Re: how to combine two fuzzy numbers

From: EarlCox (earlcox_at_earlcoxreports.com)
Date: 09/01/04 

Date: Wed, 01 Sep 2004 13:46:02 GMT

There is a difference between operators on fuzzy numbers and applying the
AND or OR operators to fuzzy sets. There is an entire framework for
performing arithmetical operations of bell-shaped, triangular-shaped and
trapezoidal-shaped fuzzy numbers based on the expectancy (degree of
uncertainty or, stated another way, the degree of elasticity) of the number.
Thus, given two crisp scalars X and Y, we can evaluate,

    about(X) + about(Y)

    about(X) - about(Y)

    about(X) / about (Y)

    about(X) * about(Y)

(where about() is, of course, an approximation hedge that converts a scalar
to a bell-shaped fuzzy number of some expectancy). These four examples are
simple arithmetic operators. There are operators that support fuzzy roots,
fuzzy power functions, etc. etc. Applying any of these operators to fuzzy
numbers requires considerations of the individual expectancies of each
number, the evaluation of similarity functions applied to each fuzzy number,
and three or four other critical parameters. Those of us who use genetic and
evolutionary programming strategies to generate fuzzy models, have long ago
learned how to combine fuzzy numbers. In any case, fuzzy arithmetic is a
difficult subject (too difficult to explain in this kind of forum).
Unfortunately, it is not a subject taken up by many academics (for reasons
that continue to elude me), but there are some reasonable references
available.

earl 

--
E a r l  C o x
Founder and President
Scianta Intelligence, LLC
Turn Knowledge Into Intelligence
www dot scianta dot com
AUTHOR:
"The Fuzzy Systems Handbook" (1994)
"Fuzzy Logic for Business and Industry" (1995)
"Beyond Humanity: CyberEvolution and Future Minds"
(1996, with Greg Paul, Paleontologist/Artist)
"The Fuzzy Systems Handbook, 2nd Ed." (1998)
"Fuzzy Logic and Genetic Algorithms for Data Mining and Exploration"
(due Early Fall 2004)
"Roger L. Bagula" <rlbtftn@netscape.net> wrote in message
news:4133B55E.1060009@netscape.net...
>
> Try: the square estimator is
> or(x,y)=Max(x,y)
> and(x,y)=Min(x,y)
> I would use "and" to get best estimate of the probable outcome (
> pessimisticly speaking)
> Which pretty says you take the lowest forcast as being the most likely.
> K Yee wrote:
> > Suppose I have 2 fuzzy estimates, x and y.  For instance, one can
imagine they
> > are independent predictions of rain by two different forecasters.
Suppose I want to
> > combine x and y into a "combined" estimate z.   What is the formula that
optimally
> > relates z to x and y?
> >
> > Know any articles about how to make optimal fuzzy combinations?
> >
> > p.s. I'm a novice, so am only interested in very simple introductory
aritcles.
>
>
> --
> Respectfully, Roger L. Bagula
> tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel:
> 619-5610814 :
> URL :  http://home.earthlink.net/~tftn
> URL :  http://victorian.fortunecity.com/carmelita/435/
>