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Fuzzy Measures and Integrals 1. Fuzzy Measure 2. Belief and Plausibility Measure 3. Possibility and Necessity Measure 4. Sugeno Measure 5. Fuzzy Integrals
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 X
Degree of evidence or belief of an element that belongs to A in X
Example Set of large numberA degree of defection of a tree
Degree of Evidence or Belief of an object that is tree
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
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
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
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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
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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
Probability Measure
• Theorem: The followings are equivalent.
1)()( .4
.3
1 if 0)(
singleton are elements focal ' of All .2
)()()()(
Baysian is 1.
ABelABel
PlBel
AAm
s Bel
BABelBBelABelBABel
Bel
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
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nn
Am
AmAAAm
xxxAXAAA
mPos
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xxxX
Possibility and Necessity Measure
• Basic Distribution and Possibility Distribution
• Ex.
.0 ,
or
)( })({
allfor })({})({)(
11
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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
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21
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2
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jk
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j
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n
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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
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n
i
n
iiing
xxxX
xfxfxfxf
Xgxfxfxfxf C
Xf
onSatisfacti of Degree Total IntegralFuzzy Sugeno
)(),...,(),( t Measuremen Objective
,..., Importance of Degree
,..., Criteria ofSet
21
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xhxhxh
ggg
xxx
