Measuring Uncertainty with Imprecision Indices Andrey Bronevich, - - PowerPoint PPT Presentation

Measuring Uncertainty with Imprecision Indices

Andrey Bronevich, Alexander Lepskiy

Technological Institute of Southern Federal University, Taganrog, RUSSIA

Various types of uncertainty

Randomness (probability theory); Nonspecificity (possibility theory); Imprecision (interval calculi, monotone

measure);

Inconsistency; Incompleteness; Fuzziness;

and so on …

Classical uncertainty measures

Shannon’s entropy (uncertainty – randomness)

Hartley’s measure (uncertainty – nonspecificity)

( ) ( )

2

( ) { } log { }

x X

S P P x P x

∈

= −∑

2

1, , ( ) log , ( ) 0, .

B B

B A H B A B A η η ⊆ ⎧ = = ⎨ ⎩

Main attributions of particular theory of uncertainty (by G. Klir)

An uncertainty function g (ex. probability); A calculus with functions g; A functional (uncertainty measures) f which

measures the amount of uncertainty associated with g (ex. Shannon’s entropy);

A methodology.

Generalized Hartley’s measure

( )

2 2 \{ }

( )log .

X

B

GH g m B B

∈ ∅

= ∑

2

( ) ,

X

B B

g m B η

∈

=∑

( ) 0, m ∅ = ( ) 0, m B ≥

2

( ) 1.

X

B

m B

∈

=

∑

Let g be a belief function: where Then

Aggregate measure of uncertainty

( )

sup ( )

P g

Au g S P

≥

=

( ) ( ), Au P S P = ( ) ( ).

B B

Au H η η =

Properties: 1) 2)

Basic notations

{ } { } { } { }

Pr Pr Pr

is the set of all set functions on 2 ; | ( ) 0 ; = | ( ) 1, be monotone measure ; is the set of all probability measures; | : , | : ;

X mon low mon up mon bel

M M g M g M g M g X g M M g M P M g P M g M P M g P M = ∈ ∅ = ∈ = = ∈ ∃ ∈ ≤ = ∈ ∃ ∈ ≥

• ,

are the sets of all belief and plausibility functions.

pl

M

Principal motivation

If then where Therefore, the “distance” between g and defines the degree of uncertainty.

low

g M ∈

Pr :

P M g P g ∃ ∈ ≤ ≤ ( ) ( ) ( ) . g A g X g A = − g

Imprecision indices

Pr 1 2 1 2 1

Let

• r

. . A functional : [0,1] is called if : 1) implies ( ) 0; 2) if then ( ) ( ) for ( ) (

low up low

M M M M f M imprecision index g M f g g g f g f g M M f g f g = = → ∈ = ≤ ≥ = ≤ Definition

( )

( ) ( )

( ) ( )

( )

2 1 1

) for ; 3) 1 for 1 for ; is called ( ( )) if extra 4) .

up low up X X k k j j j j j j

M M f M M f M M linear imprecision index f I M f g f g η η α α

= =

= = = = = ∈ =

∑ ∑

Canonical representations of linear imprecision indices

( )

2 \ :

( ) ( ) is defined by ( ) Then ( ) , ({ }) . Let ( ) ( 1) ( ) be dual Mobius transform.

X

g B B f B f mon f A B g A B A

g m B f g f B B M x x X m B g A η η µ µ µ

∈ ⊆

= ⇒ = ∈ = ∀ ∈ = −

∑ ∑

2

Let be a linear functional on then ( ) ( ) ( ) for any .

f X

B

f M f g m B g B g M

µ ∈

= ∈

∑

Proposition.

2 :

. Let be a linear functional on . a) ( ) 1 ; ( ) 0; ( ) b) ( ) 0 for all ; c) ( ) 2 \{ , }. ("avoiding sure loss" condition).

f f X f f

D low D x D X

f M m X m D f I M m D x X m D D X

µ µ µ µ ∈ ∈

= = ∈ ⇔ = ∈ ≤ ∀ ∈ ∅

∑ ∑

Theorem Corollary

2 2

( ) ( ) 1 ( ) ( ), where: 1) ( ) ( ) 0, ( ) 2 ; 2) ( )1 1 .

X X

low B X B X B

f I M f g m B g B m m X m B B m B

∈ ∈

∈ ⇔ = − ∅ = = ≥ ∀ ∈ =

∑ ∑

…through description of Möbius transform of

f

µ

( )

2 \{ }

. ( ) , where 0 , 1 , and with { } / for all . . If ( ) then 1) ( ); 2 ( )

X

low f X pl f A A f low

f I M a b b a b M x b a x X m A M X f I M µ µ η µ µ µ η µ

∈ ∅

∈ ⇔ = − > = + ∈ = ∈ = ∈ ∈ ⇔

∑

Theorem Corollary

( ) ( )

) { } ; 3) ( ) 2 \{ , };

• r 3 )

{ } .

f X f Pl

x x X m A A X B x M x X µ µ = ∀ ∈ ≥ ∀ ∈ ∅ ′ ∪ ∈ ∀ ∈

…through description of (monotone measure)

f

µ

…through description of distorted function of

{ }

( )

( )

)

( )

)

( ) ( ) ( ) ( )

1 1 1

. Let be a linear functional on and and 1/ , 1,..., . Then a) 1/ 0; ( ) b) ,1 ; c) 1 ( ) 0, , ,1 . ( ) ln ln ( ) ln ln ( ).

f i N n n n N GH low

f M P P x N i N N f I M C d t dt n t t t X X A A X GH I M µ λ λ λ λ λ µ

∞ −

= = = = ⎡ ∈ ⇐ ∈ ⎣ ⎡ − ≥ ∀ ∈ ∈⎣ = ⇒ = ⇒ ∈ Theorem Ex.

• f

µ

The algebraic structure of the set of linear imprecision indices { }

Let : ( ) .

I f low

M f I M µ = ∈

{ }

2 \{ , } ( ) 0, 2 \{ , }

. Let , ( ) , for all 2 \{ , }, 0, then is an extreme point

• f

1 are linearly independent.

X X

I A X A X X m A I A A X

M m A b A X b M µ µ η η µ

∈ ∅ > ∈ ∅

∈ = − ∈ ∅ > ⇔

∑

Theorem

The algebraic structure of the set of complementarily symmetrical imprecision indices

{ }

. We call ( ) if ( ) ( ) 2 \{ , }. . Let ( ) ( ) ( ) ( ) ( ) be primitive imprecision index. Then ( ) ( ) ( ) ( ) (such as 1 ,1 a

f f B

low X B v B B X B B

f I M complementarily symmetrical m A m A A X g g X g B g B g A A A A

µ µ

ν µ η η η ∈ = ∀ ∈ ∅ = − − + ∅ = + − ⇒ Definition Ex re linearly independent) extreme point . Let be complementarily symmetrical ( ) , ( ) 1, ( ) .

B B B B

f f B v B B B ν α α α ⇒ − ⇒ ⇔ = = ≥ ∀

∑ ∑

Theorem

The extension of imprecision indices to the set

Let ( ) ( ). . ( ) ( ).

low up

f g f g f I M f I M = ∈ ⇔ ∈ Proposition

. Let be a linear functional on then ( ) is complementarily symmetrical index on .

low up low

f M f I M M f M ∈ ∪ ⇔ Proposition

low up

M M ∪

The extension of imprecision indices to the set

• f all monotone measures

mon

M

Imp | Imp

uncertainty= imprecision inconsistency Let ( ), but then ( ) inf ( ) is the in g. If then ( ) {imprecision}=0,

low

low mon low q M q g up

f I M g M g M f g f q g M f g

∈ ≤

∪ ∈ ∈ ∉ = ∈ = ⇒ amount of imprecision

Inc | Inc

{uncertainty}={inconsistency}. By analogy, if but then ( ) inf ( ) is in g. If then ( ) {inc

up

mon up q M q g low

g M g M f g f q g M f g

∈ ≥

∈ ∉ = ∈ = ⇒ amount of inconsistency

• nsistency}=0,

{uncertainty}={imprecision}.

Properties of indices on Mmon ( ) ( ) ( ) ( ) { } { }

Pr Pr

1 2 Imp 1 Imp 2 Imp 1 Imp 2 Imp Inc Imp Inc

1) , ; 2) ( ) inf (min , ), ( ) inf (min , ); 3) is rather lower probability than upper probability if ( ) ( ) and r

M M mon

g g f g f g f g f g f g f g f g f g g M f g f g

α α

α α

∈ ∈

≥ ⇒ ≥ ≤ = = ∈ ≥

Imp Inc Imp Inc

ather upper probability then lower probability if ( ) ( ); 4) if complementarily symmetrical index on then .

low

f g f g f M f f f < − − =

Example

( )

{ }

( ) { }

1 2 3 1 2 1 2 1 | | 1 2 Imp Inc

( ) max ( ) , 1 0.5 0.5 ( ) 1 , 0.4 1 0.6 max , . Let ( ) 2 2 ( ) ( ), ( ) max ( ) ( ) | 2 .

X

i i up x X i i low low X B X

A x M X x x x N A A M g N N M v g g B g B v g g B g B B f f π π π

∈ − ∈ ∞

Π = ∈ = − Π ∈ = ∉ = − − = − ∈

∑

1 1 2 1 2

0.5 0.5 0.5 0.5(3) 0.6 0.526 0.2 0.5 0.2 0.03(3) 0.1 0.0288 v v GH v v GH N N g

∞