Using 2-Interactive Measures in Nonlinear Classifications

Within the classification algorithms based on signed fuzzy measures, an obvious limitation is the complexity of the algorithms due to the large number of interactions among attributes. For a data set with n attributes, the complexity of classification algorithm would be O (2n). But in the practical problems, the higher-order interactions among attributes are often not significant enough to the classification. So sometimes omitting them just sacrifices the accuracy a little but will save a lot of time and energy. Hence, a 2-interactive measure may be used to reduce the calculation complexity into O (n2). In this study, the Mobius Transformation and its reverse - Zeta Transformation are used to calculate the 2-interactions among attributes for the records. A comprehensive discussion on the semantic and geometric meanings of the parameters is given. The weighted Choquet integral with respect to a signed fuzzy measure serves as an aggregation tool to project the feature space onto a real axis to make the classification simple. To implement the classification, we need to determine the values of the signed fuzzy measure and the other parameters. This can be done by running an adaptive genetic algorithm based on a given training data set. The new classifier is tested by recovering the preset parameters from a set of artificial training data generated from these parameters.