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Fuzzy Logic & Approximate Reasoning

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Fuzzy Sets


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References• Journal:

– IEEE Trans. on Fuzzy Systems.– Fuzzy Sets and Systems.– Journal of Intelligent & Fuzzy Systems– International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems – ...

• Conferences:– IEEE Conference on Fuzzy Systems.– IFSA World Congress.– ...

• Books and Papers:– Z.Chi et al, Fuzzy Algorithms with applications to Image Processing and Pattern Recognition, World

Scientific, 1996.– S. N. Sivanandam, Introduction to Fuzzy Logic using MATLAB, Springer, 2007.– J.M. Mendel, Fuzzy Logic Systems for Engineering: A toturial, IEEE, 1995.– W. Siler, FUZZY EXPERT SYSTEMS AND FUZZY REASONING, John Wiley Sons, 2005– G. Klir, Uncertainty and Informations, John Wiley Sons, 2006.– L.A. Zadeh, Fuzzy sets, Information and control, 8, 338-365, 1965.– L.A. Zadeh, The Concept of a Linguistic Variable and its Application to Approximate Reasoning-I, II, III,

Information Science 8, 1975– L.A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and

fuzzy logic, Fuzzy Sets and Systems 90(1997), 111-127.– ......– http://www.type2fuzzylogic.org/– IEEE Computational Intelligence Society http://ieee-cis.org/– International Fuzzy Systems Association http://www.isc.meiji.ac.jp/~ifsatkym/– J.M. Mendel http://sipi.usc.edu/~mendel


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Contents

• Fuzzy sets.

• Fuzzy Relations and Fuzzy reasoning

• Fuzzy Inference Systems

• Fuzzy Clustering

• Fuzzy Expert Systems

• Applications: Image Processing, Robotics, Control...


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Fuzzy Sets: Outline• Introduction: History, Current Level and Further Development of

Fuzzy Logic Technologies in the U.S., Japan, and Europe • Basic definitions and terminology• Set-theoretic operations• MF formulation and parameterization

– MFs of one and two dimensions– Derivatives of parameterized MFs

• More on fuzzy union, intersection, and complement– Fuzzy complement– Fuzzy intersection and union– Parameterized T-norm and T-conorm

• Fuzzy Number• Fuzzy Relations


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History, State of the Art, and Future Development

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

1970 First Application of Fuzzy Logic in Control Engineering (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 Is Also Applied in Data and Sensor Signal Analysis. Application of Fuzzy Logic in Business and Finance.

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

1970 First Application of Fuzzy Logic in Control Engineering (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 Is Also Applied in Data and Sensor Signal Analysis. Application of Fuzzy Logic in Business and Finance.


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Types of Uncertainty and the Modeling of Uncertainty

Stochastic Uncertainty:

The Probability of Hitting the Target Is 0.8

Lexical Uncertainty:

"Tall Men", "Hot Days", or "Stable Currencies"

We Will Probably Have a Successful Business Year.

The Experience of Expert A Shows That B Is Likely to Occur. However, Expert C Is Convinced This Is Not True.

Stochastic Uncertainty:

The Probability of Hitting the Target Is 0.8

Lexical Uncertainty:

"Tall Men", "Hot Days", or "Stable Currencies"

We Will Probably Have a Successful Business Year.

The Experience of Expert A Shows That B Is Likely to Occur. However, Expert C Is Convinced This Is Not True.

Most Words and Evaluations We Use in Our Daily Reasoning Are Most Words and Evaluations We Use in Our Daily Reasoning Are Not Clearly Defined in a Mathematical Manner. This Allows Not Clearly Defined in a Mathematical Manner. This Allows Humans to Reason on an Abstract Level!Humans to Reason on an Abstract Level!


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Possible Sources of Uncertaintyand Imprecision

• There are many sources of uncertainty facing any control system in dynamic real world unstructured environments and real world applications; some sources of these uncertainties are as follows:– Uncertainties in the inputs of the system due to:

• The sensors measurements being affected by high noise levels from various sources such a electromagnetic and radio frequency interference, vibration, etc.

• The input sensors being affected by the conditions of observation (i.e. their characteristics can be changed by the environmental conditions such as wind, sunshine, humidity, rain, etc.).


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Possible Sources of Uncertaintyand Imprecision

• Other sources of Uncertainties include:– Uncertainties in control outputs which can result from the change of the actuators characteristics due to wear and tear or due to environmental changes.– Linguistic uncertainties as words mean different things to different people.– Uncertainties associated with the change in the operation conditions due to varying load and environment conditions.


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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

38.7°C38.7°C

37.2°C37.2°C

38°C38°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” !More-or-Less” Rather Than “Either-Or” !

“Strong Fever”


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Fuzzy Sets• Sets with fuzzy

boundariesA = Set of tall people


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Membership Functions (MFs)• Characteristics of MFs:

– Subjective measures– Not probability functions


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

function (MF), A, 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 {( , ( ))| }Universe 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)}


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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(


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

follows:

A x xAx X

i i

i

( ) /

A x xA

X

( ) /

X is discrete

X is continuous

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


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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 is a variable whose values are not numbers but words

or sentences in a natural or artificial language (Zadeh, 1975a, p. 201)

• Linguistic variable is characterized by [ , T(), U], in which : name of the variable, T() : the term set of , universe of discourse U

• A linguistic variable is a variable whose values are not numbers but words or sentences in a natural or artificial language (Zadeh, 1975a, p. 201)

• Linguistic variable is characterized by [ , T(), U], in which : name of the variable, T() : the term set of , universe of discourse U

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


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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


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Convexity of Fuzzy Sets

A A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21 Alternatively, A is convex if all its -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

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

• Subset:

• Complement:

• Union:

• Intersection:


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Set-Theoretic Operations


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MF Formulation

• Triangular MF: trimf x a b cx ab a

c xc b

( ; , , ) max min , ,

0

Trapezoidal MF:

Generalized bell MF: gbellmf x a b cx cb

b( ; , , )

1

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Gaussian MF: gaussmf x a b c ex c

( ; , , )

12

2

trapmf x a b c dx ab a

d xd c

( ; , , , ) max min , , ,

1 0


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MF Formulation


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MF Formulation• Sigmoidal

MF: sigmf x a b ce a x c( ; , , ) ( )

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Extensions:

Abs. difference

of two sig. MF

Product

of two sig. MF


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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:


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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)


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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)


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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.

)(A

]1,0(

)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 -level of , (0,1]

' < ' [ ] [ ]z z

, z z

z1zz z

z

,[ ] z zz

z

’

z


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

:{ ((x, y), (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}

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

under f(.) is a fuzzy set B:

nnB22B11B y/)x(y/)x(y/)x(B where yi = f(xi), i = 1 to n

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

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

B 1


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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) = x2 – 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 / 1where “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


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Interval Type-2 Fuzzy Sets:Terminology-1


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Questions