zeta-methods: The zeta transform

In kappalab: Non-Additive Measure and Integral Manipulation Functions

The zeta transform

Description

Computes the zeta transform of a set function given under the form of its Möbius transform. The zeta transform is the inverse of the Möbius transform.

Methods

object = "Mobius.capacity"

Returns an object of class capacity.

object = "Mobius.card.set.func"

Returns an object of class card.set.func.

object = "Mobius.game"

Returns an object of class game.

object = "Mobius.set.func"

Returns an object of class set.func.

References

G-C. Rota (1964), On the foundations of combinatorial theory. I. Theory of Möbius 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 Möbius 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 Mobius 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)

kappalab documentation built on Nov. 8, 2023, 1:07 a.m.