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

J. Lawry (enjl@PROBLEM_WITH_YOUR_MAIL_GATEWAY_FILE)
Fri, 11 Sep 1998 02:03:45 +0200 (MET DST)

Carlos Gershenson (carlos@jlagunez.iquimica.unam.mx) 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.
:
:
:
: >
: > >
: > > 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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:

I'm not sure that this is that good an example of a problem that cannot be
modelled by probability theory. We could after all consider the
probability that a component picked at random from a Villager was
identical to the same component (i.e. component with the same
function) in the Quest. If Villager and Quest are indeed 'similar' then
this could be 0.95.

Jonathan Lawry

-- 
Dr Jonathan Lawry,
AI Group,
Dept. Engineering Mathematics,
University of Bristol,
Queens Building,
University Walk,
Bristol, BS8 1TR, UK

Email:j.lawry@bristol.ac.uk Tel:+44 117 928 8184

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