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Fuzzy Measures and Integrals
1. Fuzzy Measure2. Belief and Plausibility Measure3. Possibility and Necessity Measure4. Sugeno Measure5. Fuzzy Integrals
Fuzzy Measures
• Fuzzy Set versus Fuzzy Measure
Fuzzy Set Fuzzy Measure
Underlying Set
Vague boundary Crisp boundary Vague boundary: Probability of fuzzy set
Representation
Membership value of an element in A
Degree of evidence or belief of an element that belongs to A in X
Example Set of large number Degree of Evidence or Belief of an object that is tree
Uncertainty vagueness: fuzzy sets ambiguity: fuzzy measures
Vagueness: associated with the difficulty of making sharp or precise distinctions in the world.
Ambiguity: associated with one-to-many relations, i.e. difficult to make a choice between two or more alternatives.
Types of Uncertainty
Fuzzy Measure vs. Fuzzy Set
Ex) Criminal trial: The jury members are uncertain about the guilt or innocence of the defendant.– Two crisp set:
1) the set of people who are guilty of the crime 2) the set of innocent people
– The concern: - Not with the degree to which the defendant is guilty. - The degree to which the evidence proves his/her membership in
either he crisp set of guilty people or in the crisp set of innocent people.
- Our evidence is rarely, if ever, perfect, and some uncertainty usually prevails.
– Fuzzy measure: to represent this type of uncertainty - Assign a value to each possible crisp set to which the element in
question might belong, signifying the degree of evidence or belief that a particular element belongs in the set.
- The degree of evidence, or certainty of the element’s membership in the set
Fuzzy Measure
• Axiomatic Definition of Fuzzy Measure
• Note:
)(lim)(limen th
...or ...either if
, of sequenceevery For y)(Continuit:g3 Axiom
)()( then , if
),(,every For ity)(Monotonic:g2 Axiom
1)( and 0)( Condition)(Boundary :g1 Axiom
]1,0[)(:
321321
iiii AgAg
AAAAAA
X
BgAgBA
XPBA
Xgg
XPg
)())(),(min( then , andA
)())(),(max( then , andA
BAgBgAgBBABA
BAgBgAgBBABA
1,0)( : xXxFuzzy Set A
set. universal : set, crisp : ere wh
]1,0[)( :
XA
XA
AgureFuzzy Meas
i
i
i
Note that
1,0:
1,0:
XPg
xA
where P(X) is a power set of X.
Belief and Plausibility Measure
• Belief Measure
• Note:
• 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.
)...()1(...)()()...( (2)
measurefuzzy a is )1(
]1,0[)(:
211
21 nn
jji
ii
in AAABelAABelABelAAABel
Bel
XBelg
)()(1)(
1)Pr()Pr()Pr()Pr()Pr()Pr(
and )Pr()Pr()Pr()Pr(
ABelABelAABel
AAAAAAAA
BABABA
Belief and Plausibility Measure
• Properties of Belief Measure
• Vacuous Belief: (Total Ignorance, No Evidence)
tree.aonly given isinterest when the0, bemay )(
tree.anot isIt tree.a isIt :Note
1)()( .3
)()( .2
)()()( 1.
ABel
AA
ABelABel
ABelBBelBA
BBelABelBABelBA
XAABel
XBel
allfor 0)(
1)(
Belief and Plausibility Measure
• Plausibility Measure
• Other Definition
• Properties of Plausibility Measure
)...()1(...)()()...( (2)
measurefuzzy a is )1(
]1,0[)(:
211
21 nn
jji
ii
in AAAPlAAPlAPlAAAPl
Pl
XPlg
)(1)(or )(1)(1)(
)(1)(
APlABelABelABelAPl
ABelAPl
)()())(1()()()(1 .2
1)()( ,1)( Since
)()()(0)()( .1
ABelAPlABelAPlAPlAPl
APlAPlAAPl
AAPlAPlAPlPlAAPl
How to calculate Belief
• Basic Probability Assignment (BPA)
• Note
1)( )2(
0)( (1)
such that ]1,0[)(:
XA
Am
m
XPm
)( and )( iprelationsh no .4
1)(y necessarilnot .3
even )( )(y necessarilnot 2.
mass.y probabilit toequalnot is 1.
AmAm
Xm
ABBmAm
m
How to calculate Belief
• Calculation of Bel and Pl
• Simple Support Function is a BPA such that
• Bel from such Simple Support Function
ABAB
BmAPlBmABel )()( )()(
sXmsAm
AX
1)( and 0)(
for which subset apick In
if 0
C if 1
and if
)(
CA
X
XCCAs
CBel
How to calculate Belief
• Bel from total ignorance
• Body of Evidence
BPA assigned the
zero.not is )(such that elements focal ofset a
where,
m
m
m
0)()( when 0)()(
1)()()( 1)()()(
allfor 0)( and 1)(
BAB
BAXB
BmPlXABmABel
XmBmAPlXmBmXBel
XAAmXm
Ex) Let the universal set X denote the set of all possible diseases
P: pneumonia, B: bronchitis, E: emphysema기관지염 기종
m Bel
P 0.05 0.05
B 0 0
E 0.05 0.05
P U B 0.15 0.2
P U E 0.1 0.2
B U E 0.05 0.1
P U B U E 0.6 1
)(
1)(XPA
Am
AB
BmABel )()(
where B: all the possible subset of A
) (
conditionboundary : 1)(
EBPX
XBel
Robot Intelligence Technology Lab.
95.06.005.01.0005.015.0
)()()()()()()(
2.0005.015.0)()()()(
05.0)()(subsets possible theall
EBPmEBmEPmBmPmBPmBPPl
BmPmBPmBPBel
PmPBel
Given Bel(·), find m(·)
where |A-B| is the size of (A-B), size: cardinality of crisp set (A-B)
Ex)
)( )()1()( || XPABBelAm BA
AB
15.005.002.0
)()1(
)()1()()1()(
) 0|||| ( 05.0)()1(
)()1()(
|}{|
|}{|||
0
||
PBel
BBelBPBelBPm
PPPBel
PBelPm
B
P
How to calculate Belief
• Dempster’s rule to combine two bodies of evidence
• Example: Homogeneous Evidence
0)(
Conflict of Degree :)()( 1
)()(
)(
: from and from Combine
21
21
2211
m
BmAmKK
BmAm
Am
mBelmBel
jBA
i
jABA
i
ji
ji
2222
1111
1)( )(
1)( )(
sXmsAm
sXmsAm
A X
A
X
AA
XA
XA
XX
)1)(1()(
)0( 1/)}1()-(1{)(
21
122121
ssXm
KKssssssAm
How to calculate Belief
• Example: Heterogeneous Evidence
2222
1111
1)( )(
1)( )(
sXmsBm
sXmsAm
A X
B
X
BA
XA
XB
XX
)1)(1()(
)0( )( ) -(1)( )-(1)(
21
121212
ssXm
KssBAmssBmssAm
BABBel
ABel
assume and on focused
on focused
2
1
XC if 1
if )1)(1(1
but if
but if
and but if
if 0
)(
21
2
1
21
CBAss
CBCAs
CBCAs
CBCACBAss
CBA
CBel
How to calculate Belief
• Example: Heterogeneous Evidence
• Example: Heterogeneous Evidence
BABBel
ABel
assume and on focused
on focused
2
1
!increasing )1()1()1()(
)1()()()(
212122121
12121
sssssssssBBel
sssssBAmAmABel
21
12
21
21
212121
2
1
1
)1()(
1
)1()(
)()()()(
assume and on focused
on focused
ss
ssBm
ss
ssAm
ssBmAmBmAm
BABBel
ABel
BA
Joint and Marginal BoE
• Marginal BPA
• Example 7.2
BARRmBmAmBAm
YPBXPAmm
XPARmAm
YyRyxXxR
RXRR
RmmR
YXPm
YX
YX
ARRX
X
YX
X
if 0)( and )()()(
)(),( allfor iff einteractiv-non are and
)( allfor )()(
B.P.A. maginal
} somefor ),(|{
: )for same( onto of projection thebe Let
0)( i.e. of elements focal ofset a is
]1,0[)(:
:
Possibility and Necessity Measure
• Consonant Bel and Pl Measure
consonant. called are measures and then the
nested, are elements focal If
PlBel
)(Am )(Bm
)}.(),(max{)(
and
)}(),(min{)(
Then evidence. ofbody consonant a be )(Let :Theorem
BPlAPlBAPl
BBelABelBABel
,m
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
– Note:
)](),(max[)(
)](),(min[)( )
)](),(max[)(
)](),(min[)(
BgAgBAg
BgAgBAgcf
BPosAPosBAPos
BNecANecBANec
1)()()](),(max[
0)()()](),(min[ .2
)(1)(
1)()( 1)()( .1
XPosAAPosAPosAPos
NecAANecANecANec
APosANec
APosAPosANecANec
Possibility and Necessity Measure
• Possibility Distribution
)(1)( )}({max)(
formula the via],1,0[: on,distributiy possibilit a
by defineduniquely becan measurey possibilitEvery :Theorem
APosANecxrAPos
Xr
Ax
on.distributi basic a called is
).( where},,..,{
tuple-n
by zedcharacteriuniquely becan measurey possibilitEvery
.1)( and allfor 0)( isThat
.},..,{ where,)(... Assume
BPA. the,by defined that Assume
. if such that on distributiy possibilit thebe
},...,,{ suppose },,...,{For
21
1
2121
2121
m
m
r
iin
n
iii
iin
ji
nn
Am
AmAAAm
xxxAXAAA
mPos
ji
xxxX
Possibility and Necessity Measure
• Basic Distribution and Possibility Distribution
• Ex.
.0 ,
or
)( })({
allfor })({})({)(
11
niii
n
ikk
n
ikkii
iiiii
AmxPl
XxxPlxPosxr
XA AmXm
xi
allfor 0)( and 1)( ignorance total:)1,...,0,0,0,0(
)1,...,1,1,1,1(
specific isanswer theondistributiy possibilitsmallest the:)0,...,0,0,0,1(
)0,...,0,0,0,1(
)1.0,0.0,0.0,0.0,4.0,3.0,0(
)2.0,3.0,3.0,3.0,3.0,7.0,1,1(
m
r
m
r
m
r
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
• Possibility Measure
• Example
)()( vFvrF
)(1)(or )()( sup APosANecvrAPos FFF
Av
F
.1)(,67.0)( ,33.0)(
.0)(,33.0)( ,67.0)(
1)()()(
}23,22,21,20,19{ },22,21,20{ },21{
23/33.022/67.021/0.120/67.019/33.0
321
321
321
121
APosAPosANec
APosAPosAPos
APosAPosAPos
AAA
rF
Summary
Fuzzy Measure
Plausibility Measure
Belief Measure
Probability Measure
Possibility Measure
Necessity Measure
Sugeno Fuzzy Measure
• Sugeno’s g-lamda measure
• Note:
.or measure Sugeno called is gThen
.1 somefor g(B)g(A)g(B)g(A))(
, with )( allFor
condition. following thesatisfying measurefuzzy a is
measureg
BAg
BAXPA, B
g
)(1
)()()(-)()()( 3.
Measurety Plausibili 1)()( then ,0 If
Measure Belief 1)()( then ,0 If
)()(1)()(
1)()()()()( 2.
measure.y probabilit a is 0 .1 0
BAg
BgAgBAgBgAgBAg
AgAg
AgAg
AgAgAgAg
AgAgAgAgAAg
g
Sugeno Fuzzy Measure
• Fuzzy Density Function
1/1)1(
......)(
},...,,{ general,In
)(}),,({
or
)()()( ))()()()()()((
)()()()(
, ,
:Note
function.density fuzzy called is })({:
},...,,{
21121
1 11
21
3212313221321321
2
21
Xx
i
nnn
j
n
jk
kjn
j
j
n
ii
n
i
g
ggggggXg
xxxX
ggggggggggggxxxg
CgBgAgCgAgCgBgBgAg
CgBgAgCBAg
CACBBA
xgg
xxxX
Sugeno Fuzzy Measure
• How to construct Sugeno measure from fuzzy density
. aconstruct
equation fromn calculatio },...,{
. ingcorrespond
aconstruct can one then given, is },...,{ If :Colloary
1/1)1()(
) (-1,in solution unique a hasequation following The :Theorem
21
21
measureg
ggg
measureg
ggg
gXg
n
n
Xx
i
i
Fuzzy Integral
• Sugeno Integral
.)(| where
,)()(
is )g( w.r.t.]1,0[:function a of integral Sugeno The :Definition
sup]1,0[
xhxF
Fggxh
Xh
X
)(xh
F
maximum theFind .
2
1
2
1
nF
F
F
n
Fuzzy Integral
• Algorithm of Sugeno Integral
ii
iiiii
ii
ii
X
n
n
gXggXgxXgXg
gxgXg
xxxX
Xgxhgxh
xhxhxh
xxxX
)()(}){()(
})({)(
giveny recursivel and },,...,,{ where
)()()(
Then
).(...)()(
thatso },...,,{Reorder
111
111
21
21
21
Fuzzy Integral
• Choquet Integral
• Interpretation of Fuzzy Integrals in Multi-criteria Decision Making
}.,....,,{ and
1)( ...)()(0 where0,)( and
),())()(())(),...((
is )g( w.r.t.]1,0[:function a of integralChoquet The :Definition
1
210
111
niii
n
i
n
iiing
xxxX
xfxfxfxf
Xgxfxfxfxf C
Xf
onSatisfacti of Degree Total IntegralFuzzy Sugeno
)(),...,(),( t Measuremen Objective
,..., Importance of Degree
,..., Criteria ofSet
21
21
21
n
n
n
xhxhxh
ggg
xxx
: evaluation value of attribute of the jth house,
where
Then, let the fuzzy measure:
where g: degree of consideration (importance) of attributes
in the evaluation process.
Ex) Evaluation of the desirability of houses
Let , where = price, = size, = facilities,
=location and = living environment, and evaluation function:
where m: # of houses, and
},,,,{ 54321 xxxxxX 1x 2x 3x
4x 5x
)( ij xh ix. 1)(0 ij xh
]1,0[)(: XPg
mjXh j ,,2,1 ],1,0[:
The desirability of the jth houses:
General linear evaluation model:
- Performs well only when the attributes of evaluation are independent and the measures of evaluation are independent.
)](),(min[max 5,,1 iiji
X
jj Hgxhghe
5
1
)(i
ijij xhwe
- Practically, price ( ) and size ( ) are not independent.
- Even if and are independent, the degree of consideration might not be independent, i.e.,
Additivity might not be true for measures.- F.I. Models are more general than the linear models.- Problem about Fuzzy Integral Evaluation Model
① How to find out the necessary attributes for evaluation.
② How to identify the fuzzy measure.
1x 2x
1x 2x
}).({})({}),({ 2121 xgxgxxg