$dC_{F}$-Integrals: Generalizing C$_{F}$-Integrals by Means of Restricted Dissimilarity Functions

The Choquet integral (CI) is an averaging aggregation function that has been used, e.g., in the fuzzy reasoning method (FRM) of fuzzy rule-based classification systems (FRBCSs) and in multicriteria decision making in order to take into account the interactions among data/criteria. Several generalizations of the CI have been proposed in the literature in order to improve the performance of FRBCSs and also to provide more flexibility in the different models by relaxing both the monotonicity requirement and averaging conditions of aggregation functions. An important generalization is the <inline-formula><tex-math notation="LaTeX">$C_{F}$</tex-math></inline-formula>-integrals, which are preaggregation functions that may present interesting nonaveraging behavior depending on the function <inline-formula><tex-math notation="LaTeX">$F$</tex-math></inline-formula> adopted in the construction and, in this case, offering competitive results in classification. Recently, the concept of d-Choquet integrals was introduced as a generalization of the CI by restricted dissimilarity functions (RDFs), improving the usability of CIs, as when comparing inputs by the usual difference may not be viable. The objective of this article is to introduce the concept of <inline-formula><tex-math notation="LaTeX">$dC_{F}$</tex-math></inline-formula>-integrals, which is a generalization of <inline-formula><tex-math notation="LaTeX">$C_{F}$</tex-math></inline-formula>-integrals by RDFs. The aim is to analyze whether the usage of <inline-formula><tex-math notation="LaTeX">$dC_{F}$</tex-math></inline-formula>-integrals in the FRM of FRBCSs represents a good alternative toward the standard <inline-formula><tex-math notation="LaTeX">$C_{F}$</tex-math></inline-formula>-integrals that just consider the difference as a dissimilarity measure. For that, we consider six RDFs combined with five fuzzy measures, applied with more than 20 functions <inline-formula><tex-math notation="LaTeX">$F$</tex-math></inline-formula>. The analysis of the results is based on statistical tests, demonstrating their efficiency. Additionally, comparing the applicability of <inline-formula><tex-math notation="LaTeX">$dC_{F}$</tex-math></inline-formula>-integrals versus <inline-formula><tex-math notation="LaTeX">$C_{F}$</tex-math></inline-formula>-integrals, the range of the good generalizations of the former is much larger than that of the latter.