  A particularly challenging context for dimensionality reduction is
multivariate circular data, i.e., data supported on a torus. Such kind of data
appears, e.g., in the analysis of various phenomena in ecology and astronomy,
as well as in molecular structures. This paper 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 (three-dimensional torus)
and molecular biology (on a seven-dimensional torus) show that ST-PCA
outperforms existing methods for the investigated datasets.
