An approach to non-linear principal components analysis using radially symmetric kernel functions

An approach to non-linear principal components using radially symmetric kernel basis functions is described. The procedure consists of two steps: a projection of the data set to a reduced dimension using a non-linear transformation whose parameters are determined by the solution of a generalized symmetric eigenvector equation. This is achieved by demanding a maximum variance transformation subject to a normalization condition (Hotelling's approach) and can be related to the homogeneity analysis approach of Gifi through the minimization of a loss function. The transformed variables are the principal components whose values define contours, or more generally hypersurfaces, in the data space. The second stage of the procedure defines the fitting surface, the principal surface, in the data space (again as a weighted sum of kernel basis functions) using the definition of self-consistency of Hastie and Stuetzle. The parameters of this principal surface are determined by a singular value decomposition and cross-validation is used to obtain the kernel bandwidths. The approach is assessed on four data sets.