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: [email protected]. 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