Extension Principle Hand Out

Extension Principle

Adriano Cruz, [email protected]

PPGI-UFRJ

September 2011

Adriano Cruz, [email protected] (PPGI-UFRJ) Extension Principle September 2011 1 / 62

Summary

1 Introduction

2 Crisp Functions, Mappings and Relations

3 Functions of Fuzzy Sets

4 Fuzzy Arithmetic

5 Interval Analysis in Arithmetic

6 Approximate Methods of ExtensionVertex MethodDSW Algorithm


Section Summary

1 Introduction



4 Fuzzy Arithmetic




Would a precise model be a contradiction?


Bibliography

Kevin M. Passino, Stephen Yurkovich, Fuzzy Control in Chapter 5, AddisonWesley Longman, Inc, USA, 1998.

Timothy J. Ross , Fuzzy Logic with Engineering Applications, John Wileyand Sons, Inc, USA, 2010.

R. R. Yager, A characterization of the extension principle, Fuzzy Sets Syst.,18, 205-217, 1986

John Yen, Reza Langari, Fuzzy Logic: Intelligence, Control and Information,Prentice Hall, USA, 1999

L. Zadeh, The concept of a linguistic variable and its application toapproximate reasoning, Part I. Inf Sci., 8, 199-249, 1975

W. Dong and H. Shah, Vertex Method for computing functions of fuzzyvariables. Fuzzy Sets Syst., 24, 65-78, 1987.

W. Dong and H. Shah and F. Wong, Fuzzy computations in risk and decisionanalysis, Civ. Eng. Syst., 2, 201-208, 1985.


Background

Consider a function y = f (x).

If we known x it is possible to determine y .

Is it possible to extend this mapping when the input, x , is a fuzzyvalue.

The extension principle developed by Zadeh (1975) and later by Yager(1986) establishes how to extend the domain of a function on a fuzzysets.


Section Summary

1 Introduction



4 Fuzzy Arithmetic




Crisp Mappings

X f(X) Y


Functions Applied to Intervals

An interval I is a crisp set, I X .

Compute the image of the interval, which is a crisp set in Y .

Presumably, sets in the power set of X can be mapped to the powerset of Y , that is f : P(X ) P(Y ).

The image B Y of a set A X can be calculated as B = f (A) orfor all x A, y = f (x)

B is defined by its characteristic value

B(y) = f (A)(y) =

y=f (x)

A(x)


Functions Applied to Intervals

x

y

I

f(I)


Functions Applied to Intervals - Example I

Consider the Universe X = {2,1, 0, 1, 2}

Consider the set A = {0, 1}

Using the Zadeh notation A = { 02 +

01 +

10 +

11 +

02}

Consider the mapping y = |4x | + 2

What is the resulting set B on the Universe Y = {2, 6, 10}


Functions Applied to Intervals - Example II

Using B(y) = f (A)(y) =

y=f (x) A(x)

and y = |4x |+ 2.

B(2) = {A(0)} = 1.

B(6) = {A(1), A(1)} = {0, 1} = 1.

B(10) = {A(2), A(2)} = {0, 0} = 0.

B = {12 +16 +

010} or B = {2, 10}.


Using Relations

It is possible to achieve the results using a relation that express themapping y = |4x |+ 2.

Lets X = {2,1, 0, 1, 2}.

Lets Y = {0, 1, 2, . . . , 9, 10}

The relation

R =

0 1 2 3 4 5 6 7 8 9 1021012

0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 00 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 1

B = A R

A = { 02 +

01 +

10 +

11 +

02} or more conveniently A = {0, 0, 1, 1, 0}


Applying the Relation

Using B(y) =

xX (A(x) R(x , y))

we find

B(y) =

{1, for y = 2, 60, otherwise

.

Or

B =

{0

0+

0

1+

1

2+

0

3+

0

4+

0

5+

1

6+

0

7+

0

8+

0

9+

0

10

}


Section Summary

1 Introduction



4 Fuzzy Arithmetic




Fuzzy Mappings

ABC

D


Starting Point

Consider two universes of discourse X and Y and a function y = f (x).

Suppose that elements in universe X form a fuzzy set A.

What is the image (defined as B) of A on Y under the mapping f ?

Similarly to the crisp definition, B is obtained as

B(y) = f (A)(y) =

y=f (x)

A(x)


Simplifying the Notation

Fuzzy vector is a convenient shorthand for calculations that usematrix relations.

Fuzzy vector is a vector containing only the fuzzy membership values.

Consider the fuzzy set:

B =

{0

0+

0.2

1+

0.3

2+

0.5

3+

0.7

4+

0.9

5+

1

6+

0

7+

0

8+

0

9+

0

10

}

The fuzzy set B may be represented by the fuzzy vector b:

b =

{0, 0.2, 0.3, 0.5, 0.7, 0.9, 1, 0, 0, 0, 0

}


Extension Principle

Suppose that f is a function from X to Y and A is a fuzzy set on Xdefined as

A = A(x1)/x1 + A(x2)/x2 + . . . + A(xn)/xn

.

The extension principle states that the image of fuzzy set A under themapping f (.) can be expressed as a fuzzy set B defined as

B = f (A) = A(x1)/y1 + A(x2)/y2 + . . .+ A(xn)/yn

where yi = f (xi )


Many-to-one mappings

If f (.) is a many-to-one mapping, then, for instance, there may existx1, x2 X , x1 6= x2, such that f (x1) = f (x2) = y

, y Y .

The membership degree at y = y is the maximum of themembership degrees at x1 and x2 more generally, we have

B(y) = max

y=f (xi )A(x)


Monotonic Continuous Functions

For each point in the interval:

Compute the image of the interval.The membership degrees are carried through.

x

y

A

B

(x)B

(x)A


Monotonic Continuous Functions Ex.

Function: y = f (x) = 0.6 x + 4.

Input: Fuzzy number - around -5.

around 5 = {0.33 +1.05 +

0.37 }.

f (around 5) = { 0.3f (3) +

1f (5) +

0.3f (7)}.

f (around 5) = { 0.30.63+4 +1

0.65+4 +0.3

0.67+4}.

f (around 5) = {0.35.8 +17 +

0.38.2}.


Monotonic Continuous Functions Ex.

xy

(x)

B

(x)A

x3 54 7621

1

2

3

4

5

6

7

8

(x)A

3 54 76

0.3

1.0

x21

7

0.3

1.0

x8.2

5.8


Non-Monotonic Continuous Functions Ex.

Function: y = f (x) = x2 6 x + 11.

Input: Fuzzy number - around -4.

around 4 = {0.32 +0.63 +

14 +

0.65 +

0.36 }.

f (around 4) = { 0.3f (2) +

0.6f (3) +

1f (4) +

0.6f (5) +

0.3f (6)}.

f (around 4) = {0.33 +0.62 +

13 +

0.66 +

0.311 }.

f (around 4) = {0.313 +0.62 +

0.66 +

0.311 }.

f (around 4) = {0.62 +13 +

0.66 +

0.311 }.


Generalizing

Suppose the input universe is composed of the Cartesian product ofmany universes.

The mapping f is defined on the power set of this universe asf : P(X1 X2 Xn) P(Y ).

Let the fuzzy sets A1,A2, . . . ,An be defined on X1,X2, . . . ,Xn

then B = f (A1,A2, . . . ,An).

The membership function of B is defined as

B(y) = maxy=f (x1,x2,...,xn)

{min [A1(x1), A2(x2), . . . , An(xn)]}

This equation is usually called the Zadehs extension principle.

If the function f is a continous-valued expression, the max operator isreplaced by the sup (supremum) which is the least upper bound.


Example

Inputs: A = {0.21 +12 +

0.74 } and B = {

0.51 +

12}

Output: f (A,B) = A B (arithmetic product).

A B =

{min(0.2, 0.5)

1+

max[min(0.2, 1),min(0.5, 1)]

2+

max[min(0.7, 0.5),min(1, 1)]

4+

min(0.7, 1)

8

}

=

{0.2

1+

0.5

2+

1

4+

0.7

8

}


Fuzzy Transform

Fuzzy transform happens when the input of a single element(nonfuzzy) maps to a fuzzy set in the output universe.

An element x in universe X is mapped to a fuzzy set B in universe Y .

B = f (x), where f is a fuzzy mapping.

If X and Y are finite f can be expressed as a fuzzy relation R or

R =

y1 y2 . . . yj . . . ymx1x2...xi...

xn

r11 r12 . . . r1j . . . r1mr21 r22 . . . r2j . . . r2m...

......

......

...ri1 ri2 . . . rij . . . rim...

......

......

...rn1 rn2 . . . rnj . . . rnm


Fuzzy Transform Singleton

For a particular singleton xi its fuzzy image is the fuzzy set Bi = f (xi)

Bi (yj ) = rij or in fuzzy vector notation

bi = {ri1, ri2, . . . , rim}.


Fuzzy Transform Generalized

For a particular fuzzy input set A its fuzzy image is B = f (A)

B(y) =

xX (A(x) R(x , y))

b = a R .

bj = maxi(min(ai , rij ))


Section Summary

1 Introduction



4 Fuzzy Arithmetic




Fuzzy Numbers

A fuzzy number is fuzzy subset of the universe of a numerical number.

A fuzzy real number is a fuzzy subset of the domain of real numbers.

A fuzzy integer number is a fuzzy subset of the domain of integers.


Examples of Fuzzy Numbers

1 2 3 4 5 6 7 8 9 10

1.0

( x)

x

1 2 3 4 5 6 7 8 9 10

1.0

( x)

x

Fuzzy Real Number 5

Fuzzy Integer Number 5


Fuzzy Arithmetic

Applying the extension principle to arithmetic operations it is possibleto define fuzzy arithmetic operations

Let x and y be the operands, z the result.

Let A, B and C denote the fuzzy sets that represent the operands x ,y and z respectively.


Fuzzy Arithmetic

Using the extension principle a fuzzy arithmetic operation denoted by {+,,,} is defined as

C (z) = maxz=xy

{min [A(x), B (y)]}


Example of Problem

We will calculate the product of two fuzzy sets defined as:

X ={01 +

0.332 +

0.663 +

14 +

0.665 +

0.336 +

07

}Y =

{02 +

0.333 +

0.664 +

15 +

0.666 +

0.337 +

08

}The result would be:

X Y =

{0

2+

0

3+

0

4+

0

5+

0.33

6+

0.33

8+

0.33

9+

0.33

10+

0.66

12

+0.33

14+

0.66

15+

0.33

16+

0.66

18+

1

20+

0.33

21+

0.66

24

+0.66

25+

0.33

28+

0.66

30+

0

32+

0.33

35+

0.33

36+

0

40

+0.33

42+

0

48+

0

49+

0

56

}


Example of Problem

0 1 2 3 4 5 6 7 8 90

0.5

1

X

(X)

0 1 2 3 4 5 6 7 8 90

0.5

1

Y

(Y)

0 10 20 30 40 50 600

0.5

1

X Y

(X

Y)


Section Summary

1 Introduction



4 Fuzzy Arithmetic




Definitions I

Let I1 and I2 two interval numbers defined ordered pair of realnumbers with lower and upper bounds.

I1 = [a, b] where a b and I2 = [c , d ] where c d .

I1 I2 = [a, b] [c , d ] where {+,,,} is another interval.

When adding or multiplying two intervals we are performing theseoperations on the infinite number of combinations of pairs from eachof the two intervals.


Definitions II

[a, b] + [c , d ] =[a + c , b + d ]

[a, b] [c , d ] =[a d , b c]

[a, b] [c , d ] =[min(ac , ad , bc , bd),max(ac , ad , bc , bd)]

[a, b] [c , d ] =[a, b]

[1

d,1

c

]since 0 / [c , d ]


Conclusions

When adding or multiplying two intervals we are performing theseoperations on the infinite number of combinations of pairs from eachof the two intervals.

We only need to find the endpoints of the intervals to find theendpoints of the solutions.


Section Summary

1 Introduction



4 Fuzzy Arithmetic




Vertex Method

The method is based on a combination of the -cut and standardinterval analysis [Dong and Shah, 1987].

The algorithm is easy to implement and can be computationallyefficient.

1

a b x0

A

I


Vertex Algorithm

Any continuous membership function can be represented by acontinuous sweep of -cut intervals from = 0+ to = 1.

Let y = f (x) be extended for fuzzy sets, or B = f (A).

A will be decomposed into a series for -cut intervals.

When f (x) is continuous and monotonic on I = [a, b] the intervalrepresenting B at a particular value of (B) is

B = f (I) = [min(f (a), f (b)),max(f (a), f (b))]


Vertex Algorithm for n-inputs

Let y = f (x1, x2, . . . , xn).

The input space is represented by n-dimensional Cartesian region.N = 2n is the number of vertices of the region.

Each of the input variables is described by an interval Ii at a specific-cut where Ii = [ai , bi ], i = 1, 2, . . . , n.

B =f (I1, I2, . . . , In) (1)

B =

[minj(f (cj )),max

j(f (cj ))

], j = 1, 2, . . . ,N (2)

where cj is the coordinate of the jth vertex representing then-dimensional Cartesian region.


Vertex Algorithm for n-inputs

The method is accurate only when the conditions of continuity andno extreme points are satisfied.

An extreme point is a point of maximum or minimum.

Extreme points should be treated as additional vertices Ek .

B =

[minj ,k

(f (cj ), f (Ek)),maxj ,k

(f (cj), f (Ek))

],

j = 1, 2, . . . ,N and k = 1, 2, . . . ,m


Example of Problem

We will use Vertex Method to determine the output of the functiony = x(2 x) to an input fuzzy set A = (x).

We will use the three -cuts: = 0+, 0.5, 1.0

The corresponding intervals are: I0+ = [0.5, 2], I0.5 = [0.75, 1.5],I1 = [1, 1].

The extreme point x = 1, y = 1 can be calculated by derivatives.

0 0.5 1 1.5 2 2.50

0.5

1

y=f(x)

x

y

0 0.5 1 1.5 2 2.50

0.5

1

A

x

(x)

0.75


Calculating

I0+ = [0.5, 2] c1 = 0.5, c2 = 2,E1 = 1

f (c1) = 0.75, f (c2) = 0, f (E1) = 1

B0+ = [min(0.75, 0, 1),max(0.75, 0, 1)] = [0, 1]

I0.5 = [0.75, 1.5] c1 = 0.75, c2 = 1.5,E1 = 1

f (c1) = 0.9375, f (c2) = 0.75, f (E1) = 1

B0.5 = [min(0.9375, 0.75, 1),max(0.9375, 0.75, 1)] = [0.75, 1]

I1 = [1, 1] c1 = 1, c2 = 1,E1 = 1

f (c1) = f (c2) = f (E1) = 1

B1 = [min(1, 1, 1),max(1, 1, 1)] = [1, 1]


Results

0 0.5 1 1.5 2 2.50

0.5

1

y=f(x)

x

y

0 0.5 1 1.5 2 2.50

0.5

1

A

x

y=(x

)

0.75

0 0.5 1 1.5 2 2.50

0.5

1

B

y

(y)


Another Example of Problem

We will use Vertex Method to calculate the product of two fuzzy setsdefined as:

X ={01 +

0.332 +

0.663 +

14 +

0.665 +

0.336 +

07

}Y =

{02 +

0.333 +

0.664 +

15 +

0.666 +

0.337 +

08

}


Interval I0+

Support for X is the interval [1, 7].

Support for Y is the interval [2, 8].

x y f()

1 2 f(a)=2

1 8 f(b)=8

7 2 f(c)=14

7 8 f(d)=56

min=2, max=56 and B0+ = [2, 56]


Interval I0.33



x y f()

2 3 f(a)=6

2 7 f(b)=14

6 3 f(c)=18

6 7 f(d)=42

min=6, max=42 and B0.33 = [6, 42]


Interval I0.66



x y f()

3 4 f(a)=12

3 6 f(b)=18

5 4 f(c)=20

5 6 f(d)=30

min=12, max=30 and B0.66 = [12, 30]


Interval I1.0



x y f()

4 5 f(a)=20

min=20, max=20 and B1.0 = [20, 20]


Vertex Method

1

10 XxY0

A

20 30 40 50 60

0.66

0.33


DSW Algorithm

It uses -cut and standard interval analysis [Dong, Shah and Wong,1985].

The algorithm:Repeat for different values of where 0 1:

Find the interval(s) in the input membership function(s) thatcorrespond to this ;Using standard binary interval operations, compute the interval for theoutput membership function for the selected ;


Example of Problem

We will use DSW Method to determine the output of the functiony = x(2 + x) to an input fuzzy set A = (x).

We will use the three -cuts: = 0+, 0.5, 1.0

The corresponding intervals are: I0+ = [0.5, 2], I0.5 = [0.75, 1.5],I1 = [1, 1].

0 0.5 1 1.5 2 2.50

2

4

6

8

y=f(x)

x

y

0 0.5 1 1.5 2 2.50

0.5

1

A

x

(x)

0.75


Calculating

I0+ = [0.5, 2]

B0+ = 2 [0.2, 2] + [0.52, 22] = [1, 4] + [0.25, 4] = [1.25, 8]

I0.5 = [0.75, 1.5]

B0.5 = 2 [0.75, 1.5] + [0.752, 1.52] = [1.5, 3] + [0.5625, 2.25] =

[2.0625, 5.25]

I1 = [1, 1]

B1 = 2 [1, 1] + [12, 12] = [2, 2] + [1, 1] = [3, 3] = 3


Results

0 0.5 1 1.5 2 2.50

5

y=f(x)

x

y

0 0.5 1 1.5 2 2.50

0.5

1

A

x

(x)

0.75

0 1 2 3 4 5 6 7 8 90

0.5

1

B

y

(y)


Another Example of Problem

We will use DSW Method to calculate the product of two fuzzy setsdefined as:

X ={01 +

0.332 +

0.663 +

14 +

0.665 +

0.336 +

07

}Y =

{02 +

0.333 +

0.664 +

15 +

0.666 +

0.337 +

08

}


Interval Arithmetic

I0+ : [1, 7] [2, 8] = [min(2, 14, 8, 56),max(2, 14, 8, 56)] = [2, 56].

I0.33 : [2, 6] [3, 7] = [min(6, 18, 14, 42),max(6, 18, 14, 42)] = [6, 42].

I0.66 : [3, 5] [4, 6] = [min(12, 20, 18, 30),max(12, 20, 18, 30)] =[12, 30].

I1 : [4, 4] [5, 5] = [min(20, 20, 20, 20),max(20, 20, 20, 20)] = [20, 20].


DSW Algorithm

1

10 XxY0

A

20 30 40 50 60

0.66

0.33


The End


IntroductionCrisp Functions, Mappings and RelationsFunctions of Fuzzy SetsFuzzy ArithmeticInterval Analysis in ArithmeticApproximate Methods of ExtensionVertex MethodDSW Algorithm

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Extension Principle Hand Out