The Generalization Negation of Probability Distribution and Its Application in Target Recognition

Thegeneralizationnegationof probabilitydistributionandits applicationintargetrecognitionbased onsensorfusion

XiaozhuanGaoandYongDeng

Abstract

InternationalJournalofDistributed SensorNetworks 2019,Vol.15(5)

TheAuthor(s)2019 DOI:10.1177/1550147719849381 journals.sagepub.com/home/dsn

Targetrecognitioninuncertainenvironmentsisahotissue.Fusionrulesareusedtocombinethesensorreportsfromdifferentsources.Inthissituation,obtainingmoreinformationto makecorrectdecisionisanessentialissue.Probabilitydistributionisoneofthemostusedmethodstorepresentuncertaintyinformation.Inaddition,thenegationofprobability distributionprovidesanewviewtorepresenttheuncertaintyinformation.Inthisarticle,theexistingnegationofprobability distributionisextendedwithTsallisentropy.Themainreasonisthatdifferentsystemshavedifferentparameter q.Some numericalexamplesareusedtodemonstrate theefficiencyoftheproposedmethod.Besides,thearticlealsodiscussesthe applicationofnegationintargetrecognitionbasedonsensorfusiontofurtherdemonstratetheimportanceofnegation.

Keywords

Probabilitydistribution,negation,Tsallisentropy,Ginientropy,sensorfusion

Datereceived:5January2019;accepted:17April2019

HandlingEditor:MohamedAbdel-Basset

Introduction

Inrecentyears,informationfusionispaidgreatattentioninmilitaryapplications.1,2 Manymethodsbased oninformationfusionhavebeenproposedtoclassify objects,3 targetrecognition4,5 anddecisionsmaking.6,7 Inmostcases,thefinalresultwhichisobtainedby informationfusionofmultiplesensorsmaybereasonable.However,therecognitionresultmaybecounterintuitiveduetohighconflict.8 Uptonow,fusionruleis stillanopenquestion.Besides,theinformationgatheredinsensorsfusingsystem9–11 existsuncertainasitis incomplete,inconsistentandpossiblyimprecise.Many methodshavebeenproposedtoobtainmoreinformation.12–14 However,duetotheexistinguncertainty,15–17 itisessentialtoobtainmoreinformationaccordingto knownknowledge.

Presentingknowledgeisanopenissue.18–20 Inmost cases,wecommonlyuse‘must’,‘may’and‘likely’to

estimatewhetheraneventwillhappenornot.Dueto theuncertainty,therearesomemethodstodealwith it.21–24 Probabilitydistributionisusedtoquantitatively describethepossibilityofoccurrenceofaresultinreal applications.25 Moreimportantly,formostcases,itis mucheasiertodescribethenegationoftheeventsthan directlydescribetheminsomecircumstances.For example,ifitisdifficulttoproveamathematicalformularigorously,however,acounterexamplecaneasily provetheformulawrong.Similarly,thereisasignificantpropertyinprobability,namelymutualexclusion.

InstituteofFundamentalandFrontierScience,UniversityofElectronic ScienceandTechnologyofChina,Chengdu,China

Correspondingauthor: YongDeng,InstituteofFundamentalandFrontierScience,Universityof ElectronicScienceandTechnologyofChina,Chengdu610054,China. Emails:dengentropy@uestc.edu.cn;prof.deng@hotmail.com

CreativeCommonsCCBY:ThisarticleisdistributedunderthetermsoftheCreativeCommonsAttribution4.0License (http://www.creativecommons.org/licenses/by/4.0/)whichpermitsanyuse,reproductionanddistributionoftheworkwithout furtherpermissionprovidedtheoriginalworkisattributedasspecifiedontheSAGEandOpenAccesspages(https://us.sagepub.com/en-us/nam/ open-access-at-sage).

Mutualexclusionmeanstheoccurrenceofonething whenitsoppositewillnothappen.Inotherwords,the probabilityofaneventwillaffectitsopposition. Therefore,itismeaningfultostudynegationofprobabilitydistribution(NPD).26 Recently,theNPDbased onGinientropywasproposedbyYagertopresent knowledgeinanewview.27

Themotivationofthisstudyistofindamoregeneralandreasonablemodetoevaluatetheuncertainty ofNPD.Ifthecorrelationsbetweenthe N elementsare strongenough,thentheextensivityofsomeentropiesis lost,whichisincompatiblewithclassicalthermodynamics.28 Tsallisentropy29 isproposedtoovercome thisdifficultywithnon-extensiveproperty.InTsallis entropy, q isnota‘tunable’parameter,butisstrictly determinedbythesystemitself.30 Asaresult,always usingGinientropyinYager’sNPD,27 whichisaspecial caseofTsallisentropywhen q = 2,isnotreasonableto manyrealsystems.Tosolvethisproblem,thearticle usesTsallisentropytomeasuretheuncertaintyof NPD,whichcanbemorereasonable.

Moreimportantly,innatureandsociety,everything hasitsnegation.Regretandexpectgiveustwoviewsto consideraproblem.Moreover,thebestalternativeisas closeasidealsolutionandisasfarasnegativesolution. Besides,NPDcanprovideanotherinformationbased onknowninformation.Thatistosay,wecananalyse thistargetfromtwosides,whichcanimprovethecorrectnessofdecision-making.ThereasonisthatNPD hasthepropertiesofimprecisionandunknown,which isbeneficialfordecision.Moreimportantly,iftheoriginalinformationishighlyconflicting,theconflictof negationmaynotbehighlyconflicting.Basedonthe discussion,itisofgreatsignificancetostudynegation. Hence,thearticlealsousesnegationtorecognizetarget basedonsensorfusion.

Therestofthisarticleisstructuredasfollows:Inthe section‘Preliminaries’,preliminariesofsomeentropies andYager’snegationmethodareintroduced.Theproposedmethodisintroducedinthesection‘Theproposedmethod’.Inthesection‘Examplesand discussion’,anumericalexampleisusedtoillustrate themethodofnegationandusingTsallisentropyto evaluatetheuncertainty.Theapplicationofnegationin targetrecognitionbasedonsensorfusionisintroduced inthesection‘Applicationofnegationbasedonsensor fusion’.Finally,someconclusionsaregiveninthesection‘Conclusion’.

Preliminaries

Inthissection,thepreliminariesofsomeentropiesand NPDwillbebrieflyintroduced.

Ginientropy

Entropyplaysaveryimportantroleinmanysystems.31–33 InYager’sNPD,theGinientropyisadopted.27

Definition1. Ginientropyisdefinedasfollows34

HG = X n i = 1 P(Ai )(1 P(Ai )) ð1Þ

where pi istheprobabilitydistributionwhichsatisfies Pn i = 1 Pi = 1.Ginientropyisexpansive,thatis, G (P1 , P2 , , Pn )= G (P1 , P2 , , Pn , 0).Itmeansaddinganelementwithzeroprobabilitydoesnotaffectthe Ginientropy.

Moreover,ifonlyforthesakeofcomparingtheuncertaintyassociatedwiththetwodistributions,theGini entropymaybepreferredtotheShannonwithsimplercalculation.35 Knowledgerepresentationisofgreatsignificance tomodernscience.Inmanyfields,ithasbeenregardedas themaindrivingforcefortheapplicationoftheorytopractice,suchas,aggregation,36 evidenceresolution,37–39 decision-making40–42 andsoon.43–45 Interestingly,probabilityis similartocoins,italsohasaoriginalsideandanegative side.46 Summarizing,negationprovidesanewviewtoinvestigatethepropertyofprobability.

Definition2. Assumingaprobabilitydistribution P = P(A1 ), P(A2 ), , P(An ) fg,theNPDwasdefined as 27 P(Ai )= 1 Ai n 1 ð2Þ

Becausetheprobabilitiesarecompletelymutually exclusive,itsnegationwasnormalized,sotheNPD satisfies 0 ł P(A1 ) ł 1 X P(Ai )= 1

Itshouldbepointedthatsincethebasicprobability assignmenthasmoreflexibilitytorepresentuncertainty, thenegationofbasicprobabilityassignmentisalsopaid attentionrecently.46

Tsallisentropy

Definition3. Givenaprobabilitydistribution P = P1 , P2 , Pn fg,theTsallisentropycanbedefined as 29

Hq P ðÞ = k q 1 1 X i Pq i !

when q ! 1,whichcorrespondstotheBoltzmannGibbsentropy,namely

H (P)= H1 (P)= X i 1 Pi lnPi ð4Þ

OnecanrewriteTsallisentropyasfollows29

Hq = k q 1 X W 1 Pi (1 Pq 1 i ) ð5Þ

where q isarealparametersometimescalledentropicindexwhichgeneralizestheusualexponentialandplays theimportantroleindescriptionofthermodynamic. k istheBoltzmannconstant(orsomeotherconvenient value,inareasoutsidephysics,suchasinformationtheory,cyberneticsandothers).28,29 Besides,when q = 2, theTsallisentropydegeneratestheGinientropy.

Theproposedmethod

Fromtheabove,itcanbeseenthatYager’smethod wouldbethespecialcase.Inordertoexpandtheapplicationofnegation,thearticleproposedamethodthat usesTsallisentropytomeasuretheuncertaintyof NPD.UsingTsallisentropytomeasuretheuncertainty ofNPDcanexpandtheapplicationofnegation

Hq = 1 q 1 X 1 Pi n 1 1 ( 1 Pi n 1 ) q 1 ð6Þ

Inthefollowingpart,thepropertyofnegation-based proposedmethodcanbediscussed.

Property1. Tsallisentropycanincreaseafternegation. Specificproofisasfollows Hq (P)= 1 q 1 3 (1 X (Pq i )) Hq (P)= 1 q 1 3 (1 X (Pq i )) when q ¼ 1 Hq (P) Hq (P)= 1 q 1 3 ( X Pq i X 1 Pi n 1 q ) D0 = Hq (P) Hq (P)+ l X N i = 1 Pi 1! ∂D0 ∂Pi = q q 1 pq 1 + q q 1 1 n 1 q 1 Pi ðÞq 1 + l Hq (P) Hq (P) ø 0 when q = 1

H1 (P) H1 (P)= X (1 Pi )ln(1 Pi )+ X Pi lnPi

D0 = H1 (P) H1 (P)+ l X N i 1 Pi 1!

∂H1 (P) H1 (P) ∂Pi = lnPi + 1 n 1 3 ln 1 Pi n 1 + l H (P) ø H (P)

Fromtheabovecalculation,itcanbeseenthat Tsallisentropycanincreaseafternegation.

Property2. ThereismaximumTsallisentropywithuniformdistributionaftermultiplenegations.

Thegeneralformulaofthenegationmethodisas follows xi + 1 = 1 n + n 3 x1 1 n 3 (n 1)i 1 ð7Þ

where xi + 1 representsthe ithnegation.Aftermultiple negations,theprobabilitydistributionisasfollows lim i!‘ xi + 1 = 1 n

Fromtheabove,itcanbeseenthatTsallisentropy reachesthemaximumaftermultiplenegations.

Besides,innature,anyoperationcancauseenergy consumption.Thechangeofentropymeanstheconsumptionofenergy,andconsumedenergycannotbe reused.Similarly,thevariationofinformationentropy isaccompaniedbytheconsumptionofinformation.Of variousentropies,Tsallisentropyisnon-extensive entropyandappliedinartificialandsocialcomplexsystems.Hence,usingTsallisentropytomeasureuncertaintyofNPDcanexpandtheapplicationsofnegation.

Examplesanddiscussion

Example

Comparing

whichattachestolowerprobabilityacquireshigherprobabilityafternegationprocess. Attractively,whathappensifnegationistakento P(xi ) ?Secondnegationismadeandtheresultsareobtained

Itisapparentthattheoriginalprobabilitydistribution P(xi )isnotequalto P(xi ).Generally,thisprocess forobtainingtheNPDisirreversiblewhen N ø 3

P(xi ) ¼ P(xi )

Itisnecessarytoexplorewhatcausesthisirreversibility.FromExample1,itcanbefoundthattheprobabilitywouldberedistributionafternegation.Hence,it shouldbeconsideredwhethertheuncertaintyhaschangedaftertakingNPD.Inthenextsection,thereare somediscussionsonwhythenegationprocessisgenerallyirreversibleandhowtomeasuretheuncertaintyof theprobabilitybasedonournegationmethod.

Furtherdiscussion

AgainconsideringtheExample1,theTable1isused toshowthechangeofprobabilityaftereachiteration ofnegationprocess.Itcanbeclearlyknownthatprobabilityisreallocatedafternegation.Gradually,the probabilitydistributionbecomesmoreandmoreclose totheuniformdistribution.Whatisthecauseofthis phenomenon?

Theconceptofentropyisderivedfromphysics.47,48 Withtheincreasingapplicationofentropy,49,50 informationentropyhasbecomeanindispensablepartof modernscientificdevelopment.51,52 Shannon53 first proposedtheconceptofinformationentropyto describetheuncertaintyandhasappliedinalotsof fields.54,55 However,asintypicalphysicalproblems, therearesomeexampleswheretheBoltzmannShannonentropyisnotsuitable.56,57 In1988,Tsallis proposedanon-extensiveentropycalledTsallis entropy.Subsequently,non-extensivestatistical mechanicswhichisGeneralizationofBoltzmannGibbsStatisticsemergedbasedonTsallisentropy. Moreimportantly,theBoltzmann-Gibbsstatisticsis recoveredasthelimitationwhen q ! 1 58 Thevalueof q ishiddeninthemicroscopicdynamicsofthesystem. Itcanbeobtainedbysomeexperiments.Arelevant improvementontheinferenceaccuracybyadopting non-extensiveentropiesisproposedbyTsallis,57 where q equals2.5obtainsabetterresult.Besides,in article,59q = 2 5 isusedtoobtainmutualinformation duringDREAM4data.Asaconsequence,usingTsallis entropytomeasuretheuncertaintyextendedthe methodofYager.27 Therefore,itisofgreatsignificance tomeasureuncertaintyofnegationwithTsallis entropy.

Let’scalculatetheuncertaintyusingTsallisentropy withdifferent q toobservehowTsallisentropychange afternegation,asshowninFigure1.Therearechanges ofTsallisentropywithfivedifferent q from 1 2 to3.It canbeeasilyseenthattheTsallisentropyincreases

Figure1. ThetrendofTsallisentropyaftereachiterationof negationprocess.

Table1. Thechangeofprobabilityaftereachiterationof negationprocess.

Frequencyofnegation P (x1 ) P (x2 ) P (x3 ) 00.20000.50000.3000 10.40000.25000.3500 20.30000.37500.3250 30.35000.31250.3375 40.32500.34370.3313 50.33750.32810.3344 60.33100.33600.3330 70.33450.33200.3335 80.33300.33400.3330 90.33330.33320.3335 100.33330.33330.3334

graduallyintheprocessofnegationandisalmost invariableafterthefifthnegation,regardlessofthe valueof q.Itisclearlyshownthattheuncertaintygraduallyincreasesinanysystemaftereachiterationof negationprocess.Whentherearemultipleiterationsof negationprocess,theprobabilitydistributionwouldbe uniformdistributionandTsallisentropyhasmaximum entropy,whichisconsistentwith Property 1and Property 2.Hence,usingTsallisentropytomeasure uncertaintyofNPDisreasonable.

Applicationofnegationbasedonsensor fusion

Targetrecognitionispaidgreatattentioninmilitary applications.60 Therearesomemethodstomakedecisions.61–63 Fusionrulescanhelpusmakebetterdecisions.64,65 However,usingexistinginformationtomake

moreaccuratedecisionisalsoanopenissue.Thenegationprovidesanewviewtoobtainmoreaccuratejudgementwiththecollectedinformation.Thespecific exampleisasfollows.

Therearethreesensorstorecognizethetargetwhich maybe A, B and C.Theresultsfromthesensorsareas follows

p1 (A)= 0 7, p1 (B)= 0 2, p1 (C )= 0 1 p2 (A)= 0:6, p2 (B)= 0:3, p2 (C )= 0:1 p3 (A)= 0 5, p3 (B)= 0 3, p3 (C )= 0 2

Dempsterruleiswidelyusedinsensordatafusion.64 Thecombinationresultsareshownasfollows

K = 0 77, p(A)= 0 91, p(B)= 0 078, p(C )= 0 087

Usingthemethodofnegation,newinformationis obtainedasfollows

p1 (A)= 0 15, p1 (B)= 0 4, p1 (C )= 0 45 p2 (A)= 0 2, p2 (B)= 0 35, p2 (C )= 0 45 p3 (A)= 0 25, p3 (B)= 0 35, p3 (C )= 0 4

Similarly,fusionruleisusedasfollows

KN = 0 8625, p(A)= 0 055; , p(B)= 0 356, p(C )= 0 375

Itiswellknownthatconflictingcoefficient K plays anessentialroleininformationfusion.8 Asaresult, analysingthedifferenceof K betweentheoriginalprobabilityanditscorrespondingnegationisnecessary.Itis easilyfoundthattheconflictafternegationbecomes bigger.Thereasonisthat p(A)= 1 p(A)represents theprobabilitythetargetisnot A,namely p(A)isthe probabilitythattargetis B or C.Besides,themethod ofnegationcontainstheuncertainclass(AorB).Hence, theconflictbecomesbiggerduetotheexistinguncertainclassafternegation.

Moreimportantly,bycomparingtheresultsbetween theoriginalandnegation,thedecisionhasmoresupporttotarget A p(A)= 0 055 reflectstheprobability thatthetargetisnot A.Thatistosay,itcanincrease theprobabilityoftarget A fromtheotherside.Hence, itcangettworesultsfromthetwosidesbasedondata, whichisbettertomakeamorereasonabledecisionin targetrecognition.

Next,consideringthechangesofentropybetween theoriginalandnegation,assumethat q = 3 asthere arethreesensors

T1 = 0 324, T2 = 0 378, T3 = 0 42, T = 0 123

T 1 = 0:421, T 2 = 0:429, T 3 = 0:439, T = 0:375

Fromtheabove,itcanbeseenthattheTsallis entropyafterfusionbecomesless,showingthatthe fusioncandecreasetheuncertaintyofinformation.

However,fusionrulesdon’tworkinextremeprobabilitydistribution.Usingnegationcanprovideanother viewtoanalysethisphenomenon.

Thereisaspecialexampletobetterexplaintheapplicationofnegation.Therearetwosensorstorecognize thetargetswhichmaybe A, B and C.Theresultsfrom thesensorsareasfollows

p1 (A)= 0 8, p1 (B)= 0, p1 (C )= 0 2 p2 (A)= 0, p2 (B)= 0 8, p3 (C )= 0 2 p2 (A)= 0, p2 (B)= 0:8, p3 (C )= 0:2

Usingfusionrules,onecangettheresultsasfollows

p(A)= 0, p(B)= 0, p(C )= 1

Fromtheresult,itcanbeseenthatthetargetmust be C as p(C )= 1.Obviously,theresultisnotveryreliable.Besides,itcanbeseenthatwhenevidenceishighly conflicting,Dempsterruledoesn’twork.However, negationcanprovideanotherinformation,whichis usefultomakedecision

p1 (A)= 0 1, p1 (B)= 0 5, p1 (C )= 0 4 p2 (A)= 0 5, p2 (B)= 0 1, p2 (C )= 0 4

Thefusionresultsareasfollows

p(A)= 0:19, p(B)= 0:19, p(C )= 0:62

Next,consideringthechangesofentropybetween theoriginalandnegation,assumethat q = 2 asthere aretwosensors

T1 = 0 4800, T2 = 0 4800, T = 0 T 1 = 0 8100, T 2 = 0 8100, T = 0 7480

Obviously,negationgivesussomenewinformation. Fromtheaboveresult,theprobabilitythattargetisnot C is0.62.Itcanbeseenthattheresultofnegationis morereasonablethantheoriginalresult.Moreimportantly,negationprovidesanotherviewtoanalyseproblemandcanbetterhandleconflictingevidence.Hence, negationisusefultomakedecision.

Inaddition,analysingtheTsallisentropybetween originalandnegation,itcanbefoundthatTsallis entropybecomesbigger.Besides,accordingtothesecondlawofthermodynamics,theentropyofanisolated systemneverdecreases.Thenegationcanincrease Tsallisentropy.Moreover,entropyisirreversible. Fromthisview,thenegationcanmakesystemsspontaneouslyevolvetowardsthermodynamicequilibrium,

thestatewithmaximumentropy.Thatistosay,negationtendstomakethesystemuniformlydistributed, andinfactitdoes.Hence,thenegationnotonlyprovidesanewwaytounderstandproblems,butalsoprovidesbetterperformanceofdecisionsupportsystem. Besides,becausetheTsallisentropyishighlyappliedin manyapplications,negationprovidesanewviewto obtaininformation.Hence,usingTsallisentropyto measuretheuncertaintycanenlargetheapplicationof negation.

Conclusion

Probabilitydistributionisefficienttorepresentknowledge.However,everythinginnatureandsocietyhasits negation,whichshowsthatnegationisveryessential. Similarly,probabilitydistributionalsohasitsnegation. Thearticleextendstheproposednegationbyusing Tsallisentropy.Besides,numericalexampleisusedto calculatetheuncertaintybydifferent q ofTsallis entropyafternegation.Itcanbefoundthatthereis maximumTsallisentropyaftertakingmanyiterations ofnegationnomatterwhatthevalueof q is,meanwhile, theprobabilitydistributionbecomesuniformdistribution.Hence,itisreasonabletochooseTsallisentropyin NPDafter q isdetermined.Fromtheabove,itcanbe knownthatthenegationcanconsumesomeinformationandincreasetheTsallisentropy.Finally,thearticle alsodiscussestheapplicationofnegationintargetrecognitionbasedonsensorfusion,whichshowsthat negationcanobtainamorereasonabledecisionwhen theconflictishigh.Insum,negationnotonlyprovides anewviewtoobtaininformationfromanotherside basedonknowninformation,butalsocaninclude impreciseclasstomakebetterdecision.Combiningthe knowninformationandtheinformationofnegation canincreasetheaccuracyofdecision-making.

Thisarticleisapreliminarystudytoobtainthe NPDbasedonuncertaintymeasurements.Thisworkis donemainlytodesignamoreefficientnegationprocess anduncertaintymeasurement,andexpandtheapplicationofnegation.Besides,itisessentialtodeterminethe uncertaintyrelatedtothenegationandstudyits properties.

Declarationofconflictinginterests

Theauthor(s)declarednopotentialconflictsofinterestwith respecttotheresearch,authorship,and/orpublicationofthis article.

Funding

Theauthor(s)disclosedreceiptofthefollowingfinancialsupportfortheresearch,authorship,and/orpublicationofthis

article:TheworkispartiallysupportedbyNationalNatural ScienceFoundationofChina(GrantNos61573290, 61503237).

ORCIDiD

YongDeng https://orcid.org/0000-0001-9286-2123

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