Nonlinear principal components

Abstract

Suppose that several variables are defined for each element of a population. Here we consider the problem of defining a reduced number of new variables or indicators which summarize the information contained in the set of original variables. One way of dealing with this problem is through the method of principal components. This method, in its original version (Hotelling, 1933), consists of finding the linear combinations of the original variables with maximum variances. Rao (1964) and Darroch (1965) showed that the principal components may be interpreted in terms of recovering the information contained in the original variables through linear functions. In this paper we generalize the method of principal components by allowing for nonlinear relationships between the new indicators and the original variables. Henry and Lazarsfeld (1968, Chap. 8) and McDonald (1962, 1967) have also considered models where the relationship between factors and variables is nonlinear. The models considered in these works assume the structural hypothesis that the errors are independent of the factors, and may therefore be considered models of factor analysis. The present approach, on the contrary, is "data analytic", and there is no necessity to assume any a priori structure. Another approach to nonlinear components was suggested by Gnanadesikan (1977). However, this latter approach is useful primarily for adjusting hypersurfaces (surfaces of maximum dimension) to a set of data, while we are here concerned with adjusting curves (surfaces of dimension 1), or surfaces of small dimension.