Fuzzy Sets: An introduction · • More on fuzzy union, intersection, and complement – Fuzzy complement – Fuzzy intersection and union ... following three properties: • 1. A

Fuzzy Logic & Approximate Reasoning

Dr. Ngo Thanh Long


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Fuzzy Sets:An introduction

TS. Ngô Thành LongKhoa Công ngh ệ Thông tin

Email: [email protected]


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Fuzzy Sets: Outline

• Introduction• Set-theoretic operations• MF formulation and parameterization• More on fuzzy union, intersection, and

complement– Fuzzy complement– Fuzzy intersection and union– Parameterized T-norm and T-conorm

• Fuzzy Number• Fuzzy Relations


Dr. Ngo Thanh Long


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History

1965 Seminal Paper “Fuzzy Logic” by Prof. Lotfi Zadeh, Faculty in Electrical Engineering, U.C. Berkeley, Sets the Fou ndation of the “Fuzzy Set Theory”

1970 First Application of Fuzzy Logic in Control Eng ineering (Europe)

1975 Introduction of Fuzzy Logic in Japan

1980 Empirical Verification of Fuzzy Logic in Europe

1985 Broad Application of Fuzzy Logic in Japan

1990 Broad Application of Fuzzy Logic in Europe

1995 Broad Application of Fuzzy Logic in the U.S.

1998 Type-2 Fuzzy Systems

2000 Fuzzy Logic Becomes a Standard Technology and I s Also Applied in Data and Sensor Signal Analysis. Applica tion of Fuzzy Logic in Business and Finance.

1965 Seminal Paper Seminal Paper ““ Fuzzy LogicFuzzy Logic ”” by Prof. by Prof. LotfiLotfi ZadehZadeh , Faculty in , Faculty in Electrical Engineering, U.C. Berkeley, Sets the Fou ndation of Electrical Engineering, U.C. Berkeley, Sets the Fou ndation of the the ““ Fuzzy Set TheoryFuzzy Set Theory ””

19701970 First Application of Fuzzy Logic in Control Enginee ring First Application of Fuzzy Logic in Control Enginee ring (Europe)(Europe)

19751975 Introduction of Fuzzy Logic in Japan Introduction of Fuzzy Logic in Japan

19801980 Empirical Verification of Fuzzy Logic in EuropeEmpirical Verification of Fuzzy Logic in Europe

19851985 Broad Application of Fuzzy Logic in JapanBroad Application of Fuzzy Logic in Japan

19901990 Broad Application of Fuzzy Logic in EuropeBroad Application of Fuzzy Logic in Europe

19951995 Broad Application of Fuzzy Logic in the U.S.Broad Application of Fuzzy Logic in the U.S.

19981998 TypeType --2 Fuzzy Systems2 Fuzzy Systems

20002000 Fuzzy Logic Becomes a Standard Technology and Is Al so Fuzzy Logic Becomes a Standard Technology and Is Al so Applied in Data and Sensor Signal Analysis. Applica tion of Applied in Data and Sensor Signal Analysis. Applica tion of Fuzzy Logic in Business and Finance.Fuzzy Logic in Business and Finance.


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Fuzzy Set Theory

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

“Strong Fever”

40.1°C40.1°C

42°C42°C

41.4°C41.4°C

39.3°C39.3°C

38.738.7°°CC

37.237.2°°CC

3838°°CC

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

““ MoreMore --oror --LessLess ”” Rather Than Rather Than ““ EitherEither --OrOr”” !!

“Strong Fever”


Dr. Ngo Thanh Long


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Fuzzy Sets• Sets with fuzzy

boundariesA = Set of tall people


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Membership Functions (MFs)


Dr. Ngo Thanh Long


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Fuzzy Sets• A fuzzy set A is characterized by a member set

function (MF), µA, mapping the elements of A to the unit interval [0, 1].

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

pairs:

Membershipfunction

(MF)

A x x x XA= ∈{( , ( ))| }µUniverse or

universe of discourseFuzzy setUniverse or

universe of discourseUniverse or

universe of discourse


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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)}

• 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)}


Dr. Ngo Thanh Long


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Fuzzy Sets with Cont. 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)}

• 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)}

2B

1050x

1

1)x(

−−−−++++====µµµµ


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Alternative Notation• A fuzzy set A can be alternatively denoted as

follows:

A x xAx X

i i

i

=∈∑µ ( ) /

A x xA

X

= ∫ µ ( ) /

X is discrete

X is continuous

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


Dr. Ngo Thanh Long


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Fuzzy Partition• Fuzzy partitions formed by the linguistic

values “young”, “middle aged”, and “old”:


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Linguistic Variables

A Linguistic Variable A Linguistic Variable Defines a Concept of Our Defines a Concept of Our Everyday Language!Everyday Language!

•A linguistic variable is a variable whose values ar e not numbers but words or sentences in a natural or artificial language (Zade h, 1975)•Linguistic variable is characterized by [x , T(x), U ], in which x : name of the variable, T(x) : the term set of x , universe of di scourse U


Dr. Ngo Thanh Long


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Fuzzy Hedges

• Suppose you had already defined a fuzzy set to describe a hot temperature.

• Fuzzy set should be modified to represent the hedges "Very" and "Fairly“: very hot or fairly hot.


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MF Terminology


Dr. Ngo Thanh Long


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Convexity of Fuzzy Sets

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

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


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Set-Theoretic Operations

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

• Subset:

• Complement:

• Union:

• Intersection:


Dr. Ngo Thanh Long


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Set-Theoretic Operations


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MF Formulation

• Triangular MF: trim f x a b cx a

b a

c x

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

−−

−−

0

Trapezoidal MF:

Generalized bell MF: gbellm f x a b cx c

b

b( ; , , ) =+

−1

1

2

Gaussian MF: gau ssm f x a b c ex c

( ; , , ) =−

−

1

2

2

σ

trapm f x a b c dx a

b a

d x

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

−−

−−

1 0


Dr. Ngo Thanh Long


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MF Formulation


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MF Formulation• Sigmoidal MF:

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

+ − −

1

1

Extensions:

Abs. differenceof two sig. MF

Productof two sig. MF


Dr. Ngo Thanh Long


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Fuzzy Complement• General requirements:

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

• Two types of fuzzy complements:– Sugeno’s complement:

– Yager’s complement:

N aa

sas( ) = −+

1

1

N a aww w( ) ( ) /= −1 1


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Fuzzy Complement

N aa

sas( ) = −+

1

1N a aw

w w( ) ( ) /= −1 1

Sugeno’s complement: Yager’s complement:


Dr. Ngo Thanh Long


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Fuzzy Intersection: T-norm• Basic requirements:

– Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a– Monotonicity: T(a, b) < T(c, d) if a < c and b < d– Commutativity: T(a, b) = T(b, a)– Associativity: T(a, T(b, c)) = T(T(a, b), c)

• Four examples:– Minimum: Tm(a, b) = min{a, b}.– Algebraic product: Ta(a, b) = a.b– Bounded product: Tb(a, b) = max{0, a+b-1}– Drastic product: Td(a, b)


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t-norm Operator

Minimum:Tm(a, b)

Algebraicproduct:Ta(a, b)

Boundedproduct:Tb(a, b)

Drasticproduct:Td(a, b)


Dr. Ngo Thanh Long


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Fuzzy Union: t-conorm or s-norm• Basic requirements:

– Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a– Monotonicity: S(a, b) < S(c, d) if a < c and b < d– Commutativity: S(a, b) = S(b, a)– Associativity: S(a, S(b, c)) = S(S(a, b), c)

• Four examples:– Maximum: Sm(a, b) = max{a, b}– Algebraic sum: Sa(a, b) = a+b-a.b– Bounded sum: Sb(a, b) = min{a+b, 1}.– Drastic sum: Sd(a, b) =


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t-conorm or s-norm

Maximum:Sm(a, b)

Algebraicsum:

Sa(a, b)

Boundedsum:

Sb(a, b)

Drasticsum:

Sd(a, b)


Dr. Ngo Thanh Long


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Fuzzy Number

• A fuzzy number A must possess the following three properties:

• 1. A must must be a normal fuzzy set,• 2. The alpha levels must be closed for

every ,• 3. The support of A, , must be

bounded.

)(αA

]1,0(∈α

)0( +A


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Fuzzy Number

1

Mem

bers

hip

func

tion is the suport of

z1 is the modal value

is an α-level of ,α (0,1]α

' < ' [ ] [ ]z zα αα α ⇔ ⊂

( ), z z− +

z+1zzα− zα

+z−

,[ ] z zz − + ≡ α α α

zɶ

α’

zɶ ∈


Dr. Ngo Thanh Long


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Fuzzy Relations• A fuzzy relation ℜ is a 2 D MF:

ℜ :{ ((x, y), µµµµℜℜℜℜ(x, y)) | (x, y) ∈∈∈∈ X ×××× Y}

Examples:� x is close to y (x & y are real numbers)� x depends on y (x & y are events)� x and y look alike (x & y are persons or objects)� Let X = Y = IR+

and R(x,y) = “y is much greater than x”The MF of this fuzzy relation can be subjectively defined as:

if X = {3,4,5} & Y = {3,4,5,6,7}

≤≤≤≤

>>>>++++++++

−−−−====µµµµ

xy if , 0

xy if ,2yx

xy)y,x(R


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Extension Principle

The image of a fuzzy set A on X

nnA22A11A x/)x(x/)x(x/)x(A µµµµ++++++++µµµµ++++µµµµ==== ⋯

under f(.) is a fuzzy set B:

nnB22B11B y/)x(y/)x(y/)x(B µµµµ++++++++µµµµ++++µµµµ==== ⋯

where yi = f(xi), i = 1 to n

If f(.) is a many-to-one mapping, then

)x(max)y( A)y(fx

B 1µµµµ====µµµµ

−−−−====


Dr. Ngo Thanh Long


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Example

– Application of the extension principle to fuzzy sets with discrete universes

Let A = 0.1 / -2+0.4 / -1+0.8 / 0+0.9 / 1+0.3 / 2

and f(x) = x 2 – 3Applying the extension principle, we obtain:B = 0.1 / 1+0.4 / -2+0.8 / -3+0.9 / -2+0.3 /1

= 0.8 / -3+(0.4V0.9) / -2+(0.1V0.3) / 1= 0.8 / -3+0.9 / -2+0.3 / 1

where “V” represents the “max ” operator, Same reasoning for continuous universes


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Transition From Type-1 toType-2 Fuzzy Sets

• Blur the boundaries of a T1 FS

• Possibility assigned—could be non-uniform

• Clean things up• Choose uniform

possibilities—interval type-2 FS


Dr. Ngo Thanh Long


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Interval Type-2 Fuzzy Sets:Terminology-1

Documents

Fuzzy Sets: An introduction · • More on fuzzy union, intersection, and complement – Fuzzy complement – Fuzzy intersection and union ... following three properties: • 1. A