Download - 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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Lambda Cuts: Examples
Solution. Given
We have
Then we can calculate
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Lambda Cuts: Examples
Solution. Given
We have
Then we can calculate
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Lambda Cuts: Examples
Solution. Given
We have
Then we can calculate
0.1 0.2 0.3 0.2 0.4
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Lambda Cuts: Examples
Solution. Given
We have
Then we can calculate
0 0 0 0
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Lambda Cuts: Examples
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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Lambda Cuts: Examples
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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Lambda Cuts: Examples
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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Lambda Cuts: Examples
Solution. Given
we have
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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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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
Weighted average method
Middle-of-maxima method First-of-maxima or last-of-maxima
!
Defuzzification Methods
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(1) Maxima
Defuzzification Methods
*( ) ( ), for allC C
z z z Z
( )z
z*z0
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(2) Centroid method
Defuzzification Methods
*( )
, 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
Defuzzification Methods
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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.
Defuzzification Methods
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(5) First-of-Maxima
Last-of-Maxima
Defuzzification Methods
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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Defuzzification: Examples
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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Defuzzification: Examples
(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
30.75