# fuzzy number input to a FIS

From: Makropoulos, Christos (c.makropoulos@ic.ac.uk)
Date: Mon Jan 22 2001 - 22:11:10 MET

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Dear sir,
Thank you for your reply. My objective is to use fuzzy numbers as input to a
FIS (in contrast to usual crisp input - fuzzification - rules-
defuzzification - crisp output. How would you process a fuzzy number (given
by a centre and spread or something equivalent) through a fuzzy inference
system? A good example is Matlab's FIS. The only idea that comes to mind is
to treat the membership functions in the FIS as a crisp mapping
fuzzy-to-fuzzy and use the extention principle in a number of a-cuts ,

i.e. "assigning membership values in a fuzzy number"

{0/(x-dx) + 0.5/(x-dx/2) + 1/x + 0.5/(x+dx/2) + 0/(x+dx)}, the
fuzzy input data value in 3 a-cuts (0, 0.5, 1)

m(x) the membership of the fuzzy number to the property in question (say
"suitability")

{0/m[(x-dx)] + 0.5/m[(x-dx/2)] + 1/m[x] + 0.5/m[(x+dx/2)] +
0/m[(x+dx)]}, the new output value?

In the case of a standard type-1 FIS like matlab's would this practically
mean that i could pass a vector [x-dx, x-dx/2, x, x+dx/2, x+dx] instead of
just a point x for every x in my data set and then just assign the [0, 0.5,
1, 0.5, 0] membership values of the original fuzzy number to the output
vector? Is there some inconsistency in my argument?

Thanks for your help
Any ideas will be appreciated
Christos

-----Original Message-----
From: WSiler@aol.com [mailto:WSiler@aol.com]
Sent: Monday, January 15, 2001 7:49 PM
To: c.makropoulos@ic.ac.uk
Subject: Re: FW: fuzzy number input to a FIS

In a message dated 1/15/01 7:44:06 AM Central Standard Time,
c.makropoulos@ic.ac.uk writes:

<< Do you pass the fuzzy number from the FIS as a series of a-cuts or is
there another way?
>>

Very definitely another way.

Our system FLOPS parameterizes fuzzy numbers and membership functions. We
can
specify three different shapes; piecewise linear (triangular or
trapezoidal),
piecewise quadratic (s-shape), and normal (Gaussian).

In our system membership functions are specified by four numbers and a
shape.
The numbers (for linear and quadratic) are the first x-value at which the
function begins to increase from 9; the first x-value at which the
membership
reaches 1; the xvalue at which the function begins to decline from zero; and

the x-value at which the function reaches zero after declining. Say these
x-values are x1, x2, x3 and x4. For normal shapes, the function has
membership of 0.5 at (x1 + x2)/2 and (x3 + x4)/2, and one at x2 and x3.

Fuzzy numbers are symmetrical about a central value, and are specified by
the
central value and a measure of dispersion. The dispersion can be specified
by
hedges (about 6, nearly 4.5) or by absolute and relative error 6 +/- 2 +/-
10%).

The FRIL shell permits specifying membership function as a set of pairs
{mu1/x1, mu2/x2, ...) or something very much like that.

I don't know what you are trying to do with fuzzy numbers. I gather from
your
original message that this has something to do with rules. If you could be
somewhat more informative about your problem I might be able to be of more
help. You might also look at my Web page, http://users/aol/com/wsiler/ where

there is a downloadable manual on building fuzzy expert systems and an
outdated demo version of our expert system shell FLOPS.

Sincerely, William Siler

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