Is it possible to directly do defuzzication a discrete fuzzy set with the requirement that the defuzzification result must be one of the discrete values in the fuzzy set?
The answer is NO.
For a continuous fuzzy set, we know the defuzzification formula according to the centroid method as follows:
We set
The defuzzification formula becomes
The above formula is exactly the formula for calculating the mathematical expected value with f(x) as the probability density function. The defuzzification value x* has the full meaning of the mean value E[X] of the fuzzy set.
Similarly the defuzzification formula according to the centroid method for discrete fuzzy sets as follows:
We set
The defuzzification formula becomes
The value x* is also the mean value E[X] of the discrete fuzzy set. However, this value generally does not match any value in the discrete fuzzy set. So what value do we get in the discrete fuzzy set to represent the defuzzification result? The value of the element with the maximum membership function value (corresponding to the highest probability) is not possible because it might be far from the mean value. Not even other values. As a result, no value can be representative with a quality for a discrete fuzzy set in the defuzzification role if only based on the fuzzy set data itself.
Maximal defuzzification method is wrong!
In this method, one takes the value of the element with the maximum membership function value (corresponding to the highest probability) as the defuzzification value. As analyzed above, this solution is wrong. That is in terms of probability. Note that fuzzy theory is a successor of classical probability theory. It can only be extended but not contrary to the principle of probability theory.
We can further illustrate in terms of fuzzy set.
Consider fuzzy set
score.high := (0/0, 0.25/1, 0.75/2, 1/3)
This fuzzy set has the mean value of (0 * 0 + 1 * 0.25 + 2 * 0.75 + 3 * 1) / (0 + 0.25 + 0.75 + 1) = 2.375VERY score.high is
(0/0, 0.0625/1, 0.5625/2, 1/3)
This fuzzy set has the mean value of (0 * 0 + 1 * 0.625 + 2 * 0.5625 + 3 * 1) / (0 + 0.625 + 0.5625 + 1) = 2.577
Two fuzzy sets score.high and VERY score.high are two different fuzzy sets with different mean values (the most important characteristic of the value of the fuzzy sets). But if the Maximal defuzzification method is applied, both sets take the value 3 - the value of the element has the highest membership function value. We can analyze similarly with VERY VERY... Many different fuzzy sets in essence cannot share the same defuzzification value!
Solution for fuzzy set classification
Instead of the defuzzification approach, we approach the classification of fuzzy sets.
Suppose we have a fuzzy term defined as follows
term SCORE(0, 3, 4) { fuzzy good = 0; fuzzy normal = 1; fuzzy bad = 2; fuzzy dangerous = 3; };
Four fuzzy sets, good, normal, bad, dangerous represent four classes. If the fuzzy set needed to classify has its mean value close to a class, it will belong to that class with the corresponding value. For example, if it belongs to the normal class, the classification value will be 1. The classification implementation is available in the Classify() function since fuzzy version 2.0.
Because the classification returns a value in the data range of the fuzzy set, this can be considered as an indirect defuzzification method.
Contact: tuyen@omarine.org
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