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.

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Preamble

Evidence Theory (Belief Functions Theory, the Dempster-Shafer theory)

Example of evidence: the value of the company shares will be in the interval A1 = [30, 40] with the belief value 0.7 or in the interval A2 = [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

F1 = {the value of the company shares will be in A = [30, 40] } has a big external conflict with the evidence F2 = {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 A1 = [30, 40] with the belief value 0.7 or in A2 = [60, 70] with the belief value 0.3 } has a big internal conflict).

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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 conflict: entropy approach [H¨

• hle 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.

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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.

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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

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Background of Evidence Theory

Background of Evidence Theory

Let X be a finite set and 2X be a powerset of X. The mass function is a set function m : 2X → [0, 1] that satisfies the conditions m(∅) = 0,

• A⊆X m(A) = 1.

Notations and terms: A ∈ 2X 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; FA = (A, 1), A ∈ 2X is called a categorical BE; FX = (X, 1) is called a vacuous BF. If Fj=(Aj, mj)∈F(X) and

j αj=1, αj∈[0, 1], then F=(A, m)∈F(X),

where A=

j Aj, m(A)= j αjmj(A). This is denoted as F= j αjFj.

In particular, we have F=

A∈A m(A)FA

∀F=(A, m)∈F(X).

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Background of Evidence Theory

Conflict Measure and Combining Rules

Let us have two BE F1 = (A1, m1) and F2 = (A2, m2). We have a question about a conflict between these BE. Historically, the conflict measure K0(F1, F2) associated with Dempster’s rule is the first among conflict measures: K0 = K0(F1, F2) =

• B∩C=∅,

B∈A1,C∈A2

m1(B)m2(C). If K0 = 1, then we have the following Dempster’s rule for combining

• f two BE:

mD(A) = 1 1 − K0

• B∩C=A m1(B)m2(C),

A = ∅, mD(∅) = 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: mDP(A) =

• B∪C=A m1(B)m2(C),

A ∈ 2X. (1)

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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 Fi = (Ai, mi), i = 1, ..., l with the help of some combining rule R: F = R(F1, ..., Fl). Therefore we can estimate the internal conflict by the formula KR

in(F) = K(F1, ..., Fl), assuming that F = R(F1, ..., Fl),

where K is some fixed (external) conflict measure, R is a fixed combining rule. Since the equation F = R(F1, ..., Fl) has many solutions then we can consider the optimization problem of finding the largest K

R in(F) and smallest KR in(F) conflicts:

K

R in(F) = arg max F =R(F1,...,Fl)

K(F1, ..., Fl), KR

in(F) = arg min F =R(F1,...,Fl)

K(F1, ..., Fl).

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Decomposition with Dempster’s Rule

Decomposition of Evidence with the Help of Dempster’s Rule

Let Sn ={(si)n

i=1 : si ≥ 0, n i=1 si =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 Fi = (Ai, mi) ∈ F(X), i = 1, 2, that satisfy the condition K0(F1, F2) =

• B∩C=∅,

B∈A1,C∈A2

m1(B)m2(C) → sup (inf) (2) with constraints (m1(B))B∈A1 ∈ S|A1|, (m2(C))C∈A2 ∈ S|A2|, K0(F1, F2) < 1, (3) (1 − K0(F1, F2)) m(A) =

• B∩C=A,

B∈A1,C∈A2

m1(B)m2(C), A ∈ A. (4)

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Restrictions on Evidencee

Restrictions on the Decomposable Set of Evidence

Note, that in the case of the general formulation KD

in(F) = 0 and this

value is achieved on the pair F1 = F, F2 = FX. In the same time we have K

D in(F) = 1 and this value achieved for such

Fi = (Ai, mi) ∈ F(X), i = 1, 2, that B ∩ C = ∅ ∀B ∈ A1, ∀C ∈ A2. Therefore, in general formulation the problem of finding K

D in(F) and

KD

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.

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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: f(F) = 1 ln |X|

• A∈A m(A) ln |A|.

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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)µf(B),

F = (A, m), where monotone set function µf(B) = f(FB), B = ∅, µf(∅) = 0 satisfies the conditions: 1) µf ({x}) = 0 for all x ∈ X; 2) µf(X) = 1; 3)

B:A⊆B (−1)|B\A|µf(B) ≤ 0 for all A = ∅, X.

The optimism (pessimism) condition of Dempster’s rule (disjunctive consensus rule) may be described by inequalities f(F) ≤ f(Fi), (f(F) ≥ f(Fi)) i = 1, 2, (5) where f : F(X) → [0, 1] is a some imprecision index.

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A Decomposed Internal Conflict for |X| = 2

A Decomposed Internal Conflict for |X| = 2

Proposition [Lepskiy 2016] Let X = {x1, x2}, F = m0FX + m1F{x1} + m2F{x2} ∈ F(X). Then KD

in(F) = 0 and

K

D in(F) =

m1m2 (1 − m1)(1 − m2) = m1m2 (m0 + m1)(m0 + m2) if m0 = 0 and K

D in(F) = 1 if m0 = 0;

the decomposition problem for disjunctive consensus rule has a solution iff √m1 + √m2 ≤ 1 and in this case KDP

in (F) = 2√m1m2,

K

DP in (F) = m0 = 1 − m1 − m2.

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A Decomposed Internal Conflict for |X| = 2

We have for F = m0FX + mF{x1} + mF{x2} ∈ F(X): KD

in(F) = 0, K D in(F) =

• m

1−m

2 , KDP

in (F) = 2m, K DP in (F) = 1 − 2m:

The internal decomposed conflict with disjunctive rule is greater than the decomposed conflict with Dempster’s rule!

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Properties of the Decomposed Conflict

Properties of the Decomposed Conflict with Dempster’s Rule. Categorical Evidence

It is intuitively clear that categorical evidence has no internal conflict. The following properties are valid for categorical evidence: KD

in(F) = 0 for every F = (A, m) ∈ F(X);

KD

in(FX) = K D in(FX) = 0 for any strict imprecise index f;

K

D in(FA) ≥ min

• 1,

1−µf (A) 1−µf (X\A)

• , if we assume that
• = 1.

In particular,

1 if A ∈ 2X\{∅, X} satisfies the condition µf(A) ≤ µf(X\A)) for l.i.i.

f then K

D in(FA) = 1;

2 K

D in(F{x}) = 1 for every x ∈ X.

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Properties of the Decomposed Conflict

Properties of the Decomposed Conflict with Dempster’s Rule. Bifocal Evidence

Let us now study the internal conflict of the bifocal evidence, i.e. F = mFA + (1 − m)FB, m ∈ (0, 1). It is intuitively clear that bifocal evidence has internal conflict, if focal elements are disjoint. Proposition If A ∩ B = ∅, then K

D in (mFA + (1 − m)FB) = 1, m ∈ (0, 1) for l.i.i f.

Example Let X = {x1, x2, x3}, m ∈ (0, 1). Then: K

D in

• mF{x1,x2} + (1 − m)F{x2,x3}
• = 0;

K

D in

• mF{x2} + (1 − m)F{x2,x3}
• = 1 − (1 − m)µf({x2, x3}).

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Properties of the Decomposed Conflict

Properties of the Decomposed Conflict with a Disjunctive Rule

Following properties are valid for an internal conflict estimated with the help of disjunctive consensus rule: if A ∈ 2X\{∅}, then KDP

in (FA) = 0; if in addition |A| > 1 then

K

DP in (FA) = 1;

K

DP in (F{x}) = 0 for every x ∈ X;

Let A, B ∈ 2X\{∅}, A ∩ B = ∅, m ∈ (0, 1). Then KDP

in (mFA + (1 − m)FB) = 0;

if, in addition A \ B = ∅ and B \ A = ∅, then K

DP in (mFA + (1 − m)FB) = 1.

Example Let X = {x1, x2} and F = mF{x1} + (1 − m)F{x1,x2}, m ∈ (0, 1). Then KDP

in (F) = 0 and K DP in (F) = 1 − m.

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Summary and Conclusion

Summary and Conclusion

In the case of Dempster’s rule it is shown that: the internal conflict of vacuous evidence is the minimum (zero); conditions are found for which internal conflict of categorical evidence has a maximal uncertainty; bifocal evidence with disjoint focal elements has the maximum potential conflict. In the case of disjunctive consensus rule it is shown that: the internal conflict of Dirac’s measure is the minimum (zero); categorical evidence with cardinality of focal element greater than

• ne has the maximum potential conflict;

bifocal evidence with intersecting but not containing each other focal elements has the maximum potential conflict. These properties show us that internal conflict estimates based on decomposition method are plausible and it can be used in applications.

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References

Bronevich A., Lepskiy A.: Imprecision indices: axiomatic, properties and

• applications. Intern. J. of General Systems. 44(7-8), 812–832 (2015)

Bronevich, A., Lepskiy, A., Penikas, H.: The Application of Conflict Measure to Estimating Incoherence of Analyst’s Forecasts about the Cost of Shares of Russian Companies. Procedia Computer Science. 55, 1113–1122 (2015) Daniel, M.: Conflict between Belief Functions: a New Measure Based on their Non-Conflicting Parts. In: Cuzzolin, F. (ed.) BELIEF 2014. LNCS, vol. 8764,

• pp. 321–330. Springer (2014)

Destercke, S., Burger, T.: Toward an axiomatic definition of conflict between belief functions. IEEE Transactions on Cybernetics. 43(2), 585–596 (2013) Jousselme, A.-L., Maupin, P.: Distances in evidence theory: Comprehensive survey and generalizations. Intern. J. of Approx. Reas. 53, 118–145 (2012) Lepskiy, A.: On internal conflict as an external conflict of a decomposition of

• evidence. In: Vejnarova, J., Kratochvil, V. (eds.) BELIEF 2016, LNAI. vol.

9861, pp. 25–34. Springer, Heidelberg (2016) Martin, A.: About Conflict in the Theory of Belief Functions. In: Denoeux, T., Masson M-H. (eds.) BELIEF, AISC, vol.164, pp.161–168. Springer (2012) Schubert, J.: The Internal Conflict of a Belief Function. In: Denoeux, T., Masson M-H. (eds.) BELIEF 2012, AISC, vol. 164, pp. 169–177. Springer (2012)

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Thanks for you attention

alex.lepskiy@gmail.com http://lepskiy.ucoz.com

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