An Introduction to Multilinear Principal Component Analysis
摘要
Principal component analysis (PCA) is a simple and very popular method in statistical data analysis. It can be done by eigenvalue decomposition of covariance matrix. Traditional PCA deals with vector variables and each observation is represented in vector form. When observations are tensor objects, such as images, videos, EEG signals over a spatial domain or gene-gene interactions (as symmetric random matrices), traditional PCA first vectorizes these tensor objects and then proceeds with the eigenvalue decomposition of a large covariance matrix. This vectorized PCA for tensor data can be difficult and inefficient. The main reason is that the estimation process of PCA is unstable when the sample size is small compared to the dimension of the vectorized data. Multilinear principal component analysis (MPCA) is a modification of PCA. It preserves the natural tensor structure of observations in searching for principal components. The main advantage of preserving the tensor structure is the parsimonious usage of parameters in specifying the principal component subspaces, which mitigates the adverse influence of high-dimensionality, and hence, leads to efficiency gain in estimation and prediction. In this article, we provide a user-friendly introduction to the basic concept and technique for MPCA. One will see the rationale for the success of MPCA, from the statistical point of view and based on some real data applications.