On Using Complex Principal Geodesic Analysis to Tackle VariabilityAmong Angular Data

Principal Component Analysis (PCA) is one of the well-known tools for modeling and visualizing a data set with correlated variables. However, it cannot be directly employed for the data taking their values in non-Euclidean space, such as angular data. An alternative option is to invoke the Complex Principal Component Analysis (CPCA) which uses the Euler formula to take the periodic feature of angles. Another possibility is the dihedral Angles Principal Geodesic Analysis (dPGA), considering the geodesic distance for the angles and then utilizing the PCA. In this paper, we propose a new method called Complex Principal Geodesic Analysis (CPGA), a combined version of dPGA and CPCA. It benefits from advantages of both tools to derive the complex covariance matrix and then invoke dimension reduction methods. The proposed method is applied to the dihedral angles of particular protein structure.