Download Transformations of fuzzy sets

CHAPTER 3
Selected Design and Processing
Aspects of Fuzzy Sets
The Development of Fuzzy Sets: Elicitation of
Membership 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.
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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.
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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
x0  arg max F ( x)
xL
a collection of solutions and their global characterization as a fuzzy set
A( x ) 
F ( x )  min F ( x )
xL
max F ( x )  min F ( x )
xL
xL
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Fuzzy set as a descriptor of feasible solutions
– Determine minimum of F
x0  arg min F ( x)
xL
a collection of solutions and their global characterization as a fuzzy set
A( x )  1 
F ( x )  min F ( x )
xL
max F ( x )  min F ( x )
xL
xL
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Fuzzy set as a descriptor of feasible solutions
If F assumes real numbers, then
– For the maximization problem
A( x ) 
F ( x )  min F ( x )
xL
max F ( x )  min F ( x )
xL
xL
– For the maximization problem
A( x )  1 
F ( x )  min F ( x )
xL
max F ( x )  min F ( x )
xL
xL
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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
membership
b
1
a
|a-b|
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Horizontal scheme of membership function estimation
– Membership value – the ratio of p/n.
– Standard deviation of the membership estimate

p / n(1  p / n)
n
and associated confidence interval determined as [p-σ, p+σ].
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Example: Horizontal
estimation
scheme
of
membership
function
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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?
p
1
X
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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:
 A( X 1 )
 A( X )
1

A
(
X
2)

M ( X k , X l )   A( X 1 )

 A( X )
n

 A( X 1 )
A( X 1 )
A( X 2 )
A( X 2 )
A( X 2 )
...
A( X n )
A( X 2 )
...
...
...
...
A( X 1 )  
1
A( X n )  
 
A( X 2 )   A( X 2 )
A( X n )    A( X 1 )
 
A( X n )   A( X n )
 
A( X n )   A( X 1 )
A( X 1 )
...
A( X 2 )
1
...
...
A( X n )
...
A( X 2 )
A( X 1 ) 
A( X n ) 

A( X 2 ) 
A( X n )  .


1 

The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Saaty’s priority approach to membership function estimation
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.
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Saaty’s priority approach to membership function estimation
Eigenvectors of reciprocal matrix
 A( X 1 ) 


 A( X i ) A( X i ) A( X i )   A( X 2 )
[ M A]i  


 A( X 1 ) A( X 2 ) A( X n )    


 A( X n )
– 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”.
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Saaty’s priority approach to membership function estimation
From reciprocal matrix to fuzzy set:
– Construct a reciprocal matrix based on expert's pairwise comparisons
– Use of scale of relative importance
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Saaty’s priority approach to membership function estimation
Experimentally constructed reciprocal matrix may not satisfy transitivity
condition resulting in some level of inconsistency.
Inconsistency index
n
lmax  n
n 1
– n = 0, if lmax =n;

– n >0, lack of consistency.
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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.
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Principle of justifiable granularity: development strategy
Reconciling conflicting criteria:
Cover most data
 A( x )
k
k
Make fuzzy set specific enough – minimize support of A, |m-a|
 A( x
max
a
max  A(xk)
min Supp(A)
a
X
data
k
)
k
|ma |
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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
c
Q
i 1
becomes minimized.
Notation:
–
U- partition matrix
–
v1, v2, …, vc – prototypes
–
||.|| - distance function
–
m- fuzzification coefficient
N
u
k 1
m
ik
|| x k  vi ||2
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Design of fuzzy sets through fuzzy clustering: Fuzzy C-Means
Partition matrix – two properties
N
0   uik  N , i  1,2,..., c
k 1
c
u
i 1
ik
 1, k  1,2,..., N
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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.
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• FCM – impact of fuzzification coefficient of geometry clusters
By changing the values of m (>1) a shape of membership functions
becomes affected.
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• Separation measure for fuzzy clusters
By quantifying among clusters, an extent of their separation is expressed
 (u1 , u2 ,..., uc )  1  c
c
c
u
i 1
i
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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
 A ( x) p( x)dx
i
X
The Development of Fuzzy Sets: Elicitation of
Membership Functions
• 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 userdriven. 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
Aggregation Operations
• Averaging operators - generalized mean
A class of operators in the form:
n
1
g(x1, x 2 ,...., x n )  p (x i ) p p  0
n i1

Aggregation Operations
• 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 Fuzzy Sets
• Transformations of numeric argument
Transformation of a point through a function
Transformations of Fuzzy Sets
• Transformations of sets
Transformation of a given set through function f
B = f(A) = {yY| y = f(x), xA}
B(y)  sup A(x)
x |y f (x )

Transformations of Fuzzy Sets
• Transformations of fuzzy sets
Transformation of a given fuzzy set through function f
B(y)  sup A(x)
x |y f (x )

Transformations of Fuzzy Sets
• Transformations of fuzzy sets: a multivariate case
Transformation of a collection of fuzzy set through function f:
B( y )  sup {min[ A1 ( x1 ), A2 ( x2 ), , An ( xn )]}
x| y  f ( x )
Transformations of Fuzzy Sets
• Fuzzy numbers and fuzzy arithmetic
Fuzzy numbers and fuzzy intervals satisfy the conditions of:
– normality;
– unimodality;
– continuity;
– boundness of support.
A(x)
A(x)
(a)
(b)
1
1
fA
fA
gA
a
0
b
c
gA
d
R
a
0
m
b
R
Transformations of Fuzzy Sets
•
Examples: Information granules of numbers, intervals, fuzzy
intervals and fuzzy numbers
A
real number
2.5
1
A
fuzzy number
about 2.5
1
2.5
2.2
R
A
real interval
[2.2, 3.0]
1
2.5 3.0
R
fuzzy interval
around [2.2, 3.0]
A
1
2.2
3.0
R
2.2
2.5 3.0
R
Transformations of Fuzzy Sets
•
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?
Transformations of Fuzzy Sets
• 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)]
a a b b
a a b b
– division: [a, b] /[ c, d ]  min  , , , , max  , , , 
c d c d
 c d c d 

Transformations of Fuzzy Sets
• Interval arithmetic and -cuts
Use α-cuts for operation A*B
(AB)= AB
and then combine the obtained results by taking a union of the obtained αcuts
A  B  U(A  B)
 [0,1]
( A  B)( x)  sup [ ( A  B) ( x)]  sup [( A  B)f ( x)]

[ 0,1]
[ 0,1]
(A  B)f (x)  (A  B) (x)
Transformations of Fuzzy Sets
• Fuzzy arithmetic and extension principle
The membership function of A*B is given in the form
( A  B )( z )  sup min( A( x ), B( y )), z  R
z  x y
Considering some t-norm, one has
( A * B )  sup ( A( x )TB( y )) , z  R
z  x* y
Depending on t-norm (minimum, drastic product) the following inequality
holds
sup ( A( x ) Td B( y ))  sup ( A( x ) T B( y ))  sup ( A( x ) Tm B( y )) z  R
z  x y
z  x y
z  x y
Transformations of Fuzzy Sets
• Examples: Fuzzy arithmetic
Depending on t-norm used, different membership functions of the result
A+B are obtained
A
B
1
1
1.5
0
3.0
4.0
2.5
1.0
X
X
A+B
Td
A+B
Tm
1
1
1.0
4.0
7.0
X
2.0
4.0
6.0
X
Transformations of Fuzzy Sets
• Fuzzy arithmetic: a fundamental result
Transformations of Fuzzy Sets
• Computing with triangular fuzzy numbers
Algebraic operations on triangular fuzzy numbers produce interesting and
practically relevant results. Consider two triangular fuzzy numbers
 x  a
m  a

 b  x
A(x)  
b  m
 0



if x  [a,m)
if x  [m,b]
otherwise
x  c
n  c

d  x
B(x)  
d  n
 0


A
membership
a
m c
if x  [c,n)
if x  [n,d]
otherwise
B
b n
d
Transformations of Fuzzy Sets
• Computing with triangular fuzzy numbers: addition
In calculations, we consider separately increasing and decreasing segments
of the membership functions of A and B
C ( z )  sup min( A( x ), B( y )), z  R
zx y
This leads to an interesting result - the sum is a triangular fuzzy number
with the membership function:
 z  (a  c)
 (m  n)  (a  c) if z  m  n

C( z)   1
if z  m  n
 (b  d)  z
 (b  d)  (m  n) if z  m  n

Transformations of Fuzzy Sets
• 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]
z  (m  a )( n  c) 2  (m  a )c  a(n  c)  ac  f1 ( )
D( z )  ( A * B)( z )  f11 ( z )
The result is not a triangular fuzzy number.
Transformations of Fuzzy Sets
• 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
z
y=(n-c) α+c
x (m  a )  a

 g1 ( )
y (n  c)  c
E ( z )  ( A / B)( z )  g11 ( )
The result is not a triangular fuzzy number.