Extension of the Fuzzy Integral for General Fuzzy Set-Valued Information

Abstract

Abstract-The fuzzy integral (FI) is an extremely flexible aggregation operator. It is used in numerous applications, such as image processing, multi-criteria decision making, skeletal ageat-death estimation and multi-source (e.g., feature, algorithm, sensor, confidence) fusion. To date, a few works have appeared on the topic of generalizing Sugeno's original real-valued integrand and fuzzy measure (FM) for the case of higher-order uncertain information (both integrand and measure). For the most part, these extensions are motivated by, and are consistent with, Zadeh's extension principle (EP). Namely, existing extensions focus on fuzzy number-(FN), i.e., convex and normal fuzzy set-(FS), valued integrands. Herein, we put forth a new definition, called the generalized FI (gFI), and efficient algorithm for calculation for FS-valued integrands. In addition, we compare the gFI, numerically and theoretically, with our non EP-based FI extension called the non-direct FI (NDFI). Examples are investigated in the areas of skeletal age-at-death estimation in forensic anthropology and multi-source fusion. These applications help demonstrate the need and benefit of the proposed work. In particular, we show there is not one supreme technique. Instead, multiple extensions are of benefit in different contexts and applications.

Fuzzy Integral for Rule Aggregation in Fuzzy Inference Systems

“…Note, that at a particular α-cut we obtain, for non-convex FSs, a discontinuous interval, e.g., an α H i that yields more than one interval, like [0, 0.1] and [0.7, 0.9], versus a single continuous interval, such as [0, 0.9]. The gFI plays off the fact that a discontinuous interval can be represented as the union of its continuous sub-intervals [9]. At each α-cut, we first decompose the discontinuous intervals into their corresponding continuous interval counterparts.…”

“…Next, we briefly review the real-valued discrete CI; see [8,9,13,14] for additional information (proofs, properties, etc. ).…”

“…In [9], we introduced the gFI, an extension of the FI based on Zadeh's extension principle (EP) for "unrestricted" (potentially non-convex and subnormal) FSs. Initially, the gFI was created for skeletal age-at-death estimation in forensic anthropology [15,16].…”

“…Initially, the gFI was created for skeletal age-at-death estimation in forensic anthropology [15,16]. Due to space considerations herein, see [9] for full mathematical detail, proofs and in-depth analysis. Algorithm 1 is an algorithmic description of how to calculate the gFI.…”

“…Herein, we investigate the role of the fuzzy integral (FI) [8], namely the Choquet integral (CI), for non-linear rule combining in fuzzy logic. In particular, we explore our two recent extensions, the generalized FI (gFI) and the non-direct FI (NDFI) [9], that are capable of aggregating any type of FS in contrast to prior extensions for interval-valued data and fuzzy numbers. The fuzzy measure (FM), introduced by Sugeno in 1974 [8], is used to model interactions (when/if available) between rules.…”