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

Carlos Gershenson (carlos@jlagunez.iquimica.unam.mx)
Wed, 26 Aug 1998 19:18:12 +0200 (MET DST)

On Mon, 17 Aug 1998, Anthony Cowden wrote:

> 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:
>
> Thanks for the automobile lead-in...
>
> 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

In the problem you propose, you would need to use "similarity". The more
similarity there is between 2 elements, the less they exculde each other.

This is why a mechanic can fix a Villager the first time he sees one.
Because it is very similar to the Quest, which he is used to.

>
> >
> > 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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"There is no Truth but that of Eternal struggle..."
-Orunlu the Keeper

Carlos Gershenson
http://132.248.11.4/~carlos/

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