Chapter 2: FUZZY SETS Introduction (2.1) Basic Definitions &Terminology (2.2) Set-theoretic Operations (2.3) Membership Function (MF) Formulation

Chapter 2: FUZZY SETS Introduction (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)

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BİL 561 Fuzzy Neural Networks Ch.2: Fuzzy setsDr. Yusuf OYSAL

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

3


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.)

4


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 {( , ( ))| }

Universe oruniverse of discourse

Fuzzy setMembership

function(MF)

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

5

Dr. Yusuf OYSAL

Fuzzy Sets with Discrete Universes

Fuzzy 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.)

6


Fuzzy Sets with Cont. Universes

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

B xx

( )

1

15010

2


7


Alternative Notation

A fuzzy set A can be alternatively denoted as follows:

A x xA

X

( ) /X is continuous

A x xAx X

i i

i

( ) /

X is discrete

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


8


Fuzzy Partition

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


9


Support(A) = {x X | A(x) > 0}

Core(A) = {x X | A(x) = 1}

Normality: core(A) A is a normal fuzzy set

Crossover(A) = {x X | A(x) = 0.5}

- cut: A = {x X | A(x) }

Strong - cut: A’ = {x X | A(x) > }


10


Crossover points

Support

- cut

Core

MF

X0

.5

1

MF Terminology


11


Convexity of Fuzzy Sets

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

A A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21

Alternatively, A is convex if all its -cuts are convex.


12


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 A(x1) = A(x2) = 0.5

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

A(x + c) = A(c – x) x X


13


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(lim and 1)x(lim set A fuzzyleft open

Ax

A-x

Ax

A-x

Ax

A-x


14


Set-Theoretic Operations (2.3)

Subset:A B A B

C A B x x x x xc A B A B ( ) max( ( ), ( )) ( ) ( )

C A B x x x x xc A B A B ( ) min( ( ), ( )) ( ) ( )

A X A x xA A ( ) ( )1

Complement:

Union:

Intersection:

15


Set-Theoretic Operations (2.3) (cont.)


MF Formulation & Parameterization (2.4)

Triangular MF: trimf x a b cx ab a

c xc b

( ; , , ) max min , ,

0

trapmf x a b c dx ab a

d xd c

( ; , , , ) max min , , ,

1 0

gbellmf x a b cx cb

b( ; , , )

1

12

2cx

2

1

e),c;x(gaussmf

Trapezoidal MF:

Gaussian MF:

Generalized bell MF:

MFs of One Dimension

17


MF Formulation & Parameterization (2.4) (cont.)

18


Change of parameters in the generalized bell MF


19


Physical meaning of parameters in a generalized bell MF


20


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


21


Sigmoidal MF:)cx(ae1

1)c,a;x(sigmf

Extensions:

Product

of two sig. MF

Abs. difference

of two sig. MF


22


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


23


Left –Right (LR) MF:

LR x cF

c xx c

Fx c

x c

L

R

( ; , , ),

,

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

c=65

a=60

b=10

c=25

a=10

b=40


24


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


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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 setY*X

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


26


Cylindrical extension

Base set A Cylindrical Ext. of A


27


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

Ry

X

YR

xY y|)y,x(maxR


28


Two-dimensional

MF

Projection

onto X

Projection

onto Y


29


Composite & non-composite MFs

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

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

“x is near 3” & “y is near 4”

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

14y

exp2

3xexp

4y2

3xexp)y,x(

22

22

A


30


These two statements are respectively defined as:

near 3 (x) = G(x;3,2) & 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|1

1)y,x(


31


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.


32


Two dimensional MFs defined by the min and max operators


33


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

34


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 aa

sas( )1

1(s > -1)

(w > 0)

35


N aa

sas( )1

1

Sugeno’s complement:

N a aww w( ) ( ) / 1 1

Yager’s complement:


36


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


37


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


38


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


39


T-norm Operator

Minimum:Tm(a, b)

Algebraicproduct:Ta(a, b)

Boundedproduct:Tb(a, b)

Drasticproduct:Td(a, b)


40


T-conorm or S-norm

The fuzzy union operator is defined by a function

S: [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


41


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


42


T-conorm or S-norm

Maximum:Sm(a, b)

Algebraicsum:

Sa(a, b)

Boundedsum:

Sb(a, b)

Drasticsum:

Sd(a, b)


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Generalized DeMorgan’s Law

T-norms and T-conorms are duals which support the generalization of DeMorgan’s law: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)


44


Parameterized T-norm and T-conorm

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

YagerSchweizer and SklarDubois and PradeHamacherFrankSugenoDombi


Documents

Chapter 2: FUZZY SETS Introduction (2.1) Basic Definitions &Terminology (2.2) Set-theoretic Operations (2.3) Membership Function (MF) Formulation