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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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