Chapter 2: FUZZY SETS513].pdfCSE 513 Soft Computing Dr. Djamel Bouchaffra Ch.2: Fuzzy sets 6 Convexity of Fuzzy Sets A fuzzy set A is convex if for any λin [0, 1],µAAA(( ))min((),())λxx

CSE 513 Soft Computing Dr. Djamel Bouchaffra

Ch.2: Fuzzy sets 1

Chapter 2: FUZZY SETSIntroduction (2.1)

Basic Definitions &Terminology (2.2)

Set-theoretic Operations (2.3)

Membership Function (MF) Formulation & Parameterization (2.4)

More on Fuzzy Union, Intersection & Complement (2.5)

Jyh-Shing Roger Jang et al., Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence, First Edition, Prentice Hall, 1997

Introduction (2.1)

Sets with fuzzy boundaries

A = Set of tall people

Heights5’10’’

1.0

Crisp set A

Membershipfunction

Heights5’10’’ 6’2’’

.5

.9

Fuzzy set A1.0


Ch.2: Fuzzy sets 2

Membership Functions (MFs)

Characteristics of MFs:Subjective measuresNot probability functions

MFs

Heights

.8

“tall” in Asia

.5 “tall” in the US

5’10’’.1 “tall” in NBA

Introduction (2.1) (cont.)

Basic definitions & Terminology (2.2)

Formal definition:A fuzzy set A in X is expressed as a set of ordered

pairs:

A x x x XA= ∈{( , ( ))| }µ

Universe oruniverse of discourse

Fuzzy set Membershipfunction(MF)

A fuzzy set is totally characterized by aA fuzzy set is totally characterized by amembership function (MF).membership function (MF).


Ch.2: Fuzzy sets 3

Fuzzy Sets with Discrete UniversesFuzzy set C = “desirable city to live in”X = {SF, Boston, LA} (discrete and non-ordered)C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}(subjective membership values!)

Fuzzy set A = “sensible number of children”X = {0, 1, 2, 3, 4, 5, 6} (discrete universe)A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}(subjective membership values!)

Basic definitions & Terminology (2.2) (cont.)

Fuzzy Sets with Cont. Universes

Fuzzy set B = “about 50 years old”X = Set of positive real numbers (continuous)B = {(x, µB(x)) | x in X}

µ B x x( ) =

+−

1

1 5010

2



Ch.2: Fuzzy sets 4

Alternative Notation

A fuzzy set A can be alternatively denoted as follows:

A x xAX

= ∫ µ ( ) /X is continuous

A x xAx X

i ii

=∈

∑ µ ( ) /X is discrete

Note that Σ and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.


Fuzzy Partition

Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:



Ch.2: Fuzzy sets 5

Support(A) = {x ∈ X | µA(x) > 0}

Core(A) = {x ∈ X | µA(x) = 1}

Normality: core(A) ≠ ∅ ⇒ A is a normal fuzzy set

Crossover(A) = {x ∈ X | µA(x) = 0.5}

α - cut: Aα = {x ∈ X | µA(x) ≥ α}

Strong α - cut: A’α = {x ∈ X | µA(x) > α}


Crossover points

Support

α - cut

Core

MF

X0

.5

1

α

MF Terminology



Ch.2: Fuzzy sets 6

Convexity of Fuzzy Sets

A fuzzy set A is convex if for any λ in [0, 1],

µ λ λ µ µA A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21+ − ≥Alternatively, A is convex if all its α-cuts are convex.


Fuzzy numbers: a fuzzy number A is a fuzzy set in IR that satisfies normality & convexity

Bandwidths: for a normal & convex set, the bandwidth is the distance between two unique crossover points

Width(A) = |x2 – x1|With µA(x1) = µA(x2) = 0.5

Symmetry: a fuzzy set A is symmetric if its MF is symmetric around a certain point x = c, namely

µA(x + c) = µA(c – x) ∀x ∈ X



Ch.2: Fuzzy sets 7

Open left, open right, closed:

0)x(lim)x(lim set A fuzzy closed

1)x(lim and 0)x(lim set A fuzzyright open

0)x(limand1)x(limset A fuzzyleft open

AxA-x

AxA-x

AxA-x

=µ=µ⇔

=µ=µ⇔

=µ=µ⇔

+∞→∞→

+∞→∞→

+∞→∞→


Set-Theoretic Operations (2.3)

Subset:A B A B⊆ ⇔ ≤µ µ

C A B x x x x xc A B A B= ∪ ⇔ = = ∨µ µ µ µ µ( ) max( ( ), ( )) ( ) ( )

C A B x x x x xc A B A B= ∩ ⇔ = = ∧µ µ µ µ µ( ) min( ( ), ( )) ( ) ( )

A X A x xA A= − ⇔ = −µ µ( ) ( )1

Complement:

Union:

Intersection:


Ch.2: Fuzzy sets 8

Set-Theoretic Operations (2.3) (cont.)

MF Formulation & Parameterization (2.4)

Triangular MF: trim f x a b cx ab a

c xc b

( ; , , ) max min , ,=−−

−−

0

trapm f x a b c dx ab a

d xd c

( ; , , , ) max min , , ,=−−

−−

1 0

gbellm f x a b cx c

b

b( ; , , ) =+

−

1

12

2cx21

e),c;x(gaussmf

σ−

−=σ

Trapezoidal MF:

Gaussian MF:

Generalized bell MF:

MFs of One Dimension


Ch.2: Fuzzy sets 9

MF Formulation & Parameterization (2.4) (cont.)

Change of parameters in the generalized bell MF



Ch.2: Fuzzy sets 10

Physical meaning of parameters in a generalized bell MF


Gaussian MFs and bell MFs achieve smoothness, they are unable to specify asymmetric Mfs which are important in many applications

Asymmetric & close MFs can be synthesized using either the absolute difference or the product of two sigmoidal functions



Ch.2: Fuzzy sets 11

Sigmoidal MF: )cx(ae11)c,a;x(sigmf −−+

=

Extensions:

Productof two sig. MF

Abs. differenceof two sig. MF


A sigmoidal MF is inherently open right or left & thus, it is appropriate for representing concepts such as “very large”or “very negative”

Sigmoidal MF mostly used as activation function of artificial neural networks (NN)

A NN should synthesize a close MF in order to simulate the behavior of a fuzzy inference system



Ch.2: Fuzzy sets 12

Left –Right (LR) MF:

L R x cF

c xx c

Fx c

x c

L

R

( ; , , ),

,α β

α

β

=

−

<

−

≥

Example: F x xL ( ) max( , )= −0 1 2 F x xR ( ) exp( )= −3

c=65a=60b=10

c=25a=10b=40


The list of MFs introduced in this section is by no means exhaustive

Other specialized MFs can be created for specific applications if necessary

Any type of continuous probability distribution functions can be used as an MF



Ch.2: Fuzzy sets 13

MFs of two dimensions

In this case, there are two inputs assigned to an MF: this MF is a twp dimensional MF. A one input MF is called ordinary MF

Extension of a one-dimensional MF to a two-dimensional MF via cylindrical extensions

If A is a fuzzy set in X, then its cylindrical extension in X*Y is a fuzzy set C(A) defined by:

C(A) can be viewed as a two-dimensional fuzzy set

∫ µ=Y*X

A )y,x(|)x()A(C


Cylindrical extensionBase set A Cylindrical Ext. of A



Ch.2: Fuzzy sets 14

Projection of fuzzy sets (decrease dimension)

Let R be a two-dimensional fuzzy set on X*Y. Then the projections of R onto X and Y are defined as:

and

respectively.

x|)y,x(maxRX

RyX ∫

µ=

∫

µ=

YRxY

y|)y,x(maxR


Two-dimensionalMF

Projectiononto X

Projectiononto Y



Ch.2: Fuzzy sets 15

Composite & non-composite MFs

Suppose that the fuzzy A = “(x,y) is near (3,4)” is defined by:

This two-dimensional MF is compositeThe fuzzy set A is composed of two statements:

“x is near 3” & “y is near 4”

( )

)1,4;y(G*)2,3;x(G 1

4yexp2

3xexp

4y2

3xexp)y,x(

22

22

A

=

−−

−−=

−−

−−=µ


These two statements are respectively defined as:µ near 3 (x) = G(x;3,2) & µ near 4 (x) = G(y;4,1)

If a fuzzy set is defined by:

it is non-composite.

A composite two-dimensional MF is usually the result of two statements joined by the AND or OR connectives.

5.2A |4y||3x|11)y,x(

−−+=µ



Ch.2: Fuzzy sets 16

Composite two-dimensional MFs based on min & max operations

Let trap(x) = trapezoid (x;-6,-2,2,6)trap(y) = trapezoid (y;-6,-2,2,6)

be two trapezoidal MFs on X and Y respectively

By applying the min and max operators, we obtain two-dimensional MFs on X*Y.


Two dimensional MFs defined by the min and max operators



Ch.2: Fuzzy sets 17

More on Fuzzy Union, Intersection & Complement (2.5)

Fuzzy complement

Another way to define reasonable & consistent operations on fuzzy sets

General requirements:

Boundary: N(0)=1 and N(1) = 0Monotonicity: N(a) > N(b) if a < bInvolution: N(N(a) = a

More on Fuzzy Union, Intersection & Complement (2.5) (cont.)

Two types of fuzzy complements:

Sugeno’s complement:

(Family of fuzzy complement operators)

Yager’s complement:

N a aww w( ) ( ) /= −1 1

N a asas

( ) = −+

11 (s > -1)

(w > 0)


Ch.2: Fuzzy sets 18

N a asas

( ) = −+

11

Sugeno’s complement:

N a aww w( ) ( ) /= −1 1

Yager’s complement:


Fuzzy Intersection and Union:

The intersection of two fuzzy sets A and B is specified in general by a function

T: [0,1] * [0,1] → [0,1] with

This class of fuzzy intersection operators are called T-norm (triangular) operators.

( )

T. function the for operator binary a is * where

)x(*)x()x(),x(T)x(~

B~

ABABA µµ=µµ=µ ∩



Ch.2: Fuzzy sets 19

T-norm operators satisfy:

Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = aCorrect generalization to crisp sets

Monotonicity: T(a, b) < T(c, d) if a < c and b < dA decrease of membership in A & B cannot increase a membership in A ∩ B

Commutativity: T(a, b) = T(b, a)T is indifferent to the order of fuzzy sets to be combined

Associativity: T(a, T(b, c)) = T(T(a, b), c)Intersection is independent of the order of pairwise groupings


T-norm (cont.)

Four examples (page 37):

Minimum: Tm(a, b) = min (a,b) = a ∧ b

Algebraic product: Ta(a, b) = ab

Bounded product: Tb(a, b) = 0 V (a + b – 1)

Drastic product: Td(a, b) =

<==

1ba, if 0,1 a if b,1b if ,a



Ch.2: Fuzzy sets 20

T-norm Operator

Minimum:Tm(a, b)

Algebraicproduct:Ta(a, b)

Boundedproduct:Tb(a, b)

Drasticproduct:Td(a, b)


T-conorm or S-norm

The fuzzy union operator is defined by a functionS: [0,1] * [0,1] → [0,1]

wich aggregates two membership function as:

where s is called an s-norm satisfying:

Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = aMonotonicity: S(a, b) < S(c, d) if a < c and b < dCommutativity: S(a, b) = S(b, a)Associativity: S(a, S(b, c)) = S(S(a, b), c)

( ) )x()x()x(),x(S B~

ABABA µ+µ=µµ=µ ∪



Ch.2: Fuzzy sets 21

T-conorm or S-norm (cont.)

Four examples (page 38):

Maximum: Sm(a, b) = max(a,b) = a V b

Algebraic sum: Sa(a, b) = a + b - ab

Bounded sum: Sb(a, b) = 1 ∧ (a + b)

Drastic sum: Sd(a, b) =

>==

0ba, if 1,0a if b,0b if ,a


T-conorm or S-norm

Maximum:Sm(a, b)

Algebraicsum:

Sa(a, b)

Boundedsum:

Sb(a, b)

Drasticsum:

Sd(a, b)



Ch.2: Fuzzy sets 22

Generalized DeMorgan’s Law

T-norms and T-conorms are duals which support the generalization of DeMorgan’slaw:

T(a, b) = N(S(N(a), N(b)))S(a, b) = N(T(N(a), N(b)))

Tm(a, b)Ta(a, b)Tb(a, b)Td(a, b)

Sm(a, b)Sa(a, b)Sb(a, b)Sd(a, b)


Parameterized T-norm and T-conorm

Parameterized T-norms and dual T-conorms have been proposed by several researchers:

YagerSchweizer and SklarDubois and PradeHamacherFrankSugenoDombi


Chapter 2: FUZZY SETSIntroduction (2.1)Introduction (2.1) (cont.)Basic definitions & Terminology (2.2)Basic definitions & Terminology (2.2) (cont.)Basic definitions & Terminology (2.2) (cont.)Basic definitions & Terminology (2.2) (cont.)Basic definitions & Terminology (2.2) (cont.)Basic definitions & Terminology (2.2) (cont.)Basic definitions & Terminology (2.2) (cont.)Basic definitions & Terminology (2.2) (cont.)Basic definitions & Terminology (2.2) (cont.)Basic definitions & Terminology (2.2) (cont.)Set-Theoretic Operations (2.3)Set-Theoretic Operations (2.3) (cont.)MF Formulation & Parameterization (2.4)MF Formulation & Parameterization (2.4) (cont.)More on Fuzzy Union, Intersection & Complement (2.5)More on Fuzzy Union, Intersection & Complement (2.5) (cont.)More on Fuzzy Union, Intersection & Complement (2.5) (cont.)More on Fuzzy Union, Intersection & Complement (2.5) (cont.)More on Fuzzy Union, Intersection & Complement (2.5) (cont.)More on Fuzzy Union, Intersection & Complement (2.5) (cont.)T-norm OperatorMore on Fuzzy Union, Intersection & Complement (2.5) (cont.)More on Fuzzy Union, Intersection & Complement (2.5) (cont.)T-conorm or S-normMore on Fuzzy Union, Intersection & Complement (2.5) (cont.)More on Fuzzy Union, Intersection & Complement (2.5) (cont.)

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Chapter 2: FUZZY SETS513].pdfCSE 513 Soft Computing Dr. Djamel Bouchaffra Ch.2: Fuzzy sets 6 Convexity of Fuzzy Sets A fuzzy set A is convex if for any λin [0, 1],µAAA(( ))min((),())λxx