Decomposition of Evidence and Internal Conflict Alexander Lepskiy - PowerPoint PPT Presentation

Decomposition of Evidence and Internal Conflict Alexander Lepskiy National Research University - Higher School of Economics, Moscow, Russia The Fifth International Conference on Information Technology and Quantitative Management - ITQM 2017, December 8-10, 2017, New Delhi, India Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 1 / 21

Preamble Aggregation of Uncertain Information Application of aggregation image processing; multi sensor fusion; aggregation of experts information; pattern recognition, etc. Tools for presentation of uncertain information probability theory; fuzzy sets; possibility theory; evidence theory, etc. Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 2 / 21

Preamble Evidence Theory (Belief Functions Theory, the Dempster-Shafer theory) Example of evidence : the value of the company shares will be in the interval A 1 = [30 , 40] with the belief value 0.7 or in the interval A 2 = [35 , 45] with the belief value 0.3. The characteristics of evidence : uncertainty; reliability; conflict 1 external conflict between bodies of evidence ( E.g. the evidence F 1 = { the value of the company shares will be in A = [30 , 40] } has a big external conflict with the evidence F 2 = { the value of the company shares will be in B = [60 , 70] } ); 2 internal conflict of one evidence ( E.g. the evidence F = { the value of the company shares will be in A 1 = [30 , 40] with the belief value 0.7 or in A 2 = [60 , 70] with the belief value 0.3 } has a big internal conflict). Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 3 / 21

Preamble Estimation of External and Internal Conflicts There are several approaches to the estimation of external conflict : axiomatic approach [Martin 2012, Destercke & Burger 2013, Bronevich et al. 2015]; metric approach [Jousselme et al. 2001, Jousselme & Maupin 2012, Liu 2006]; structural approach [Martin 2012]; algebraic approach [Lepskiy 2013] etc. There are the following approaches to the estimation of internal conflic t: entropy approach [H¨ ohle 1982, Yager 1983, Klir& Ramer 1990 etc]; axiomatic approach [Harmanec 1995, Bronevich & Klir 2010]; auto-conflict approach [Osswald & Martin 2006, Daniel 2010]; decompositional approach [Schubert 2012, Roquel et al. 2014, Lepskiy 2016] etc. Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 4 / 21

Preamble The Main Idea of Decompositional Approach The following assumption is the basis of decompositional approach . Evidence with a great internal conflict has been obtained as a result of aggregation information from several different sources with the help of some combining rule. Then the (external) conflict of the decomposed set of evidence can be regarded as an internal conflict of the original evidence. The decomposition result is ambiguous. Therefore we can talk only about the upper and lower estimates of the internal conflict in this case. In addition, it is necessary to introduce some additional restrictions in order to the result is not trivial or degenerate. Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 5 / 21

Preamble Outline of Presentation Background of Evidence Theory Decomposition of Evidence A Decomposed Internal Conflict for | X | = 2 Properties of the Decomposed Conflict Summary and Conclusion Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 6 / 21

Background of Evidence Theory Background of Evidence Theory Let X be a finite set and 2 X be a powerset of X . The mass function is a set function m : 2 X → [0 , 1] that satisfies the conditions m ( ∅ ) = 0, � A ⊆ X m ( A ) = 1. Notations and terms : A ∈ 2 X is called a focal element , if m ( A ) > 0; A = { A } be a set of all focal elements of evidence; F = ( A , m ) is called a body of evidence (BE) ; F ( X ) be a set of all BE on X ; F A = ( A, 1), A ∈ 2 X is called a categorical BE ; F X = ( X, 1) is called a vacuous BF . If F j =( A j , m j ) ∈F ( X ) and � j α j =1, α j ∈ [0 , 1], then F =( A , m ) ∈F ( X ), where A = � j A j , m ( A )= � j α j m j ( A ). This is denoted as F = � j α j F j . In particular, we have F = � A ∈A m ( A ) F A ∀ F =( A , m ) ∈F ( X ). Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 7 / 21

Background of Evidence Theory Conflict Measure and Combining Rules Let us have two BE F 1 = ( A 1 , m 1 ) and F 2 = ( A 2 , m 2 ). We have a question about a conflict between these BE. Historically, the conflict measure K 0 ( F 1 , F 2 ) associated with Dempster’s rule is the first among conflict measures: � K 0 = K 0 ( F 1 , F 2 ) = m 1 ( B ) m 2 ( C ) . B ∩ C = ∅ , B ∈A 1 ,C ∈A 2 If K 0 � = 1, then we have the following Dempster’s rule for combining of two BE: 1 � A � = ∅ , m D ( ∅ ) = 0 . m D ( A ) = B ∩ C = A m 1 ( B ) m 2 ( C ) , 1 − K 0 Dubois and Prade’s disjunctive consensus rule is a dual rule to Dempster’s rule in some sense. This rule is defined by a formula: � A ∈ 2 X . m DP ( A ) = B ∪ C = A m 1 ( B ) m 2 ( C ) , (1) Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 8 / 21

Decomposition of Evidence The General Idea of Decomposition In general case we can assume that some evidence describing with the help of BE F = ( A , m ) has a great internal conflict, if its information source is a heterogeneous. In this case we can consider that the BE F = ( A , m ) is a result of combining of several BE F i = ( A i , m i ), i = 1 , ..., l with the help of some combining rule R : F = R ( F 1 , ..., F l ). Therefore we can estimate the internal conflict by the formula K R in ( F ) = K ( F 1 , ..., F l ) , assuming that F = R ( F 1 , ..., F l ) , where K is some fixed (external) conflict measure, R is a fixed combining rule. Since the equation F = R ( F 1 , ..., F l ) has many solutions then we can consider the optimization problem of finding the R in ( F ) and smallest K R largest K in ( F ) conflicts: R K R K in ( F ) = arg max K ( F 1 , ..., F l ) , in ( F ) = arg min K ( F 1 , ..., F l ) . F = R ( F 1 ,...,F l ) F = R ( F 1 ,...,F l ) Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 9 / 21

Decomposition with Dempster’s Rule Decomposition of Evidence with the Help of Dempster’s Rule i =1 : s i ≥ 0 , � n Let S n = { ( s i ) n i =1 s i =1 } be a n -dimensional simplex. Then optimization problems for Dempster’s rule and l = 2 have the following formulation. We have to find the BE F i = ( A i , m i ) ∈ F ( X ), i = 1 , 2, that satisfy the condition � m 1 ( B ) m 2 ( C ) → sup K 0 ( F 1 , F 2 ) = (inf) (2) B ∩ C = ∅ , B ∈A 1 ,C ∈A 2 with constraints ( m 1 ( B )) B ∈A 1 ∈ S |A 1 | , ( m 2 ( C )) C ∈A 2 ∈ S |A 2 | , K 0 ( F 1 , F 2 ) < 1 , (3) � (1 − K 0 ( F 1 , F 2 )) m ( A ) = m 1 ( B ) m 2 ( C ) , A ∈ A . (4) B ∩ C = A, B ∈A 1 ,C ∈A 2 Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 10 / 21

Restrictions on Evidencee Restrictions on the Decomposable Set of Evidence Note, that in the case of the general formulation K D in ( F ) = 0 and this value is achieved on the pair F 1 = F , F 2 = F X . D In the same time we have K in ( F ) = 1 and this value achieved for such F i = ( A i , m i ) ∈ F ( X ), i = 1 , 2, that B ∩ C = ∅ ∀ B ∈ A 1 , ∀ C ∈ A 2 . D Therefore, in general formulation the problem of finding K in ( F ) and K D in ( F ) is not meaningful. A similar situation will be for the disjunctive consensus rule . Restrictions on the decomposable set of evidence : structural restrictions; conflict restrictions; restrictions associated to combining rules; mixed restrictions. Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 11 / 21

Restrictions on Evidencee Restrictions Associated to Combining Rules Dempster’s rule is an optimistic rule . If one evidence argues that the true alternative belongs to the set A , and the other – to the set B , then after combination of evidence in accordance with Dempster’s rule we get that the true alternative belong to the set A ∩ B . Similarly, the disjunctive consensus rule is pessimistic rule , because we get that the true alternative belong to the set A ∪ B in the above situation. These limitations can be taken into account with the help of imprecision index f : F ( X ) → [0 , 1]. For example, it may be the normalized generalized Hartley measure : 1 � A ∈A m ( A ) ln | A | . f ( F ) = ln | X | Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 12 / 21

Restrictions on Evidencee In general form linear imprecision index (l.i.i.) has the following presentation [Bronevich & Lepskiy 2007]: � f ( F ) = B ∈A m ( B ) µ f ( B ) , F = ( A , m ) , where monotone set function µ f ( B ) = f ( F B ), B � = ∅ , µ f ( ∅ ) = 0 satisfies the conditions: 1) µ f ( { x } ) = 0 for all x ∈ X ; 2) µ f ( X ) = 1; B : A ⊆ B ( − 1) | B \ A | µ f ( B ) ≤ 0 for all A � = ∅ , X . 3) � The optimism (pessimism) condition of Dempster’s rule (disjunctive consensus rule) may be described by inequalities f ( F ) ≤ f ( F i ) , ( f ( F ) ≥ f ( F i )) i = 1 , 2 , (5) where f : F ( X ) → [0 , 1] is a some imprecision index. Alexander Lepskiy (HSE) Decomposition of Evidence ITQM 2017 13 / 21