Generalizations, extensions and applications for principal component analysis

Principal component analysis (PCA) is one of the most important dimension reduction technique. It is widely used in many applications including economics, finance and medical research. In this research, several novel generalizations of PCA are proposed to adapt the technique to more complicated scenarios. In the first project, we propose a principal surface model for manifold-like datasets in 3D space. In the second part, a new concept of graphical intra-class correlation coefficient (GICC) is defined and a Markov Chain Monte Carlo Expectation-Maximization (mcmcEM) algorithm is used for likelihood optimization. In the third part, we propose multilevel binary principal component analysis (MBPCA) models for finding the principal components of multilevel binary dataset. A variational expectation maximization algorithm is used for likelihood optimization