Re: Fuzzy relations vs. Mamdani model


Subject: Re: Fuzzy relations vs. Mamdani model
WSiler@aol.com
Date: Sun Nov 19 2000 - 13:24:07 MET


In a message dated 11/17/00 8:05:43 AM Central Standard Time,
pimvdb@cs.utwente.nl writes:

<<
 WSiler@aol.com wrote:
> Actually, the fuzzy relation involved a fuzzy implication operator, and is
> completely unworkable for any true fuzzy implication operator (one which
> collapses to the classical implication for crisp values).
 
 Why unworkable?
>>

The answer is simple. Since computers are discrete, we give a discrete
evaluation. We are given the rule "If A = A' then B' = B", to be evaluated as
follows.

We first construct the implication matrix A(i) -> B(j), where A(i)->B(j) is
defined as any of the implication operators defined in Klir and Yuan's
Chapter 11, "Approximate Reasoning". For our demonstration below, we include
the Classical, Gaines-Reischer, Godel, Wu, S, SS, SG and GS operators which
they define as valid fuzzy implication operators. We also include the Mamdani
operator, which works, even though it is not a valid fuzzy implication
operator.

We select A and APr (A') vectors which overlap but do not coincide, a very
common case. We get these results:

Test of some fuzzy implications
Data:
 i a' a b
 1 0.00 0.00 0.00
 2 0.00 0.13 0.13
 3 0.00 0.25 0.25
 4 0.00 0.38 0.38
 5 0.00 0.50 0.50
 6 0.13 0.63 0.63
 7 0.25 0.75 0.75
 8 0.38 0.88 0.88
 9 0.50 1.00 1.00
10 0.63 0.88 0.88
11 0.75 0.75 0.75
12 0.88 0.63 0.63
13 1.00 0.50 0.50
14 0.88 0.38 0.38
15 0.75 0.25 0.25
16 0.63 0.13 0.13
17 0.50 0.00 0.00
18 0.38 0.00 0.00
19 0.25 0.00 0.00
20 0.13 0.00 0.00

I b b'C b'M b'GR b'G b'S b'SS b'SG b'GS b'Wu
 1 0.00 0.75 0.00 0.50 0.50 0.75 0.75 0.50 0.50 0.75
 2 0.13 0.75 0.13 0.63 0.63 1.00 1.00 0.63 0.63 0.75
 3 0.25 0.75 0.25 0.75 0.75 1.00 1.00 0.75 0.75 0.75
 4 0.38 0.75 0.38 0.88 0.88 1.00 1.00 0.88 0.88 0.88
 5 0.50 0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
 6 0.63 0.75 0.63 1.00 1.00 1.00 1.00 1.00 1.00 1.00
 7 0.75 0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00
 8 0.88 0.88 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00
 9 1.00 1.00 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00
10 0.88 0.88 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00
11 0.75 0.75 0.75 1.00 1.00 1.00 1.00 1.00 1.00 1.00
12 0.63 0.75 0.63 1.00 1.00 1.00 1.00 1.00 1.00 1.00
13 0.50 0.75 0.50 1.00 1.00 1.00 1.00 1.00 1.00 1.00
14 0.38 0.75 0.38 0.88 0.88 1.00 1.00 0.88 0.88 0.88
15 0.25 0.75 0.25 0.75 0.75 1.00 1.00 0.75 0.75 0.75
16 0.13 0.75 0.13 0.63 0.63 1.00 1.00 0.63 0.63 0.75
17 0.00 0.75 0.00 0.50 0.50 0.75 0.75 0.50 0.50 0.75
18 0.00 0.75 0.00 0.50 0.50 0.75 0.75 0.50 0.50 0.75
19 0.00 0.75 0.00 0.50 0.50 0.75 0.75 0.50 0.50 0.75
20 0.00 0.75 0.00 0.50 0.50 0.75 0.75 0.50 0.50 0.75

Note that except for the Mamdani operator, the result for B' is non-zero
everywhere from minus to plus infinity. This is clearly a useless result.
However, the Mamdani operator gives a quite reasonable and useful result.

Willilam Siler

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