Abstract:

Kernel discriminants are greatly appreciated because 1) they permit to establish nonlinear boundaries between classes and 2) they offer the possibility of visualizing graphically the data vectors belonging to different classes. One such method, called Generalized Discriminant analysis (GDA) was proposed by Baudat and Anouar (2000). GDA operates on a kernel matrix of size N x N, (N denotes the sample size) and is for large N prohibitive. Our aim was to find out how this method works in a real situation, when dealing with relatively large data. We considered a set of predictors of erosion risk in the Kefallinia island categorized into five classes of erosion risk (together N=3422 data items). Direct evaluation of the discriminants, using entire data, was computationally demanding. Therefore, we sought for a representative sample. We found it by a kind of sieve algorithm. It appeared that using the representative sample, we could greatly speed up the evaluations and obtain discriminative functions with good generalization properties. We have worked
with Gaussian kernels which need one declared parameter SIGMA called kernel
width. We found that for a large range of parameters the GDA algorithm gave
visualization with a good separation of the considered risk classes.