Polynomial Kernel Discriminant Analysis for 2D visualization of classification problems

In multivariate classification problems, 2D visualization methods can be very useful to understand the data properties whenever they transform the n -dimensional data into a set of 2D patterns which are similar to the original data from the classification point of view. This similarity can be understood as that a classification method works similarly on the original n -dimensional and on the 2D mapped patterns, i.e., the classifier performance should not be much lower on the mapped than on the original patterns. We propose several simple and efficient mapping methods which allow to visualize classification problems in 2D. In order to preserve the structure about the original classification problem, the mappings minimize different class overlap measures, combined with different functions (linear, quadratic and polynomic of several degrees) from ℝ^n to ℝ^2 . They are also able to map into ℝ^2 new data points (out of sample), not used during the mapping learning. This is one of the main benefits of the proposed methods, since few supervised mappings offer a similar behavior. For 71 data sets of the UCI database, we compare the SVM performance using the original and the 2D mapped patterns. The comparison also includes other 34 popular supervised and unsupervised methods of dimensionality reduction, some of them used for the first time in classification. One of the proposed methods, the Polynomial Kernel Discriminant Analysis of degree 2 (PKDA2), outperforms the remaining mappings. Compared to the original n -dimensional patterns, PKDA2 achieves 82