# Re: Transfoming probability distributions into fuzzy sets - can anyone help?

WSiler (wsiler@aol.com)
Mon, 17 Aug 1998 20:10:26 +0200 (MET DST)

>While I agree that we should not rule out a relationship between fuzzy
>sets and probability ( indeed I am a strong advocate of probabilistic
>semantics for fuzzy sets) I do not agree that we should take probability
>distributions of random variables (normalised or not) as membership
>functions of fuzzy sets. The former quantify uncertainty regarding the
>value of a random variable and the other vagueness of definition.
>
It is certainly true that "probability distributions quantify uncertainty
regarding the value of a random variable", to say that "[membership functions
of fuzzy sets characterize] vagueness of definition" is a quite unnecessary
restriction on fuzzy sets. Having worked on real-world applications of fuzzy
expert systems for some fifteen years now, I consider that fuzzy sets can
characterize uncertainty of whatever origin, including both vagueness and
values of random variables among many others.

To assert that a normal distribution characterizes a numeric random variable
subject to a large number of small errors amounts to a tautology, parameterized
perhaps as a mean and variance. However, I can (and often do) characterize that
same variable as a bell-shaped fuzzy number, paramaterized perhaps as central
value and a hedge "roughly". There is no vagueness here, just an uncertainty as
to precise value. In an expert system, "roughly 2" is a heck of a lot more
useful than "2 +/- 25%".

A list of the kinds of uncertainty which can be fruitfully represented by fuzzy
quantities (e.g. truth values of scalars, fuzzy numbers, membership functions,
truth values of rules, truth values of members of a discrete fuzzy set,...)
would probably be quite long. If I'm not sure that a car is a Ford or a
Chevrolet, that uncertainty is easily represented by the grades of membership
in a discrete fuzzy set of car makes, for example.

I'm not sure what latitude FRIL offers in the kinds of things which can be
represented by fuzzy quantities, but I surely hope it covers more than vague
definitions.

William Siler

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