Generalization of the Fuzzy Integral for discontinuous interval- and non-convex interval fuzzy set-valued inputs

The Fuzzy Integral (FI) is a powerful approach for non-linear data aggregation. It has been used in many settings to combine evidence (typically objective) with the known “worth” (typically subjective) of each data source, where the latter is encoded in a Fuzzy Measure (FM). While initially developed for the case of numeric evidence (integrand) and numeric FM, Grabisch et al. extended the FI to the cases of continuous intervals and normal, convex fuzzy sets (i.e., fuzzy numbers). However, in many real-world applications, e.g., explosive hazard detection based on multi-sensor and/or multi-feature fusion, agreement based modeling of survey data, anthropology and forensic science, or computing with respect to linguistic descriptions of spatial relations from sensor data, discontinuous interval and/or non-convex fuzzy set data may arise. The problem is no theory and algorithm currently exists for calculating the FI for such a case. Herein, we propose an extension of the FI to discontinuous interval- and convex normal Interval Fuzzy Set (IFS)-valued integrands (with a numeric FM). Our approach arises naturally from analysis of the Extension Principle. Further, we provide a computationally efficient approach to computing the proposed extension based on the union of the FIs on the combinations of continuous sub-intervals and we demonstrate the approach using examples for both the Choquet FI (CFI) and Sugeno FI (SFI).