From: earlcox@earlcoxreports.com
Bill,
You are correct, but a full spectrum of fuzzy arithmetic and fuzzy
comparison operators is necessary for any robust fuzzy expert system. As an
example you need to be able to include statements like:
if A is much greater than B
if A is very much greater than B
if A is somewhat less than somewhat near(B)
if A is not much greater than B
where "A" and "B" are either scalars or fuzzy sets (any competent fuzzy
expert system should automatically apply a "Near" or "About" hedge to a
scalar (or a scalar expression) and thus convert it into a fuzzy number).
The techniques for comparing fuzzy numbers with varying degrees of overlap
and with varying topologies are not simple, and I am still amazed that so
few practitioners of fuzzy system modeling are not familiar with them. As
you know, I have been building fuzzy business models since 1979 with fuzzy
number comparators.
And, of course, Silvert is absolutely correct, the underlying mechanics of
fuzzy comparison and crisp comparison operators are very different (since
fuzzy compartors must take into account the latent entropy of the operation,
that is, a fuzzy comparator enforces closure -- the result of A is much
greater than B (as an example) is a new fuzzy set whose expectancy (width)
is dependent on the explicit or implicit similarity function applied to the
two target sets.)
Just a note
Earl
"William Siler" wrote in message
news:49b9df3d.0312141524.45fc34c4@posting.google.com...
> "William Silvert" wrote in message
news:<3fdca4ff$0$24722$a729d347@news.telepac.pt>...
> > I find questions like this disturbing. Why should we assume that there
is an
> > exact correspondence between operations on fuzzy sets and on crisp sets?
> >
> > Either ab, or a and b overlap. If they overlap you can compare the
> > means, the medians, the ranges, or anything else. There are more
dimensions
> > to the comparison of two fuzzy numbers, why ignore that?
> >
> > Bill Silvert
> >
> Bill silvert is right in asserting that "There are more dimensions to
> the comparison of two fuzzy numbers". In my experience the main use of
> these approximate comparisons is when we are comparing a scalar to a
> fuzzy number. Then the comparison scheme I outlined in my previous
> communication is of great utility. For example, in a fuzzy expert
> system for detecting and classifying alarm conditions for a patient in
> an intensive care unit, we continuously compare an input value
> (pulmonary artery catheter blood pressure) to a "normal range", where
> the normal range is a fuzzy number that varies from patient to
> patient, and slowly over time for a specific patient. If the current
> pressure is less than the normal, it means something is going wrong;
> the confidence that it is really less is a quantitative indication as
> to how wrong it is. If the pressure returns qualitatively to the
> normal range, and stays there for a reasonable length of time, it
> indicates that the problem has been corrected. A similar set of rules
> provides detection of several different problems. In a program to get
> the minerals in a rock sample, we compare fuzzy peaks in its x-ray
> diffraction spectrum with fuzzy peaks in a set of library spectra for
> single minerals. In a program to classify regions in an echocardiogram
> of a beating heart, we compare the area, x-centroid and y-centroid of
> an echocardiogram with fuzzily defined lbrary values. And so on. In
> short, the full array of approximate numerical comparisons for fuzzy
> numbers are very useful in the real world.
>
> William Siler
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