Indices for Introspection on the Choquet Integral

Fuzzy measures (FMs) encode the worth (or importance) of different subsets of information sources in the fuzzy integral (FI). It is well-known that the Choquet FI (CFI) often reduces to an elementary aggregation operator for different selections of the FM. However, FMs are often learned from training data or they are derived from the densities (worth of just the singletons). In these situations an important question arises; what is the resultant CFI really doing? Is it aggregating data relative to an additive measure, a possibility measure, or something more complex and unique? Herein, we introduce new indices (distance formulas) and fuzzy sets that capture the degree to which the CFI is behaving like a set of known aggregation operators. This has practical application in terms of gaining a deeper understanding into a given problem, guiding new learning methods and evaluating the CFI’s benefit.