UN ALGORITMO GENÉTICO PARA SELECCIÓN DE KERNEL EN ANÁLISIS DE COMPONENTES PRINCIPALES CON KERNELS

Abstract

Principal Component Analysis with Kernels (KPCA) is an extension of Principal Component Analysis (PCA which is basically a PCA on the original data after they were sent, via a non-linear transformation, to a space called the feature space. The key for a successful KPCA is to extract directions of maximum variability in the transformed data and then identify these directions with patterns of maximum variability of the original data. However, there are situations for which KPCA is not sufficient to detect these directions of maximum variability. In this work we address this problem: we build a convex space of kernels obtained from the set of all convex linear combinations of a fixed set of kernels. In this space we find the optimal kernel defined by that which produces the largest percentage of explained variance by a KPCA. This optimization problem consists of finding the coefficients of the convex linear combination of the optimal kernel. We solve the convex optimization problem with a genetic algorithm. The proposal is illustrated producing a ranking of the 210 municipalities in the State of Veracruz using 10 socioeconomic variables. The proportion of explained variance by the first component of a PCA is 56%. With our proposal, the first principal component of the ACPK extracts 99% of the variability in the feature space