There is one last piece to using fuzzy sets (see Fuzzy Set Theory and Fuzzy Correlation and Inference). How do you get a representative value from a fuzzy set? Obviously, it is hard to set a faucet to slow or a car to fast. We need a specific value that hopefully best represents the fuzzy set in its domain. This is a process called defuzzification.
The easiest method would be just to arbitrarily pick a point from the fuzzy set, perhaps the point in the domain where the fuzzy set reaches its maximum. This is easy, but is clearly not the best. What if a fuzzy set looks like the following:
Fuzzy Variable: Temperature
Fuzzy Values: [(pi 30 5)] or [(pi 60 10)](*)
1.0000 * *
0.9500
0.9000
0.8500 * *
0.8000
0.7500
0.7000
0.6500 * * * *
0.6000
0.5500
0.5000
0.4500
0.4000
0.3500 * *
0.3000
0.2500
0.2000
0.1500 * * * *
0.1000
0.0500
0.0000************* ******** ***************
|----|----|----|----|----|----|----|----|----|----|
0.00 20.00 40.00 60.00 80.00 100.00
Universe of Discourse: From 0.00 to 100.00
Which maximum point is the "best" for the fuzzy set: 30 or 60? A better method would be to take the average of the maximum points, in this case 45. This method is very common and is called mean of maxima defuzzification.
However, there are even better methods for defuzzification. Consider the following fuzzy set:
Fuzzy Variable: Temperature
Fuzzy Values: [warm > 0.6500] or [very [extremely hot]](*)
1.0000 *****
0.9500
0.9000 *
0.8500 *
0.8000
0.7500 *
0.7000
0.6500 *****************
0.6000 * * *
0.5500 * *
0.5000 * *
0.4500 * * *
0.4000 * *
0.3500 * *
0.3000 *
0.2500 *
0.2000 **
0.1500 **
0.1000 **
0.0500 **
0.0000*
|----|----|----|----|----|----|----|----|----|----|
0.00 20.00 40.00 60.00 80.00 100.00
Universe of Discourse: From 0.00 to 100.00
The mean of maxima defuzzification method would give a value of 90. Clearly, though, the bulk of this fuzzy set is in the hump in the middle. It would seem that there must be a better defuzzification method which would factor in all parts of a fuzzy set. A common defuzzification in control domains is the moment, or center of gravity, defuzzification method because it accounts for the entire fuzzy set shape and tends to smooth out the fuzzy region. This method finds the center of gravity where half of the "weight" of the fuzzy set is on the left and half on the right. The moment defuzzification method figures out the area under the shape of the fuzzy set and gets the center point. For the fuzzy set above, the moment defuzzification method returns 61.39.
Next, Fuzzy Expressions
