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 functi on involves a nonlinear transformation of the principal components. Th e procedure is shown to be closely related to the exploratory projecti on pursuit (EPP) algorithm proposed by Friedman (1987), and thus its u sefulness in clustering and spatial differentiation applications is ap parent. An efficient gradient ascent algorithm is proposed for impleme ntation, based upon the use of a stochastic approximation. The inheren t 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 (19 36) Iris data. An industrial application of the technique is also pres ented. Necessary conditions for convergence are shown.