Unifying Perspective for Gappy Proper Orthogonal Decomposition and Probabilistic Principal Component Analysis

Abstract

In aerospace engineering, various problems such as restoring impaired experimental flow data can be handled by gappy proper orthogonal decomposition. Similar to gappy proper orthogonal decomposition, probabilistic principal component analysis can approximate missing data with the help of an expectation-maximization algorithm, yielding an expectation-maximization algorithm for probabilistic principal component analysis (expectation-maximization principal component analysis). Although both gappy proper orthogonal decomposition and expectationmaximization principal component analysis address the same missing-data-estimation problem, their antithetical formulation perspectives hinder their direct comparison; the development of the former is deterministic, whereas that of the latter is probabilistic. To effectively differentiate both methods, this research provides a unifying leastsquares perspective to qualitatively dissect them within a unified least-squares framework. By virtue of the unifying least-squares perspective, gappy proper orthogonal decomposition and the expectation-maximization principal component analysis turn out to be similar in that they are twofold: basis and least-squares coefficient evaluations. On the other hand, they are dissimilar because the expectation-maximization principal component analysis, unlike gappy proper orthogonal decomposition, dispenses with either a gappy norm or a proper orthogonal decomposition basis. To illustrate the theoretical analysis of both methods, numerical experiments using simple and complex data sets quantitatively examine their performance in terms of convergence rates and computational cost. Finally, comprehensive comparisons, including theoretical and numerical aspects, establish that the expectationmaximization principal component analysis is simpler and thereby more efficient than gappy proper orthogonal decomposition.