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

Anthony Cowden (cowden@sonalysts.com)
Mon, 17 Aug 1998 21:57:55 +0200 (MET DST)

WSiler wrote:
>
> >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.

Bill:

To help me understand some of the points raised, allow me to pose a
problem:

I own a Mercury Villager mini-van, which is made in the same factory as
the Nissan Quest (in Ohio, by the way), and most of the parts are
identical and interchangeable. As you might assume, they look very
similar. Now, if I see 2 mini-vans in a parking lot, and they appear to
be a Villager/Quest, but I can't tell from the distance I am at, than
the probability that the one on the left is a Villager is .5, and the
probability that it is a Quest is .5 (the same goes for the one on the
right).

Now, if I walk out into the parking lot and inspect the 2 vehicles, I
find that the one on the left is a Quest and the one on the right is
also a Quest. The probability now is 0.0 that either one is a
Villager. But what about the membership in the set (classification,
identity, whatever) of Villager? I would say that the Quest has a
membership of .95 in the set of Villager (and vice versa). How does
probability help explain to a mechanic that he can fix a Villager if he
has only ever fixed Quests before?

Tony

>
> 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
>

```--
*********************************************************************
Anthony Cowden, Manager, Fuzzy Systems Solutions
Sonalysts, Inc.
Fuzzy Systems Solutions: http://www.sonalysts.com/fuzzy.html
Fuzzy Query (TM): http://www.sonalysts.com/fq.html

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