Information Fusion with Belief Functions: a comparison of Proportional Conflict Redistribution PCR5 and PCR6rules for Networked Sensors Roman Ilin Sensors Directorate Air Force Research Laboratory Wright Patterson AFB, OH, USA rilin325@gmail.com Erik Blasch Information Directorate Air Force Research Laboratory Rome, NY, USA erik.blasch@gmail.com AbstractWecompareseveralbelieffusionmethods, includingtheproportionalconflictredistributionrules(PCR5 andPCR6)formultiplesources.ThePCRfusionofevidence methodshaveshownimprovementovertheclassicalDempster-ShaferandBayesianfusiontechniquesinthepresenceof conflicting information. The PCR6 rule shows improvement over PCR5whenthenumberofsourcesincreases.UsingHasse graphicaldiagrams,wehighlightthecomparisonbetweenthe methods.Toourknowledge,thisisthefirstsuchcomparison betweenPCR5andPCR6withmorethantwosources.The resultspointtowardatransitionbetweenPCR5andPCR6at three sources. KeywordsProbability,BeliefFunctions,ConflictingData, Sensor Networks, Hasse Diagram I.INTRODUCTIONOverthepastdecade,therehavebeennumerous discussionsontheproportionalconflictredistributionrules (PCR)inevidentialbelieffunctions.Acomprehensive analysisofbelieffunctionsincludesBayesianmethods, Dempster-Shafertheory(DST),andDezert-Smarandache theory(DSmT),astheoriginofthePCRrules[1].Themost current PCR5 and PCR6 rules address the limitation of the DS ruleforcombiningpotentiallyhighlyconflictingsourcesof evidencebyredistributingthebeliefmass.Intheliterature, few investigations directly compare the performance of PCR5 versus PCR6 and if so, only for small number of sources. Hongfeietal.[2]comparedPCR1-PCR6withtwo sensorsforthreeclassclassificationsandfoundequivalent performancewhilesuggestingPCR6fwithadistribution proportionvariationbasedontheglobalbasicbelief assignment(gbba).InanotherstudybySmarandacheand Dezert [3], PCR5 and PCR6 arecompared for two priors and three classes with additions of a PCR5 fusion-conditional rule.The origins of the PCR rules development begin with the classic paper in 2005 [4] for two sources. In 2006, a summary ofDSmTwasprovidedin[5]andMartinandOsswald presentedPCR6[6]formultiplesources.Together,in2008, thesegroupscombinedforasummaryofqualitativeand quantitative belief combination rules [7]. ThebeliefanalysisusingPCRhasbeenappliedtomany areas showing promise for different applications [8], [9], [10]. OneareaofinterestistargettrackingwhichbeganwithDS methods[11],[12],[13].Tracking,classification,and identificationincludedtheJoint-BeliefProbabilityData AssociationFilter(JBPDAF)withimageryintelligence (IMINT) with a DST classification. A update to DST included PCR5trackingbyDezertandTchamovaetal.[14]and Dambrevilleetal.usingthePCR6[15].Pannetieretal. combinedthePCRmethodsfortrackingandclassification with IMINT [16] along with interactive multiple model (IMM) approaches[17].Othercasesincludetheparticlefilter(PF) withthePCR5[18].Finally,arecentupdateincludedthe Tchamova, J. Dezert T-Conorm-Norm (TCN) as comparisons with thePCR5 fusion rule, Dempster's rule,andupdateswith the fuzzy fusion rules [19]. TherelationshipsbetweenDST,DSmT,andBayesian approaches are demonstrated in Figure 1, which is based on J. DezertsDSmTtutorial.Eachsourceassignsconditional probabilitiestopossibletargetclasses,givenobserved measurements. In the Bayesian approach, the probabilities are fuseddirectly.DSTandDSmTintroduceadditionallevelsof uncertaintyrepresentationandmethodsforconflict redistribution. Figure1ComparisonofBayesian,Dempster-Shafer,andPCR5Fusion TheoriesAlong with the comparisons and applications of the PCR5 and PCR6 (although individually), there were developments in the PCR assessments. In particular, the relationship between DST 18th International Conference on Information FusionWashington, DC - July 6-9, 2015978-0-9964527-1-72015 ISIF 2084andBayesianmethodswasinvestigatedin[20]withthe following conclusion: Onlyintheveryparticularcasewherethebasicbelief assignmentstocombineareBayesianandwhentheprior informationisuniform(orvacuous),theDempstersruleremains consistentwiththeBayesfusionrule.,Dempstersruleis incompatiblewiththeBayesruleanditisnotageneralizationof the Bayes fusion rule Recently,thePCR6wascomparedtoanaveragingrule[21]. Asensitivitycomparisonwasthenmade[22]overthemany rulesincludingDempsters rule, Yagers rule, Floreas robust combination rule (RCR), disjunctive rule, Dubois and Prades rule,proportionalconflictredistributionrule(PCR5),and meanrule[8];howeverPCR6wasnotincludedinthe comparison. Thereviewherepresentsaninterestingdevelopment.In mostcases,eitherthePCR5orthePCR6wereusedand typicallycomparedtoDSTmethods.Innocases,werethey comparedtogetherfortrackingandclassification.Workby Blasch et al. [1] aimed to compare DST, Bayes, and PCR. The comparisondemonstratedthebenefitsofPCRoverDSTand Bayes, but the study was limited as the PCR5 and PCR6 rules are the same for two source fusion.In this study,weseek to compare PCR5with PCR6with three and more sources, and thus add to the discussion of PCR rules.Therestofthepaperisasfollows.Sect.IIdiscusses beliefformalisms.SectIIIintroducesPCR5andSect.IV PCR6.Sect.Vdoesacomparison.Finally,Sect.VIandVII providediscussionandconclusions.Inadditiontothe performance comparison, we utilized graphical illustrations of PCR5 and PCR6 rules with Hasse diagrams in order to clarify the differences between the two. II.BELIEF FUNCTIONS FORMALISM Dempster-Shafer theory assumes a finite set u of possible answers to a question, referred to as the frame of discernment. Theeventscorrespondtosubsetsof u,withtheiruncertainty quantifiedbythebasicprobabilityassignment,orbpa(also referredtoasbasicbeliefassignment,orbba),whichisa function m: 2H - |u,1] such that m() = u, (1) and m(A)ALH= 1. (2) Thekeyideahereisthatnotallsubsetsof u encodeinferred information; somearebasic events and can beassigned basic (elementary)probabilitymass.Thosesubsetsthatweassign non-zerom()arecalledfocalelements.Thisstructureis called a body of evidence. In the most general case all subsets of u (allelementsofthepowersetof u)mayserveasfocal elements,howeverinpracticethesetoffocalelementsisa subset of the powerset. DSmTgeneralizestheframeofdiscernment u toconsist ofpossiblyoverlappingsets.Overlappingsetshighlightsthat thefullsetofpossibleeventsincludesallpossible intersectionsandunionsoftheelementsofu -whichisa largersetofeventsthanthatofDST.Inthisworkweonly consider the Dempster model. A belief function Bcl: 2H - |u,1] is computed from the bpa as follows: Bcl(A) = m(B)BLA, (3) whereA L u.Thebpacanberecoveredfromthebelief function using the following formula, m(A) =(-1)|A\B|Bcl(B)BLA. (4) Thusthereisaone-to-onecorrespondencebetween m and Bcl. The belief functions satisfy the following properties: Bcl() = u, Bcl(u) = 1, Bcl __An=1_ (-1)|I|+1Bcl __AeI_.Ic{1,2..n],I= (5) Thelast propertyshows that belief functions arenon-additive [8] even in the case of two disjoined subsets A and B: Bcl(A U B) Bcl(A) + Bcl(B),incontrasttoadditivityof probability functions. If, and only if, the focal elements of bpa are1-elementsubsets(singletons)of u,does Bcl become equivalent and reduces to a probability function. Acombinationofevidencecomingfromdifferentsources isakeyfeatureofDST.Giventwobodiesofevidencewith bpas m1and m2 that are combined or fused, the conjunctive rule is given by the following formula: m1r2(A) = m1(B)m2(C)BrC=A. (6) The result of (6) needs to be normalized in order to satisfy (2).Dividing m1r2 by 1-K achieves normalization, with K = m1(B)m2(C)BrC=. (7) Formulas (6) and (7) correspond to the Dempsters rule of combination. The un-normalized version (6) of the Dempsters rule is used in Transferable Belief Models [23]. Thequantity Kis referred to as thetotal conflicting mass. The classical Dempsters rule is inconsistent in the presence of conflictingevidence,ashasbeenshownrecentlyin[20], althoughtheinconsistencieshavebeenpointedoutbyearlier researchers,startingwithLotfiZadeh.Therefore,other combinationruleshavebeenproposed.Weconsiderthe ProportionalConflictRedistributionrules(PCR5andPCR6 2085rules)proposedin[4],[6].ThemainideabehindthePCR rulesistoconsidereachterminthetotalconflictingmass separatelyandredistributethemassofthetermtotheevents involvedinthistermsconflict.Thenextsectionsprovidea detailed description of these rules. III.PROPORTIONAL CONFLICT REDISTRIBUTION 5 (PCR5) ThegeneralformulasforPCR5andPCR6aregivenin[4], and[6]respectively.Duetolargenumberofindicesand nestedsummations,theyaredifficulttounderstand.Several examplesoftheseformulasforsimplecasesaregiveninthe earlierpublications,forexample[5].Here,weattemptto make therulesmoreunderstandable by illustrating themwith diagrams. We consider the fusion of an arbitrary number of sources, S e {1,2, .].ThefirststepinbothPCR5andPCR6isto computetheconjunctivecombinationofallsources,whichis the generalization of (6), for all subsets of u, A L u: mr(A) =m1(X1)m2(X2) .mS(XS)X1rX2r.rXS=A. (8) The total conflicting mass is given as follows, K =m1(X1)m2(X2) .mS(XS)X1rX2r.rXS=. (9) Eachtermin(9)isreferredtoasapartialconflicting mass,andisdenotedby m(X1 r X2 r .r XS).Eachpartial conflicting mass is redistributed to its subsets in proportion to the basic probability mass already assigned to these subsets. Inordertoillustratetherules,wewillutilizeagraphical representationofacollectionofsets,calledHassediagrams. In general, Hasse diagrams represent partially ordered sets, or posets [24]. Poset is a set with relation of partial order defined onit.Thisrelationisusuallyreferredtoaslessthan, denotedby .Onthediagrams,theelementsofaposetare shownascirclesandtherelatedelementsareconnectedby lines. In the case of sets, two sets are related if one of them is asubsetoftheother-inotherwordstherelationofpartial order is the set inclusion relation. EXAMPLE 1: ConsiderthecaseofS=3sources.Theframeofdiscernment contains only 2 elements, A and B, u = {A, B], with the basic probability mass assignments for each source given in Table 1. There are, of course, 4 elements in the powerset of u. TABLE 1 {A]{B]{A, B] m10.60.4 m20.30.7 m30.10.9 Theconflictingmassisobtainedbylistingallset combinationsfromthe3sourcesthatresultinempty intersection and have non-zero mass assignments. In this case, thereare43=64combinationsofsets(4fromeachofthe3 sources).Outofthese,wehaveonlythefollowing3partial conflicts (Table 2): TABLE 2 Partial Conflict TermsValues m1({A])m2({B])m3({B])0.018 m1({A])m2({A, B])m3({B])0.042 m1({A])m2({B])m3({A, B])0.162 Total Conflicting Mass:0.222

Considerthefirstpartialconflict.Theredistributionofthe mass (0.018) using PCR5 rule is illustrated in Fig. 2. Figure2RedistributionofthefirstpartialconflictingmassfromTable2 accordingtoPCR5. XA and XB arethemassesaddedtothesetsAandB respectively. ThemassesXA andXB arecomputedusingthefollowing formulas, with curvy brackets removed for clarity: K = m1(A)m2(B)m3(B)XA =m1(A)m1(A) + m2(B)m3(B)KXB =m2(B)m3(B)m1(A) + m2(B)m3(B)K (10)ThekeyfeatureofPCR5isthatthemassassignments comingfrom several sourcesand assigned to thesame subset (B in this example) are multiplied, and after this multiplication theconflictingmassisdividedproportionallytothemassin each unique conflicting subset. Thus in this example we divideK into two parts proportional to m1(A) and m2(B)m3(B). Consider the next partial conflict (row 2 in Table 2). The redistributionofthemass(0.042)usingPCR5ruleis illustrated in Fig. 3. Figure3RedistributionofthesecondpartialconflictingmassfromTable2 accordingtoPCR5. XA and XB arethemassesaddedtothesetsAandB respectively. 2086Here, the masses XA and XB are computed using the following formulas (again with curvy brackets removed for clarity): K = m1(A)m2(A, B)m3(B)XA =m1(A)m1(A) + m3(B)KXB =m3(B)m1(A) + m3(B)K (11)Inordertounderstandthisredistribution,weinvokethe conceptofcanonicalformoftheset(see[8],Vol.2,page 209).Inthiscase,c(A r (A U B) r B) = A r B ,which clarifiesthatonlythesets A and B remaininginthecanonical formareinconflictwitheachother.Theproportionsofthe conflicting mass K are computed with respect to the canonical form,whichiswhy m2(A, B) isnotusedintheproportions. Massm2(A, B)isusedhowever,inthecomputationofK itself.Thethirdpartialconflictisresolvedanalogously,as shown in Fig. 4 and the formulas below. Figure4RedistributionofthethirdpartialconflictingmassfromTable2 accordingtoPCR5. XA and XB arethemassesaddedtothesetsAandB respectively. The redistributed massed are given as follows. K = m1(A)m2(A, B)m3(B)XA =m1(A)m1(A) + m3(B)KXB =m3(B)m1(A) + m3(B)K (12)At this point we observe that the first partial conflict was also redistributedaccordingtothecanonicalform,exceptwehad to remember to multiply the masses assigned to the same sets by different sources. The general procedure for PCR5 appears to be as follows: The PCR5 Procedure 1.Compute conjunctive consensus (8), (9) 2.List all partial conflicts 3.For each partial conflict, a.computecanonicalformandthecorresponding masses (multiplying some of them) b.Redistributetheconflictwithrespecttothesetsin the canonical formEXAMPLE 2: Consider another example that illustrates the PCR5 procedure. The set u consists of 3 elements, u = {A, B, C]. The following bodiesofevidencearegiveninTable3.Therearenow8 elements in the powerset of u. TABLE 3 {A]{B]{A, B]{C]{A, C]{B, C]{A, B, C] m10.20.8 m20.30.7 m30.30.7 Thefollowingtableliststhepartialconflictsforthis example. TABLE 4 Partial Conflict TermsValues m1({A])m2({B])m3({B, C])0.018 m1({A])m2({A, B, C])m3({B, C])0.042 m1({A])m2({B])m3({A, B, C])0.042 Total Conflicting Mass:0.102 Considerthefirstconflict.Thediagramdepictingthesets involved in the conflict is given in Fig. 5. Figure5RedistributionofthefirstpartialconflictingmassfromTable4 accordingtoPCR5. XA and XB arethemassesaddedtothesetsAandB respectively. Inthiscase,thecanonicalformofthesetsinvolvedinthe partial conflict is c(A r B r (B U C)) = A r B. The set B U C disappearsfromthecanonicalform.SinceB U Cwas absorbed by B, and the two sets are not identical, and we do notmultiplytheirweights.Thustheredistributionisdoneas follows: K = m1(A)m2(B)m3(B, C)XA =m1(A)m1(A) + m2(B)KXB =m2(B)m1(A) + m2(B)K (13)In the second conflict, we obtain the following diagram: 2087 Figure6RedistributionofthesecondpartialconflictingmassfromTable4 accordingtoPCR5. XA and XBC arethemassesaddedtothesetsAandBC respectively. Thecanonicalformforthisconflictisc(A r (B U C) r(A U B U C)) = A r (B U C).Therearenomasses that need tobemultiplied.Thus,theredistributioniscomputedas follows: K = m1(A)m2(A, B, C)m3(B, C)XA =m1(A)m1(A) + m3(B, C)KXBC=m2(B, C)m1(A) + m3(B, C)K (14) Next we discuss PCR6 in a similar graphical manner. IV.PROPORTIONAL CONFLICT REDISTRIBUTION 6 (PCR6) ThePCR6rulewasproposedassimplificationofPCR5[6]. ThegeneralformulaforPCR6isalsocomplexandcanbe difficult to understand. Herewepresent PCR6 as a procedure illustratedwith diagrams andtables.Theprocedure for PCR6 can be summarized as follows. The PCR6 Procedure 1.Compute conjunctive consensus (8), (9) 2.List all partial conflicts 3.Foreachpartialconflict,redistributetheconflicttothesets involvedintheconflictproportionallyrelativetotheir masses. Thetransformationtocanonicalformisabsenthereandalso themultiplicationofthemassassignedtothesamesetsby different sources. Figure7RedistributionofthefirstpartialconflictingmassfromTable2 according to PCR6. XA and XB1 +XB2 are the masses added to the sets A and B respectively. Consider the first example fromSection III, given in Table 1. TheredistributioncanbeillustratedbythediagraminFig.7, corresponding to the first partial conflict listed in Table 2. As we can see, themass that is moving to set B consists of two parts. The formulas are given below. K = m1(A)m2(B)m3(B)XA =m1(A)m1(A) + m2(B) +m3(B)KXB1 =m2(B)m1(A) + m2(B) + m3(B)KXB1 =m3(B)m1(A) + m2(B) + m3(B)K (15)Consider the diagram in Fig.8, for the second conflict in Table 2. Figure8RedistributionofthesecondpartialconflictingmassfromTable2 according to PCR6. XA and XB and XAB are the masses added to the sets A, B, and AB respectively. ThedifferencebetweenPCR5andPCR6ismoreprominent heresincetheset A U B receivessomemassincaseofPCR6 anditdidnotreceiveanymassinthecaseofPCR5.The formulas are as follows. K = m1(A)m2(A U B)m3(B)XA =m1(A)m1(A) +m2(A U B) + m3(B)KXAB=m2(A U B)m1(A) + m2(A U B) +m3(B)KXB =m3(B)m1(A) + m2(BA U B) +m3(B)K (16)Inlightoftheseexamples,thegeneralprocedurefor PCR6 becomes clear. HavingpresentedPCR5andPCR6usingthediagrams, wenowseekacomparisonofmultiplesensorsovermultiple sets. The method supports both classification and tracking. V.PERFORMANCE COMPARISONS In this section, we compare the performance of several fusion rulesusingsimulatedscenarios.Thesimulationis implemented for an arbitrary number of sensors S and possible targetsN.Weassumethateachsensorcanbeassigneda confusion matrix that characterizes its classification accuracy. 2088Ifthetrueobjectis oI andthesensormadeadecision o],the corresponding confusion matrix entry is defined as CH(i, ]) =Pi (o]|o). This implies that, given that thesensor declared o], wegettheprobabilitymassassignedtothesetofpossible targetsas Pi (o|o]).ThismasscanbecomputedusingBayes formula, but we need to know the a priori target probabilities:

Pi(oI|oj) = Pi(oj|oI) P(oI)Z.(17) Thedenominator Z isthenormalizationfactor.Thus,the rowsofconfusionmatrixcontaintheprobabilitydistribution ofthetargetdeclarationgiventhetruetargetclass.Ateach stepofthesimulation,wesamplefromthedistributiongiven by Pi ( |o).WethenusetheBayesformula(17)tocompute theposteriorprobabilitiesforeachtarget.Theseprobabilities are fused directly in the Bayesian fusion approach. In order to applythebelieffunctionbasedapproaches,weselectthe argument with the maximum posterior probability as the most likely target and assign themaximum posterior probabilityas the probability mass for this target. The rest of the probability massisassignedtothesureevent u.Thebodiesofevidence correspondingtoeachsensorarethencombinedwitheach otherandwiththebodyofevidencefromtheprevioustime step. The simulation procedure is outlined below. The generic FuSE function in step 8 is substituted by a particular rule, such as PCR6, etc. Line 9 computes pignistic probabilities used for decision making, as described in [25]. Multi-sensor fusion simulation Given:N number of targets S number of sensors Kmux number of time steps in the simulation CHs confusion matrices, s = 1. . S Ik-target type for time step k, k = 1. . Kmux

m0(u) = 1, the initial bodies of evidence is total ignorance 1.For each time step k 2.For each sensor s 3.oskis a sample from Pi(osk|Ik, s) CHs(, Ik) 4.Compute Pi (otk|osk) for otk = 1. . N 5.otk = aig maxotk Pi (otk|osk) max a posteriori target6.Create body of evidence usk, such that 7. msk(ot) = maxot Pi (otk|osk)msk(u) = 1 -maxot Pi (otk|osk) 8.Combine previous fused body of evidence with the new bodies of evidence, mk = FuSE(mk-1, m1k, . . , mSk) 9.Compute pignistic probabilities from mk, Bctk 10.Make target declaration based on the maximum Bctk

A. Case 1: Tracking and Classification: 2 sources AsampleresultofsimulationisshowninFig.9.Thetrue classoftheobservedtargetchangesovertimeandthus conflictisintroducedbetweenthesubsequentmeasurements. Inaddition,falsetargetdeclarationscauseconflictaswell. Theabilityofdifferentfusionmethodstorecoverfromthese conflicts in a timely manner can be observed. Wesetthenumberoftimestepsto120,andperform50 independent runs. The average values over 50 runs are shown in the Fig. 9, which is made for target class 1.

Figure 9 Target tracking and classification simulation, in case of N=2 possible targets and S=2 sources.The Truth value equals to 0 when the true target class is not 1 and it equals to 1 when the true target class is 1. The plot showshowtheprobabilitymassassignedtotargetclass1by thefusedbodyofevidencechangesovertime,for5 combinationrules.Aswecansee,thePCR5andPCR6rules arethebestathandlingconflictinginformationwhichis reflected in the short transition time when the true target class switches (steps 20, 50, 70, and 80). Theclassificationaccuracyofthefusionrulesis estimatedbydividingthenumberofstepswithcorrecttarget classdeclarationbythetotalnumberoftimesteps.Asitcan beexpectedfromFig.9,theaccuracyofPCR5andPCR6is higher than the accuracy of DST and the other rules. B. Tracking and Classification: S sources Inthenextexperiments,weranthesimulationswith increasingnumberofsources.Theresultsobtainedforthe2 class case are shown in Fig. 10. Figure10ClassificationaccuracyoftargettrackingandclassificationinsimulatedscenarioswithS=1,2,3,4,and 5sources.AVGsimplyaveragesthe bpas. 2089These results show an interesting trend. As the number of sourcesincreases,theperformanceofPCR5significantly deteriorates.Theotherrulesshowconsistentperformance, with PCR6 resulting in the highest accuracy. In our analysis, we used a variety of confusion matrices to representthesourceclassificationusedforthesimulations; Table5showssomeexamplesofrandomlygenerated confusion matrices. TABLE 5 CM1 0.89920.1008 0.06470.9353 CM2 0.87630.1237 0.07170.9283 CM3 0.73640.2636 0.19080.8092 CM4 0.62710.3729 0.30710.6929 CM5 0.68560.3144 0.27560.7244 InFig.11,weshowanotherresultsofasimilar experiment, where the number of time steps was set to 30 and the number of sources varied from 1 to 7. The cross-over point between PCR5 and PCR6 isbetween 3 and 4 sources.Again, theperformanceofPCR5decreasestobasicallychancelevel for 6 and 7 sources (and less than DST). Figure11ClassificationaccuracyoftargettrackingandclassificationinsimulatedscenarioswithS=1through7sources,for2targets.AVGsimply averages the bpas. Note that in our simulation, the number of masses of evidence that need to be combined equals to the number of sources plus oneaswecombinethecurrenttimestepevidencewiththe fused body of evidence from the next step. This explains why theperformanceofPCR5andPCR6isexactlythesamefor onesource,sincetheproceduresbecomeidenticalwhen combining two bodies of evidence [6]. VI.DISCUSSION Inordertodemonstratetheresultspresentedinthisworkwe have accomplished the following tasks. 1. Investigated PCR5 and PCR6 rules and presented them in graphical form. Although this is not a novel contribution, webelievethatitsignificantlyimprovesthereadability of PCR rules by non-experts in the area. 2. Implemented PCR5 and PCR6 for S sources and N targets. OurimplementationofPCR6wascomparedagainst ArnaudMartinstoolboximplementationtoyield identicalnumericresults.However,ourimplementation issignificantlyfasterthanMartins.Wewereunableto find alternative PCR5 implementation to compare to our own.3.VerifiedthatPCR5andPCR6givethesameresultsfor S=2andanynumberoftargetsN.Thisagreementwith thetheoryboostedourconfidenceinourPCR5 implementation. 4.Implementedasimulationofmultiplesourcetarget trackingandclassificationforarbitrarynumberof sources and possible targets. 5.ComparedclassificationaccuracyofPCR5,PCR6,DST, Average, and Bayesfusion rulesforN=2 target tracking and classification problem for the number of sources S=1 through 7. Whiletherehasnotbeenastudyofthiskindreported,we foundthatPCR5isnotwellsuitedformanysources.An obviousconcernisthattheimplementationneedstobe validated; however, as per the literature, the trend is consistent where PCR5 is not suggested for many sources. What is thus a contributionisthatPCR5istypicallyusedwith2sourcesof whichwehaveshownthatitisalsousefulforthreesources. Itsperformanceformorethanthreesourcesislowerthan PCR6.Wesuggestthatfurtherindependentinvestigationsbe conducted to compare PCR5 and PCR6. We also note that the averagefusionruledemonstratedperformancecomparableto PCR6.ThisconcurswiththerecentworkcomparingPCR6 and average fusion rule [21]. VII.CONCLUSIONS Inthisstudy,wehavecomparedseveralfusionmethodsand demonstratedtheimprovedperformanceofPCR5andPCR6 overtheclassicalDempster-ShaferandBayesian methodologies.Additionally,weutilizedHassediagramsto improve the understandability of the PCR methods. Through a comparativestudy,theadvantageofPCR6overPCR5for morethanthreesourceswaspresented.Futureworkincludes updatingthecomparisonwithdevelopmentsinPCRsuchas thefuzzy-combinationrules,combinationofour implementation with latest techniques in tracking and situation modeling[26][27],andutilizationofourimplementations with real data.ACKNOWLEDGMENT Support for this research was provided by the Air Force Office ofScientificResearch,DepartmentoftheAirForce,grant number14RY04CORaswellas13RI02CORfromthe DDDASprogram.WeutilizedtheMatlabtoolboxfrom ArnaudMartinforDSTcomputationsandPCR6verification, whichwasappreciated. InitialPCR5developmentswerefrom Jean Dezert. REFERENCES 2090 [1] E. Blasch, J. Dezert and B. Pannetier, "Overview of Dempster-Shafer and Belief Function Tracking Methods," in Signal Processing, Sensor Fusion, and Target Recognition XXII, Proc. of SPIE, 2013.[2] L. Hongfei, J. Hongbin, T. Kangsheng and F. 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