generalization of p-PCA for manifold-valued data, specifically shapes, incorporating both non-linear geometry and phylogenetic covariance.

Findings

Simulation results on the sphere show accurate estimators and fast convergence.

Application to mammal jaw shapes demonstrates practical utility.

Method effectively captures phylogenetic and shape structure in data.

Abstract

Phylogenetic PCA (p-PCA) is a version of PCA for observations that are leaf nodes of a phylogenetic tree. P-PCA accounts for the fact that such observations are not independent, due to shared evolutionary history. The method works on Euclidean data, but in evolutionary biology there is a need for applying it to data on manifolds, particularly shapes. We provide a generalization of p-PCA to data lying on Riemannian manifolds, called Tangent p-PCA. Tangent p-PCA thus makes it possible to perform dimension reduction on a data set of shapes, taking into account both the non-linear structure of the shape space as well as phylogenetic covariance. We show simulation results on the sphere, demonstrating well-behaved error distributions and fast convergence of estimators. Furthermore, we apply the method to a data set of mammal jaws, represented as points on a landmark manifold equipped with the LDDMM metric.