Clustering on the torus by conformal prediction

Abstract

Motivated by the analysis of torsion (dihedral) angles in the backbone of proteins, we investigate clustering of bivariate angular data on the torus $[-\mathit{\pi },\mathit{\pi })\times [-\mathit{\pi },\mathit{\pi })$. We show that naive adaptations of clustering methods, designed for vector-valued data, to the torus are not satisfactory and propose a novel clustering approach based on the conformal prediction framework. We construct several prediction sets for toroidal data with guaranteed finite-sample validity, based on a kernel density estimate and bivariate von Mises mixture models. From a prediction set built from a Gaussian approximation of the bivariate von Mises mixture, we propose a data-driven choice for the number of clusters and present algorithms for an automated cluster identification and cluster membership assignment. The proposed prediction sets and clustering approaches are applied to the torsion angles extracted from three strains of coronavirus spike glycoproteins (including SARS-CoV-2, contagious in humans). The analysis reveals a potential difference in the clusters of the SARS-CoV-2 torsion angles, compared to the clusters found in torsion angles from two different strains of coronavirus, contagious in animals.