is.monotone-methods: 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 set.func, card.set.func or Mobius.set.func.

Arguments

Methods

object = "Mobius.set.func", verbose = "logical", epsilon = "numeric": Returns an object of class logical. If verbose=TRUE, displays the violated monotonicity constraints, if any.
object = "card.set.func", verbose = "logical", epsilon = "numeric": Returns an object of class logical. If verbose=TRUE, displays the violated monotonicity constraints, if any.
object = "set.func", verbose = "logical", epsilon = "numeric": Returns an object of class logical. If verbose=TRUE, displays the violated monotonicity constraints, if any.

Details

For objects of class set.func or card.set.func, the monotonicity constraints are considered to be satisfied (cf. references hereafter) if the following inequalities are satisfied $$\mu(S \cup i) - \mu(S) \ge -epsilon$$ for all $S$ and all $i$. For objects of class Mobius.set.func, it is required that a similar condition with respect to the Möbius representation be satisfied (cf. references hereafter).

References

A. Chateauneuf and J-Y. Jaffray (1989), Some characterizations of lower probabilities and other monotone capacities through the use of Möbius inversion, Mathematical Social Sciences 17:3, pages 263--283.

M. Grabisch (2000), The interaction and Möbius 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.

Examples

Run this code

## 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)