Scaled Torus Principal Component Analysis

A particularly challenging context for dimensionality reduction is multivariate circular data, that is, data supported on a torus. Such kind of data appears, for example, in the analysis of various phenomena in environmental sciences and astronomy, as well as in molecular structures. This article introduces Scaled Torus Principal Component Analysis (ST-PCA), a novel approach to perform dimensionality reduction with toroidal data. ST-PCA finds a data-driven map from a torus to a sphere of the same dimension and a certain radius. The map is constructed with multidimensional scaling to minimize the discrepancy between pairwise geodesic distances in both spaces. ST-PCA then resorts to principal nested spheres to obtain a nested sequence of subspheres that best fits the data, which can afterwards be inverted back to the torus. Numerical experiments illustrate how ST-PCA can be used to achieve meaningful dimensionality reduction on low-dimensional torii, particularly with the purpose of clusters separation, while two data applications in astronomy (on a three-dimensional torus) and molecular biology (seven-dimensional torus) show that ST-PCA outperforms existing methods for the investigated datasets. Supplementary materials for this article are available online.降维任务中极具挑战性的一类场景是多元循环数据，即支撑于环面（torus）之上的数据。此类数据可见于环境科学、天文学等领域的各类现象分析，以及分子结构研究当中。本文介绍了缩放环面主成分分析（Scaled Torus Principal Component Analysis，ST-PCA），一种适用于环面数据的新型降维方法。ST-PCA可构建从环面到相同维度、固定半径球面的数据驱动映射：该映射通过多维标度法构建，以最小化两个空间内两两测地距离的偏差。随后，ST-PCA借助主嵌套球面（principal nested spheres）得到一组最贴合数据的嵌套子球面序列，后续可将该序列逆映射回环面空间。数值实验验证了ST-PCA可在低维环面上实现具备实际意义的降维，尤其适用于簇分离任务；而天文学领域（基于三维环面）与分子生物学领域（基于七维环面）的两项实际数据应用表明，针对所研究的数据集，ST-PCA的表现优于现有方法。本文的补充材料可在线获取。