Fuzzy Sets and Systems

Lecture 1(Introduction)

Bu- Ali Sina UniversityComputer Engineering Dep.

Spring 2010

Fuzzy sets and system

• Introduction and syllabus• References• Grading

Fuzzy sets and systemSyllabus

Course Outline

Theory

• Theory of fuzzy sets, from crisp sets to fuzzy sets, basic concepts anddefinitions,

• Fuzzy operations, t-norms, t-conorms, aggregation operations

• Fuzzy arithmetic, fuzzy numbers, linguistic variables

• Fuzzy relations, fuzzy equivalence, fuzzy relational equations

• Fuzzy measures, possibility theory, Dempster-Shafer theory ofevidence,

• Fuzzy logic, multi-valued logic, fuzzy qualifiers

• Uncertainty-based information, uncertainty measures, entropy,nonspecificity,

Course Outline

Application• Construction of fuzzy sets and operations from experts or data-

sample

• Approximate reasoning, fuzzy expert systems,

• Fuzzy systems, rule-based, data-based, and knowledge basedsystems

• Fuzzy control, design of fuzzy controllers

• Fuzzy modeling, fuzzy regression

• Fuzzy clustering, fuzzy pattern recognition, cluster validity

• Fuzzy information retrieval and fuzzy databases

• Fuzzy decision making, fuzzy ranking, information and fuzzy fusion

Fuzzy sets and systemReferences

Recommended references (Journals)

• IEEE Trnas. On Fuzzy Systems

• Fuzzy sets and systems

• Int. Jou. Of Uncertainty, Fuzziness and Knowledgebased systems

• Jour. Of Intelligent and Fuzzy systems

• Fuzzy optimization and decision making

Grading Policies

Exams 50%– Midterm 50% (25% of total) about 15/2/89– Final 50% (25% of total)

Final Project (25%)– One project (deadline is about 31/6/88)

Seminar (15%)– Every body present a seminar (select a subject

until 15/1/89)Home works (10%)

– 5 home works

Fuzzy Sets and Systems

Class HoursSunday 10-13

Send your homework tohkhotanloo@yahoo.com with subjectFuzzy

• Introduction–What is a Fuzzy set?–Why Fuzzy?–Application

• Fuzzy sets–Introduction to Crisp sets–Fuzzy sets theory– Definition– Membership function

Outline

What is a fuzzy set?

�A set is a collection of its members.

�The notion of fuzzy sets is an extensionof the most fundamental property of sets.

�Fuzzy sets allows a grading of to what extentan element of a set belongs to that specific set.

What is a fuzzy set?

Let us observe a (crisp) reference set

X = {1; 2; 3; 4; 5; 6; 7; 8; 9; 10}

The (crisp) subset C of X,

C = {x | 3 < x < 8} , c={4,5,6,7}

The set F of big numbers in X

F = {10; 9; 8; 7; 6; 5; 4; 3; 2; 1}

Why Fuzzy?

Precision is not truth. Henri Matisse

So far as the laws of mathematics refer to reality, they are notcertain. And so far as they are certain, they do not refer toreality.

Albert Einstein

As complexity rises, precise statements lose meaning and meaningfulstatements lose precision.

Lotfi Zadeh

Why Fuzzy?� Complex, ill-defined processes difficult for description and analysis by exact

mathematical techniques

�Approximate and inexact nature of the real word; vague concepts easily dealt with

by humans in daily life

�Thus, we need other technique, as supplementary to conventional quantitative

methods, for manipulation of vague and uncertain information, and to create

systems that are much closer in spirit to human thinking. Fuzzy logic is a

strong candidate for this purpose.

Fuzzy and Probability , Randomness and Fuzziness

� Fuzzy is not just another name for probability.

� The number 10 is not probably big! ...and number 2 is not probably notbig.

� Uncertainty is a consequence of non-sharp boundaries between thenotions/objects, and not caused by lack of information.

� Randomness refers to an event that may or may not occur.

� Randomness: frequency of car accidents.

� Fuzziness refers to the boundary of a set that is not precise.

� Fuzziness: seriousness of a car accident.

Example : A fuzzy set of tall man

Another Example: Age groups

Introduction

� Fuzzy set theory was initiated by Zadeh in the early1960s

� L. A. Zadeh, Fuzzy sets. Information and Control, Vol. 8,pp. 338-353. (1965).

� http://wwwisc.cs.berkeley.edu/zadeh/papers/Fuzzy%20Sets-1965.pdf

� L. A. Zadeh, Outline of a new approach to the analysis ofcomplex systems and decision processes, IEEETransactions on Systems, Man and Cybernetics SMC-3,28-44, 1973.

• Fuzzy Logic

• Fuzzy Controlo �euro - Fuzzy Systemo Intelligent Controlo Hybrid Control

• Fuzzy Pattern Recognition

• Fuzzy Modeling

IntroductionIntroductionApplications Domain

Crisp Set theory

Basic concepts

Set: a collection of items

To Represent sets– List method A={a, b, c}

– Rule method C = { x | P(x) }

– Family of sets {Ai | i ∈ I }

– Universal set X and empty set ∅

A ⊆ B : x ∈ A implies that x ∈ BA = B : A ⊆ B and B ⊆ AA ⊂ B : A ⊆ B and A ≠ B

Power set

• All the possible subsets of a given set X iscall the power set of X, denoted by P(X) = {A|A ⊆X}

• |P(X) | = 2^n when |X| = n

• X={a, b, c}• P(X) = {∅, a, b, c, {a, b}, {b, c}, {a, c}, X}

Set Operations

Union

A∪B = {x|x∈A ∨ x∈B}µA∪B(x) = µA(x) ∨ µB(x) = Max{µA(x),µB(x)}

Set operationsSet Operations

Intersection

A∩B = {x|x∈A ∧ x∈B}µA∩B(x) = µA(x) ∧ µB(x) = Min{µA(x),µB(x)}

Set operations

Complement

A′ = {x|x∉A ∧ x∈X}µA′ (x) = 1- µA(x)

Set Operations

Set Operations

Difference

A�B = {x|x ∈ A ∧ x ∉ B}µA�B (x) = µA(x)� µB(x)

Basic properties of set operations

function

A function from a set A to a set B isdenoted by f: A→B

– Many to one– One-to-one

Characteristic function

Let A be any subset of X, the characteristic function of A, denotedby χ, is defined by

Characteristic function of the set of real numbersfrom 5 to 10

Real numbers

Total ordering: a ≤ bReal axis: the set of real number ℜ (x-

axis)Interval: [a,b], (a,b), (a,b]One-dimensional Euclidean space

two-dimensional Euclidean spaceThe Cartesian product of two real number

– ℜ ×ℜ– Plane– Cartesian coordinate, x-y axes

Cartesian product– A ×B = {<a, b> | a∈A and b ∈B}

Convexity

A subset of Euclidean space A is convex, if linesegment between all pairs of points in the setA are included in the set.

partitionGiven a nonempty set A, a family of disjoint subsets of A is

called a partition of A, denoted by Π(A), if the union ofthese subsets yields the set A.

Π(A) = {Ai | i ∈I, ∅≠Ai ⊆A} ⇔Ai ∩ Aj = ∅ for each pair i ≠ j(i,j ∈I)

and∪ i ∈IAi=A

1, 2, 3, 4, 5

4, 5

3, 4, 51, 2

1, 2, 3

4, 51, 2 3

partition

partition

partition refinement

Fuzzy SetsFuzzy SetsFuzzy SetsFormal definition:

A fuzzy set A in X (universal set) is expressed as aset 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).

Membership functionsMembership functionsAssign to each element x of X a number A(x)

A: X → [0, 1]

The degree of membership

Discrete Fuzzy SetsDiscrete Fuzzy SetsDiscrete Fuzzy Sets

Fuzzy set C = “desirable city to live in”X = {SF, Boston, LA} (discrete and nonordered)C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}

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)}

Continuous Fuzzy SetsContinuous Fuzzy SetsContinuous Fuzzy Sets

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

Fuzzy Sets representations

• List form• Tabular form• Rule form• Membership form

List representationsList representation of very high educatedB = 0/0 + 0/1 + 0/2 + 0.1/3 + 0.5/4 +0.8/5 + 1/6OrB={<0,0>,<1,0>,<2,0>,<3,0.1>,<4,0.5>,<5,0.8>,<6,

1>}

ORGeneral notation A=∑A(x)/x

Tabular Representation

16

0.85

0.54

0.13

02

01

00

membershiplevel

Rule form

M = {x ɛ X | x meets some conditions};

where the symbol | denotes the phrase "suchthat".

M = {x ɛ X | x is old man};

Membership form

MF TerminologyMF TerminologyMF Terminology

MF

X

.5

1

0Core

Crossover points

Support

α - cut

α

Basic Concepts

The support of a fuzzy set A in the universal set X is a crispset that contains all the elements of X that have nonzeromembership values in A, that is,

A fuzzy singleton is a fuzzy setwhose support is a single point in X.

Basic conceptsA crossover point of a fuzzy set is a point in X whose membership

value to A is equal to 0:5.

The height, h(A) of a fuzzy set A is the largest membershipvalue attained by any point. If the height of a fuzzy set is

equal to one, it is called a normal fuzzy set, otherwise it is

subnormal.

An α- cut of a fuzzy set A is a crisp set thatcontains all the elements in X that havemembership value in A greater than or equal to α.

Aα

Basic ConceptsA strong α-cut of a fuzzy set

A is a crisp set α+A thatcontains all the elements inX that have membershipvalue in A strictly greaterthan α.

We observe that the strong α-cut 0+A is equivalent to the supportsupp(A).

The 1-cut 1A is often called the core of A.

Note! Sometimes the highest non-empty α-cut is calledthe core of A. (in the case of subnormal fuzzy sets, this isdifferent).The word kernel is also used for both of the above definitions.

Aah )(

Basic concepts

The set of all levels α ɛ [0, 1] thatrepresent distinct α-cuts of a givenfuzzy set A is called a level set of A.

Main types of membership functions (MF):(a) Triangular MF is specified by 3 parameters {a,b,c}:

(b) Trapezoidal MF is specified by 4 parameters {a,b,c,d}:

>≤≤≤≤

<

=

cxif0,cxbifb),-(cx)-(cbxaifa),-(ba)-(x

axif,0

)c,b,a:x(trn

≥<≤

<≤<≤

<

=

dxif0,dxcifc),-(dx)-(d

cxbif1,bxaifa),-(ba)-(x

axif0,

d)c,b,a,:x(trp

MF functionsMF functions

>≤≤≤≤

<

=

cxif0,cxbifb),-(cx)-(cbxaifa),-(ba)-(x

axif,0

)c,b,a:x(trn

(c) Gaussian MF is specified by 2 parameters {a,δ}:

(d) Bell-shaped MF is specified by 3 parameters {a,b,δ}:

(e) Sigmoidal MF is specified by 2 parameters {a,b}:

δ=δ 2

2a)-(x-exp)a,:x(gsn

2b

a-x1

1)b,a,:x(bllδ

+

=δ

b)-a(x-e11b)a,:x(sgm

+=

MF functionsMF functions

MF FormulationMF FormulationMF FormulationSigmoidal MF:

sigm f x a b ce a x c( ; , , ) ( )=

+ − −

1

1

Extensions:

Abs. differenceof two sig. MF

Productof two sig. MF

disp_sig.m

MF FormulationMF FormulationMF Formulation

L-R MF:LR x c

Fc x

x c

Fx c

x c

L

R

( ; , , )

,

,

α βα

β

=

−

<

−

≥

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

difflr.m

c=65a=60b=10

c=25a=10b=40