https://github.com/cran/kappalab
Tip revision: 04797bea4a817fcf6daff673777d0c2fc5ae2c45 authored by Ivan Kojadinovic on 07 November 2023, 20:20:02 UTC
version 0.4-12
version 0.4-12
Tip revision: 04797be
Choquet.integral-methods.Rd
\name{Choquet.integral-methods}
\docType{methods}
\alias{Choquet.integral}
\alias{Choquet.integral-methods}
\alias{Choquet.integral,Mobius.game,numeric-method}
\alias{Choquet.integral,card.game,numeric-method}
\alias{Choquet.integral,game,numeric-method}
\title{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 \code{game}, \code{card.game} or
\code{Mobius.game}. If the integrand is not positive, this function
computes what is known as the \emph{asymmetric Choquet integral}.}
\section{Methods}{
\describe{
\item{object = "Mobius.game", f = "numeric" }{The Choquet integral of
\code{f} is computed from the \enc{Möbius}{Mobius} transform of a game.}
\item{object = "game", f = "numeric" }{The Choquet integral of
\code{f} is computed from a game.}
\item{object = "card.game", f = "numeric" }{The Choquet integral of
\code{f} is computed from a cardinal game.}
}}
\references{
G. Choquet (1953), \emph{Theory of capacities}, Annales de l'Institut
Fourier 5, pages 131-295.
D. Denneberg (2000), \emph{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), \emph{Fuzzy Measures and
Integrals: Theory and Applications}, Physica-Verlag.
M. Grabisch and Ch. Labreuche (2002), \emph{The symmetric and asymmetric Choquet
integrals on finite spaces for decision making}, Statistical Papers 43, pages
37-52.
M. Grabisch (2000), \emph{A graphical interpretation of the Choquet integral}, IEEE
Transactions on Fuzzy Systems 8, pages 627-631.
J.-L. Marichal (2000), \emph{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), \emph{Some quantities represented by the Choquet
integral}, Fuzzy Sets and Systems 56, pages 229-235.
Murofushi and M. Sugeno (2000), \emph{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.
}
\seealso{
\code{\link{game-class}}, \cr \code{\link{Mobius.game-class}},
\cr \code{\link{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)
}
\keyword{methods}