Lecture05-Defuzzification

8/12/2019 Lecture05-Defuzzification

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Lecture 05

Defuzzification

"


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vDefuzzification means the fuzzy-to-crispconversions. The fuzzy results generated cannot be used to the applications.

It is necessary to convert the fuzzy quantities into crisp quantities

for further processing.

vDefuzzification has the capability to reduce a fuzzy set to a crisp single-valued quantity or as a set.


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Outline

vLambda Cuts for Fuzzy Sets

for Fuzzy Relations

vDefuzzification Methods Maxima method

Centroid method

Weighted average method

Middle-of-maxima method

First-of-maxima or last-of-maxima


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Lambda Cuts for Fuzzy Sets

vConsider a fuzzy set Its lambda cut denoted by

is a crisp set:

the value of lambda cut set isx, when the membership value

corresponding tox is greater than or equal to the specified.

This lambda cut set can also be called as alpha cut

set.

(0 1)


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Lambda Cuts for Fuzzy Sets

vProperties of Lambda Cut Sets

The standard set of operations on fuzzy sets is similar to the

standard set operations on lambda cut sets.


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Lambda Cuts for Fuzzy Relations

vConsidering a fuzzy relation A fuzzy relation can be converted into a crisp relation

by depending the lambda cut relation of the fuzzy

relation as:

v Properties of Lambda Cut Relations


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Lambda Cuts: Examples

Example 1. Two fuzzy sets P

and Q

aredefined onx as follows:

Find the following cut sets


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Solution. Given

We have

Then we can calculate


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Solution. Given

We have



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Solution. Given

We have


0.1 0.2 0.3 0.2 0.4


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Solution. Given

We have


0 0 0 0


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Example 2. The fuzzy setsA

and B

aredefined in the universeX = {0, 1, 2, 3}, with the

following membership functions:

Define the intervals alongx-axis corresponding

to the cut sets for each fuzzy setA and Bfor following values of = 0.2, 0.5, 0.6.

2


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Solution. The membership degrees for each elements:

Then

That is,

2 ( 3)x +2 ( 5)x x + 1/ 3 4 / 7 3 / 4

B

B


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Example 3. For the fuzzy relation:

find the cut relations for the following values

of = 0+, 0.2, 0.9, 0.5.


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Solution. Given

we have


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Outline

vLambda Cuts for Fuzzy Sets

for Fuzzy Relations

vDefuzzification Methods

Maxima method

Centroid method


Middle-of-maxima method

First-of-maxima or last-of-maxima


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Defuzzification Methods

vThe output of an entire fuzzy process can beunion of two or more fuzzy sets.


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vDefinition 1. Defuzzification is a process to select arepresentative element from the fuzzy output inferredfrom the fuzzy rule-based system.

vThere are various methods used for defuzzifying

the fuzzy output functions: Maxima method

Centroid method


Middle-of-maxima method First-of-maxima or last-of-maxima

!



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(1) Maxima


*( ) ( ), for allC C

z z z Z

( )z

z*z0


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(2) Centroid method


*( )

, for all( )

C

C

z zdzz z Z

z dz

=

%

( )z

z*

z0


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(3) Weighted average method This method can be used only for symmetrical

output membership functions

Weighting each membership function in the obtained

output by its largest membership value



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(4) Middle-of-maxima method The present of the maximum membership need not

be unique!

i.e., the maximum membership need not be a single point, it

can be a range.



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(5) First-of-Maxima

Last-of-Maxima


max

0z


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Defuzzification: Examples

Example 4. For the given membership functionas shown, determine the defuzzified output

value using different defuzzification methods.

z

0.7

0.5

1

1 2 3 4 5 6

1

z

0.5

1

1 2 3 4 5 6

2


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0.7

0.5

1

1 2 3 4 5 6

1

z

0.5

1

1 2 3 4 5 6

2

0 0

0.7

0.5

1

1 2 3 4 5 6

1 2AU

0a

bc

d e

f

0.35 0 2

0.7 2 2.7

( ) 2 2.7 3

1 3 4

0.5 3 4 6

z z

z

z z z

z

z z


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(2) Centroid method

(1) Maxima method

Not applicable since there is no a single maximum

point.

0.35 0 2

0.7 2 2.7

( ) 2 2.7 3

1 3 4

0.5 3 4 6

z z

z

z z z

z

z z


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Review Questions

1. Define defuzzification process.

2. What is the necessity to convert the fuzzy quantities into

crisp quantities?

3. State the method lambda cuts employed for the conversion of

the fuzzy set into crisp.

4. How is lambda cut method employed for a fuzzy relation?5. How does the maximum method convert the fuzzy quantity

to crisp quantity?

6. In what way does the Centroid method perform the

defuzzification process?

7. Compare the methods employed for defuzzification process

on the basis of accuracy and time consumption.


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Exercise Problems

1. Two fuzzy setsA and B both defined on x are as

follows:

Find


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Exercise Problems

2. The fuzzy setA , B , C are all defined on the universe

X = [0, 5] with the following membership functions:

(a) Sketch the membership functions

(b) Define the intervals alongx-axis corresponding to the

cut sets for each of the fuzzy setsA , B , C for the

following values of. = 0.2, = 0.5, = 0.9.


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Exercise Problems

4. By using maxima and centroid methods, convert fuzzyoutput to a crisp value z! for the following graph

z

0.5

1

1 2 3 4 5 60

1

2


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Exercise Problems

5. Convert fuzzy value z to precise value z! for the

following graph by using weighted average methodand middle-ofmaxima method.

The membership functions in the above graph are symmetrical functions.

0.5

1

1 2 3 4 5 60 7 8 9 10

1

2

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