The M<c3><b6>bius transform

Description

Computes the M<c3><b6>bius transform of a set function. The M<c3><b6>bius transform is the inverse of the zeta transform.

Methods

object = "capacity"

Returns an object of class Mobius.capacity.

object = "card.set.func"

Returns an object of class Mobius.card.set.func.

object = "game"

Returns an object of class Mobius.game.

object = "set.func"

Returns an object of class Mobius.set.func.

References

G-C. Rota (1964), On the foundations of combinatorial theory. I. Theory of M<c3><b6>bius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2, pages 340-368.

A. Chateauneuf and J-Y. Jaffray (1989), Some characterizations of lower probabilities and other monotone capacities through the use of M<c3><b6>bius inversion, Mathematical Social Sciences 17, pages 263-283.

M. Grabisch, J-L. Marichal and M. Roubens (2000), Equivalent representations of set functions, Mathematics of Operations Research 25:2, pages 157-178.

See Also

Mobius.capacity-class,
Mobius.card.set.func-class,
Mobius.game-class,
Mobius.set.func-class,
capacity-class,
card.set.func-class,
game-class,
set.func-class,
zeta-methods.

Examples

## a capacity
mu <- capacity(0:15)
mu

## its Mobius transform
a <- Mobius(mu)
a

## its zeta transform
zeta(a)

## a similar example with a game object
mu <- game(c(0,-2:12))
mu
Mobius(mu)
zeta(Mobius(mu))

## a similar example with a set.func object
mu <- set.func(-7:8)
mu
Mobius(mu)
zeta(Mobius(mu))

## a similar example with a card.set.func object
mu <- card.set.func(-3:4)
mu
Mobius(mu)
zeta(Mobius(mu))

Results