An Evidence Clustering DSmT Approximate Reasoning Method Based on Convex Functions Analysis GUO Qiang1, HE You1 , GUAN Xin2 , DENG Li3, PAN Lina4, JIAN Tao1 (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: GUO Qiang, Tel numbers: +8615098689289, E-mail address: gq19860209@163.com. Abstract Withtheincreasingnumberoffocalelementsinframeofdiscernment,computationalcomplexityof DSmT(Dezert-SmarandacheTheory)increasesexponentially,whichblocksthewideapplicationanddevelopmentof DSmT. To solvethis problem, anew evidence clustering DSmT approximatereasoning method is proposed in this paperbasedonconvexfunctionsanalysis.Thecomputationalcomplexityofthemethodinthispaperincreases linearly instead of exponentially with the increasing number of focal elements indiscernment framework. First, the method clusters the belief masses of focal elements in each evidence. Then, the first step results are obtained by the proposed DSmT approximate convex functions formula. Finally, themethod gets the approximatefusion results by normalization method. The results of simulation show that the approximate fusion results of the method in this paper hashigherEuclideansimilaritywiththeexactfusionresultsofDSmT+PCR5,andneedlesscomputational complexitythantheexistingapproximatemethods.Especially,inthecaseoflargedataandcomplexfusion problems, the method in this paper can get highly accurate results and need low computation complexity. Keywords:Evidenceclustering;Approximatereasoning;Informationfusion;Convexfunctionsanalysis; Dezert-Smarandache Theory 1.Introduction1 As a novel key technologe with vigorous development, information fusion can integrat multiple-source incomplete information and reduce uncertainty of information which always has the contradiction and redundancy.Information fusioncanimproverapidcorrectdecisioncapacityofintelligentsystemsandhasbeensuccessfullyusedinthe militaryandeconomyfields,thusgreatattentionhasbeenpaidtoitsdevelopmentandapplicationbyscholarsin recent years1-6. As information evironment becomes more and more complex, greater demands for efficient fusion of highlyconflictevidencearebeingplacedon informationfusion.DSmTisaneweffectivemethodforthefusion problemofuncertain,impreciseandhighlyconflictevidence,jointlyproposedbyFrenchscientistDr.JeanDezert and American mathematician Florentin6. DSmT can be considered as an extension of the classical Dempster-Shafer. *ManuscriptClick here to view linked References DSmT is able to solve complex static or dynamic fusion problems beyond the limits of the DST framework, specially whenconflictsbetweensourcesbecomelargeandwhentherefinementoftheframeoftheproblemunder consideration, denotedO, becomes inaccessible because of the vague, relative and imprecise nature of elements of O6-8.Recently,DSmThasbeenappliedtomanyfields,suchas,imageprocessing,RobotsMapReconstruction, Target Type Tracking, Sonar imagery and Radar targets classification and so on6,9-12. The bottleneck problem to block thewideapplicationanddevelopmentofDSmTisthatwiththeincrementoffocalelementnumberinframeof discernment, computational complexity increases exponentially. In order to solve this problem, many scholars presents approximate reasoning method of evidence combination in D-S framework13-15. But these methods almost can not satisfy the small amount of computational complexity and less lossofinformationrequirementsatthesametime.Dr.LiXindeandotherscholarsproposedafastapproximate reasoning method in hierarchical DSmT16-18. However, when processing highly conflict evidence by the method, the belief assignments of correct main focal elements transfer to the other focal elements, which leads to low Euclidean similarity of the results in this case. AimingatreducingthecomputationalcomplexityofDSmTandobtainingaccurateresultsinanycase,anew evidence clustering DSmT approximate reasoning method is proposed in this paper. In section 2, information fusion methodofDSmT+PCR5isintroducedbriefly.Insection3,MathematicalanalysisofDSmc+PCR5formulais conducted,whichdiscoverseveryconflictmassproductsatisfiesthepropertiesofconvexfunction,obtainsthe approximateconvexfunctionformulaofDSmT+PCR5andanalysesapproximateconvexfunctionformulaerrors. Then a new approximate reasoningDSmTmethod is proposed by analysis ofapproximate convex functionformula errors in section 4. In section 5, analysis of computation complexity of DSmT+PCR5 and the method in this paper is carriedout.Theresultsofsimulationshowthattheresultsofthemethodproposedinthispaperhavehigher EuclideansimilaritywiththeexactfusionresultsofDSmT+PCR5,andlowercomputationalcomplexitythan existing DSmT approximate method18 in section 6. 2.InformationfusionmethodofDSmT+PCR5 Instead of applying adirecttransfer of partial conflicts onto partial uncertaintiesaswith DSmH, theidea behind theProportionalConflict Redistribution(PCR) rule19is to transfer (total or partial) conflictingmasses to non-empty sets involved in the conflicts proportionally with respect to the masses assigned to them by sources as follows6: 1. calculation the conjunctive rule of the belief masses of sources; 2. calculation the total or partial conflicting masses; 3.redistributionofthe(totalorpartial)conflictingmassestothenon-emptysetsinvolvedintheconflicts proportionally with respect to their masses assigned by the sources. ThewaytheconflictingmassisredistributedyieldsactuallyseveralversionsofPCRrules.PCR5isthemost mathematically exact redistribution method of conflicting mass. This rule redistributes the partial conflicting mass to the elements involved in the partial conflict, considering the conjunctive normal form of the partial conflict. It does a betterredistributionoftheconflictingmassthanDempstersrulesincePCR5goesbackwardsonthetracksofthe conjunctive rule and redistributes the conflicting mass only to the sets involved in the conflict and proportionally to their masses put in the conflict. The PCR5 formula for the combination of two sources (s = 2) is given by6: PCR5[ ] 0, \ m X G XOC = e2 21 2 2 1PCR5 12\ 1 2 2 1( ) ( ) ( ) ( )[ ] ( ) [ ]( ) ( ) ( ) ( )Y G XX Ym X mY m X mYm X m Xm X mY m X mY|Oe== + ++ +(1) whereallsetsinvolvedinformulasareincanonicalformandwhereGOcorrespondstoclassicalpowerset2Oif Shafers model is used, or to a constrained hyper-power setDOif any other hybrid DSm model is used instead,or to thesuper-powersetSOiftheminimalrefinement refO ofO isused; 12( ) ( ) m X m X = correspondstothe conjunctive consensus onX between the2 S = sources and where all denominators are different from zero. If a denominator is zero, that fraction is discarded. 3.MathematicalanalysisofDSmc+PCR5formula As shown in formula (1), 2 21 2 2 1\ 1 2 2 1( ) ( ) ( ) ( )[ ]( ) ( ) ( ) ( )Y G XX Ym X mY m X mYm X mY m X mY|Oe=++ +has symmetry. Due to the symmetry, analyse one item21 21 2( ) ( )( ) ( )m X mYm X mY +. Let 1 2( ) , ( ) mX a mY b = = , get 2 22 1 21 2( ) ( ) 1[1 ( )]( ) ( )m X mY a ba am X mY a b a b= = + + +, then 22 1 21 2/ 1 2 1 2( ) ( ) 1 1 1[ ] [ ( )], , ,( ) ( )nY G X nX Ym X mYa n a b b b Ym X mY a b a b a b|Oe== + + + =+ + + +(2) Let 1( ) f xa x=+, dueto( ) f x iscontinuousfunction on[0,1],hassecondorderderivativeson(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+ + ++ + + s , the equality holds iff 1 2 nx x x = = = . The approximate convex function formula is given by: 1 2 1 21 1 1( ) /n nna x a x a x a x x x n+ + + = + A+ + + + + + +(3) 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 nA = + + + + ++ + + + + + + + + + + + +(4) analysis of thei item in equality(4). 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 = + + + + + + +. By taylor expansion theorem, 2 3 00 0 0 0 0 00''( ) 1 1 '''( )'( )( ) ( ) ( ) , ( , ) ( , )2 3!i i i i iif x ff x x x x x x x x x x xa x a xoo = + + e+ +or . Then equality (4) is transformed to 0 1 0 2 0 1 01 2 1 212 2 2 2 01 0 2 0 1 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=+ + + = + + + + + + + + + ++ + + + + 0 1 0 2 0 1 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= =+ + + = + + + + + + + + 2 3 3 3 4 4 4 0 1 20 1 0 2 0 0 1 0 2 0 01'''( ) '''( ) '''( ) ''( )( ) [( ) ( ) ( ) ] ( ) ( ) ( )3! 4! 4! 4!nni n nif x f f fo x x x x x x x x x x x x x xo o o= = + + + + + + + Analysis of( ) , 2, 3, ,mf x m= . 11 1( )m mmf x ma x a x| | | |= = ||+ +\ . \ ., then 2 111 1 0 00 0 0 00 0210 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 | | | | = || + +\ . \ .| | | | | |= ||| | + +\ . \ . \ . If 0x x s ,0 0 0x x x x x a = < + , then 11 0 00 0( ) ( )( ) ( )( 1)! !m mm mf x f xx x x xm m > . If 0 02, , 2 m x x x a x = > < + ,001 1( ) 0( 1)x xm a x| | > | +\ .. So if 02 , 1, 2, ,ix a x i n < + = , 11 0 00 0( ) ( )( ) ( ) , 2( 1)! !m mm mi if x f xx x x x mm m > >. Namely, 2 3 0 0 00 0 0''( ) '''( ) ( )( ) ( ) ( )2 3! !mmi i if x f x f xx x x x x xm > > > . 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.Thenthesumofhethirdorderitemsismuchsmallerthanthesumofthesecondorderitmesif 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 < + = . So, 202 10 0 311 2 1 2 1 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==+ + + ~ = < ++ + + + + + + . Then 202 2 2 2 2 1 2 1 0 1 1 20 31 1 2 1 2 1 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 == + + ++ + + ~ =+ + + + + + + +. From the above analysis, the errors are related to 201( )niix x=and 2302( )aa x +. By the properties of 2302( )aa x +, if the mean point 0xincreases, 2302( )aa x + decreases quickly accordingly. When the cluster set { }ix is not particularly divergent, 201( )niix x= is much smaller than divergent cluster. So get the conclsion that if the distribution of the cluster set { }ix is concentrated and the mean point 0xis large, the erros can be smaller. Based on the above analysis, for reducing approximate errors and remaining lower computing complex, an evidence clustering method is proposed as follows: 1) Force the mass assignments of focal elements in the evidece 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 the focal element ix with the maximal 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