We describe a method to construct orthonormal basis coordinates which maximizes over redundant dictionaries (frames) of biorthogonal bases the class separability index or distances among classes. The method proposes an algorithm which consists of biorthogonal expansions over two redundant
dictionaries. In multiclassification problems, embedded classes are often present, we show how the biorthogonality of the expansion can really help to construct a coordinate system which characterizes the classes. The algorithm
is created for training wavelet networks in order to provide an efficient coordinate system maximizing the Cross Entropy function between two complementary classes. The algorithm works with a preliminary extracted features with shrinkage technique in order to reduce the dimensionality of the problem. In particular, our attention is pointed out for a practical time frequency monitoring, detection and classification of transients in rail vehicle systems. We want to distinguish transients as inrush current and no inrush current and among
them between the two complementary classes: dangerous inrush current and no dangerous inrush current. The proposed algorithm could be used on line in order to recognize the dangerous transients in real time and thus shut-down the vehicle. We show how, with a limited number of wavelets and with few iterations on the compressed data, good and fast performances are achieved. Simulations using real measured data on the vehicle line are included to illustrate and to analyse the effectiveness of the proposed method.