Measure Identification for the Choquet Integral: A Python Module

Abstract

Abstract Fuzzy integrals are common concepts which are used to aggregate input values in practical applications. Aggregation of inputs using fuzzy integrals opens up numerous possibilities for modeling interaction, redundancy, and synergy of inputs. However, fuzzy integrals need a fuzzy measure to start this aggregation process. This situation pushes us into the fuzzy measure identification process. This process becomes difficult due to the monotony condition of the fuzzy measure and the exponential increase on the number of measure parameters. There are in the literature many ways to determine fuzzy measures. One of them is learning from data. In this paper, our aim is to introduce a new fuzzy measure identification tool to learn measures from empirical data. It is a Python module which finds the measure that minimizes the difference between the computed and expected outputs of the Choquet integral. In addition, we study some properties of the learning process. In particular, we consider k-additive fuzzy measures and belief functions as well as arbitrary fuzzy measures. Using these variety of measures we examine the effect of k and noisy data on the learning process.

Keywords

• Fuzzy measure identification
• k-additive fuzzy measure
• Belief functions
• Möbius transform
• Python