Sugeno fuzzy integral generalizations for Sub-normal Fuzzy set-valued inputs

In prior work, Grabisch put forth a direct (i.e., result of the Extension Principle) generalization of the Sugeno fuzzy integral (FI) for fuzzy set (FS)-valued normal (height equal to one) integrands and number-based fuzzy measures (FMs). Grabisch´s proof is based in large on Dubois and Prade´s analysis of functions on intervals, fuzzy numbers (thus normal FSs) and fuzzy arithmetic. However, a case not studied is the extension of the FI for sub-normal FS integrands. In prior work, we described a real-world forensic application in anthropology that requires fusion and has sub-normal FS inputs. We put forth an alternative non-direct approach for calculating FS results from sub-normal FS inputs based on the use of the number-valued integrand and number-valued FM Sugeno FI. In this article, we discuss a direct generalization of the Sugeno FI for sub-normal FS integrands and numeric FMs, called the Sub-normal Fuzzy Integral (SuFI). To no great surprise, it turns out that the SuFI algorithm is a special case of Grabisch´s generalization. An algorithm for calculating SuFI and its mathematical properties are compared to our prior method, the Non-Direct Fuzzy Integral (NDFI). It turns out that SuFI and NDFI fuse in very different ways. We assert that in some settings, e.g., skeletal age-at-death estimation, NDFI is preferred to SuFI. Numeric examples are provided to stress important inner workings and differences between the FI generalizations.