Sugeno integral

In mathematics, the Sugeno integral, introduced by Michio Sugeno as a fuzzy integral in work on fuzzy measures at the Tokyo Institute of Technology, is a type of integral with respect to a fuzzy measure.

Let $(X, Ω)$ be a measurable space and let $h : X \to [0, 1]$ be an $Ω$ -measurable function.

The Sugeno integral over the crisp set $A \subseteq X$ of the function $h$ with respect to the fuzzy measure $g$ is defined by:

\int_{A} h (x) \circ g = sup_{E \subseteq X} [min (min_{x \in E} h (x), g (A \cap E))] = sup_{α \in [0, 1]} [min (α, g (A \cap F_{α}))]

where $F_{α} = {x | h (x) \geq α}$ .

The Sugeno integral over the fuzzy set $\tilde{A}$ of the function $h$ with respect to the fuzzy measure $g$ is defined by:

\int_{A} h (x) \circ g = \int_{X} [h_{A} (x) \land h (x)] \circ g

where $h_{A} (x)$ is the membership function of the fuzzy set $\tilde{A}$ .

Usage and Relationships

Sugeno integral is related to h-index.

↑ Sugeno, Michio (1972). "Fuzzy Measure and Fuzzy Integral". Transactions of the Society of Instrument and Control Engineers. 8 (2): 218–226. doi:10.9746/sicetr1965.8.218.
↑ Sugeno, Michio (1974). Theory of fuzzy integrals and its applications (Doctoral thesis). Tokyo Institute of Technology.
↑ Mesiar, Radko; Gagolewski, Marek (December 2016). "H-Index and Other Sugeno Integrals: Some Defects and Their Compensation". IEEE Transactions on Fuzzy Systems. 24 (6): 1668–1672. Bibcode:2016ITFS...24.1668M. doi:10.1109/TFUZZ.2016.2516579. ISSN 1941-0034.