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) = − ∑ ( ) ( ) S P ( ) P { } log x P { } x 2 x X ∈ � Hartley’s measure (uncertainty – nonspecificity) ⎧ 1, B A , ⊆ = ⎨ H ( ) log B , ( ) A η = η 2 � B B 0, B A . ⎩

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 = ∑ g m B η ( ) , Let g be a belief function: X B B 2 ∈ ∑ m ∅ = ( ) 0, m B ≥ ( ) 0, m B ( ) 1. where = X B 2 ∈ = ∑ ( ) Then GH g m B ( )log B . 2 X B 2 \{ } ∈ ∅

Aggregate measure of uncertainty ( ) Au g sup ( ) S P = P g ≥ Au P ( ) S P ( ), = Properties: 1) Au ( ) H ( ). 2) η = η B B

Basic notations X � M is the set of all set functions on 2 ; { } � M g M g | ( ) 0 ; = ∈ ∅ = 0 { } � M = g M | g X ( ) 1, g be monotone measure ; ∈ = mon 0 � M is the set of all probability measures; Pr { } � M g M | P M : g P , = ∈ ∃ ∈ ≤ low mon Pr { } � M g M | P M : g P ; = ∈ ∃ ∈ ≥ up mon Pr � M , M are the sets of all belief and plausibility bel pl functions.

Principal motivation g M P M Pr : g P g ∈ ∃ ∈ ≤ ≤ If then where low g A ( ) g X ( ) g A ( ) . = − g Therefore, the “distance” between g and defines the degree of uncertainty.

Imprecision indices Let M M or M M . = = low up Definition . A functional : f M [0,1] is called → imprecision index if : 1) g M implies ( ) f g 0; ∈ = Pr 2) if g g then ( f g ) f g ( ) for M M ≤ ≥ = 1 2 1 2 low ( ) f g ( ) f g ( ) for M M ; ≤ = 1 2 up ( ) ( ) ( ) 3) f 1 for M M f 1 for M M ; η = = η = = low up X X is called linear imprecision index ( f I M ( )) if extra ∈ ( ) ∑ ∑ k k ( ) 4) f g f g . α = α j j j j j 1 j 1 = =

Canonical representations of linear imprecision indices ∑ ⇒ g m ( ) B f g ( ) is defined by = η g X B B 2 ∈ ( ) f ( ) B η = µ f B Then ( ) B M , ({ }) x 0 x X . µ ∈ µ = ∀ ∈ f mon f ∑ A B \ g Let m ( ) B ( 1) g A ( ) = − A B : A ⊆ be dual Mobius transform. Proposition. Let be a linear functional on f M ∑ µ then ( ) f g m ( ) ( ) for any B g B g M . = f ∈ X B 2 ∈

µ …through description of Möbius transform of f Theorem . Let be a linear functional on f M . ∑ µ µ a) m ( X ) 1 ; m ( D ) 0; f = f = X D 2 ∈ ∑ µ f I M ( ) b) m ( D ) 0 for all x X ; ∈ ⇔ = ∈ f low D x D : ∈ µ X c) m ( D ) 0 D 2 \{ , X }. ≤ ∀ ∈ ∅ f Corollary ("avoiding sure loss" condition). ∑ f I M ( ) f g ( ) 1 m B g B ( ) ( ), ∈ ⇔ = − low X B 2 ∈ X where: 1) ( m ) m X ( ) 0, ( ) m B 0 B 2 ; ∅ = = ≥ ∀ ∈ ∑ 2) m B ( )1 1 . = B X X B 2 ∈

µ …through description of (monotone measure) f Theorem . f I M ( ) a b , ∈ ⇔ µ = µ − η low f X where b 0 , a 1 b , M and with > = + µ ∈ pl ( ) { } x b a / for all x X . µ = ∈ ∑ Corollary . If m A ( ) then µ = η f X A A 2 \{ } ∈ ∅ 1) M ( X ); µ ∈ f 0 ( ) 2 ) { } x 0 x X ; µ = ∀ ∈ f f I M ( ) ∈ ⇔ X low 3) ( ) m A 0 A 2 \{ , X }; ≥ ∀ ∈ ∅ ( ) or 3 ) B { } x M x X . ′ µ ∪ ∈ ∀ ∈ f Pl

µ …through description of distorted function of f Theorem . Let be a linear functional on f M and ( ) { } � P and P x 1/ N , i 1,..., N . Then µ = λ = = f i ( ) a) 1/ N 0; λ = ) ⎡ 1 ∞ f I M ( ) b) C ,1 ; ∈ ⇐ λ ∈ ⎣ N n 1 ) − ⎡ ( ) n n 1 c) 1 d ( ) t dt 0, n � , t ,1 . − λ ≥ ∀ ∈ ∈ ⎣ N ( ) ( ) ⇒ Ex. ( ) t ln t X ln X λ = ( ) ( ) ⇒ ( ) A ln A ln X GH I M ( ). µ = ∈ GH low

The algebraic structure of the set of linear imprecision indices { } Let M : f I M ( ) . = µ ∈ I f low ∑ Theorem . Let M , m A ( ) b , µ ∈ µ = η − η I X A X A 2 \{ , X } ∈ ∅ X for all A 2 \{ , X }, b 0, then is an extreme point ∈ ∅ > µ { } of M 1 are linearly independent. ⇔ m A ( ) 0, > I A X A 2 \{ , X } ∈ ∅

The algebraic structure of the set of complementarily symmetrical imprecision indices Definition . We call f I M ( ) complementarily ∈ low µ µ X symmetrical if m ( ) A m ( ) A A 2 \{ , X }. f = f ∀ ∈ ∅ Ex . Let ( ) g g X ( ) g B ( ) g B ( ) g ( ) be primitive ν = − − + ∅ B imprecision index. Then ⇒ { } ( ) A ( ) A ( ) A ( ) A (such as 1 ,1 µ = η + η − η v B B X B B B ⇒ ⇒ a re linearly independent) extreme point ν − B Theorem . Let be complementarily symmetrical f ⇔ ∑ ∑ f ( ) B v , ( ) B 1, ( ) B 0 B . = α α = α ≥ ∀ B B B

The extension of imprecision indices to the set M M ∪ low up Let ( ) f g f g ( ). = Proposition . f I M ( ) f I M ( ). ∈ ⇔ ∈ low up Proposition . Let be a linear functional on f M then f I M ( M ) is f ∈ ∪ ⇔ low up complementarily symmetrical index on M . low

The extension of imprecision indices to the set M of all monotone measures mon uncertainty= imprecision inconsistency ∪ Let f I M ( ), g M but g M then ∈ ∈ ∉ low mon low f ( ) g inf f q ( ) is the amount of imprecision in g. = Imp q M | q g ∈ ≤ low ⇒ If g M then f ( ) g 0 {imprecision}=0, ∈ = up Imp {uncertainty}={inconsistency}. By analogy, if g M but g M then ∈ ∉ mon up f ( ) g inf f q ( ) is amount of inconsistency in g. = Inc q M | q g ∈ ≥ up ⇒ If g M then f ( ) g 0 {inc onsistency}=0, ∈ = low Inc {uncertainty}={imprecision}.

Properties of indices on M mon ⇒ ( ) ( ) ( ) ( ) 1) g g f g f g , f g f g ; ≥ ≥ ≤ 1 2 Imp 1 Imp 2 Imp 1 Imp 2 { } 2) f ( ) g inf f (min , g ), = α Imp M α ∈ Pr { } f ( ) g inf f (min , g ); = α Inc M α ∈ Pr 3) g M is rather lower probability than ∈ mon upper probability if f ( ) g f ( ) and r g ather ≥ Imp Inc upper probability then lower probability if f ( ) g f ( ); g < Imp Inc 4) if f complementarily symmetrical index on M − low then f f f . − = Imp Inc

Example ( ) A max ( ) x M , Π = π ∈ X x x x i i up x X ∈ 1 2 3 ( ) 1 0.5 0.5 N A ( ) 1 A M , π = − Π ∈ 1 i i low 0.4 1 0.6 π { } g max N N , M . = ∉ 2 1 2 low ∑ 1 − ( ) | X | Let ( ) v g 2 2 g B ( ) g B ( ), = − − 1 X B 2 ∈ { } X v ( ) g max g B ( ) g B ( ) | B 2 . = − ∈ ∞ f f Imp Inc v v GH v v GH 1 1 2 ∞ N 0.5 0.5 0.5 0 0 0 1 N 0.5(3) 0.6 0.526 0 0 0 2 g 0.2 0.5 0.2 0.03(3) 0.1 0.0288