Randomness of Shapes and Statistical Inference on Shapes via the Smooth Euler Characteristic Transform

TL;DR

This paper develops a mathematical framework for modeling the randomness of shapes and introduces statistical inference methods using the smooth Euler characteristic transform, validated through simulations and real primate molar data.

Contribution

It establishes the theoretical foundations for shape randomness modeling and proposes chi-squared based algorithms for hypothesis testing on shapes.

Findings

Algorithms effectively detect shape differences in primate molar data.

Simulation studies validate the mathematical derivations.

Framework bridges multiple mathematical and statistical fields.

Abstract

In this article, we establish the mathematical foundations for modeling the randomness of shapes and conducting statistical inference on shapes using the smooth Euler characteristic transform. Based on these foundations, we propose two chi-squared statistic-based algorithms for testing hypotheses on random shapes. Simulation studies are presented to validate our mathematical derivations and to compare our algorithms with state-of-the-art methods to demonstrate the utility of our proposed framework. As real applications, we analyze a data set of mandibular molars from four genera of primates and show that our algorithms have the power to detect significant shape differences that recapitulate known morphological variation across suborders. Altogether, our discussions bridge the following fields: algebraic and computational topology, probability theory and stochastic processes, Sobolev spaces and functional analysis, analysis of variance for functional data, and geometric morphometrics.