Re: Question: Fuzzy set theory & theory of evidence

Cliff Joslyn (joslyn@lanl.gov)
Sat, 11 Oct 1997 11:40:44 +0200 (MET DST)

Johan wrote:
> Uncertainty, non-statistical in nature, can be modelled by 'newly'
> developed formalisms as fuzzy set theory and the Dempster-Shafer theory of
> evidence. How are thse two theories related? Can they be seen as
> competitive, or are they complementary, relating to different 'kinds of
> uncertainty?

To a certain extent, they are complementary. Fundamentally, they are
different in that a fuzzy set is an unconstrained (except to be in
[0,1]) weighting of elements, whereas a DS "body of evidence" (also
known as a random set) begins with an additively constrained weighting
on subsets. However, the so called "one-point trace" of a body of
evidence is a fuzzy set, and Goodman has shown that there are
isomorphisms between classes of random sets and the classes of fuzzy
sets which are their one-point traces. This gets a bit technical, but
suffice it to say that DS and fuzzy occupy two important places in the
overall world of non-probabilistic information theory.

For general background, see:

Klir, George and Folger, Tina: (1987) Fuzzy Sets, Uncertainty, and
Information, Prentice Hall

For the gory details, see:

Joslyn, Cliff: (1996) "Aggregation and Completion of Random Sets with
Distributional Fuzzy Measures", Int. J. of Uncertainty, Fuzziness, and
Knowledge-Based Systems, v. 4:4, pp. 307-329.

Goodman, IR: (1994) "New Characterization of Fuzzy Logic Operators
Producing Homomorphic-Like
Relations with One-Point Coverages of Random Sets", in: Advances in
Fuzzy Theory and Technology, ed. PP Wang, pp.133-159, Bookwrights Press,
Raleigh NC,

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