Defuzzification: What, Why and How?
Fuzzification converts the crisp input into a fuzzy value. Defuzzification converts the fuzzy output of the fuzzy inference engine into a crisp value so that it can be fed to the controller. The fuzzy results generated can not be used in an application, where a decision has to be taken only on crisp values. A controller can only understand the crisp output. So it is necessary to convert the fuzzy output into a crisp value.
There is no systematic procedure for choosing a good defuzzification strategy. The selection of defuzzification procedure depends on the properties of the application
Rule base:
Consider the following two rules in the fuzzy rule base.
R1: If x is A then y is C
R2: If x is B then y is D
A pictorial representation of the above rule base is shown in the following figures
What is the crisp output for an input say xâ ?
Defuzzification methods:
Lambda Cut Method
Maxima Methods
- Height method
- First of maxima (FoM)
- Last of maxima (LoM)
- Mean of maxima (MoM)
Weighted average method
Centroid methods
- Center of gravity method (CoG)
- Center of sum method (CoS)
- Center of area method (CoA)
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Lambda Cut Method:
This Lambda-cut set AÎģ is also called the alpha-cut set.
Lambda-cut method is applicable to derive the crisp value of a fuzzy set or fuzzy relation.
In this method, a fuzzy set A is transformed into a crisp set AÎģ for a given value of Îģ (0 ⤠Îģ ⤠1) as,
AÎģ = { x | ÎŧA(x) âĨ Îģ }
Example – 1: Lambda-cut for Fuzzy Set
A = { (x1, 1.0), (x2, 0.5) , (x3, 0.3) , (x4, 0.4) }
For Îģ = 1: A1= { x1 }
For Îģ = 0.5: A0.5= { x1, x2 }
For Îģ = 0.4: A1= { x1, x2, x4 }
Example – 2: Lambda-cut for Fuzzy Relation
Let us define RÎģ = { (x, y)Â |Â ÎŧR(x, y) âĨ Îģ } as a Îģ cut relation of the fuzzy relation R.
Properties of Îģ cut sets:
If A and B are two fuzzy sets, defined with the same universe of discourse, then
( A âĒ B )Îģ = AÎģ âĒ BÎģ
( A ⊠B)Îģ = AÎģ ⊠BÎģ
( A‘)Îģ â ( AÎģ)’, except for the value of Îģ = 0.5
For any value Îģ1 âĨ Îģ2 implies AÎģ1 â AÎģ2
Test Your Knowledge:
For data given in the table, apply the lambda-cut method and find the following:
1. P0.2, Q0.3
2. ( P âĒ Q )0.6
3. ( P âĒ P‘ )0.8
4. ( P ⊠Q)0.4
1. {x2, x3, x4, x5}, {x1, x2, x3, x5}
2. {x1, x2, x3, x5}
3. {x1, x2}
4. {x5}