kappalab (version 0.4-0)

zeta-methods: The zeta transform

Description

Computes the zeta transform of a set function given under the form of its M�bius{Mobius} transform. The zeta transform is the inverse of the M�bius{Mobius} transform.

References

G-C. Rota (1964), On the foundations of combinatorial theory. I. Theory of M�bius{Mobius} 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�bius{Mobius} 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

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

Examples

## the M�bius transform of a capacity
a <- Mobius.capacity(c(rep(0,15),1),4,4)
a

## its zeta transform
zeta(a)

## let us check
Mobius(zeta(a))

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