Theory of multivalent delta-fuzzy measures and its application

The well known fuzzy measures, Lambda-measure and P-measure, have only one formulaic solution, the former is not a closed form, and the latter is not sensitive enough. In this paper, a novel fuzzy measure, called Delta-measure, is proposed. This new measure proves to be a multivalent fuzzy measure which provides infinitely many solutions to closed form, and it can be considered as an extension of the above two measures. In other words, the above two fuzzy measures can be treated as the special cases of Delta-measure. For evaluating the Choquet integral regression models with our proposed fuzzy measure and other different ones, a real data experiment by using a 5-fold cross-validation mean square error (MSE) is conducted. The performances of Choquet integral regression models with fuzzy measure based on Delta-measure, Lambda-measure and P-measure, respectively, a ridge regression model and a multiple linear regression model are compared. Experimental result shows that the Choquet integral regression models with respect to Delta-measure based on Gamma-support outperforms other forecasting models.