Chinese Journal of Electronics Vol.27, No.4, July 2018 A New PCR Combination Rule for Dynamic Frame Fusion∗ JIN Hongbin1, LI Hongfei2,3, LAN Jiangqiao1 and HAN Jun1 (1. Air Force Early Warning Academy, Wuhan 430019, China) (2. Shanghai Jiao Tong University, Shanghai 200240, China) (3. No.95174 Unit of PLA, Wuhan 430040, China) Abstract — Dynamic frame fusion which is based on with time taking in account the non-existential integrity hybridDSmmodelisanimportantproblemininformation constraints to establish new model of frame[3]. fusion. But the traditional combination rules are mainly In the classical Dempster-Shafer theory (DS under fixed discernment frame (Shafer model and free theory)[4−6], the main research focus on the static fusion DSm model) responding to static model. A new method for dynamic proportionalconflict redistributionrules (dy- for nonexistence of non-existential integrity constraints namicPCRrules)basedonhybridDSmmodelisproposed concept, as computation problem[7], static combination for the shortness of classical dynamic PCR rules. In the rule[8], combination of unreliable evidence[9−11], eviden- new dynamicPCR rule, combinationinvolvedwithempty tial classification method[12,13]. But the dynamic frame set is defined as one kind to obtain more reasonable re- sults. For the redistribution weight, the conjunction Basic fusion is more practical than static fusion[14]. The the- belief assignment (BBA) and conflict redistribution BBA atrical foundation of DSm theory is hybrid DSm model are both taken into account to raise the fusion precision. anddynamic constraintcondition.So DSm theory is per- The effectiveness of revised dynamic PCR rule is studied fect for the definition, representation and procession of andsimulatedinbothaspectoffusionaccuracyandcalcu- lation. dynamic frame fusion problem[2]. Constraint condition Key words — Dynamic frame fusion, Hybrid DSm changes with time to generate new hybrid DSm model. model, Dynamic proportionalconflict redistribution rule. As the example[3], lets consider the set of three targets I. Introduction at a given time k to be Θk = {θ1,θ2,θ3}. And one receives a new information at time k+1 confirming that Combination rule is the core part of Dezert- onetarget,saytargetθ3 hasbeendestroyed.Theproblem Smarandache theory (DSm theory)[1,2]. The effectiveness one need to solve is how to combine efficiently evidence ofDSmtheoryfusionsystemdependsonthepropertiesof takingintoaccountthisnewnon-existentialintegritycon- combinationrule.Andthecombinationruleshouldsatisfy straintθ3 ≡Ø in the new model ofthe frame to establish the applicationrequirementsofusers.In mostofclassical the most threatening and surviving targets belonging to problem using belief functions, the frame of discernment Θk+1 ={θ1,θ2}. Θ = {θ1,θ2,...,θn} is considered static. This means the Smarandache and Dezert propose a series of combi- set of elements in the frame (assumed to be non-empty nation rules to solve the fusion problem under dynamic and distinct) and the underlying integrity constraints of frame.Thesecombinationrulesaredynamicproportional theframearefixedandremainthesamewithtime.These conflict redistribution combination rules (dynamic PCR fusion problems are called static fusion. In some appli- rules)[15] and Dezert-Smarandache hybrid combination cation however, like in target tracking and battlefield rules (DSmH rules)[2]. The combination rules have three surveillance for example, the use of such invariant frame steps in evidence processing. Firstly, these combination is not very appropriate because it can truly change with rules define non-existential sets caused by the modifica- time depending on the changing process of the events. In tion of discernment frame. Secondly, the Basic belief as- the dynamic frame fusion, frame of discernment changes signment (BBA) of the non-existential sets is transferred ∗Manuscript Received Mar.9, 2016; Accepted May 21, 2016. This work is supported by the National Natural Science Foundation of China(No.61102168) andMilitaryinnovationFoundation(No.X11QN106). (cid:2)c 2018ChineseInstitute ofElectronics.DOI:10.1049/cje.2018.04.008 822 Chinese Journal of Electronics 2018 toemptysetØ.Lastly,theBBAisredistributedforthere- is determined by both the elements in the discernment lationshipemptysetØandelementsindiscernmentframe frame and non-existential integrity constraints. The dis- tocompletethedynamicframefusion.ThedynamicPCR cernmentframechangeswiththenon-existentialintegrity rulesare moreeffective thandynamic DSmH rulesshown constraints. Hybrid DSm model fits the reality because in Ref.[15]. But there are two kinds of drawbacks for dy- somesubsetsofΘ cancontainelementsknowntobetruly namicPCRrules: a)the unreasonableredistribution,the exclusivebutalsotrulynonexistingatallatagiventime. redistribution of empty set Ø is similar with the redis- tribution of existential sets; b) insufficiency use of infor- mation, some useful information is not involved in the redistributionweight.Thecontributionofthispaperisto propose a revised method for dynamic PCR rule based on hybrid DSm model. In the revised method, combina- tion involved with empty set Ø is defined as one kind Fig.1.The model of fusion problems. (a) Free DSm model; to obtain more reasonable results. For the redistribution (b)Shafermodel;(c)HybridDSmmodel weight, the conjunction BBA and conflict redistribution The traditionalcombinationrules areunder fixeddis- BBAarebothtakinginto accountto raisethe fusionpre- cernment frame (Shafer model and free DSm model) re- cision.The revisedPCRrule makesbetter use ofexisting sponding to static fusion. But the dynamic frame fusion information without adding more calculation. The effec- and hybrid DSm model are consistent with the practi- tiveness of revised PCR rule is shown in the simulation. cal situation. Smarandache and Dezert propose a series II. Hybrid DSm Model of combination rule to solve the problem under dynamic frame fusion. These combination rules are dynamic PCR rules and DSmH rules. Firstly, the dynamic PCRa rule is DSm theory is motivated by expanding the frame of analyzed in the paper. Then, a new revised PCR rule is discernmenttoallowforpresumedsingletonsinDStheory to actually have a well-defined intersection[3]. The frame proposedtoovercomethe shortnessofPCRarule.Lastly, thecalculationisdefinedandstudiedtoevaluatethecom- of discernment changes with time according to the dy- bination rules. namic constraint condition, which is the foundation of 1. Revisedmethod of dynamic frame PCR rule dynamic frame fusion. Smarandache and Dezert analyze dynamic PCR rules DSm theory contains Shafer model, free DSm model andhybridDSmmodel.Dependingontheintrinsicnature and DSmH rules in Ref.[15] to draw a conclusion oftheelementsofthefusionproblemunderconsideration, that PCRa rule is the most effective rule. PCRa rule: it can however happen that free DSm model and Shafer mPCRa(Ø)=0 and ∀A∈GΘ\Ø model do not fit the reality because some subsets of Θ mPCRa(A) cancontainelementsknowntobetrulyexclusivebutalso =m12(A) (cid:3) (cid:4) twrourlkyinngononexdisytninagmaictfaulslioant aprgoibvelenmtiwmheer(septehceialflryamweheΘn + (cid:2) m1(A)2m2(X) + m2(A)2m1(X) varieswith time with the revisionofthe knowledge avail- X∈GΘ\Ø m1(A)+m2(X) m2(A)+m1(X) able). These integrity constraints are then explicitly and X(cid:2)∩A=Ø formally introduced into the free DSm model Mf(Θ) in + [m1(A)m2(X)+m2(A)m1(X)] (1) X∈Ø (cid:2) order to adapt it properly to fit as close as possible with m1(X)m2(Y) the reality and permit to construct a hybrid DSm model X,Y∈Ø M(Θ). The hybrid DSm model M(Θ) can deal with the +m12(A)· (cid:2) situationofsets whichmightbecome empty attime tk or m12(Z) Z∈GΘ\Ø new sets/elements that might arise in the frame at time tk+1[3]. The model in Fig.1(c) is the hybrid DSm model TheredistributionstrategyforBBAnotinvolvedwith with θ1∩θ3 = Ø and θ2∩θ3 = Ø. The free DSm model empty set Ø is the same as that of classicalPCR rule[16]. and Shafer model can be considered as two special case And PCRa redistributes the BBA involved with empty of hybrid DSm model. set Ø to non-empty sets, as two kinds. Firstly, the redis- tributionstrategyofintersectionbetweennon-empty sets III. Combination Rule of Dynamic Frame Fusion and empty set Ø is defined as S1. Secondly, the redistri- bution strategy of intersection between empty set Ø and ThediscernmentframesofShafermodelandfreeDSm empty set Ø is defined as S2. (cid:2) modelaredeterminedbythe elementsinthe discernment S1 = [m1(A)m2(X)+m2(A)m1(X)] (2) frame.WhilethediscernmentframeofhybridDSmmodel X∈Ø A New PCR Combination Rulefor DynamicFrame Fusion 823 (cid:2) m1(X)m2(Y) the non-existentialsets whichare not suitable for mutual X,Y∈Ø processing. In addition, the BBA of conflict redistribu- S2 =m12(A)· (cid:2) (3) m12(Z) tion can express the condition of the support to certain Z∈GΘ\Ø element.ButtheBBAofconflictredistributionleavesout ButtherearetwokindsofproblemsfordynamicPCR of consideration of PCRa rule. rules: a) the unreasonable redistribution, the redistribu- To overcome the shortness of PCRa rule, a revised tion of empty set Ø is similar with the redistribution of PCR rule is proposed. For the problem a), intersection existential sets in classical PCR rule[16]; b) insufficiency betweennon-emptysetsandemptysetØS1 andintersec- use of information, some useful information is not in- tion between empty set Ø and empty set Ø S2 uniformly volved in the redistribution weight. The PCRa rule re- process. For the problem b), redistribution BBA of con- distributes the empty sets via using the same method of flict set for non-empty sets is added into redistribution processing unknown set Θ. The unknown set Θ is the weight(besides conjunctionBBA inPCRa rule).Revised universal set of elements in discernment frame suitable PCR rule is called PCRd for short: mPCRd(Ø) = 0 and for mutual processing. But empty sets Ø are defined as ∀A∈GΘ\Ø . ⎡ ⎤ mPCRd(A)=mt(A)+ (cid:2)mt(A) ⎣ (cid:2) [m1(A)m2(X)+m2(A)m1(X)]+(cid:2) m1(X)m2(Y)⎦ mt(X) X∈Ø,A∈GΘ\Ø X,Y∈Ø X∈GΘ\Ø (4) (cid:3) (cid:4) And mt(U)=m12(U)+ (cid:2) mm11(U(U))+2mm22(P(P)) + mm22(U(U))+2mm11(P(P)) (5) U,P∈GΘ\Ø P∩U=Ø RevisedPCRruleusesthe informationinintersection TherevisedPCRruleandclassicaldynamicPCRrules more effective. And the redistribution of empty set Ø is are simulated and analyzed with two examples. The ex- more reasonable.The fusion resultof revisedPCRrule is ample1isaclassicalcaseinSmarandache’spaper[15].The better than that of PCRa rule. example 2 is a dynamic frame example with overall pro- 2. Analysis of calculation cess.ThefusionresultandcalculationofrevisedPCRrule The calculation is taken into account besides the fu- (PCRd) and dynamic PCR rules (PCRa, PCRb, PCRc) sion result to evaluate the combination rules. It is lack of are analyzed below. applicabilityforcombinationrulewhichhasperfectfusion 1. Example 1 (classical case in Smarandache’s result and high calculation.In the areaof BBA combina- paper[15]) tion,evidencecombinationcanbe consideredasthecom- InTable1,therevisedPCRruledistributesmoreBBA position of multiplication operation (division operation) to single elements than other three methods. The calcu- which is atom operation. lation of revised PCR rule and PCRa rule is 46, higher Definition 1 In hybrid DSm model, lets consider than that of PCRb rule and PCRc rule. n focal elements Θ = {θ1,θ2,...,θn} in the k evidence, Table 1. Fusion result and calculationof the set of(cid:9)foca(cid:9)l elements is GΘ, the number of focal ele- Smarandache’s classical case ments is (cid:9)GΘ(cid:9)(cid:9), fo(cid:9)r the evidence mj(Ajl), j = 1,2,...,k, θ1 θ2 θ3≡Ø θ1∪θ2 Calculation l = 1,2,...,(cid:9)GΘ(cid:9), Ajl ⊆ Θ, The weighted sum of mul- Prior:m1(.) 0.2 0.4 0.3 0.1 – tiplication operation times (division operation) for cer- Prior:m2(.) 0.3 0.1 0.4 0.2 – mPCRa(.) 0.420 0.452 0 0.128 46 taincombinationruleistheBBAcombinationcalculation mPCRb(.) 0.404 0.436 0 0.160 30 which is expressed as O(f(n,k)). mPCRc(.) 0.364 0.396 0 0.240 26 Theredistributioncontentandredistributionweightis mPCRd(.) 0.417 0.457 0 0.126 46 adjusted in the revised PCR rule. The conjunction BBA 2. Example 2 (dynamic frame example with andredistributionBBA ofconflict set for non-empty sets overall process) which have been obtained in the pre-processing are used Lets assume a fusion system which runs on hybrid in revised PCR rule. So the calculation of revised PCR DSm model. The system starts at time t = 1, the dis- rule is same as that of PCRa rule. The revised PCR rule cernment frame Θ = {θ1,θ2,θ3,θ2∩θ3}. There is new has better fusion result without more calculation. So the information to confirm θ2∩θ3 ≡Ø in t=3, the discern- revised PCR rule is more effective in the aspect of calcu- ment frame update as Θ = {θ1,θ2,θ3} (m3(θ2∩θ3) also lation analysis. appearsinoriginalBBA).Thereisnewinformationadded IV. Simulation and Analysis toconfirmθ3 ≡Øint=9,thediscernmentframeupdate 824 Chinese Journal of Electronics 2018 as Θ = {θ1,θ2} .We assume that the target is 1 in the simulation. The sensor obtain the incorrect identification result (BBA) caused by interference in time t = 4 and t = 8.The BBA and non-existential integrity constraints at different moments are shown in Table 2. Table 2. BBA and non-existential integrity constraintsat different moments Time θ1 θ2 θ3 θ2∩θ3 Constraints mt=1(.) 0.5 0.2 0.2 0.1 – mt=2(.) 0.6 0.1 0.2 0.1 – mt=3(.) 0.7 0.1 0.1 0.1 θ2∩θ3≡Ø mt=4(.) 0.2 0.7 0.1 0 θ2∩θ3≡Ø mmtt==56((..)) 00..76 00..11 00..11 00..12 θθ22∩∩θθ33≡≡ØØ Fig.2.ThetendencyoffusionresultBBAforθ1 mt=7(.) 0.6 0.2 0.1 0.1 θ2∩θ3≡Ø The calculationoffourcombinationrulesis simulated mt=8(.) 0.3 0.4 0.2 0.1 θ2∩θ3≡Ø and analyzed in Table 4. mt=9(.) 0.6 0.1 0.2 0.1 θ2∩θ3≡θ3≡Ø mt=10(.) 0.8 0.1 0.1 0 θ2∩θ3≡θ3≡Ø Table 4. The calculationof four combination rules The BBA is fused by three PCR rules and revised 1 2 3 4 5 6 7 8 9 Total PCRrule.The dynamicfusionresultsareshowninTable mPCRa 50 54 54 54 54 54 54 24 24 422 3. The tendency of fusion result BBA for θ1 displays in mPCRb 50 51 51 51 51 51 51 22 22 400 Fig.2. The figure shows four key points of BBA for θ1. mPCRc 50 46 50 50 50 50 50 26 26 398 mPCRd 50 54 54 54 54 54 54 24 24 422 Table 3. Dynamic fusion result Numberof 3. Analysis of simulation θ θ θ θ ∩θ θ ∪θ ∪θ 1 2 3 2 3 1 2 3 fusion Example 1 is classical case in Ref.[15]. The character mPCRa(.)0.6862 0.0583 0.0986 0.1569 0 1 mPCRb(.) 0.6862 0.0583 0.0986 0.1569 0 of single element focusing is shown in example 1. A com- mPCRc(.) 0.6862 0.0583 0.0986 0.1569 0 plete simulation with dynamic non-existential integrity mPCRd(.) 0.6862 0.0583 0.0986 0.1569 0 constraints is given in example 2. The original BBA is mPCRa(.)0.8919 0.0465 0.0616 0 0 responding to the time BBA obtained (Time). In Table 2 mPCRb(.) 0.8820 0.0516 0.0664 0 0 2, we define the time of combination as number of fusion mPCRc(.) 0.8767 0.0463 0.0613 0 0.0157 mPCRd(.) 0.9203 0.0327 0.0470 0 0 (Number). Number 1 means the first time of fusion, and mPCRa(.)0.6254 0.3499 0.0247 0 0 so on. The fusion begins in Time 1, the BBA of Time 1 3 mPCRb(.) 0.6180 0.3556 0.0264 0 0 and that of Time 2 are fused to obtain fusion result of mPCRc(.) 0.6153 0.3585 0.0262 0 0 Number 1. The constraints of θ2 ∩θ3 ≡ Ø and θ3 ≡ Ø mPCRd(.) 0.6462 0.3338 0.0200 0 0 areaddedinTime3andTime9.The dataofTime4and mPCRa(.)0.7881 0.1895 0.0224 0 0 mPCRb(.) 0.7836 0.1935 0.0229 0 0 Time 8 are conflicting evidence. From Table 2 and Fig.1, 4 mPCRc(.) 0.7817 0.1954 0.0229 0 0 revised PCR rule is better than other three combination mPCRd(.) 0.8179 0.1609 0.0212 0 0 rule in the overall process. There are not constraints in mPCRa(.)0.8697 0.1073 0.0230 0 0 Number 1. So the fusion results of four rule are same. 5 mmPPCCRRbc((..)) 00..88666597 00..11019181 00..00223332 00 00 The Number 6 and Number 9 are the fusion result of mPCRd(.) 0.9061 0.0714 0.0225 0 0 low conflicting condition shown in Fig.3(b) and Fig.3(d) mPCRa(.)0.8961 0.0841 0.0197 0 0 respectively. The fusion result of θ1 mPCRa(θ1) reaches 6 mPCRb(.) 0.8946 0.0857 0.0197 0 0 0.9167and0.9767inNumber6andNumber9better2.3% mPCRc(.) 0.8939 0.0862 0.0199 0 0 and1.7% than PCRa rule (the best in the three dynamic mPCRd(.) 0.9167 0.0651 0.0182 0 0 mPCRa(.)0.7781 0.1707 0.0512 0 0 PCR rules). The Number 2 and Number 7 are the fusion 7 mPCRb(.) 0.7769 0.1717 0.0514 0 0 result of high conflicting condition shownin Fig.3(a) and mPCRc(.) 0.7763 0.1722 0.0515 0 0 Fig.3(c) respectively. The fusion result of θ1 mPCRa(θ1) mPCRd(.) 0.7800 0.1679 0.0521 0 0 reach0.6462and 0.7780in Number 2 and Number 7 bet- mPCRa(.)0.8945 0.1055 0 0 0 ter3.3%and0.25%thanPCRarule(thebestinthethree mPCRb(.) 0.8867 0.1133 0 0 0 8 mPCRc(.) 0.8786 0.1059 0 0 0.0155 dynamic PCR rules). Revised PCR rule has best fusion mPCRd(.) 0.9282 0.0718 0 0 0 resultinprocessinglowconflictingevidenceandhighcon- mPCRa(.)0.9601 0.0399 0 0 0 flicting evidence. mPCRb(.) 0.9571 0.0429 0 0 0 9 The calculation of four combination rules is shown in mPCRc(.) 0.9568 0.0417 0 0 0.0015 mPCRd(.) 0.9767 0.0233 0 0 0 Table4.ThePCRcrulehasthelowestcalculationas398. A New PCR Combination Rulefor DynamicFrame Fusion 825 Fig.3.Key points of BBA for θ1 in fusion process. (a) Fusion results of fusion Number 2; (b) Fusion results of fusion Number6;(c)FusionresultsoffusionNumber7;(d)FusionresultsoffusionNumber9 The calculation of PCRb rule is lager as 400. The PCRa The two kinds of combination rules can be mixed, which rule has the largest calculation as 422 in three classical is the further researchemphasis for our team. rules. Revised PCR rule is based on the calculated BBA ofPCRa rule to obtainthe same calculationas PCRa.In References 9 times fusion, calculation of revised PCR rule is larger 5.5% and 6% than PCRb rule and PCRc rule, which is [1] J.Dezert,“Foundationsforanewtheoryofplausibleandpara- acceptable. doxical reasoning”, Information and Security Journal, Vol.12, No.1,pp.26–30, 2002. In conclusion, revised PCR rule improves the way of [2] J. Dezert, “An introduction to DSmT for information fu- distribution based on the using the information of PCRa sion”,NewmathematicsandNaturalComputation,Vol.8,No.3, rule effectively. The fusion result of revised PCR rule is pp.343–359, 2012. better than PCR rules in both high conflicting condition [3] F. Smarandache, and J. Dezert, Advances and Applications of DSmT For Information Fusion Vol.3, American Research andlowconflictingcondition.Thecalculationremainsthe Press,Rehoboth, USA,2009. same as PCRa rule without extra calculation. 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[11] M. Khodabandeh, and A. Mohammad-Shahri, “Uncertainty 826 Chinese Journal of Electronics 2018 evaluation for an ultrasonic data fusion based target differen- LI Hongfei is a postdoctoral re- tiation problem usinggeneralized aggregated uncertainty mea- searcherinShanghaiJiaoTongUniversity. sure”,Measurement, Vol.59,No.1,pp.139–144, 2015. His research interests include information [12] L. M. Jiao, Q. Pan and X. X. Feng, “An evidential k-nearest fusion,Uncertaintyreasoningandevidence neighborclassificationmethodwithweightedattributes”,Proc. theory. of the 16th international conference on Information fusion,Is- tanbul,Turkey,pp.145–150, 2013. [13] Z.G.Liu,J.DezertandQ.Pan,“Combinationofsourcesofevi- dencewithdifferentdiscountingfactorsbasedonanewdissimi- laritymeasure”,DecisionSupportSystems,Vol.52,pp.133–141, 2011. [14] Z.G. Liu, J. Dezert and G. Mercier, “Dynamic evidential rea- LANJiangqiao isaprofessorofAir soning for change detection in remote sensing images”, IEEE Force Early Warning Academy. His recent TransactionsonGeoscienceandRemoteSensing,Vol.50,No.5, research interests include military intelli- pp.1955–1967, 2012. genceandinformationfusion. [15] F. Smarandache and J. Dezert, “Extended PCR rules for dy- namic frames”, Proc. of the 15th International Conference on Information Fusion,Singapore,pp.263–270, 2012. [16] F. Smarandache and J. Dezert, Advances and Applications of DSmTforInformationFusionVol.2,AmericanResearchPress, Rehoboth, USA,2006. JIN Hongbin is currently an asso- HAN Jun (corresponding au- ciate professor in Air Force Early Warn- thor) is a lecturer of Air Force Early ing Academy. His research interests in- Warning Academy. His recent research cludeinformationfusion, target identifica- interests include radar signal process- tion, and efficiency evaluation for C4ISR ing and combat application of radar. system.(Email: [email protected]) (Email: [email protected])