Sugeno integral

In mathematics, the Sugeno integral, named after M. Sugeno, is a type of integral with respect to a fuzzy measure. Let $(X,\mathrm{\Omega })$ be a measurable space and let $h:X\to [0,1]$ be an $\mathrm{\Omega }$-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:

$\underset{A}{\int }h(x)\circ g=\underset{E\subseteq X}{\sup }[\min (\underset{x\in E}{\min }h(x),g(A\cap E))]=\underset{\alpha \in [0,1]}{\sup }[\min (\alpha ,g(A\cap F_{\alpha }))]$

where $F_{\alpha }=\{x|h(x)\ge \alpha \}$.

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

$\underset{A}{\int }h(x)\circ g=\underset{X}{\int }[h_A(x)\wedge h(x)]\circ g$

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

Usage and Relationships

Sugeno integral is related to h-index.

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 MR0874969