Re: Transfoming probability distributions into fuzzy sets -

Anthony Cowden (cowden@sonalysts.com)
Wed, 26 Aug 1998 19:36:39 +0200 (MET DST)

At 11:19 AM 8/19/98 -0500, Carlos Gershenson wrote:
>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.

Yes, I agree with this approach.

What I was trying to demonstrate in my example was that after we test the
probability (i.e., walk up to the car and see that it is one and not the
other) we are left with a binary classification of the car (it is (1) a
Villager and (0) a Quest). There is no longer any probability in this
case; the binary truth has been arrived at. And that's fine, really. That
is what probability is really good at.

With fuzzy sets, we can encode more information, and in a way (I don't
believe) that we can not with probability: we can encode the degree to
which we found not only a Villager, but also a Quest.

Well, I may be mixing apples and oranges a little bit in this whole
discussion, but I thought it was an interesting case to discuss to
illuminate some points...

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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>
>"There is no Truth but that of Eternal struggle..."
> -Orunlu the Keeper
>
> Carlos Gershenson
> http://132.248.11.4/~carlos/
>
>
>
*********************************************************************
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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