Sugeno integral
In mathematics, the Sugeno integral, named after M. Sugeno,[1] is a type of integral with respect to a fuzzy measure. Let [math]\displaystyle{ (X,\Omega) }[/math] be a measurable space and let [math]\displaystyle{ h:X\to[0,1] }[/math] be an [math]\displaystyle{ \Omega }[/math]-measurable function.
The Sugeno integral over the crisp set [math]\displaystyle{ A \subseteq X }[/math] of the function [math]\displaystyle{ h }[/math] with respect to the fuzzy measure [math]\displaystyle{ g }[/math] is defined by:
- [math]\displaystyle{ \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] }[/math]
where [math]\displaystyle{ F_\alpha = \left\{x | h(x) \geq \alpha \right\} }[/math].
The Sugeno integral over the fuzzy set [math]\displaystyle{ \tilde{A} }[/math] of the function [math]\displaystyle{ h }[/math] with respect to the fuzzy measure [math]\displaystyle{ g }[/math] is defined by:
- [math]\displaystyle{ \int_A h(x) \circ g = \int_X \left[h_A(x) \wedge h(x)\right] \circ g }[/math]
where [math]\displaystyle{ h_A(x) }[/math] is the membership function of the fuzzy set [math]\displaystyle{ \tilde{A} }[/math].
Usage and Relationships
Sugeno integral is related to h-index.[2]
References
- ↑ 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. https://ieeexplore.ieee.org/document/7378290.
- Gunther Schmidt (2006) Relational measures and integration, Lecture Notes in Computer Science # 4136, pages 343−57, Springer books
- M. Sugeno & T. Murofushi (1987) "Pseudo-additive measures and integrals", Journal of Mathematical Analysis and Applications 122: 197−222 MR0874969
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