credibility theory baoding liu uncertainty theory laboratory department of mathematical sciences...

Credibility Theory

Baoding LiuUncertainty Theory Laboratory

Department of Mathematical Sciences

Tsinghua University

It is a new branch of mathematics that studies the behavior of fuzzy phenomena.

Baoding Liu Tsinghua University

http://orsc.edu.cn/~liu

Uncertainty Theory & Uncertain Programming U T L A BFashion of Mathematics

2300 Years Ago: Euclid: “Elements”, First Axiomatic System

1899: Hilbert: Independence, Consistency, Completeness

1931: K. Godel: Incompleteness Theorem

1933: Kolmogoroff: Probability Theory

2004: B. Liu: Credibility Theory



Uncertainty Theory & Uncertain Programming U T L A B

Why I do not possibility measure?

(a) Possibility is not self-dual, i.e., Pos{ }+Pos{ } 1.

(b) I will spend "about $300": (200,300,400).

In order to cover my expenses with maximum chance,

how much needed?

Pos{ } 1

cA A

x x

A self-dual measure is absolutely ne

300

eded!




Five Axioms

1 2

Axiom 1. Cr{ }=1.

Axiom 2. Cr{ } Cr{ } whenever .

Axiom 3. Cr is self-dual, i.e., Cr{ } Cr{ } 1.

Axiom 4. Cr 0.5=sup Cr{ } if Cr{ } 0.5 for each .

Axiom 5. For each ( ), we have

Cr{

c

i i i i i

n

A B A B

A A

A A A i

A P

A

1 1

11

1 1( , , ) ( , , )

1 1( , , )( , , )

sup min Cr { }, if sup min Cr { } 0.5

}1 sup min Cr { } 0.5, if sup min Cr { } 0.5

Independence (Yes) Consistency (?) Completeness

.

(Ab

n n

cnn

k k k kk n k nA A

k k k kk n k nAA

solutely No)




Credibility Subadditivity Theorem

A credibility measure

Liu (UT, 2004)

Credibil

is additive if and onl

ity measure is subadditive, i.e.,

Cr{ } Cr{

y if

there are at most two elements in un

}+

iversal set.

Cr{ }.A B A B




Credibility Semicontinuity Laws

1 2

Liu (UT, 2004)

Theorem: Let ( , ( ),Cr) be a credibility space,

and , , ( ). Then

lim Cr{ } Cr{ }

if one of the following conditions is satisfied:

(a) Cr{ } 0.5 and ; (b) lim Cr{ } 0.5 an

ii

i ii

P

A A P

A A

A A A A

d ;

(c) Cr{ } 0.5 and ; (d) lim Cr{ } 0.5 and .

i

i i ii

A A

A A A A A A




Credibility Extension Theorem

*

* *

Li and Liu (2005)

If Cr{ } satisfies the credibility extension condition,

supCr{ } 0.5,

Cr{ }+supCr{ } 1 if Cr{ } 0.5,

then Cr{ } has a unique extension to a credib

ility

measure on P( ),

supCr{ }, if supCr{ } 0.5Cr{A}=

1 supCr{ } 0.5, if supCr{ } 0.5.c

A A

AA




Fuzzy Variable

A fuzzy variable is a function from a credibility

space ( ,P( ),Cr) to the set of real nu

Membership functi

mbers.

( ) 2Cr 1.

A function : [0,1]

i

on:

:

Definition :

Sufficient and Necessary Condit

x

io

x

n

s a membership function iff sup ( ) 1.x




Credibility Measure by Membership Function

Let be a fuzzy variable with membership

function . Then

1Cr{ } sup ( ) 1 sup ( ) .

2 cx A x A

A x x

Liu and Liu (IEEE TFS, 2002)




Independent Fuzzy Variables

1 2

11

1 2

Zadeh (1978), Nahmias (1978), Yager (1992), Liu (2004)

Liu and Gao (2005)

The fuzzy variables , , , are independent if

Cr { } min Cr{ }

for any sets , , , of .

m

m

i i i ii m

i

m

B B

B B B




Theorem: Extension Principle of Zadeh

1 2

1 2

1 2

1 2

1( , , , )

Let , , , be independent fuzzy variables with

membership functions , , , , respectively.

Then the membership function of ( , , , ) is

( ) sup min

It is only a

( ).

pplic

n

n

n

n

i ii nx f x x x

f

x x

able to independent fuzzy variables.

It is treated as a theorem, not a postulate.




Expected Value

0

0

Liu and Liu (IEEE TFS, 2002)

Let be a fuzzy variable. Then the expected value of

is defined by

E[ ] Cr{ }d Cr{ }d

provided that at least one of the two integrals is finite.

Yage

r r r r

r (1981, 2002): discrete fuzzy variable

Dubois and Prade (1987): continuous fuzzy variable




Why the Definition Reasonable?

0

0

(i) Since credibility is self-dual, the expected value

E[ ] Cr{ }d Cr{ }d

is a type of Choquet integral.

(ii) It has an identical form with random case,

E[ ] Pr{

r r r r

r

0

0}d Pr{ }d .r r r




Credibility Distribution

Liu (TPUP, 2002)

The credibility distribution : ( , ) [0,1]

of a fuzzy variable is defined by

( ) Cr{ | ( ) }.x x




A Sufficient and Necessary Condition

Liu (UT, 2004)

A function : [0,1] is a credibility distribution

if and only if it is an increasing function with

lim ( ) 0.5 lim ( )

lim ( ) ( ) if lim ( ) 0.5 or ( ) 0.5.x x

y x y x

x x

y x y x




Entropy (Li and Liu, 2005)

1

What is the degree of difficulty of predicting the specified value

that a fuzzy variable will take?

[ ] (Cr{ })n

ii

H S x

( ) ln (1 ) ln(1 )S t t t t t




Random Phenomena Fuzzy Phenomena

(1654) (1965

Probability Credibility

Three Axioms Five Axioms

Sum "+" Maximiza

)

Probability Theory Credibility Theory

(1933) (2004)

tion " "

Product " " Minimization " "




Essential of Uncertainty Theory

Probability Theory Function Theory

Credibility Theory

Measure Theory

Two basic problems?

[1] Measure o

f Unio

{ } }

+

{n: A B A

[2] Measure of Produ

{ } " "

{ } { } { } " "

{ } { } { } " "

{ } { } { }

t

"

:

"

c

B

A B A B

A B A B

A B A B




What Mathematics Made?

(+, )-Axiomatic System: Probability Theory

( , )-Axiomatic System: Credibility Theory

( , )-Axiomatic System: Nonclassical Credibility Theory

(+, )-Axiomatic System: Inconsistent




Fuzzy ProgrammingFuzzy Programming

max ( , )

subject to:

( , ) 0, 1,2, ,

- Man proposes

- God disposes

It is not a mathematical model!

j

f x

g x j m

x




The Simplest

Given two fuzzy variables and ,

which one is greater?

The Most FundamentalProblem




Expected Value Criterion

[ ] [ ].

Objective: max ( , ) max [ ( , )]

Constraint: ( , ) 0 [ ( , )] 0j j

E E

f x E f x

g x E g x




pjxgE

xfE

j ,,2,1,0)],([

:subject to

)],([max

Liu and Liu (IEEE TFS, 2002)Find the decision with maximum expected returnsubject to some expected constraints.

Fuzzy Expected Value Model




Optimistic Value Criterion

sup sup( ) ( )

Objective: max ( , ) (Undefined)

max max : Cr{ ( , ) }

Constraint: ( , ) 0 Cr ( , ) 0

x

x f

j j

f x

f f x f

g x g x




(Maximax) Chance-Constrained Programming

(Maximax) Chance-Constrained Programming

Liu and Iwamura (FSS, 1998)Maximize the optimistic value subject to chance constraints.

ffx

maxmax

pjxg

fxf

f

j ,,2,1,0),(Cr

}),({Cr

:subject to

max




Pessimistic Value Criterion

inf inf( ) ( )

Objective: max ( , )

max min : Cr{ ( , ) }

Constraint: ( , ) 0 Cr ( , ) 0

x

x f

j j

f x

f f x f

g x g x




(Minimax) Chance-Constrained Programming

(Minimax) Chance-Constrained Programming

Liu (IS, 1998)Maximize the pessimistic value subject to chance constraints.

pjxg

fxf

f

j

fx

,,2,1,0),(Cr

}),({Cr

:subject to

minmax




Credibility Criterion

Cr Cr

Remark: Different choice of produces different ordership.

r r

r




Fuzzy Dependent-Chance ProgrammingFuzzy Dependent-Chance Programming

j

max Cr{ ( , ) 0, 1,2, , }

subject to:

g ( , ) 0, 1,2, ,

kh x k q

x j p

Liu (IEEE TFS, 1999)Find the decision with maximum chance to meet the event in an uncertain environment.




Classify Uncertain Programming via Graph

Information

PhilosophySingle-Objective PMOP

GPDP

MLP

Structure

EVM CCP DCP

Random FuzzyFuzzy random

FuzzyStochastic

Maximax Minimax




Last Words

[1] Liu B., Foundation of Uncertainty Theory.[2] Liu B., Introduction to Uncertain

Programming.

If you want an electronic copy of my book, or source files of hybrid intelligent algorithms,please download them from


credibility theory baoding liu uncertainty theory laboratory department of mathematical sciences...

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