Sugeno integral

In mathematics, the Sugeno integral, named after M. Sugeno,^[1] is a type of integral with respect to a fuzzy measure.

Let $(X,\Omega )$ be a measurable space and let $h:X\to [0,1]$ be an $\Omega$ -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}}\left[\min \left(\min _{x\in E}h(x),g(A\cap E)\right)\right]={\sup _{\alpha \in [0,1]}}\left[\min \left(\alpha ,g(A\cap F_{\alpha })\right)\right]

where $F_{\alpha }=\left\{x|h(x)\geq \alpha \right\}$ .

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}\left[h_{A}(x)\wedge h(x)\right]\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.^[2]

^ Sugeno, M. (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. doi:10.1109/TFUZZ.2016.2516579. ISSN 1941-0034.