A nonlinear extension of principal component analysis for clustering and spatial differentiation

The limitations in the use of linear principal component analysis (PCA) for identifying morphological features of data sets are discussed. A nonlinear extension of PCA is introduced to provide this capability. It differs from ordinary PCA methods only in that the objective function involves a nonlinear transformation of the principal components. The procedure is shown to be closely related to the exploratory projection pursuit (EPP) algorithm proposed by Friedman (1987), and thus its usefulness in clustering and spatial differentiation applications is apparent. An efficient gradient ascent algorithm is proposed for implementation, based upon the use of a stochastic approximation. The inherent computational advantages in the suggested implementation over other EPP methods are evident, given that EPP methods require the estimation of a density function at every iteration. The nonlinear extension is evaluated by using several well-known datasets, including Fisher's (1936) Iris data. An industrial application of the technique is also presented. Necessary conditions for convergence are shown.