Nonlinear principal component analysis—Based on principal curves and neural networks

Abstract

Many applications of principal component analysis (PCA) can be found in recently published papers. However principal component analysis is a linear method, and most engineering problems are nonlinear. Sometimes using the linear PCA method can be inadequate when nonlinearities are involved in the data. So there is a need for a nonlinear principal component analysis (NLPCA) method. The principal curve [Hastie and Stuetzle (1989) J. Am. Stat. Ass. 84 (406), 502–516] is a generalization of a linear principal component, but when applied to data sets the algorithm does not yield an NLPCA model in the sense of principal loadings. In this paper we present an NLPCA method which integrates the principal curve algorithm and neural networks. The results of both simulated and real problems show that the method is excellent for solving nonlinear principal component problems. Potential applications of NLPCA are discussed.