Quasi-linear PCA: low order spline's approach to non-linear principal components

Abstract

Nonlinear Principal Components Analysis (PCA) addresses the nonlinearity problem by relaxing the linear restrictions on standard PCA. A new approach on this subject is proposed in this paper, quasi-linear PCA. Basically, it recovers a spline based algorithm designed for categorical variables and introduces continuous variables into the framework without the need of a discretization process. By using low order spline transformations the algorithm is able to deal with nonlinear relationships between variables and report dimension reduction conclusions on the nonlinear transformed data as well as on the original data in a linear PCA fashion. The main advantages of this approach are; the user do not need to care about the discretization process; the relative distances within each variables' values are respected from the start without discretization losses of information; low order spline transformations allow recovering the relative distances and achieving piecewise PCA information on the original variables after optimization. An example applying our approach to real data is provided below.