fuzzy alpha cuts

8/10/2019 fuzzy alpha cuts

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Advances in Intelligent and Soft Computing, Vol. 27

First course on fuzzy theory and applicationsKwang Hyung Lee


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Why fuzzy set theory

Uncertainty

Zadeh (1965)

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Chapter 1

FUZZY SETS

Jin-Ping Chen


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Sets

Crisp set membership function

membership degree: {0,1} Fuzzy set

membership function: user specify membership degree: [0,1]

1, if and only if( )0, if and only if A

x A x x A

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Expression of crisp set

1 2{ , ,..., }n A x x x

1 2{ | satisfies , , ..., }n A x x p p p

{1,5,9,12,17} A

{ | 3 and 10} A x x x

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Expression of fuzzy set

{( , ( ))} A A x x

1

( ) /n

A i ii

A x x

{(2,1.0),(3,0.5)} A

1.0 0.52 3

A

( ) / A A x x 2

1where ( )

1 A x

x

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Expansion of fuzzy set

Type-n fuzzy set The value of membership degree might include

uncertainty Type-2 fuzzy set

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Level-2 fuzzy set elements are fuzzy sets

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9

cold

16 17 18 19 20 21 ..

warm


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Expanding concepts of fuzzy set

Support Normalized fuzzy set

-cut set Level set Convex fuzzy set Fuzzy number Magnitude of fuzzy set Subset of fuzzy set

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Example of fuzzy set

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{5,15,25,35, 45,55,65,75,85} X


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Support

Support of A

example

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support( ) { | ( ) 0} A A x X x

support(young) {15,25,35,45,55}

support(adult) {15,25,35, 45,55,65,75,85}support(infant)


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Height

The maximum value of the membership degree

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( ) 1height young

( ) 0height infant


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Normalized fuzzy set

Normalized fuzzy set height is 1 young, adult, and senior are normalized fuzzy sets

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-cut

-cut set

Example

If ,15

{ | ( ) } A A x X x

0.2young {12, 25,35, 45}0.8young {25,35}

0.6senior {65,75,85}

' ' A A

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-cut

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-cut


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-cut


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Level set

Level set

Example

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{ | , ( ) , 0} A A x X x

young {0,0.1,0.2,0.4,0.8,1}

senior {0,0.1,0.2,0.6,1}


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Convex fuzzy set

If all -cut sets are convex, the fuzzy set withthese -cut sets is convex

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Convex fuzzy set

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( ) min( ( ), ( ))

where (1 ) , , , [0,1] A A At r s

t r s r s R


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Non-convex fuzzy set

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Relative scalar cardinality

Example

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| ||| ||| | A A X

| senior | 0.1 0.2 0.6 1 1 2.9

| X | 9| | 2.9

|| || 0.32| | 9

senior senior

X


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Relation of fuzzy sets A and B are equivalent

A is a subset of B

A is a proper subset of B

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( ) ( ) A B A B iff x x

( ) ( ) A B A B iff x x

X x x x B A ),()( BAand iff B A B A


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Subset of fuzzy set

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Standard operation of fuzzy set

Complement Union

Intersection

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Example

Complement

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( ) 1 ( ), A A x x x X

{(5,0),(15,0.1),(25,0.9),(35,1),(45,1),...,,(85,1)} A

{(5,1),(15,0.9),(25,0.1)} A


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Union

Example

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( ) max( ( ), ( )), A B A B x x x x X

" " " "

{(15,0.2),(25,1),(35,1),(45,1),(55,1),(65,1),(75,1),(85,1)}

young adult


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Intersection

Example

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( ) min( ( ), ( )), A B A B x x x x X

" " " "

{(15,0.1), (25,0.9), (35,0.8), (45,0.4), (55,0.1)}

young adult


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Thanks for your attention!

fuzzy alpha cuts

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