Informatica

Structurization of the sample covariance matrix reduces the number of the parameters to be estimated and, in a case the structurization assumptions are correct, improves small sample properties of a statistical linear classifier. Structured estimates of the sample covariance matrix are used to decorellate and scale the data, and to train a single layer perceptron classifier afterwards. In most from ten real world pattern classification problems tested, the structurization methodology applied together with the data transformations and subsequent use of the optimally stopped single layer perceptron resulted in a significant gain in comparison with the best statistical linear classifier – the regularized discriminant analysis.

There exist two principally different approaches to design the classification rule. In classical (parametric) approach one parametrizes conditional density functions of the pattern classes. In a second (nonparametric) approach one parametrizes a type of the discriminant function and minimizes an empirical classification error to find unknown coefficients of the discriminant function. There is a number of asymptotic expansions for an expected probability of misclassification of parametric classifiers. Error bounds exist for nonparametric classifiers so far. In this paper an exact analytical expression for the expected error EPN of nonparametric linear zero empirical error classifier is derived for a case when the distributions of pattern classes are spherically Gaussian. The asymptotic expansion of EPN is obtained for a case when both the number of learning patterns N and their, dimensionality p increase infinitely. The tables for exact and approximate expected errors as functions of N, dimensionality p and the distance δ between pattern classes are presented and compared with the expected error of the Fisher's linear classifier and indicate that the minimum empirical error classifier can be used even in cases where dimensionality exceeds the number of learning examples.

Small training sample effects common in statistical classification and artificial neural network classifier design are discussed. A review of known small sample results are presented, and peaking phenomena related to the increase in the number of features and the number of neurons is discussed.