Classification by nonlinear integral projections

Abstract

A new method based on nonlinear integral projections for classification is presented. The contribution rate of each combination of the feature attributes, including each singleton, toward the classification is represented by a fuzzy measure. The non-additivity of the fuzzy measure reflects the interactions among the feature attributes. The weighted Choquet integral with respect to the fuzzy measure serves as an aggregation tool to project the feature space onto a real axis optimally according to an error criterion, and the classifying attribute is properly numericalized on the axis simultaneously making the classification simple. To implement the classification, we need to determine the unknown parameters, the values of fuzzy measure and the weight function. This can be done by running an adaptive genetic algorithm on the given training data. The new classifier is tested by recovering the preset parameters from a set of artificial training data generated from these parameters. It also performs well on several real-world data sets. Beyond discriminating classes, this method can also learn the scaling requirements and the respective importance indexes of the feature attributes as well as the relationships among them. A comprehensive discussion on the semantic and geometric meanings of the parameters is given. Moreover, we show how these parameters' values can be used for short-listing important feature attributes to reduce the complexity (dimensions) of the classification problem. Our new method also compares favorably with other methods on some well-known real-world benchmarks.