A basic belief assignment can have up to 2^n focal elements, and combining
them with a simple conjunctive operator will need O(2^2n) operations. This
article proposes some techniques to limit the size of the focal sets of the
bbas to be combined while preserving a large part of the information they
carry. The first section revisits some well-known definitions with an
algorithmic point of vue. The second section proposes a matrix way of building
the least committed isopignistic, and extends it to some other bodies of
evidence. The third section adapts the k-means algorithm for an unsupervized
clustering of the focal elements of a given bba.Comment: Belief 2012, Compi\`egne : France (2012

Osswald, Christophe

English

arXiv

International audienceA basic belief assignment can have up to 2n focal elements, and combining them with a simple conjunctive operator will need O(22n ) opera- tions. This article proposes some techniques to limit the size of the focal sets of the bbas to be combined while preserving a large part of the information they carry. The first section revisits some well-known definitions with an algorithmic point of vue. The second section proposes a matrix way of building the least committed isopignistic, and extends it to some other bodies of evidence. The third section adapts the k-means algorithm for an unsupervized clustering of the focal elements of a given bba

HAL-Université de Bretagne Occidentale

Controling the number of focal elementsChristophe OsswaldTo cite this version:Christophe Osswald. Controling the number of focal elements. Thierry Denoeux, Marie-He´le`neMasson. Belief 2012, May 2012, Compie`gne, France. Springer, pp.135-143, 2012, <10.1007/978-3-642-29461-7>. <hal-00800702>HAL Id: hal-00800702https://hal-ensta-bretagne.archives-ouvertes.fr/hal-00800702Submitted on 14 Mar 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Controling the number of focalelementsSome combinatorial considerationsChristophe OsswaldAbstract A basic belief assignment can have up to 2n focal elements, andcombining them with a simple conjunctive operator will need O(22n) opera-tions. This article proposes some techniques to limit the size of the focal setsof the bbas to be combined while preserving a large part of the informationthey carry.The first section revisits some well-known definitions with an algorithmicpoint of vue. The second section proposes a matrix way of building the leastcommitted isopignistic, and extends it to some other bodies of evidence. Thethird section adapts the k-means algorithm for an unsupervized clustering ofthe focal elements of a given bba.Key words: Basic belief assignments, Combinatorial complexity, Focal ele-ments, k-means, Pignistic probability, Body of evidence, Least commitment1 General considerations on basic belief assignmentsLet the finite set X = {x1, . . . , xn} be our frame of discernment. The size ofX will be noted n = |X |. The set of all the subsets of X will be noted 2X .Definition 1. [Shafer(1976)] The application m from 2X to [0, 1] is a basicbelief assignment (bba) if :∑A⊆Xm(A) = 1 (1)The constraint of closed world is modeled by m(∅) = 0. If m(∅) is greaterthan 0, we either have an open world or a conflict within the information.Christophe OsswaldENSTA Bretagne, Lab-STICC UMR 3192 e-mail: Christophe.Osswald@ensta-bretagne.fr12 Christophe OsswaldDefinition 2. Let m be a bba on X . A ⊆ X is a focal element of m ifm(A) > 0. The focal set of m is composed of all its focal elements :F (m) = {A ⊆ X | m(A) > 0} (2)The size of m is noted |m| = Card(F (m)).Of course, |m| 6 2n. In most applications, |m| will be very small comparedto 2n when a bba is constructed from a source’s information, but after somesteps of combination, this limit can be reached.Definition 3. Let m be a bba on X . The most usual bodies of evidence are :• The belief:bel(A) =∑B⊆A,B 6=∅m(B) =∑B⊆A,B 6=∅,B∈F (m)m(B) (3)• The plausibility:pl(A) =∑B∩A 6=∅m(B) =∑B∩A 6=∅,B∈F (m)m(B) (4)• The commonality:q(A) =∑B⊇Am(B) =∑B⊇A,B∈F (m)m(B) (5)• The pignisitic probability, which is additive (knowing betP({x}) for allx ∈ X is sufficient):betP(A) =11−m(∅)∑B⊆X|A ∩B||B|m(B) =11−m(∅)∑B∈F (m)|A ∩B||B|m(B) (6)When the context is not obvious, the bba used to define the body of evidencewill be placed as an index : betPm(A) instead of betP(m).In the definition 3, the first expression concerns all the subsets of X , andthe second expression concerns only the focal elements. Therefore, if f iseither of the bodies of evidence, and A a subset of X , a natural implemen-tation of the equation brings an algorithm which calculates f(A) in O(2n)operations with the first expression. As the second expression only browsesthe focal set of m, its complexity is O(|m|), for the same result.The most popular combination operator is the non-normalized conjunctiverule, also known as Smet’s rule. It is a quite simple operator to implement;it is associative, and therefore allows to combine many sources.Controling the number of focal elements 3Definition 4. Let m1 and m2 be two bbas on X . The conjunctive combina-tion of m1 and m2 is a bba on X , m1 ⊕m2, defined by :(m1 ⊕m2)(A) =∑B⊆X,C⊆X,B∩C=Am1(B)m2(C) =∑B∈F (m1),C∈F (m2),B∩C=Am1(B)m2(C) (7)The cost for calculating B ∩ C is O(n). The first expression brings an al-gorithm in O(n22n)operations for calculating (m1 ⊕m2)(A), and O(n23n)for determining m1 ⊕ m2. The second expression brings an algorithm inO (n|m1||m2|) operations for calculating (m1 ⊕ m2)(A) = (m1 ⊕ m2)(A),and O (n2n|m1||m2|) for determining m1 ⊕m2.Smets [Smets(2002)] proposed a nice implementation in O(n2n) operationsfor transformations between bba and commonality. The conjunctive combina-tion of the commonality functions is a simple multiplication, which is linear,but on vectors having a size of 2n.The expression (7), nor the commonality, can prevent us from makingoperations on non-focal elements of m1⊕m2. Let the bba be implemented byan adaptive structure that contains information only for its focal elements. Ahashtable is a convenient way for it. The algorithm 1 uses only O(n|m1||m2|)to build m1 ⊕m2.The size ofm∩ is at most |m1||m2|. The algorithm coming from (7) needs tobe executed for all the subsets of X , but the algorithm 1 only works on the fo-cal elements ofm∩, and does not compute useless intersections [Smets(1994)].Using a hashtable for the focal elements, with a hashcode calculation in O(n)operations, the conjunctive combination takes O(n|m1||m2|) operations.Data: bbas m1, m2Result: bba m∩forall B ∈ m1 doforall C ∈ m2 doif B ∩ C ∈ m∩ thenm∩(B ∩ C)← m∩(B ∩ C) +m1(B)m2(C)elseAdd B ∩ C to m∩m∩(B ∩ C)← m1(B)m2(C)Algorithm 1: Conjunctive combinationHowever, the very nature of the combination operator brings a combina-torial explosion of the focal set. Let mi be the bba defined by mi(X) =12and mi(X\{xi}) =12 : |mi| = 2. Let m∩ be the conjunctive combination of allthose bbas : m∩ = m1 ⊕ . . .⊕mn. For any A ⊆ X , m∩(A) =12n . Therefore,F (m∩) = 2X and |m∩| = 2n.The objective of the following sections will be to guarantee that the sizeof a bba cannot be too large, and to respect its nature as much as possible.4 Christophe Osswald2 Linear algebra for bbasThe definition 3 builds the bodies of evidence bel, pl, betP and q as lineartransformations of m. Considering a bba m on X and an integer K, ourobjective will be to build an bba m′ on X such that |m′| 6 K and fm′(A) =fm(A) for some bodies of evidence f and some subsets A of X .Within this section, we forbid ∅ to be a focal element of m, and we donot allow it to become a focal element of m′. As convenient consequences, wehave bel(A) 6 betP(A) 6 pl(A), bel(X) = 1, and pl(X) = 1.A popular and efficient way to build a bba from a probability or anothersource of uncertain information is to build a least committed bba having thesame pignistic probability than the source [Smets(1990)].Definition 5. Let m be a bba on X . A bba m′ is an isopignisitic of m if∀x ∈ X, betPm(x) = betPm′(x) (8)The bba m′ is the least committed isopignistic of m if for any isopignistic m′′of m and for any A ⊆ X , plm′(A) > plm′′(A).The algorithm 2 builds the least committed isopignistic in O(n2 + n|m|)operations. It contains at most n focal elements.Data: bba m on XResult: bba m′ on Xforall x ∈ X doCalculate p[i] = betP(x)A← X ; k ← |X |while max(p) 6= 0 doi ← argmin(p)m′(A)← kp[i]forall j ∈ p dop[j]← p[j]− p[i]Delete element i from pA ← A\{xi}; k ← k − 1Algorithm 2: Building the least committed isopignisticIf we calculate betP(x) for all x ∈ X , and order the elements of X suchthat pi = betP(xi) > betP(xi+1) = pi+1, the focal elements of the leastcommitted isopignistic are a subset of the Ai = {x1, . . . , xi}.We havepi = betP(xi) =n∑k=i1km′(Ak) (9)Let p be the vector of the pi and y be the vector of the m′(Ai). Wehave p = Bety with Bet a n× n matrix, triangular and inversible. ThereforeControling the number of focal elements 5y = Bet−1p, withBet =1 1213 · · ·1n−11n0 1213 · · ·1n−11n... 0 13 · · ·1n−11n.... . .. . ........ . . 1n−1...0 · · · · · · · · · 0 1n, Bet−1 =1 −1 0 · · · · · · 00 2 −2 0...0 0 3 −3. . ........ . .. . .. . . 0.... . . (n−1) −(n−1)0 · · · · · · · · · 0 n(10)As Bet−1 is a triangular band matrix, we can compute all the m′(Ai) frompi in O(n) operations.With O(n|m|) operations for computing betP, O(n lnn) operations forsorting X , O(n) operations for building the sets Ai (with an adapteddata structure) and O(n) operations for solving the linear system, buildingthe least committed isopignistic costs O(n(lnn + |m|)) operations. Usually,|m|  lnn, and the cost of the least committed isopignistic is not greaterthan the cost of computing betP(x) for the elements of X .The interval [bel(A), pl(A)], containing betP(A), can be interpretated asan uncertainty on A [Janez and Appriou(1996)]. For singletons, bel is trivial:bel(x) = m(x). For sets of size n− 1, pl is trivial: pl(X\{x}) = 1−m({x}).Considering the non-trivial bodies of evidence on the sets of interest {x1},. . . , {xn}, B1 = X\{x1}, . . . , Bn = X\{xn}, we search a bba m′ with thosefocal elements, forming a vectory = (m′({x1}), . . . ,m′({xn}),m′(B1), . . . ,m′(Bn))T(11)which verifies:∀i ∈ b1, nc, plm′({xi}) = plm({xi}) (12)∀i ∈ b1, nc, belm′(Bi) = belm(Bi) (13)We have:plm′({xi}) = m′({xi}) +∑j 6=im′(Bj) (14)belm′(Bi) =∑j 6=im′({xj}) +m′(Bi) (15)As ∀i, plm′({xi}) + belm′(Bi) =∑im′({xi}) +∑i m′(Bj), there are onlyn+1 independent equations among the 2n listed above: we cannot guaranteeto kep at the same time plm({xi}) and belm(Bi) on those 2n focal elements.As q(Bi) = m(Bi) +m(X) and q({xi}) = pl({xi}), introducing common-ality does not bring any new independent equation.6 Christophe Osswald2.1 Mixing Bet with other bodies of evidenceHere we search a bba with 2n focal elements which is an isopignistic of m andrespects an other body of evidence on some focal elements. In the followingexamples, we allow the Ai obtained in section 2 to be focal elements, and wecomplete them with ({xi})i∈b1,nc or the (Bi)i∈b1,nc.With plausibility, we should use the focal elements ({xi})i∈b1,nc. Webuild a vectory = (m′({x1}), . . . ,m′({xn}),m′(A1), . . . ,m′(An))T(16)The constraints are:betP(xi) = m′({xi}) +n∑k=i1km′(Ak) (17)pl({xi}) = m′({xi}) +n∑k=im′(Ak) (18)As A1 = {x1}, we cannot have m′(A1) 6= m′({x1}); we have only 2n−1 focalelements. We drop the term m′({x1}) in y, and the constraint on pl({xi}) toobtain a matrix P such thatPy = (plm({x2}), . . . , plm({x2}), betP(x1), . . . , betP(xn))T (19)The matrix P4 and more generally Pn are:P4 =0 0 0 1 1213141 0 0 0 1213140 1 0 0 0 13140 0 1 0 0 0 141 0 0 0 1 1 10 1 0 0 0 1 10 0 1 0 0 0 1, Pn =0 · · · 0In−1BetnIn−10... Un−10(20)where Betn is matrix obtained in the section 2 and Un−1 the upper triangular(n−1)×(n−1) matrix full of 1.The matrix Pn is inversible, and we can solve this system in O(n3) oper-ations. Overall, we can reduce the focal set of m to 2n− 1 focal elements inO(n(n2 + |m|)) operations, respecting betP and pl on the singletons.With commonality, we obtain the same results : q({xi}) = pl({xi}).With belief, we should use (Bi)i∈b1,nc as focal elements instead of ({xi}).As bel(Bi)+pl({xi}) = 1, we obtain another – but similar – (2n−1)×(2n−1)inversible matrix.Controling the number of focal elements 73 Optimatization by k-means[Denoeux and Yaghlane(2002)] proposed to reduce a bba by adapting the sin-gle linkage hierarchical clustering algorithm to coarsen its focal set. Anotherinteresting family of unsupervized clustering algorithm are the k-means tech-niques, born from the ISODATA method of [Ball and Hall(1965)]. One canadapt this method to find a subset K of 2X limited in size: |K| 6 k.Usual k-means does not guarantee an optimal choice of centers: findingthem is equivalent to the minimum-k center, which is a NP-Complete prob-lem [Garey and Johnson(1979)]. The convergence of the k-means algorithmis guaranteed, but only to a local minimum of the intra-cluster variance.Data: bba m, integer k with k 6 |m|Result: bba mkLet C[1], . . . , C[k] be k focal elements of m [1]repeatforall j 6 k do C[j]← ∅forall A ∈ m do [2]C[argmin(dist(A,Cj))]← C[argmin(dist(A,Cj))] ∪ {A}forall j 6 k do C[j] ← center of C[j]; [3]until ending condition reached [4]forall j 6 k domk(C[j])←∑A∈C[j] m(A)Algorithm 3: k-means, in a general way that applies to focal elements.[1] It is natural to initialize the algorithm with the k focal elements withthe greatest masses. But, as the algorithm converges – if it converges – toa local minimum, it should be a good idea to execute various instances,with random starting sets.[2] The focal element A is affected to the center C[j] such thatdist(A,C[j]) =∣∣∣(A ∩ C[j])∪(A ∩C[j])∣∣∣ (21)is minimal. It corresponds to a natural L1 distance based on an exclusiveor. In case of equal distances to different centers, it is possible to:• choose a random one (the algorithm is no longer deterministic)• use a lexicographical order (elements are no longer equivalent)• try to build balanced clusters (the underlying problem is NP-complete)[3] The usual k-means technique uses the geometrical barycenter of thefocal sets of C[j] seen as points of [0, 1]n : C[j] ←∑A∈C[j] m(A)A.It would build fuzzy focal elements, which is not the way the definition 2accepts them. Therefore, we put x in the new C[j] if and only if :∑A∈C[j],x∈Am(A) >∑A∈C[j],x 6∈Am(A) (22)8 Christophe Osswald[4] As we “move” the centers of the classes to the nearest sharp subset ofX , the total intra-cluster variance is not necessarily decreasing. Therefore,the ending condition must include a maximum steps number, and/or testthe cycles it should encounter.A step of the algorithm 3 costs O(kn|m|) operations. A reasonable numberof steps before ending the loop is k, and we obtain an algorithm in O(k2n|m|)operations. If we want to compare this approach with the ones of the section2.1, we should use k = 2n− 1, and get an algorithm in O(n3|m|) operations.4 ConclusionIn a general way, dealing with basic belief assignments on large frames ofdiscernment need a proper encoding of the focal sets. We propose to usehashtables for this purpose, but this not the only way. We propose two cate-gories of methods for restricting any bba to a bba modest in focal set size.We extend the principle of isopignistic to other bodies of evidence to builda bba with only 2n−1 focal elements, respecting both the pignistic probabilityand another body of evidence of the original bba. We first determine the valueof the bodies of evidence on some simple elements, and then determine therestricted focal set. A linear equation gives the restricted bba.Trying to restrict the focal set to a number of respresentative elementsleads to a NP-Complete problem. We adapt the k-mean algorithm to builda heuristical solution. It is more expensive, but it does not need to define apriori a focal set, and can adapt to more situations.References[Ball and Hall(1965)] Ball GH, Hall DJ (1965) Isodata, a novel methed of data analysisand pattern classification. Tech. rep., Stanford Research Institute[Denoeux and Yaghlane(2002)] Denoeux T, Yaghlane AB (2002) Approximating the com-bination of belief functions using the fast moebius transform in a coarsened frame.International Journal of Approximate Reasoning 31(1–2):77–101[Garey and Johnson(1979)] Garey MR, Johnson DS (1979) Computers and intractability– a guide to the theory of NP -Completeness. Freeman[Janez and Appriou(1996)] Janez F, Appriou A (1996) The´orie de l’Evidence et cadres dediscernement non exhaustifs. Traitement du Signal 13(3):237–250[Shafer(1976)] Shafer G (1976) A mathematical theory of evidence. Princeton UniversityPress[Smets(1990)] Smets P (1990) Constructing the pignistic probability function in a contextof uncertainty. Uncertainty in Artificial Intelligence 5:29–39[Smets(1994)] Smets P (1994) The transferable belief model. Artificial Intelligent 66:191–234[Smets(2002)] Smets P (2002) Matrix Calculs for Belief Functions. International Journalof Approximate Reasoning 31:1–30

Controling the number of focal elements

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Controling the number of focal elements

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