Exploring the variability of DNA molecules via principal geodesic analysis on the shape space

Abstract

Most of the linear statistics deal with data lying in a Euclidean space. However, there are many examples, such as DNA molecule topological structures, in which the initial or the transformed data lie in a non-Euclidean space. To get a measure of variability in these situations, the principal component analysis (PCA) is usually performed on a Euclidean tangent space as it cannot be directly implemented on a non-Euclidean space. Instead, principal geodesic analysis (PGA) is a new tool that provides a measure of variability for nonlinear statistics. In this paper, the performance of this new tool is compared with that of the PCA using a real data set representing a DNA molecular structure. It is shown that due to the nonlinearity of space, the PGA explains more variability of the data than the PCA.

Keywords:

• nonlinear statistics
• statistical shape analysis
• DNA modeling
• principal component analysis
• principal geodesic analysis

Acknowledgement

We wish to thank an anonymous referee for comments and suggestions that helped us improve this paper. This research was in part supported by a grant from IPM (NO. 89620011).