Misplaced Pages

Sugeno integral

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

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

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

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

∫ A h ( x ) ∘ g = sup E ⊆ X [ min ( min x ∈ E h ( x ) , g ( A ∩ E ) ) ] = sup α ∈ [ 0 , 1 ] [ min ( α , g ( A ∩ F α ) ) ] {\displaystyle \int _{A}h(x)\circ g={\sup _{E\subseteq X}}\left={\sup _{\alpha \in }}\left}

where F α = { x | h ( x ) ≥ α } {\displaystyle F_{\alpha }=\left\{x|h(x)\geq \alpha \right\}} .

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

∫ A h ( x ) ∘ g = ∫ X [ h A ( x ) ∧ h ( x ) ] ∘ g {\displaystyle \int _{A}h(x)\circ g=\int _{X}\left\circ g}

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

Usage and Relationships

Sugeno integral is related to h-index.

References

  1. Sugeno, M. (1974) Theory of fuzzy integrals and its applications, Doctoral. Thesis, Tokyo Institute of Technology
  2. 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.
Categories: