A high order cumulants based multivariate nonlinear blind source separation method

Abstract

This article addresses the problem of identifying multiple linear and nonlinear patterns from multivariate noisy data represented by an additive model. Following the proposed nonlinear model, the blind source separation (BSS) criterion, as a function of high-order cumulants, is shown to produce a block-structured joint cumulant matrix by an orthogonal rotation. An intuitive interpretation of this criterion is to rotate the elements of whitened principal component analysis (PCA) scores such that they are as independent as possible. The resulting optimal joint cumulant matrix contains diagonal "blocks" that correspond to the linear and nonlinear patterns caused by independent sources, from which linear patterns are recognized as in linear BSS. The nonlinear patterns are identified by extracting their lower-dimensional manifolds via the principal curves method and then transforming back to the original data space. As illustrated in the experimental study, the estimated linear and nonlinear patterns will provide more accurate diagnosing of the root causes that contribute to the observed variability in multivariate manufacturing.