Fuzzy Measures and
Integrals
1.
2.
3.
4.
5.
Fuzzy Measure
Belief and Plausibility Measure
Possibility and Necessity Measure
Sugeno Measure
Fuzzy Integrals
Fuzzy Measures
• Fuzzy Set versus Fuzzy Measure
Fuzzy Set
Underlying
Set
Vague boundary
Fuzzy Measure
Crisp boundary
Vague boundary:
of fuzzy set
Probability
Representation
Membership value of an
element in X
Degree of evidence or
belief of an element that
belongs to A in X
Example
Set of large number
A degree of defection of
a tree
Degree of Evidence or
Belief of an object that is tree
Fuzzy Measure
• Axiomatic Definition of Fuzzy Measure
g : P( X )  [0,1]
Axiomg1 : (BoundaryCondition)g ()  0 and g ( X )  1
Axiomg2 : (Monotonic
ity)For every A, B  P( X ),
if A  B, then g ( A)  g ( B)
Axiomg3 : (Continuity) For everysequence of X ,
if either A1  A2  A3  ... or A1  A2  A3  ...
then limi  g ( Ai )  g (limi  Ai )
• Note:
A  B  A and A  B  B, thenmax(g ( A), g ( B))  g ( A  B)
A  B  A and A  B  B, thenmin(g ( A), g ( B))  g ( A  B)
Belief and Plausibility Measure
• Belief Measure
g : Bel( X )  [0,1]
(1) Bel is a fuzzy measure
(2) Bel( A1  A2  ... An )   Bel( Ai )   Bel( Ai  A j )  ...  (1) n 1 Bel( A1  A2  ... An )
i
i j
• Note:
Pr(A  B)  Pr(A)  Pr(B)  Pr(A  B) and
Pr(A  A )  Pr(A)  Pr(A )  Pr(A  A )  Pr(A)  Pr(A )  1
Bel( A  A )  1  Bel( A)  Bel( A )
• Interpretation:
Degree of evidence or certainty factor of an element in X that belongs to the
crisp set A, a particular question. Some of answers are correct, but we don’t
know because of the lack of evidence.
Belief and Plausibility Measure
• Properties of Belief Measure
1. A  B    Bel( A  B)  Bel( A)  Bel( B)
2. A  B  Bel( B)  Bel( A)
3. Bel( A)  Bel( A )  1
Note: A  It is a tree. A  It is not a tree.
Bel( A ) may be 0, when theinterestis given only a tree.
• Vacuous Belief: (Total Ignorance, No Evidence)
Bel( X )  1
Bel( A)  0 for all A  X
Belief and Plausibility Measure
• Plausibility Measure
g : Pl( X )  [0,1]
(1) Pl is a fuzzy measure
(2) Pl( A1  A2  ... An )   Pl( Ai )   Pl( Ai  A j )  ...  (1) n 1 Pl( A1  A2  ... An )
i
i j
• Other Definition
Pl( A)  1  Bel( A )
Pl( A )  1  Bel( A )  1  Bel( A) or Bel( A)  1  Pl( A )
• Properties of Plausibility Measure
1. Pl( A  A )  Pl()  0  Pl( A)  Pl( A )  Pl( A  A )
Since Pl( A  A )  1, Pl( A)  Pl( A )  1
2. 1  Pl( A)  Pl( A )  Pl( A)  (1  Bel( A))  Pl( A)  Bel( A)
How to calculate Belief
• Basic Probability Assignment (BPA)
m : P( X )  [0,1] such that
(1) m()  0
(2)
 m( A)  1
A X
• Note
1. m is not equal to probability mass.
2. not necessarily m( A)  m( B) even B  A
3. not necessarily m( X )  1
4. no relationship m( A) and m( A )
How to calculate Belief
• Calculation of Bel and Pl
Bel( A) 
 m(B)
B A
Pl( A) 
 m(B)
B A
• Simple Support Function is a BPA such that
In X pick a subset A for which
m( A)  s  0 and m( X )  1  s
• Bel from such Simple Support Function
s if A  C and C  X

Bel(C )   1
if C  X
0
if A  C

How to calculate Belief
• Bel from total ignorance
m( X )  1 and m( A)  0 for all A  X
Bel( X ) 
 m( B )  m( X )  1
Pl( A) 
B X
Bel( A) 
 m( B )  0
when A  X
B A
 m( B )  m( X )  1
A B  
Pl() 
 m( B )  0
 B 
• Body of Evidence
, m where
  a set of focalelementssuch thatm() is not zero.
m  theassigned BPA
How to calculate Belief
• Dempster’s rule to combine two bodies of evidence
Combine Bel1 from m1 and Bel2 from m2 :
 m ( A )m ( B )
m( A) 
Ai  B j  A
1
i
2
j
1 K
K
 m ( A )m ( B ) : Degree of Conflict
Ai  B j 
1
i
2
j
m()  0
• Example: Homogeneous Evidence
m1 ( A)  s1 m1 ( X )  1  s1
m2 ( A)  s2 m2 ( X )  1  s2
m( A)  {s1s2  s1 (1- s2 )  s2 (1  s1 )}/ 1  K ( K  0)
m( X )  (1  s1 )(1  s2 )
X
A X
XX
A
A A
A X
A
X
How to calculate Belief
• Example: Heterogeneous Evidence
Bel1 focused on A
Bel2 focused on B and assume A  B  
X
A X
XX
m1 ( A)  s1 m1 ( X )  1  s1
B
A B
B X
m2 ( B)  s2 m2 ( X )  1  s2
m( A)  (1- s2 )s1
A
m( B)  s2 (1- s1 ) m( A  B)  s2 s1 ( K  0)
m( X )  (1  s1 )(1  s2 )
0 if A  B  C
s s if A  B  C but A  C and B  C
1 2
s1 if A  C but B  C
Bel(C )  
s2 if A  C but B  C
1  (1  s1 )(1  s2 ) if A  B  C

1 if C  X
X
How to calculate Belief
• Example: Heterogeneous Evidence
Bel1 focused on A
Bel2 focused on B and assume A  B
Bel( A)  m( A)  m( A  B)  s1 (1  s2 )  s1s2  s1
Bel( B)  s1s2  s1 (1  s2 )  s2 (1  s1 )  s2  s1 (1  s2 ) increasing!
• Example: Heterogeneous Evidence
Bel1 focused on A
Bel2 focused on B and assume A  B  
 m ( A)m ( B)  m ( A)m ( B)  s s
A B  
1
m( A) 
2
s1 (1  s2 )
1  s1s2
1
2
m( B ) 
1 2
s2 (1  s1 )
1  s1s2
Probability Measure
• Theorem: The followings are equivalent.
1. Bel is Baysian
Bel( A  B)  Bel( A)  Bel( B)  Bel( A  B)
2. All of Bel' s focalelementsare singleton
m( A)  0 if A  1
3. Bel  Pl
4. Bel( A)  Bel( A )  1
Joint and Marginal BoE
• Marginal BPA
m : P( X  Y )  [0,1]
R is a set of focalelementsof m i.e. m( R)  0
Let RX be theprojectionof R onto X (same for RY ) :
RX  {x  X | ( x, y )  R for some y  Y }
maginalB.P .A.
m X ( A) 
 m( R) for all A  P( X )
R:R X  A
m X and mY are non - interactive iff for all A  P( X ), B  P(Y )
m( A  B)  m X ( A)  mY ( B) and m( R)  0 if R  A  B
• Example 7.2
Possibility and Necessity Measure
• Consonant Bel and Pl Measure
If focalelementsare nested,
then theBel and Pl measuresare called consonant.
m( A)
m(B )
T heorem: Let (,m) be a consonantbody of evidence.T hen
Bel( A  B)  min{Bel( A), Bel( B)}
and
Pl( A  B)  max{Pl( A), Pl( B)}.
Possibility and Necessity Measure
• Necessity and Possibility Measure
– Consonant Body of Evidence
• Belief Measure -> Necessity Measure
• Plausibility Measure -> Possibility Measure
– Extreme case of fuzzy measure
Nec( A  B)  min[Nec( A), Nec( B)]
Pos( A  B)  max[Pos( A), Pos( B)]
cf ) g ( A  B)  min[g ( A), g ( B)]
g ( A  B)  max[g ( A), g ( B)]
– Note: 1. Nec( A)  Nec( A )  1 Pos( A)  Pos( A )  1
Nec( A)  1  Pos( A )
2. min[Nec( A), Nec( A )]  Nec( A  A )  Nec()  0
max[Pos( A), Pos( A )]  Pos( A  A )  Pos( X )  1
Possibility and Necessity Measure
• Possibility Distribution
T heorem: Everypossibility measurecan be uniquely defined by
a possibility distribution, r : X  [0,1], via theformula
Pos( A)  max{r ( x)}
xA
Nec( A)  1  Pos( A )
For X  {x1 , x2 ,...xn }, suppose r  {1 ,  2 ,..., n }
be thepossibility distribution such that i   j if i  j.
Assume that Pos defined by m, theBP A.
Assume A1  A2  ...  An ( X ), where Ai  {x1 , x2 ,..xi }.
n
T hatis m( A)  0 for all A  Ai and  m( Ai )  1.
i 1
Everypossibility measurecan be uniquely characterized by
n - tuple
m  {1 ,  2 ,.. n }, where i  m( Ai ).
m is called a basic distribution.
Possibility and Necessity Measure
• Basic Distribution and Possibility Distribution
i  r ( xi )  Pos({xi })  Pl({xi }) for all xi  X
n
n
k i
k i
i  Pl({xi })   m( Ak )  k
or
i  i  i 1 ,  n1  0.
• Ex.
r  (1,1,0.7,0.3,0.3,0.3,0.3,0.2)
m  (0,0.3,0.4,0.0,0.0,0.0,0.1)
r  (1,0,0,0,...,0)
m  (1,0,0,0,...,0) : thesmallest possibility distribution  theanswer is specific xi
r  (1,1,1,1,...,1)
m  (0,0,0,0,...,1) : totalignorancem( X )  1 and m( A)  0 for all A  X
Fuzzy Set and Possibility
• Interpretation
– Degree of Compatibility of v with the concept F
– Degree of Possibility when V=v of the proposition p: V is F
rF (v)  F (v)
• Possibility Measure
PosF ( A)  sup rF (v) or NecF ( A)  1  PosF ( A)
vA
• Example
rF  0.33 / 19  0.67 / 20  1.0 / 21 0.67 / 22  0.33 / 23
A1  {21}, A2  {20,21,22}, A1  {19,20,21,22,23}
Pos( A1 )  Pos( A2 )  Pos( A3 )  1
Pos( A1 )  0.67, Pos( A2 )  0.33, Pos( A3 )  0.
Nec( A1 )  0.33, Pos( A2 )  0.67, Pos( A3 )  1.
Summary
Fuzzy
Measure
Plausibility
Measure
Probability
Measure
Belief
Measure
Necessity
Measure
Possibility
Measure
Sugeno Fuzzy Measure
• Sugeno’s g-lamda measure
g is a fuzzy measuresatisfyingthefollowingcondition.
For all A, B  P( X ) with A  B  ,
g ( A  B)  g(A)  g(B)  g(A)  g(B) for some   1.
T heng is calledSugeno measureor g   m easure.
• Note:
1.   0  g 0 is a probability measure.
2. g ( A  A )  g ( A)  g ( A )  g ( A) g ( A )  1
 g ( A)  g ( A )  1  g ( A) g ( A )
 If   0, then g ( A)  g ( A )  1  Belief Measure
 If   0, then g ( A)  g ( A )  1  Plausibility Measure
g ( A)  g ( B) - g ( A  B)  g ( A) g ( B)
3. g ( A  B) 
1  g ( A  B )
Sugeno Fuzzy Measure
• Fuzzy Density Function
X  {x1 , x2 ,..., xn }
g i : g ({xi }) is called fuzzy densit y funct ion.
Not e:
A  B  , B  C  , A  C   
g ( A  B  C )  g ( A)  g ( B )  g (C ) 
 ( g ( A) g ( B)  g ( B ) g (C )  g ( A) g (C ))  2 g ( A) g ( B) g (C )
or
g ({x1 , x2 , x3 })  g 1  g 2  g 3   ( g 1 g 2  g 2 g 3  g 1 g 3 )  2 g 1 g 2 g 3
In general, X  {x1 , x2 ,..., xn }
n 1
n
g( X )   g  
j
j 1
n
g
j 1 k  j 1
j
g k 2   ...n 1 g 1 g 2 ...g n


   (1  g i )  1 /   1
 xi X

Sugeno Fuzzy Measure
• How to construct Sugeno measure from fuzzy density
T heorem: T hefollowingequation has a unique solutionin (-1,)


g ( X )    (1  g i )  1 /   1
 xi X

Colloary: If {g 1 , g 2 ,...g n } is given, thenone can constructa
corresponding g   m easure.
{g 1 , g 2 ,...g n }   calculation from equation
 constructa g   m easure.
Fuzzy Integral
• Sugeno Integral
Definition: T heSugeno integralof a functionh : X  [0,1] w.r.t.g() is
  g ( F ),
 h( x)  g  sup

[ 0 ,1]
where F  x | h( x)   .
h(x)
1  F 
 2  F 
1

 Find t hemaximum

 n  F n 
2
.
F
X
Fuzzy Integral
• Algorithm of Sugeno Integral
Reorder X  {x1 , x2 ,..., xn } so that
h( x1 )  h( x2 )  ...  h( xn ).
T hen
 h(x)  g  h( x )  g ( X )
i
i
X
where X i  {x1 , x2 ,..., xi }, and recursively given
g  ( X 1 )  g  ({x1})  g 1
g  ( X i )  g  ( X i 1  {xi })  g  ( X i 1 )  g i  g  ( X i 1 ) g i
Fuzzy Integral
• Choquet Integral
Definition: T heChoquet integralof a function f : X  [0,1] w.r.t.g() is
n
Cg ( f ( x1 ),...f ( xn ))   ( f ( xi )  f ( xi 1 )) g ( X i ),
i 1
and f ( x0 )  0, where0  f ( x1 )  f ( x2 )  ...  f ( xn )  1
and X i  {xi , xi 1 ,....,xn }.
• Interpretation of Fuzzy Integrals in Multi-criteria
Decision Making
Set of Criteria x1 , x2 ,...xn 


Degree of Importance g 1 , g 2 ,...g n
ObjectiveMeasurement  h( x1 ), h( x2 ),...,h( xn )
Sugeno Fuzzy Integral T otalDegree of Satisfaction