Inference on 3D Procrustes means: tree bole growth, rank-deficient diffusion tensors and perturbation models

TL;DR

This paper extends the CLT for manifold means to Procrustes and Ziezold means, enabling new statistical tests and analysis of diffusion tensors and tree growth patterns.

Contribution

It generalizes the CLT for extrinsic and intrinsic means on manifolds, including rank-deficient tensors, and applies these results to diffusion imaging and forestry.

Findings

Procrustes mean shows inconsistency under perturbation models

CLT extension allows for new one-sample tests on manifold means

Application reveals tree stem evolution towards cylinders with age

Abstract

The Central Limit Theorem (CLT) for extrinsic and intrinsic means on manifolds is extended to a generalization of Fr\'echet means. Examples are the Procrustes mean for 3D Kendall shapes as well as a mean introduced by Ziezold. This allows for one-sample tests previously not possible, and to numerically assess the `inconsistency of the Procrustes mean' for a perturbation model and `inconsistency' within a model recently proposed for diffusion tensor imaging. Also it is shown that the CLT can be extended to mildly rank deficient diffusion tensors. An application to forestry gives the temporal evolution of Douglas fir tree stems tending strongly towards cylinders at early ages and tending away with increased competition.