expect.Choquet.unif-methods {kappalab}

Expectation and standard deviation of the Choquet integral in the uniform and normal cases

Description

Methods for computing the expectation and standard deviation of the Choquet integral in the standard uniform and standard normal cases.

Methods

object = "game"

Returns the expectation or the standard deviation of the Choquet integral.

References

J-L. Marichal and I. Kojadinovic (2007), The distribution of linear combinations of lattice polynomials from the uniform distribution, submitted.

See Also

game-class, Mobius.game-class.

Examples

## a capacity
mu <- capacity(c(0,0.1,0.6,rep(0.9,4),1))

## the expectation and the standard deviation
## of the Choquet integral in the uniform case
expect.Choquet.unif(mu)
sd.Choquet.unif(mu)

## the same but empirically
m <- 10000
ch <- numeric(m)
for (i in 1:m) {
     f <- runif(3) 
     ch[i] <- Choquet.integral(mu,f)
}
mean(ch)
sd(ch)

## the expectation and the standard deviation
## of the Choquet integral in the normal case
expect.Choquet.norm(mu)
sd.Choquet.norm(mu)
expect.Choquet.norm(Mobius(mu))

## the same but empirically
for (i in 1:m) {
     f <- rnorm(3) 
     ch[i] <- Choquet.integral(mu,f)
}
mean(ch)
sd(ch)