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 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