Sugeno integral explained

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} } \left[\min\left(\min_{x\in E} h(x), g(A\cap E)\right)\right]= \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]

References

• 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

Notes and References

1. Sugeno, M. (1974) Theory of fuzzy integrals and its applications, Doctoral. Thesis, Tokyo Institute of Technology
2. Mesiar . Radko . Gagolewski . Marek . H-Index and Other Sugeno Integrals: Some Defects and Their Compensation . IEEE Transactions on Fuzzy Systems . December 2016 . 24 . 6 . 1668–1672 . 10.1109/TFUZZ.2016.2516579 . 1941-0034.