| is.monotone-methods {kappalab} | R Documentation |
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.
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 U i) - mu(S) >= -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).
logical. If verbose=TRUE,
displays the violated monotonicity constraints, if any.logical. If
verbose=TRUE, displays the violated monotonicity
constraints, if any.logical. If verbose=TRUE,
displays the violated monotonicity constraints, if any.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.
Mobius.set.func-class,
card.set.func-class,
set.func-class.
## a monotone set function mu <- set.func(c(0,1,1,1,2,2,2,3)) mu is.monotone(mu) ## the Möbius 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)