Torus Principal Component Analysis with an Application to RNA Structures

TL;DR

This paper introduces a novel torus-PCA method that respects the cyclic topology of data on tori, using sphere deformation and nested sphere analysis, with applications to RNA structure data.

Contribution

The paper presents a new torus-PCA technique that preserves cyclic topology and avoids overfitting, improving analysis of data on tori such as RNA structures.

Findings

Successfully applied to RNA data sets, providing interpretable principal components.

Validated against benchmark RNA data, demonstrating improved structure insights.

Offers a robust method for PCA on toroidal data with singularity management.

Abstract

There are several cutting edge applications needing PCA methods for data on tori and we propose a novel torus-PCA method with important properties that can be generally applied. There are two existing general methods: tangent space PCA and geodesic PCA. However, unlike tangent space PCA, our torus-PCA honors the cyclic topology of the data space whereas, unlike geodesic PCA, our torus-PCA produces a variety of non-winding, non-dense descriptors. This is achieved by deforming tori into spheres and then using a variant of the recently developed principle nested spheres analysis. This PCA analysis involves a step of small sphere fitting and we provide an improved test to avoid overfitting. However, deforming tori into spheres creates singularities. We introduce a data-adaptive pre-clustering technique to keep the singularities away from the data. For the frequently encountered case that the residual variance around the PCA main component is small, we use a post-mode hunting technique for more fine-grained clustering. Thus in general, there are three successive interrelated key steps of torus-PCA in practice: pre-clustering, deformation, and post-mode hunting. We illustrate our method with two recently studied RNA structure (tori) data sets: one is a small RNA data set which is established as the benchmark for PCA and we validate our method through this data. Another is a large RNA data set (containing the small RNA data set) for which we show that our method provides interpretable principal components as well as giving further insight into its structure.