Singular Value Decomposition and Its Visualization

Abstract

Singular value decomposition (SVD) is a useful tool in functional data analysis (FDA). Compared to principal component analysis (PCA), SVD is more fundamental, because SVD simultaneously provides the PCAs in both row and column spaces. We compare SVD and PCA from the FDA view point, and extend the usual SVD to variations by considering different centerings. A generalized scree plot is proposed to select an appropriate centering in practice. Several useful matrix views of the SVD components are introduced to explore different features in data, including SVD surface plots, image plots, curve movies, and rotation movies. These methods visualize both column and row information of a two-way matrix simultaneously, relate the matrix to relevant curves, show local variations, and highlight interactions between columns and rows. Several toy examples are designed to compare the different variations of SVD, and real data examples are used to illustrate the usefulness of the visualization methods.