1 An Evidence Clustering DSmT Approximate Reasoning Method Based on Convex Functions Analysis Qiang GUO 1, You HE1, Xin GUAN2, Li DENG3, Lina PAN 4, Tao JIAN 1 (1. Research Institute of Information Fusion, Naval Aeronautical and Astronautical University, Yantai Shandong 264001,China; 2. Electronics and Information Department, Naval Aeronautical And Astronautical University, Yantai Shandong 264001,China; 3. Department of Armament Science and Technology, Naval Aeronautical and Astronautical University, Yantai Shandong 264001,China; 4. Department of Basic Science, Naval Aeronautical and Astronautical University, Yantai Shandong 264001, China) Corresponding author: Qiang GUO, Tel numbers: +8615098689289, E-mail address: gq19860209@163.com. Abstract ThecomputationalcomplexityofDezert-SmarandacheTheory(DSmT)increasesexponentiallywiththelinear incrementofelementnumberinthediscernmentframe,anditlimitsthewideapplicationsanddevelopmentof DSmT.Inordertoefficientlyreducethecomputationalcomplexityandremainhighaccuracy,anewEvidence ClusteringDSmTApproximateReasoningMethodfortwosourcesofinformationisproposedbasedonconvex functionanalysis.Thisnewmethodconsistsofthreesteps.First,thebeliefmassesoffocalelementsineach evidenceareclusteredbytheEvidenceClusteringmethod.Second,theun-normalizedapproximatefusionresults areobtainedusingtheDSmTapproximateconvexfunctionformula,whichisacquiredbasedonthemathematical analysis of Proportional Conflict Redistribution 5 (PCR5) rule in DSmT. Finally, thenormalization step is applied. Thecomputationalcomplexityofthisnewmethodincreaseslinearlyratherthanexponentiallywiththelinear growthoftheelements.Thesimulationsshowthattheapproximatefusionresultsofthenewmethodhavehigher EuclideansimilaritytotheexactfusionresultsofPCR5basedinformationfusionruleinDSmTframework (DSmT+PCR5),anditrequireslowercomputationalcomplexityaswellthantheexistingapproximatemethods, especially for the case of large data and complex fusion problems with big number of focal elements. Keywords:Evidenceclustering;Approximatereasoning;Informationfusion;Convexfunctionsanalysis; Dezert-Smarandache Theory 1.Introduction Asanovelkeytechnologywithvigorousdevelopment,informationfusioncanintegratemultiple-source incompleteinformationandreduceuncertaintyofinformationwhichalwayshasthecontradictionandredundancy. Information fusion can improve rapid correct decision capacity of intelligent systems and has been successfully used in the military and economy fields, thus great attention has been paid to its development and application by scholars inrecentyears[1-9].Asinformationenvironmentbecomesmoreandmorecomplex,greaterdemandsforefficient fusionofhighlyconflictinganduncertainevidencearebeingplacedoninformationfusion.Belieffunctiontheory (also called evidence theory) referred by Dezert-Smarandache theory (DSmT) [9] and Dempster-Shafer theory (DST) [10,11]canwelldealwiththeuncertainandconflictinformation.DSmT,jointlyproposedbyDezertand *ManuscriptClick here to view linked References2 Smarandache,isconsideredasthegeneralextensionofDST,sinceitbeyondstheexclusivenesslimitationof elements in DST. DSmT can obtain precise results for dealing with complex fusion problems in which the conflict is highandtherefinementoftheframeisnotaccessible[9].Recently,DSmT(belieffunctiontheory)hasbeen successfullyappliedinmanyareas,suchas,MapReconstructionofRobot[12,13],DecisionMakingSupport[14], TargetTypeTracking[15,16],ImageProcessing[17],SonarImagery[18],DataClassification[19-21],Clustering [22,23],andsoon.Particularly,theveryrecentcredalclassificationmethods[12,15,20]workingwithbelief functionshavebeenintroducedbyLiu,Dezert,etalfordealingwithuncertaindata,andtheobjectisallowedto belongtoanysingletonclassandsetofclasses(calledmeta-class)withdifferentbeliefmasses.Bydoingthis,the credalclassifiersareabletowellcapturetheuncertaintyofclassificationandalsoefficientlyreducetheerrors. However,themainproblemoftheapplication(e.g.classificationtask)ofDSmTisthatwhenthefocalelements number increases linearly, computational complexity increases exponentially. Many approximate reasoning methods of evidence combination in DST framework were presented in [24-26]. But thesemethodscannotsatisfythesmallamountofcomputationalcomplexityandlesslossofinformation requirementsatthesametime.Inrecentyears,therearesomeimportantarticles[27-34]dealingwiththe computationalcomplexityofthecombinationalgorithmsformulatedinDSmTframeworkindifferentways. Djiknavorian [27] has proposed a novelmethod and a Matlab program to reduce the DSmT hybrid rule complexity. For manipulating the focal elements easily, Martin [28] proposed a Venn diagram codification, which can reduce the DSmTcomplexitybyonlyconsideringthereducedhyper-powerset rDOafterintegratingtheconstraintsinthe codificationatthebiginning.Abbas[29,30]hasproposedaDSmTbasedcombinationschemeformulti-class classificationwhichalsoreducesthenumberoffocalelements.Li[31]hasproposedamethodforreducingthe informationfusioncomplexity,whichisdifferentfromtheabovemethodsbyreducingthecombinedsources numbersinsteadofreducingthenumberoffocalelements.Liandotherscholars[32-34]alsoproposedan approximatereasoningmethodforreducingthecomplexityoftheProportionalConflictRedistribution5(PCR5) based information fusion rule within DSmT framework. However, when processing highly conflict evidences by the methodin[32],thebeliefassignmentsofcorrectmainfocalelementstransfertotheotherfocalelements,which leads to low Euclidean similarity of the results in this case. AimingatreducingthecomputationalcomplexityofPCR5basedinformationfusionrulewithinDSmT framework(DSmT+PCR5)andobtainingaccurateresultsinanycase,anewEvidenceClusteringDSmT ApproximateReasoningMethodfortwosourcesofinformationisproposedinthispaper.InSection2,thebasics knowledgeonDST,DSmTandthedissimilaritymeasuremethodofmultievidencesareintroducedbriefly.In Section3,mathematicalanalysisofPCR5formulaisconducted,whichdiscoverseveryconflictmassproduct satisfies the properties of convex function. A new DSmT approximate convex function formula is proposed and error analysisoftheproposedformulaisalsopresented.Basedontheerroranalysis,anEvidenceClusteringmethodis proposed as the preprocessing step and the normalization method is applied as the final step of the proposed method forreducingtheapproximateerror.Theprocessoftheproposedmethodisgiven,thenanalysisofcomputation complexity of DSmT+PCR5 and the proposed method are presented. In Section 4, the results of simulation show that the approximate fusion results of the method proposed in this paperhave higher Euclidean similarity with the exact fusionresultsofDSmT+PCR5,andlowercomputationalcomplexitythanexistingDSmTapproximatereasoning method in [32]. The conclusions are given in Section 5. 3 2.Basicknowledge In this section, we will give an overview of the basics knowledge on DST and DSmT, which are closely related to our work in this paper . 2.1. Dempster-Shafer Theory (DST) Letusconsideradiscernmentframe 1 2{ , , , }nu u u O= containingnelements 1 2, , ,nu u u ,whichisthe refinementofthediscernmentbasedontheShafersmodel.Thebasicbeliefassignment(bba)isdefinedoverthe power-set2Owhich consists of all subsets ofO. For example, if onehas 1 2 3{ , , } u u u O= , the power setis given by1 2 3 1 2 1 3 2 3 1 2 32 { , , , , , , , } u u u u u u u u u u u uO= C, and the bba(.) : 2 [0,1] mO on the power set is defined by [10,11] ( ) 0,i im X X = = C(1) 2 ,1( ) 1iiX i nmXOe s s=(2) Theelement iX iscalledfocalelementsifitholds( ) 0imX > .Dempstersruleisoftenusedforthe combination of multiple sources of evidence represented by bbas in Shafersmodel, and it requires that the bbas mustbeindependent.Thebbaoftheithsourceofevidenceisdenoted im .TheDempstercombinationruleis definedbyEquation(3)andtheconflictinDempstercombinationrule,denotedbyC,isdefinedbyEquation(4) [10,11] DS 1 2,1( ) ( ) ( )1i ji jX X Zi jm Z m X m X ZC= == _ O(3) 1 2, ,( ) ( )i ji ji jX X i jX XC m X m X_O ==C= (4) OnecanseethatalltheconflictingbeliefsChasbeenredistributedtootherfocalelements.Dempstersrule usually produces veryunreasonableresults in thefusion of high conflicting information dueto theredistribution of conflictingbeliefs.Inordertosolvethisproblem,manyalternativecombinationruleslikeProportionalConflict Redistribution1-6(PCR1-6) rules [36,38,39] have been developed. 2.2. Dezert-Smarandache Theory (DSmT) DSmT[29]overcomestheexclusivenesslimitationinShafersmodel.Inmanyfusionproblems,thehypotheses canbevagueinrealityandtheelementsarenotpreciselyseparatedwhichdontsatisfytheShafersmodel.The hyper-powersetdenotedbyDOisbuiltbyapplyingoperatorandtotheelementsinO[35,36].Letus considerasimpleframeofdiscernemnt1 2{ , } u u O= ,thenonegets 1 2 3 1 2 1 2{ , , , , , } D u u u u u u uO= C .Thebbain DSmT is defined over the hyper-power set as(.) : [0,1] m DO . In the combination of multiple sources of evidence, there exist two models in DSmT [35,36]: 1) free combination modeland2)hybridcombinationmodelwhichisoftenusedinrealapplicationbecauseittakesintoaccountsome integrityconstraints.Inhybridcombinationrule,ittransferspartialconflictingbeliefstothecorresponding intersected elements, but this increases the uncertainty of fusion results. The Proportional Conflict Redistribution1-6 4 (PCR1-6)rules[36,38,39]providesproperconflictredistributionways,andtheyproportionallytransferconflicting masses to the involved elements. The difference of PCR1-6 rulesmainly lies in the redistribution of conflicts, and PCR5 is considered as themost preciseredistributionway[36,38,39].ThecombinationoftwoindependentsourcesofevidencesbyPCR5ruleis given as follows [36,38,39] 1 2 1 2, a n d ,( ) ( ) ( )iiY Z G Y ZY Z Xm X m Y m ZOe = C== (5) 2 21 2 2 11 2and1 2 2 1PCR5( ) ( ) ( ) ( )( )and ( ) ( ) ( ) ( )( )0ji ji j i ji i iX G i j i j i jiX Xim X m X m X m Xm X X G Xm X m X m X m Xm XXOOe ==C( + + e = C (+ + (= = C(6) whereGOcanbeenseenasthepowerset2O,thehyper-powersetDOandthesuper-powersetSO,if discernment of the fusion problem satisfies the Shafers model, the hybrid DSm model, and the minimal refinement refO ofOrespectivelyandwherealldenominatorsaremorethanzeroandthefractionisdiscardedwhenthe denominator of it is zero [36,38,39]. Nevertheless, PCR5 rule still has some disadvantages, such as, firstly, it is not associative in the fusion of multiple (morethan2)sourcesofevidences,sothecombinationordermayhaveinfluenceontheresults,secondly,its computationalcomplexityincreasesexponentially,whenthefocalelementsnumberincreases.Ourresearchinthis paper is mainly for reducing the complexity of PCR5 within DSmT framework. 2.3. The dissimilarity measure method of multi evidences The dissimilarity measure method of multi evidences and several Evidence Support Measure of Similarity(ESMS) functionshavebeengivenin[31,40].TheoftenusedEuclideanESMSfunctionandJousselmeESMSfunctionare briefly recalled. 1) Euclidean ESMS function 1 2( , )ESim mmLet 1 2{ , , , }, 1nn u u u O= > ,GObe the cardinality ofGO, 1( ) m and 2( ) m be two bbas. The Euclidean ESMS function is defined by [31] | |21 2 1 211( , ) 1 ( ) ( )2GE i iiSim mm m X m XO== (7) 2) Jousselme ESMS function 1 2( , )JSim mmThe Jousselme ESMS function [31] is defined based on the Jousselme et al. measure [40] 1 2 1 2 1 21( , ) 1 ( ) ( )2TJSim mm m m D m m = (8) where[ ]ijD D = is aG GO O positively definite matrix, and/ij i j i jD X X X X = with,i jX X GOe . SomemoreESMSfunctionscanbeseenin[31]fordetails. 1 2( , )ESim mm isconsideredwiththefastest convergence speed [31], and it is adopted here as the dissimilarity measure for comparison of the method proposed in this paper with the other methods. 3.AnEvidenceClusteringDSmTApproximateReasoningMethod 5 3.1 Mathematical analysis of PCR5 formula As shown in Equation (6), 2 21 2 2 1and1 2 2 1( ) ( ) ( ) ( )( ) ( ) ( ) ( )ji ji j i jX G i j i j i jX Xm X m X m X m Xm X m X m X m X Oe ==C ( + (+ + ( has symmetry. Due to the symmetry, one item 21 21 2( ) ( )( ) ( )i ji jm X m Xm X m X+is analyzed. Let 1( )imX a = and 2( )jm X x =get 221 2 21 2( ) ( )11( ) ( )i ji jm X m Xaxa am X m X a x a x ( | |= = |(+ + +\ . .(9) Let 1 2 2, , , { ( ) | , and}n j j i jx x x m X i j X G X XOe = e = Cthen 21 2 2and1 2 1 2( ) ( )1 1 1( ) ( )ji jX G i j i j nX Ym X m Xa n am X m X a x a x a x Oe ==C (( | |= + + + ((|+ + + + (( \ . .(10) Let 1( ) f xa x=+, since( ) f x iscontinuousfunction on(0,1),ithasasecondorderderivativeson(0,1),and''( ) 0 f x > on(0,1), ( ) f x is a convex function. So( )1 21 21( ) ( ) ( )nnx x xf x f x f x fn n+ + + | |+ + + > |\ ., the equation holds iff 1 2 nx x x = = = . The approximate convex function formula is given by 1 21 2 1 21 1 1, 0, 0 iff ( ) /nn nnx x xa x a x a x a x x x n+ + + = + A A > A = = = =+ + + + + + +.(11) Let 1 2 i nx x x x s s s s s , carry out analysis of convex function formula errors 1 1 22 1 2 1 21 1( ) /1 1 1 1( ) / ( ) /nn n na x a x x x na x a x x x n a x a x x x n (A = (+ + + + + ((+ + + ((+ + + + + + + + + + .(12) Analysis of thei item in Equation (12). Let 1 2 0( ) /nx x x n x + + + = , then 1 2 01 1 1 1( ) /i n ia x a x x x n a x a x = + + + + + + +.(13) By Taylor expansion theorem 2 3 00 0 0 0 0 00''( ) 1 1 '''( )'( )( ) ( ) ( ) , ( , ) or ( , )2 3!i i i i iif x ff x x x x x x x x x x xa x a xoo = + + + e+ +,(14) then Equation (14) is transformed to 6 | |0 1 0 2 0 01 2 1 22 2 2 2 01 0 2 0 0 011 1 1'( ) ( ) ( ) ( )( ) /''( )( ) ( ) ( ) ( )2nn nnn iinf x x x x x x xa x a x a x a x x x nf xx x x x x x ox x=+ + + = + + + + + + + + + + ( + + + + + . (15) Since| |0 1 0 2 0 0'( ) ( ) ( ) ( ) 0nf x x x x x x x + + + = , then 2 2 00 01 11 2 1 2''( ) 1 1 1( ) ( )( ) / 2n ni ii in nf x nx x o x xa x a x a x a x x x n= =+ + + = + + + + + + + + .(16) where 2 3 3 3 4 4 0 1 20 1 0 2 0 0 1 0 2 0140'''( ) ''''( ) ''''( )( ) ( ) ( ) ( ) ( ) ( )3! 4! 4!''''( )( )4!ni ninnf x f fox x x x x x x x x x x xfx xo oo= ( = + + + + + + + +. Analysis of ( )( ) , 2, 3, ,mf x m = ( ) ( 1)( )1 1( )m mmf x ma x a x| | | |= = ||+ +\ . \ .,(17) then ( 2) ( 1)( 1) ( )1 1 0 00 0 0 00 0( 2)10 00 0( ) ( ) 1 1 1 1( ) ( ) ( ) ( )( 1)! ! ( 2)! ( 1)!1 1 1 1( ) 1( 2)! ( 1)m mm mm m m mmmf x f xx x x x x x x xm m m a x m a xx x x xm a x m a x | | | | = || + +\ . \ . ( | | | |= (|| + + ( \ . \ . .(18) If 0x x s ,0 0 0x x x x x a = < + , then ( 1) ( )1 0 00 0( ) ( )( ) ( )( 1)! !m mm mf x f xx x x xm m > .(19) If 0 02, , 2 m x x x a x > > < + , then 001 1( ) 01x xm a x| | > | +\ ..(20) So if 02 , 1, 2, ,ix a x i n < + = , ( 1) ( )1 0 00 0( ) ( )( ) ( ) , 2( 1)! !m mm mi if x f xx x x x mm m > >,(21) namely, ( )2 3 0 0 00 0 0''( ) '''( ) ( )( ) ( ) ( )2 3! !mmi i if x f x f xx x x x x xm > > > .(22) Neglect the fourth order item errors and more order item errors. Forthethirdorderitemisoddnumberitem,foreach, 1, 2, ,ix i n = , 33 00( )( )3!if xx x canbepositiveand negative.Thenthesumofthethirdorderitemsismuchsmallerthanthesumofthesecondorderitmesif 02 , 1, 2, ,ix a x i n < + = . Neglect the third order item and more order item errors if 02 , 1, 2, ,ix a x i n < + = . 7 So, 202 10 0 311 2 1 2 0( )1 1 1( ) , 2( ) / 2 2( )ni nii iin nx xn Mx x x a xa x a x a x a x x x n a x==+ + + ~ = < ++ + + + + + + .(23) Then202 2 2 2 2 1 2 0 1 1 20 31 1 2 1 2 0( )''( )( )( ) / 2 2( )ni nn n iii n nx xx x x x f x x xa a a x x aa x a x a x a x x x n a x==| |( + + ++ + + ~ = |(+ + + + + + + +\ . . (24) From the above analysis, the errors are related to 201( )niix x=and 2302( )aa x +. Bythepropertiesof 2302( )aa x +,ifthemeanpoint 0x increases, 2302( )aa x +decreasesquicklyaccordingly. When the cluster set{ }ix is not particularly divergent, 201( )niix x=is much smaller than divergent cluster. So get theconclusionthatifthedistributionoftheclusterset{ }ix isconcentratedandthemeanpoint 0x islarge,the errors can be smaller. Basedontheaboveerroranalysis,forreducingapproximateerroroftheDSmTapproximateconvexfunction formula, a new Evidence Clustering method is proposed as follows 1) Force the mass assignments of focal elements in the evidence to two sets by the standard of 2n. 2) If 2ixn> , ix is forced to one set, denoted by{ }Lix , and the sum of mass assignments for{ }Lix is denoted by LS , the number of points in{ }Lix is denoted by Ln ; otherwise, ix is forced to the other set, denoted by{ }Six . 3) If{ }Si ix x e , pick thefocal element ix with themaximal value max ix ; if max2(1 )LiLSxn n>, ix is forced to one set{ }Lix . 4) Go on the step 3), untill max2(1 )LiLSxn n