Fuzzy Measures &
Measures of Fuzziness
Presented by : Armin
Contents
 Introduction
 Measure & Integral
 Fuzzy Measure
 Fuzzy Integral ( Choquet & Sugeno)
 Measure of fuzziness
Fuzzy measures & measures of Fuzziness
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Introduction
The measure is one of the most important concepts in
mathematics and so is the integral with respect to the
measure. They have many applications in engineering,
and their main characteristic is the additivity. This
characteristic is very effective and convenient, but
often too inflexible or too rigid. As a solution to the
rigidness problem the fuzzy measure was proposed.
Fuzzy measures & measures of Fuzziness
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Measure
 Set functions :
A function defined on a family of sets is called a set function.
 Additive :
 Monotone :
 ( AUB)   ( A)   ( B)
if A  B :  ( A)   ( B)
 Normalized :
min{ ( A) | A  X }  0 and max{ ( A) | A  X }  1
Fuzzy measures & measures of Fuzziness
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Measure
 Measure
 A measure on X :
a non-negative additive set function defined on 2x
 A normalized measure : Probability measure
 An additive set function : Signed measure
Fuzzy measures & measures of Fuzziness
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Measures

A measure measures the size of sets.
 Counting
measure mc : |A|
 Probability : tossing a die
 δ x0 : Dirac measure on X focused on x0

Null set
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Measures & Integral
f
This is a set! and the
integral is measuring
the “size” of this set!
Integrals are like
measures! They
measure the size of a
set. We just describe
that set by a function.
Therefore, integrals
should satisfy the
properties of
measures.
A
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Measures & Integral
 Integral
:

 Integral
fdm 
 f ( x).m({x})
xX
over A :
 fdm   f 1
A
dm   fdmA
A
X: finite set, m : signed measure on X, f : function on X
Fuzzy measures & measures of Fuzziness
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Measures & Integral
Fuzzy measures & measures of Fuzziness
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Measures & Integral
Fuzzy measures & measures of Fuzziness
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Measures & Integral
Fuzzy measures & measures of Fuzziness
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Fuzzy Measures [Sugeno 1977]
Fuzzy measure is a set function g defined on  that:
1. g(∅) = 0, g(X) = 1.
2. If A, B   and A  B then g(A)  g(B).
3. If An   and A1  A2  ... Then
lim g ( An )  g (lim An )
n 
Fuzzy measures & measures of Fuzziness
n 
12
Fuzzy measure
Fuzzy measure is:
•Non-additive
•Non-monotonic
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Fuzzy measure
Monotone and non-additive
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Fuzzy measure

General Discussion
 Fuzzy
set: a value is assigned to each element of
the universal set signifying its degree of
membership in a particular set with unsharp
boundaries.
 Fuzzy measure: assign a value to each crisp set
signifying the degree of evidence or belief that a
particular element belongs in the set.
g:
- -  0,1
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Fuzzy Measures

Fuzzy Set versus Fuzzy Measure
Fuzzy Set
Underlying
Set
Representation
Example
Fuzzy Measure
Vague boundary
Crisp boundary
Vague boundary: Probability of
fuzzy set
Membership value of an
element in X
Degree of evidence or
belief of an element that belongs to
A in X
Set of large number
Degree of Evidence or
A degree of defection of a tree Belief of an object that is tree
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Fuzzy integral (Choquet)

An extension of the ordinary integral
C 

n
fd    (ai  ai 1 ). ({x | f ( x)  ai })
i 1
Each worker xi works f (xi ) hours a day.

{
{
{
f (xi ) = f (x1) + [f (x2 )- f (x1 )] + … +[f (xi )- f (xi-1 )]
X
X\{x1}
Xn
Group A produces the amount μ(A) in one hour
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Fuzzy integral (Choquet)
a4
a3
0
a2
a1
a0  0
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Fuzzy integral (Sugeno)



Is defined only for normalized fuzzy measures
μ : a normalized fuzzy measure
f : a function with range {a1, a2, a3,…an}
∫ f o μ = Max{Min{a , μ({x | f(x) >= a })}}
i

i
If μ is a 0-1 fuzzy measure
∫ f o μ = (c) ∫ f dμ
min f(x) <= ∫ f o μ <= max f(x)
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Fuzzy integral (Sugeno)
μ
: a normalized fuzzy measure
 f : a function with range {a1, a2, a3,…an}
 f ( x)d    ( x | f (x)   ).d
    max
    min
Fuzzy measures & measures of Fuzziness
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Sugeno Integral & Choquet Integral
 Choquet
Integral:
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Sugeno Integral & Choquet Integral
 Sugeno
Integral :
Student A
18
16
i=1 : min(0.5 , 1) = 0.5
i=2 : min(0.8,0.5) = 0.5
i=3 : min(0.9 , 0.45) = 0.45
10
Lit
Phys
Math
Max(0.5 , 0.5 , 0.45) = 0.5
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Sugeno Integral & Choquet Integral
 Sugeno
Integral for aggregation :
 Choquet
Integral for aggregation :
Fuzzy measures & measures of Fuzziness
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Measures of fuzziness

Indicates the degree of fuzziness of a fuzzy set
 De
Luca and Termini :
based on entropy of fuzzy set and Shannon function
 Higashi
and Klir + Yager:
based on the degree of distinction between the fuzzy set
and its complement
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Measures of fuzziness
The measure of fuzziness d(A) :

if A is a crisp set
d(A) = 0

if μ(x) = ½ for every x
if B is crisper than A
if  is complement of A
d(A) = the maximum value
d(A) ≥ d(B)
d(Â) = d(A)


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Measures of fuzziness[De Luca & Termini 1972]

The entropy as a measure of fuzzy set A={(x, μ(x))}
is defined as :
d  A  H  A  H A 
n
H ( A)   K   A xi  ln A xi 
i 1

“μ(A)” is the membership function of the fuzzy set A

n is the number of element in support of A and K is a positive constant
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Measures of fuzziness[De Luca & Termini 1972]

Example:

Let A=“ Integers close to 10”
A={(7, .1), (8, .5), (9, .8), (10, 1), (11, .8), (12, .5), (13, .1)}

Let k=1 , so
d(A) = .325 + .693 + .501 + 0 + .501 + .693 + .325 = 3.038

Let B=“ Integers quite close to 10”
B={(6, .1), (7, .3), (8, .4), (9, .7), (10, 1), (11, .8), (12, .5), (13, .3), (14, .1)}

Let k=1 , so
d(A) = .325 + .611 + .673 + .611 + 0 + .501 + .693 + .611 + .325 = 4.35
A is crisper than B .
Fuzzy measures & measures of Fuzziness
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Measures of fuzziness [Yager 1979]

Requirement of distinction between A and its
complement is not satisfied by fuzzy sets :
A A  X
A A  
 So
,any measure of fuzziness should be measure
of this lack
the measure is based on the distance
between a fuzzy set and its complement
Fuzzy measures & measures of Fuzziness
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Measures of fuzziness [Yager 1979]





Distance :
D p A, A   [ |  A  xi    A  xi  | ]1/ p
Let S=supp(A) :
Dp S , S  S
p
 
A measure of fuzziness :
1/ p
f p  A  1 
Dp A, A 
|| sup( A) ||
p=1 : Hamming metric
D1 A, A   | 2 A xi  1 |
p=2 : Euclidean metric
D2 A, A  
| 2 x  1 | 
2 1/ 2
A
Fuzzy measures & measures of Fuzziness
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Measures of fuzziness [Yager 1979]

Example (cont. slide 13):
 Let
A=“ Integers close to 10”
 Let B=“ Integers quite close to 10”
 For p=1:
D1 A, A   3.8
 
D1 B, B  4.6
 For
supp(A)  7
supp(B)  9
f1  A  1 
3 .8
 0.457
7
4.6
f1 B   1 
 0.489
9
p=2:
D2 A, A   1.73
 
D2 B, B  1.78
1
2
supp(A)  2.65
1
2
supp(B)  3
Fuzzy measures & measures of Fuzziness
f 2  A  1 
1.73
 0.347
2.65
1.78
f 2 B   1 
 0.407
3
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Reference

“Fuzzy Measures and Fuzzy Integrals”, Toshiaki Murofushi
and Michio Sugeno.

“Fuzzy measures and integrals”, Radko Mesiar.

“Fuzzy set Theory and Its Applications”, H.J.Zimmermann.
“The Evolution of the Concept of Fuzzy Measure”, Luis
Garmendia

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