KPCA-based training of a kernel fuzzy classifier with ellipsoidal regions

In a fuzzy classifier with ellipsoidal regions, a fuzzy rule, which is based on the Mahalanobis distance, is defined for each class. Then the fuzzy rules are tuned so that the recognition rate of the training data is maximized. In most cases, one fuzzy rule per one class is enough to obtain high generalization ability. But in some cases, we need to partition the class data to define more than one rule per class. In this paper, instead of partitioning the class data, we map the input space into the high dimensional feature space and then generate a fuzzy classifier with ellipsoidal regions in the feature space. We call this classifier kernel fuzzy classifier with ellipsoidal regions. To speed up training, first we select independent training data that span the subspace in the feature space and calculate the kernel principal components. By this method, we can avoid using singular value decomposition, which leads to training speedup. In the feature space, training data are usually degenerate. Namely, the space spanned by the mapped training data is a proper subspace. Thus, if the mapped test data are in the complementary subspace, the Mahalanobis distance may become erroneous and thus the probability of misclassification becomes high. To overcome this problem, we propose transductive training: in training, we add basis vectors of the input space as unlabelled data; and in classification, if mapped unknown data are not in the subspace we expand the subspace so that they are included. We demonstrate the effectiveness of our method by computer simulations.