himpunan fuzzy i

Fuzzy SetsFuzzy Sets

Referensi : Neuro Fuzzy and Soft Computing J.-S. Roger Jang

Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets

Intelligent System

OutlineOutlineIntroductionBasic definitions and terminologySet-theoretic operationsMF formulation and parameterization

MFs of one and two dimensionsDerivatives of parameterized MFs

More on fuzzy union, intersection, and complement

Fuzzy complement Fuzzy intersection and unionParameterized T-norm and T-conorm

Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets 2


Conventional (Boolean) Set Theory:Conventional (Boolean) Set Theory:

Fuzzy Set TheoryFuzzy Set Theory

© INFORM 1990-1998 Slide 3

“Strong Fever”

40.1°C40.1°C

42°C42°C

41.4°C41.4°C

39.3°C39.3°C

38.7°C

37.2°C

38°C

Fuzzy Set Theory:Fuzzy Set Theory:

40.1°C40.1°C

42°C42°C

41.4°C41.4°C

39.3°C39.3°C

38.7°C

37.2°C

38°C

“More-or-Less” Rather Than “Either-Or” !

“Strong Fever”

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

Membership Functions (MFs)

Characteristics of MFs:○ Subjective measures○ Not probability functions


MFs

Heights

.8

“tall” in Asia

.5 “tall” in the US

5’10’’.1 “tall” in NBA

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 set Membershipfunction

(MF)

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

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!)

Dr. Djamel Bouchaffra 7

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

1 5010

2

Basic definitions & Terminology

Alternative Notation

A fuzzy set A can be alternatively denoted as follows:


A x xAX

( ) /X is continuous

A x xAx X

i ii

( ) /

X is discrete

Note that 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”:



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) > }



Crossover points

Support

- cut

Core

MF

X0

.5

1

MF Terminology

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.

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



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

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:

Set-Theoretic Operations


MF Formulation & Parameterization

Triangular MF:

Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch.2: Fuzzy sets

trim f x a b cx a

b a

c x

c b( ; , , ) max m in , ,

0

trapm f x a b c dx a

b a

d x

d c( ; , , , ) m ax m in , , ,

1 0

gbellm f x a b cx c

b

b( ; , , )

1

1

2

2cx21

e),c;x(gaussmf

Trapezoidal MF:

Gaussian MF:

Generalized bell MF:

MFs of One Dimension

Change of parameters in the generalized bell MF




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



○ 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



○ Left –Right (LR) MF:


LR x c

Fc x

x c

Fx c

x c

L

R

( ; , , )

,

,

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



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