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

Fred A Watkins (fwatkins@hyperlogic.NO_SPAM.com)
Mon, 17 Aug 1998 20:56:41 +0200 (MET DST)

wsiler@aol.com (WSiler) wrote:

>>Just a general comment that I think it is best to keep fuzzy logic and
>>probability theory as far removed as possible!
>
>I realize than many or most fuzzy persons share your viewpoint here, but I do
>not. Both probability distributions and membership functions look from
>different vantage points ar pretty much the same thing - uncertain numbers,
>whatever the cause of the uncertainty may be. A very few persons have worked
>seriously on the relationship between the two, which I think is unfortunate.

There are mathematical relations (see below) because both fields grind over
the same territory: R, the reals (usually, at least, because the input to the
system is commonly a *measurement*) and I, the unit interval. But the key point
of difference is philosophical. Fuzziness is ambiguity occasioned by the use
of words; since words want *definition*, and definitions are impossible to
obtain in the realm of experience, to use a word to refer to external reality
is to be imprecise. The degree of imprecision is measured as degrees of fuzziness.
On the other hand, probability describes lack of knowledge of *well defined*
events. To summarize: fuzziness is lack of information in *meanings*, probability
is lack of information in *occurrences". Of course, "well defined events" are,
strictly speaking, unavailable because of the difficulty in applying definitions
to events; simplification is always required. Finally, if one wants to investigate
probability as a description of *belief*, OK, but now the treatment is a mathematical
abstraction (admitting "well-defined events") that has unclear relation to reality.
A famous remark by Einstein captures this idea very well.

>There is, in any event, a quick (if incomplete and inaccurate) answer to the
>question. Probability density functions have area one; fuzzy numbers and
>membership functions usually have max value one. Simply normalize the
>probability distribution to a max value of one, and there is your fuzzy number
>or membership function. If the probability distribution is discrete, do the
>same thing, and there is your discrete fuzzy set.

Let's tighten up just a bit. The typical definition for a *distribution function"
over the real line is

F(x) = P((-infinity, x])

where P is the underlying probability measure. Since F is defined in terms of P,
it inherits its min and max from P (i.e., 0 and 1). Since the measures of sets
in a nested sequence of measurable sets are monotone, so is F, since for
any x < y we have (-infinity, x] is included in (-infinity, y].

The above implies that any distribution *function* is in fact a fuzzy set, because
a fuzzy set is simply a map from the "universe" (here it's R) into I = [0, 1],
without other conditions.

But distribution functions are not particularly valuable linguistically. For example,
while a Gaussian about zero might describe "(approximately) zero", the distribution function
that density generates (the "erf" function) has no meaningful relation to the word "zero".

What most people want to do is start with probability *densities*. A density f (over R)
is a non-negative integrable function such that Integral_over_R(f) = 1. Given a density
f, one recovers its distribution F by integration:

F(x) = Integral_from_-infinity_to_x(f(t) dt)

It's not always possible to obtain a density from a distribution because the derivative
of the distribution may not exist. Further assumptions can be made to help with this.

Anyway, the idea now is to relate densities to fuzzy sets. As previously remarked, a
density is integrable, but integrability does not imply boundedness. This means that
the suggested normalization is not generally applicable. But when a density has a
finite supremum (least upper bound) this normalization is possible and the normalized
density is a fuzzy set *by definition*. On the other hand, a fuzzy set need not be
integrable (remember "erf"?) and so a fuzzy set cannot in general be converted into
a probabilty density without loss of data (say, by truncation). Moreover, a fuzzy set
need not be monotone and so is not in general a distribution function.

To summarize: a fuzzy set derives a probability density precisely when that set is
integrable; a probability density derives a fuzzy set precisely when it is bounded.

>The question of the circumstances under which that simple answer is wrong is
>interesting, and comments on this would be welcome. Am I wide open?

It's really pretty straightforward. Hope this helps.

Fred A Watkins, Ph.D.
HyperLogic Corporation
PO Box 300010
Escondido, CA 92030-0010 USA
voice: +1 760 746 2765 x 9117
fax: +1 760 746 4089
email: fwatkins@hyperlogic.SNIP_THIS_OUT.com

Decode email address to contact me.

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