chapter 3 selected design and processing aspects of fuzzy sets

CHAPTER 3

Selected Design and Processing Aspects of Fuzzy Sets

The Development of Fuzzy Sets: Elicitation ofMembership Functions

• Elicitation of membership functions is of significant relevance to conceptual and algorithmic developments of fuzzy sets.

• A number of general approaches:

– horizontal approach;

– vertical approach;

– Saaty priority method (analytical hierarchy process, AHP);

– fuzzy clustering.


• Semantics of fuzzy sets – some general observations:

– Fuzzy sets as meaningful (semantically sound) constructs;

– The number of fuzzy sets used to describe some variable (construct) limited to 7+/-2 terms (membership functions);

– Fuzzy sets require calibration – adjustment of membership functions depending on the context in which fuzzy sets are used.


• Fuzzy set as a descriptor of feasible solutions

The intent is to describe a collection of solutions to a given optimization problem by characterizing then through degrees of feasibility as optimal solutions.

– Determine maximum of F where F assumes positive values

a collection of solutions and their global characterization as a fuzzy set

)(maxarg0 xFxx L

)(min)(max

)( min )()(

xFxF

xFxFxA

xx

x

LL

L



– Determine minimum of F

a collection of solutions and their global characterization as a fuzzy set

)(minarg0 xFxx L

)(min)(max

)( min )(1)(

xFxF

xFxFxA

xx

x

LL

L



If F assumes real numbers, then

– For the maximization problem

– For the maximization problem

)(min)(max

)( min )()(

xFxF

xFxFxA

xx

x

LL

L

)(min)(max

)( min )(1)(

xFxF

xFxFxA

xx

x

LL

L


• Fuzzy set and a notion of typicality

Issue of gradual typicality captured through membership degrees

In geometry: ideal geometric figures (circle, ellipse, square...)

Perception of geometry of ellipsoide:

• (a) higher differences |a-b|, less typicality of the figure

• (b) ratios a/b and the departure from an ideal shape where a/b=1

a

b

|a-b|

membership 1


• Horizontal scheme of membership function estimation

– Identify a collection of elements of the universe of discourse X and query a panel of n experts: does x belong to concept A?

– Count the number of ‘yes” responses (p) and calculate the ratio of p/n.

p/n

X


• Horizontal scheme of membership function estimation

– Membership value – the ratio of p/n.

– Standard deviation of the membership estimate

and associated confidence interval determined as [p-σ, p+σ].

nnpnp )/1(/


• Example: Horizontal scheme of membership function estimation


• Vertical scheme of membership function estimation

Determination of successive α-cuts and a formation of fuzzy set

– Query a panel of n experts: what are the elements of X which belong to fuzzy set A at a degree not lower than α?

1

p

X


• Saaty’s priority approach to membership function estimation

Determination of membership function through a series of pairwise comparisons of elements of X with regard to their preference vis-a-vis a given concept – fuzzy set.

Consider that for elements X1, X2, …, Xn, we have the membership grades A(X1), A(X2), …, A(Xn). Organize them in a form of a reciprocal matrix:

.

1...)()(

)()(

...)()(...1

)()(

)()(...

)()(1

)()(...

)()(

)()(

......)()(...

)()(

)()(

)()(...

)()(

)()(

),(

21

2

1

2

1

2

1

21

2

2

2

1

2

1

2

1

1

1

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XAXA

XX

nn

n

n

n

nnn

n

n

lkM



Properties of reciprocal matrices:

– The diagonal values are equal to 1;

– Entries symmetrically positioned with respect to the diagonal satisfy condition if multiplicative reciprocality that is M(Xk,Xl)=1/ M(Xl,Xk).

– Transitive property: M(Xk,Xl) M(Xk,Xl)= M(Xk,Xl) for all indexes i, j, k.



Eigenvectors of reciprocal matrix

– The i-th element of the above vector is equal to nA(Xi).

– Overall MA=nA

– A is the eigenvector of M associated with the largest eigenvalue of M equal to “n”.

)(

)()(

)()(

)()(

)()(][ 2

1

21

n

n

iiii

XA

XAXA

XAXA

XAXA

XAXAA

M



From reciprocal matrix to fuzzy set:

– Construct a reciprocal matrix based on expert's pairwise comparisons

– Use of scale of relative importance



Experimentally constructed reciprocal matrix may not satisfy transitivity condition resulting in some level of inconsistency.

Inconsistency index

– = 0, if max =n;

– >0, lack of consistency.

max n

n 1


• Principle of justifiable granularity

Experimental data captured in a form of a certain information granule – fuzzy sets.

The resulting fuzzy set is required to satisfy:

– Sufficient level of experimental evidence – to be as high as possible.

– Sufficient specificity – to be as high as possible.

These two requirements are in conflict.


• Principle of justifiable granularity: development strategy

Reconciling conflicting criteria:

Cover most data

Make fuzzy set specific enough – minimize support of A, |m-a|

X

max A(xk)

min Supp(A)

data

a

||

)(max

am

xAk

k

a

k

kxA )(


• Design of fuzzy sets through fuzzy clustering: Fuzzy C-Means

Grouping n-dimensional data located in Rn into c clusters – fuzzy sets so that an objective function

becomes minimized.

Notation:

– U- partition matrix

– v1, v2, …, vc – prototypes

– ||.|| - distance function

– m- fuzzification coefficient

2

11

|||| ik

N

k

mik

c

i

uQ vx


• Design of fuzzy sets through fuzzy clustering: Fuzzy C-Means

Partition matrix – two properties

ciNuN

kik 1,2,..., ,0

1

Nkuc

iik 1,2,..., ,1

1


• FCM – minimization problem

Minimize objective function with respect to:

– prototypes;– partition matrix.

Use of Lagrange multipliers in the optimization with respect to partition matrix.

Prototypes easily determined when using Euclidean distance function.


• FCM – impact of fuzzification coefficient of geometry clusters

By changing the values of m (>1) a shape of membership functions becomes affected.


• Separation measure for fuzzy clusters

By quantifying among clusters, an extent of their separation is expressed

c

ii

cc ucuuu

121 1),...,,(


• Fuzzy equalization

Select membership functions A1, A2, …, Ac in such a way so that their expected values are made equal for all fuzzy sets, that is

xxpxAi )d()(X


• Design of membership functions: main guidelines

– Highly visible, well-articulated semantics, keep the number of terms in the range 7+/-2.

– Different views at fuzzy sets associated with their underlying estimation techniques.

– Fuzzy sets are context-dependent and require calibration (when applied to a certain problem).

– Two main categories of estimation techniques - data–driven and user-driven. Also some hybrid approaches are anticipated.

Aggregation Operations

• Aggregation operator regarded as a mapping satisfying conditions:

– monotonicity

g(x1, x2,…, xn) > g(y1, y2,…, yn), if xi > yj

– boundary conditions

g(0, 0, ….. , 0) = 0 and g(1, 1, … .., 1) = 1


• Averaging operators - generalized mean

A class of operators in the form:

g(x1, x2,....,xn ) 1n

(x i)p

i1

n

p p 0


• Examples of generalized means

Transformations of Fuzzy Sets

• Extension principle

Different ways of mapping input (number, set, fuzzy set) through a given function f


• Transformations of numeric argument

Transformation of a point through a function


• Transformations of sets

Transformation of a given set through function f

B = f(A) = {yY| y = f(x), xA}

B(y) supx |yf (x )

A(x)


• Transformations of fuzzy sets

Transformation of a given fuzzy set through function f

B(y) supx |yf (x )

A(x)


• Transformations of fuzzy sets: a multivariate case

Transformation of a collection of fuzzy set through function f:

)]}(,),(),({min[sup)( 2211)(|

nnfy

xAxAxAyB xx


• Fuzzy numbers and fuzzy arithmetic

Fuzzy numbers and fuzzy intervals satisfy the conditions of:

– normality;

– unimodality;

– continuity;

– boundness of support.

1

A(x)

R a b c d 0

1

A(x)

R a m b 0

(a) (b)

fA

gA

fA

gA


• Examples: Information granules of numbers, intervals, fuzzy intervals and fuzzy numbers

1 1

2.5 2.5

1 1

2.2

2.2 3.0

3.0 2.5 2.2 3.0

real number 2.5

fuzzy number about 2.5

real interval [2.2, 3.0]

fuzzy interval around [2.2, 3.0]

R R

R R

A A

A A


• Examples:

Consider that you traveled for 2 hours at speed of about 110 km/hr. What was the distance you traveled? The speed is described in the form of some fuzzy set S whose membership function is given.

The next example is a more general version of the above problem.

You traveled at speed of about 110 km/hr for about 3 hours. What was the distance traveled? We assume that both the speed and time of travel are described by fuzzy sets.

In a certain manufacturing process, there are five operations completed in series. Given the nature of the manufacturing activities, the duration of each of them can be characterized by fuzzy sets T1, T2,…, and T5. What is the time of realization of this process?


• Interval arithmetic and -cuts

Basic arithmetic operations on intervals:

– addition: [a,b] + [c,d] = [a + c, b + d]

– subtraction: [a,b] - [c,d] = [a - d, b - c ]

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

– division:

db

cb

da

ca

db

cb

da

cadcba ,,,max,,,,min],/[],[


• Interval arithmetic and -cuts

Use α-cuts for operation A*B

(AB)= AB

and then combine the obtained results by taking a union of the obtained α-cuts

A B (A B) [0,1]U

)]()[(sup)]()([sup))((]1,0[]1,0[

xBAxBAxBA f

(A B)f (x)(A B) (x)


• Fuzzy arithmetic and extension principle

The membership function of A*B is given in the form

Considering some t-norm, one has

Depending on t-norm (minimum, drastic product) the following inequality holds

R

zyBxAzBAyxz

)),(),(min(sup))((

R

zyTBxABAyxz

,))()((sup)*(*

))()((sup yBTxA dyxz

))()((sup yBTxAyxz

R

zyBTxA myxz

))()((sup


• Examples: Fuzzy arithmetic

Depending on t-norm used, different membership functions of the result A+B are obtained

1 1

3.0

1 1

2.0 4.0

0 1.5 2.5 4.0 1.0

4.0 7.0 1.0

A+B Tm

A+B Td

6.0

X X

X X

A B


• Fuzzy arithmetic: a fundamental result


• Computing with triangular fuzzy numbers

Algebraic operations on triangular fuzzy numbers produce interesting and practically relevant results. Consider two triangular fuzzy numbers

A(x)

x am a

if x [a,m)

b xb m

if x [m,b]

0 otherwise

B(x)

x cn c

if x [c,n)

d xd n

if x [n ,d]

0 otherwise

B A

a m c b n d

membership


• Computing with triangular fuzzy numbers: addition

In calculations, we consider separately increasing and decreasing segments of the membership functions of A and B

This leads to an interesting result - the sum is a triangular fuzzy number with the membership function:

R

zyBxAzCyxz

)),(),(min(sup)(

nmzifn)(md)(b

zd)(bnmzif

nmzifc)(an)(m

c)(az

zC 1)(


• Computing with triangular fuzzy numbers: multiplication

As before we consider separately increasing and decreasing segments of the membership functions of A and B. For the increasing parts of the membership functions:

x=(m-a)α+a y=(n-c) α+c

z=xy=[(m-a) α+a][(n-c) α+c]

The result is not a triangular fuzzy number.

)()()())(( 12 faccnacamcnamz

)())(*()( 11 zfzBAzD


• Computing with triangular fuzzy numbers: division

As before we consider separately increasing and decreasing segments of the membership functions of A and B. For the increasing parts of the membership functions

x=(m-a)α+a y=(n-c) α+c

The result is not a triangular fuzzy number.

)()()(

1 g

ccnaam

yxz

)())(/()( 11 gzBAzE

chapter 3 selected design and processing aspects of fuzzy sets

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