EE-646

Lecture-4Types of Membership Functions

Basic Definitions

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Symmetric MF

A fuzzy set is symmetric if its membershipfunction (MF) is symmetric about a certainpoint x = c and we write

( ) ( );A Ac x c x x Xµ µ+ = − ∀ ∈

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Decreasing MF (Open Left)

A fuzzy set is open left or decreasing ifmembership values continuously decrease(from 1) as we increase x

lim ( ) 1 & lim ( ) 0A Ax xx xµ µ

→−∞ →+∞= =

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Increasing MF (Open Right)

A fuzzy set is open right or increasing ifmembership values continuously increase (upto 1) as we increase x

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lim ( ) 0 & lim ( ) 1A Ax xx xµ µ

→−∞ →+∞= =

Approximating MF (Closed Fuzzy Set)

A fuzzy set is closed if values on both endsdecrease to zero i.e.

lim ( ) lim ( ) 0A Ax xx xµ µ

→−∞ →+∞= =

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Types of Membership Functions

• Increasing Type (Γ-function, S-function)• Decreasing Type (L or Z-function)• Approximation Type (Triangular function,

Trapezoidal function, Gaussian Function, Bellfunction)

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Γ- Function

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( )0,

: , 1,

,

xx x

x x

αα β β

α α ββ α

<Γ = ≥ − ≤ <

−

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S - Function

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( )

2

2

0,

2 ,: , ,

1 2 ,

1,

x

x xS x

x x

x

α

α α βγ α

α β γα β γ

γ αγ

<

− ≤ < − = −

− ≤ < − ≥

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By symmetry, we can reduce the no. of parameters and we can take

2α γβ +

=

µ (x)

x

L or Z - Function

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( )

1,

: , ,

0,

xxL x x

x

ααα β α β

β αβ

< −= ≤ < − ≥

µ(x)

x

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Triangular Function

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( )( ) ( )( ) ( )

0 ,/ ,

or : , ,/ ,

0 ,

xx x

xx x

x

αα β α α β

α β γγ γ β β γ

γ

≤ − − < ≤∆ Λ =

− − < ≤ >

µ (x)

x

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Trapezoidal or Π-Function

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( )( ) ( )

( ) ( )

0 ,/ ,

: , , , 1 ,/ ,

0 ,

xx x

x xx x

x

αα β α α β

α β γ δ β γδ δ γ γ δ

δ

≤ − − < ≤Π = < ≤ − − < ≤ >

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µ(x)

xα β γ δ

Gaussian Function

• Also known as normalized distributionfunction. It is defined as

• It can be used as inc, dec or approx. typefunction by controlling only two parameters

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( )21Gaussian : , exp

2c

cx xx x σσ

− = −

Gaussian Function

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xc

x

µ(x)

Generalized Bell Function

• Crossover points are c ± a• BW is 2a• Flat on top

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( ) 21Bell : , ,

1bx a b c

x ca

=−

+

Generalized Bell Function

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Effect of Change in parameters

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Sigmoidal Function

• Used extensively in ANN theory• Please see yourself

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Today’s Task

• Find out the MATLAB commands for thesefunctions and generate some samplefunctions

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