Choquet.integral-methods {kappalab}

Choquet integral

Description

Computes the Choquet integral of a discrete function with respect to a game. The game can be given either under the form of an object of class game, card.game or Mobius.game. If the integrand is not positive, this function computes what is known as the asymmetric Choquet integral.

Methods

object = "Mobius.game", f = "numeric"

The Choquet integral of f is computed from the Möbius transform of a game.

object = "game", f = "numeric"

The Choquet integral of f is computed from a game.

object = "card.game", f = "numeric"

The Choquet integral of f is computed from a cardinal game.

References

G. Choquet (1953), Theory of capacities, Annales de l'Institut Fourier 5, pages 131-295.

D. Denneberg (2000), Non-additive measure and integral, basic concepts and their role for applications, in: M. Grabisch, T. Murofushi, and M. Sugeno Eds, Fuzzy Measures and Integrals: Theory and Applications, Physica-Verlag, pages 42-69.

M. Grabisch, T. Murofushi, M. Sugeno Eds (2000), Fuzzy Measures and Integrals: Theory and Applications, Physica-Verlag.

M. Grabisch and Ch. Labreuche (2002), The symmetric and asymmetric Choquet integrals on finite spaces for decision making, Statistical Papers 43, pages 37-52.

M. Grabisch (2000), A graphical interpretation of the Choquet integral, IEEE Transactions on Fuzzy Systems 8, pages 627-631.

J.-L. Marichal (2000), An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria, IEEE Transactions on Fuzzy Systems 8:6, pages 800-807.

Murofushi and M. Sugeno (1993), Some quantities represented by the Choquet integral, Fuzzy Sets and Systems 56, pages 229-235.

Murofushi and M. Sugeno (2000), Fuzzy measures and fuzzy integrals, in: M. Grabisch, T. Murofushi, and M. Sugeno Eds, Fuzzy Measures and Integrals: Theory and Applications, Physica-Verlag, pages 3-41.

See Also

game-class,
Mobius.game-class,
card.game-class.

Examples

## a normalized capacity
mu <- capacity(c(0:13/13,1,1))

## and its Mobius transform
a <- Mobius(mu)

## a discrete positive function f
f <- c(0.1,0.9,0.3,0.8)

## the Choquet integral of f w.r.t mu
Choquet.integral(mu,f)
Choquet.integral(a,f)

## a similar example with a cardinal capacity
mu <- uniform.capacity(4)
Choquet.integral(mu,f)