SequentialAdaptiveCombination ofUnreliable SourcesofEvidenceZhun-gaLiuQuanPanYong-meiChengJean Dezert Abstract-Intheories of evidence, several methods have beenproposedtocombine agroupof basic belief assignments al-together at a given time. However, in some applications indefenseorinroboticstheevidencesfromdifferentsourcesareacquiredonlysequentiallyandmust beprocessedinreal-timeand the combination result needs to be updated the most recentinformation. An approach for combining sequentially unreliablesourcesof evidenceispresentedinthispaper. Thesourcesofevidence are not considered as equi-reliable in the combinationprocess, and no prior knowledge on their reliability is required.The reliability of each source is evaluated on the y by a distancemeasure, whichcharacterizesthevariationbetweenonesourceof evidence with respect to the others. If the source is consideredas unreliable,then itsevidence is discountedbefore enteringinthe fusion process. Dempsters rule of combination and its mainalternatives including Yagers rule, Dubois and Prade rule, andPCR5areadaptedtoworkunderdifferent conditions. Inthispaper, weproposetoselectthemostadaptedcombinationruleaccording to the value of conicting belief before combining theevidence. Thelastpartofthispaperisdevotedtoanumericalexample to illustrate the interest of this approach.Keywords: evidence theory, combination rule, evidencedistance, conicting belief.I. INTRODUCTIONEvidence theories1are widely applied in the eld of infor-mation fusion. A particular attention has been focused on howtoefcientlycombinesourcesof evidencealtogether at thesame time (static approach), and many rules aside Dempstersrule have been proposed [1], [2], [6], [9]. In many applicationshowever, theevidences fromdifferent sources areacquiredsequentiallybydifferent sensors or humanexperts andthebelief updating and decision-making need to be taken in real-time which requires a sequential/dynamic approach rather thana static approach of the fusion problem.Usuallytheevidences arisingfromdifferent independentsources are often considered equally reliable in the combina-tionprocess, whenthepriorknowledgeabout thereliabilityof each source is unknown. However, all the sources ofevidence to be combined can have different reliabilities in realapplications. If the sources of evidence are considered as equi-reliable, theunreliableonesmaybringaverybadinuenceoncombinationresult, andevenleadstoinconsistentresultsand wrong decisions. Thus, the reliability of each source mustbetakenaccount inthefusionprocess as best as possibletoprovide a useful andunbiasedresult. Inthis work, wepropose to evaluate on the y the relability of the sources tocombinebasedonanevidential distance/reliabilitymeasure.Fromthisreliabilitymeasure, onecandiscount accordinglytheunreliablesourcesbeforeapplyingaruleofcombinationof basic belief assignments (bbas).Many rules, like Dempsters rule [7] and its alternatives canbeusedtocombinesourcesofevidencesexpressedbybbasandtheyall havetheir drawbacksandadvantages(see[8],Vol. 1, for a detailed presentation). Dempsters rule, is usuallyconsideredwelladaptedforcombiningtheevidencesinlowconict situations and it requires acceptable complexity whenthedimensionoftheframeofdiscernment isnot toolarge.Dempsters rule however provides counter-intuitive behaviorswhenthe sources evidences become highlyconicting. Topalliate this drawback, several interesting alternatives havebeen proposed when Dempsters rule doesnt work well,mainly: Yagers rule[9], Dubois andPraderule(DPrule)[2], and PCR5 (proportional conict redistribution rule no 5)[8] developedinDSmTframework. The difference amongDempstersruleanditsmainalternativesmainlyliesinthedistribution of the conicting belief m() which is generallyusedtocharacterizethetotalamountofconict[4]betweensources. Inthispaper, weproposetoselect theproper ruleof combination based on the value of the total degree ofconict m(). The last part of this paper presents a numericalexample toshowhowthe approachof sequential adaptivecombination of unreliable sources of evidence works.II. PRELIMINARIESA. Basics of Dempster-Shafer theory (DST)DST[7] is developedinShafers model. Inthis model,a xed set ={1, 2, . . . , n} is called the frame ofdiscernment of fusionproblem. All the elements inaremutuallyexclusiveandexhaustive. Theset ofall subsetsof1DST (Dempster-Shafer Theory) [7] or DSmT (Dezert-Smarandache Theory) [8].Originally published as Liu Z., Pan Q., Cheng Y., Dezert J., Sequential adaptive combination of unreliable sources of evidence, In proc. of International Workshop on Belief Functions, Brest, France, April 2-4, 2010, and reprinted with permission.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 423iscalledthepower set of , andit isdenoted2. Forinstance, if = {1, 2, 3}, then2= {, 1, 2, 3, 1 2, 1 3, 2 3, 1 2 3}. Abasicbeliefassignment(bba), also called mass of belief, is a mapping m : 2 [0, 1]associated to a given body of evidence B such thatm() = 0and A2 m(A) =1. Thecredibility(alsocalledbelief)of Ais dened by Bel(A) = B2BAm(B). Thecommonality functionq(.) and the plausibility functionPl(.)are also dened by Shafer in [7]. The functionsm(.),Bel(.),q(.) andPl(.) are in one-to-one correspondence.Let m1(.) and m2(.) betwobbas providedbytwoin-dependent bodiesofevidence B1and B2overtheframeofdiscernment . The fusion/combination ofm1(.) withm2(.),denoted m(.) = [m1 m2](.) is obtained in DSTwithDempsters rule of combination as follows:m() = 0m(A) =

X1X2=Am1(X1)m2(X2)

X1X2=m1(X1)m2(X2)A = , A 2(1)Thedegreeofconictbetweenthebodiesofevidence B1and B2is dened bym() =

X1,X22X1X2=m1(X1)m2(X2) (2)Dempsters rule can be directly extended to the combinationof S independent and equally reliable sources. It is a commu-tative and associative rule of combination and it preserves theneutral impact of thevacuousbelief assignment denedbymvba() = 1.B. Main alternatives to Dempsters ruleDempsters rule yields counterintuitive results when theevidences highly conict because of its way of assigning themassofconictingbelief m(). Thus, alotofalternativesto Dempsters rule have been proposed for overcoming limita-tions of Dempsters rule. The main alternative rules includingYagers rule [9], DP rule [2] and PCR5 [8] are briey recalled. Yagersrule:Yageradmitstheconictingbeliefisnotreliable. Som()istransferredtothetotal ignorancein Yagers rule. It is given bym() = 0 and forA = ,A 2bym(A) =

X,Y 2XY =Am1(X)m2(Y ), forA = m() = m1()m2() +

X,Y 2XY =m1(X)m2(Y )(3) Dubois &Prade rule: This ruleassumes that if twosources of evidence are in conict, one of them is rightbut we dont know which one. Thus, ifX Y= , thenthe mass committed to the setX Yby the conjunctiveoperatorshouldbetransferredtoX Y . Accordingtothisprinciple, DPruleisdenedbym()=0andforA = andA 2bym(A) =

X,Y 2XY =Am1(X)m2(Y )+

X,Y 2XY =XY =Am1(X)m2(Y ) (4) PCR5rule:PCR5transfersthepartialconictingmassto the elements involved in the conict, and it is consid-eredasthemost mathematicallyexact redistributionofconicting mass to nonempty sets following the logic ofthe conjunctive rule. PCR5 is dened bym() = 0 andA = , A 2bym(A) =

X1,X22X1X2=Am1(X1)m2(X2)+

X22X2A=[m1(A)2m2(X2)m1(A) +m2(X2) +m2(A)2m1(X2)m2(A) +m1(X2)] (5)The details, examples and the extension of PCR5 formula(5) forS> 2 sources are given in [8].C. Discounting source of evidenceWhenthesourcesofevidencesarenotconsideredequallyreliable, it is reasonable to discount each unreliable source si,i=1, 2, . . . , Sbyareliabilityfactor i [0, 1]. Followingtheclassical discountingmethod[7], anewdiscountedbbam

(.)isobtainedfromtheinitial bbam(.)providedbytheunreliable sourcesias followsm

(A) = i m(A), A = m

() = 1

A2A=m

(A)(6)i = 1 means that the total condence in the source si, and theoriginal bba doesnt need to be discounted. i = 0 means thatthe source is si is totally unreliable and its bba is revised as avacouous bba m

() = 1, which will have a neutral impact inthe fusion process. In practice, the discounting method can beused efciently if one has a good estimation of the reliabilityfactorofeachsource. Weshowinthenextsectionhowonecan evaluate the relability of a source.III. EVALUATING THE RELIABILITY OF EACH SOURCEWithout prior knowledgeonthereliabilityof thesourcesofevidence, weproposetoevaluatethereliabilityfactorsofeachsourcebasedonthedistancebetweenthebbafromagivensourcesiwithrespect totheothers. Ifthebbaofthegiven source, say si varies too much with respect to the others,thissourceofevidenceisconsiderednotreliableanditwillbediscountedbeforetobecombined. Wewill showfurtherhowthediscounting/reliabilityfactor canbeestimated. WeimplicitlyassumeherethatthefollowingprincipleTruthisreected by the majority of opinions holds.Advances and Applications of DSmT for Information Fusion. Collected Works. Volume 424In [3], Jousselme et al. have proposed the following distancemeasuredJ(m1, m2)betweentwobbas2m1m1(.)andm2m2(.) dened on the same power set 2:dJ(m1, m2) =

12(m1 m2)TD(m1 m2) (7)where D is a 2|| 2||positive matrix whose elements aredenedasDij |AiBj||AiBj|whereAiandBjareelementsofthepowerset 2. dJ(m1, m2) [0, 1] isadistancewhichmeasures the similarity between m1 and m2 considering boththe values and the relative specicity of focal elements of eachbba.The total degree of conictm() obtained from all focalelements which are incompatible doesnt actually capture thesimilarity between bbas as shown by Martin et al. in [5].If Npiecesofevidencem1, m2, . . . , mNarecombinedsequentially, two approaches similarly with [5] could be usedtomeasure the variationbetweenmjandthe others. OneconsiderstheaveragevaluedJbetweenmjandtheotherswhich is given bydj11(mj) =1j 1j1

i=1dJ(mj, mji) (8)The other one is simply dened asdj12(mj) = dJ(mj, mj11) (9)where mj11 mj11(.) is obtained by the sequentialcombininationof the bbas m1(.), m2(.),. . . , mj1(.), i.e.mj11(.) = (((m1 m2) m3) mj1)(.) with a fusionrule such as Dempsters rule, Yagers rule, DP rule, PCR5, etc.Thesecondmeasure, dj12(mj), reectsonlythedifferencebetweenmjandthecombinedbbamj11andthus cannotpreciselymeasurethesimilaritybetweenmjandtheotherindividual evidences m1(.), m2(.), . . . , mj1(.) because someinformation on specicities of these individual bbas has beenlost forever through the fusion process. The following exam-ples will show the distinction between these two methods.Example 1: Lets consider theframeof discernment ={A, B, C}, Shafers model and the same following bbasm1(.) : m1(A) = 0.5, m1(B) = 0.2m1(A B) = m1(C) = m1() = 0.1m2(.) : m2(A) = 0.5, m2(B) = 0.2m2(A B) = m2(C) = m2() = 0.1...mj(.) : mj(A) = 0.5, mj(B) = 0.2mj(A B) = mj(C) = mj() = 0.1The difference betweenmj(.), forj 2, and all the bbasmi(.), for i < j according to formula (8) gives dj11(mj) = 0,which shows correctly that mj(.) is identical to the other bbasmi(.), fori