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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