https://github.com/cran/kappalab
Tip revision: 085a64b3faed6893ba219fbdb8aa3e2e0d3f56f1 authored by Ivan Kojadinovic on 18 July 2015, 00:00:00 UTC
version 0.4-7
version 0.4-7
Tip revision: 085a64b
is.monotone-methods.Rd
\name{is.monotone-methods}
\docType{methods}
\alias{is.monotone}
\alias{is.monotone-methods}
\alias{is.monotone,Mobius.set.func-method}
\alias{is.monotone,card.set.func-method}
\alias{is.monotone,set.func-method}
\title{Test method}
\description{Tests whether a set function is monotone with respect to set
inclusion. The set function
can be given either under the form of an object of class
\code{set.func}, \code{card.set.func} or \code{Mobius.set.func}.
}
\section{Methods}{
\describe{
\item{object = "Mobius.set.func", verbose = "logical", epsilon =
"numeric" }{
Returns an object of class \code{logical}. If \code{verbose=TRUE},
displays the violated monotonicity constraints, if any.}
\item{object = "card.set.func", verbose = "logical", epsilon =
"numeric" }{
Returns an object of class \code{logical}. If
\code{verbose=TRUE}, displays the violated monotonicity
constraints, if any.}
\item{object = "set.func", verbose = "logical", epsilon = "numeric" }{
Returns an object of class \code{logical}. If \code{verbose=TRUE},
displays the violated monotonicity constraints, if any.}
}}
\details{
For objects of class \code{set.func} or \code{card.set.func}, the
monotonicity constraints are considered to be satisfied
(cf. references hereafter) if the following inequalities are satisfied
\deqn{\mu(S \cup i) - \mu(S) \ge -epsilon}{mu(S U i) - mu(S) >=
-epsilon}
for all \eqn{S} and all \eqn{i}.
For objects of class \code{Mobius.set.func}, it is
required that a similar condition with respect to the \enc{M鐽ius}{Mobius}
representation be satisfied (cf. references hereafter).
}
\references{
A. Chateauneuf and J-Y. Jaffray (1989), \emph{Some characterizations of
lower probabilities and other monotone capacities through the use of
\enc{M鐽ius}{Mobius} inversion}, Mathematical Social Sciences 17:3, pages
263--283.
M. Grabisch (2000), \emph{The interaction and \enc{M鐽ius}{Mobius} representations of fuzzy
measures on finites spaces, k-additive measures: a survey}, in:
Fuzzy Measures and Integrals: Theory and Applications, M. Grabisch,
T. Murofushi, and M. Sugeno Eds, Physica Verlag, pages 70-93.
}
\seealso{
\code{\link{Mobius.set.func-class}},
\cr \code{\link{card.set.func-class}},
\cr \code{\link{set.func-class}}.
}
\examples{
## a monotone set function
mu <- set.func(c(0,1,1,1,2,2,2,3))
mu
is.monotone(mu)
## the Mobius representation of a monotone set function
a <- Mobius.set.func(c(0,1,2,1,3,1,2,1,2,3,1),4,2)
is.monotone(a)
## non-monotone examples
mu <- set.func(c(0,-7:7))
is.monotone(mu,verbose=TRUE)
a <- Mobius(mu)
is.monotone(a,verbose=TRUE)
}
\keyword{methods}
%\keyword{ ~~ other possible keyword(s)}