Information Fusion with Topological Event Spaces Roman Ilin Sensors Directorate Air Force Research Laboratory Wright Patterson AFB, OH, USA rilin325@gmail.com Jun Zhang Department of Psychology & Department of Mathematics University of Michigan Ann Arbor, MI, USA junz@umich.edu Abstractwedevelopanovelinformationfusionscheme based on topological event space, viewed asa distributive lattice. Wediscusstheadvantagesoftopologicalmodelingandcompare ourapproachtotheexistingBayesian,Dempster-Shafer,and Dezert-Smarandacheapproaches.Theproposedschemeis describedindetailandillustratedwithanexampleoffusionof three sensors in the presence of missing information. KeywordsInformationFusion,Probability,LatticeTheory, DistributiveLattice,Topology,TopologicalEventSpace,Belief Functions,SensorNetworks,Dempster-Shafer,Proportional Conflict Redistribution, Target ClassificationI.INTRODUCTIONClassical information fusion methods utilize probability theory tomodelbothindividualsensorperformanceandtherulesof their combination. The Dempster-Shafer theory (DST) of belief functionsprovidesanalternativetoclassicalprobabilityby assigningbasicprobabilitymasstosubsetsofthegroundset, whichmodelsthesetofelementaryevents,asopposedto assigningtosingletonsintheBayesianmethods[1].DST eliminates the need to assign a priori probabilities to the ground set.Further development ofthebelieffunctiontheoryresulted intheDezert-Smarandachetheory(DSmT)[2],[3].Bayesian and DST methods perform poorly in the presence of conflicting evidence[4][5][6],hencetheintroductionofseveral ProportionalConflictRedistribution(PCR)rulesthatare applicabletobothDSTandDSmT.Theadvantageof DST/DSmToverBayesianmethodscomesatapriceof computing with an exponentially larger event space - the set of all subsets of the ground set (the powerset). The event space of DSmT is even larger as it includes not only the subsets but also alltheirunionsandintersections.Thecomputationalload increases rapidly with the size of the problem. This work studies information fusion and uncertainty modeling in topological event spaces [7], [8], [9]. One of the advantages of using a topological event space is its smaller size, compared totheBooleaneventspaceusedbyBayesianandDST approaches, yet still forms a closed system for event calculus.Thetopologicalconceptsofopenandclosedsets,boundary andinterior, providerichermodelingenvironments,whichare suitableforrationaldecisionmakingsincetheyadmit probability functions [10]. Topological event spaces have been used to model human decision making and to explain results of experimentsinvolvinghumanjudgmentsofprobabilities[9]. Ourgoalistoformulateafusionframeworkbasedon topologicaleventspacesthatcanbeutilizedinawidevariety ofapplications.Aninitialformulationofsuchaframework briefly reported in [11].Here we provide detailed descriptions of the steps involved in implementing the system and illustrate a possible application of our framework to the fusion of feature detectors with missing information.Therestofthispaperisorganizedasfollows.Weprovide theoreticalbackgroundontopologicaleventspacesinSec.II. Beliefsandprobabilitiesontopologicalspacesarebest explainedthroughconnectionwiththelatticetheory,andthis connection is the subject of Sec. III. The background on belief functionsonBooleaneventspacesandarbitrarylatticesis coveredinSec.IV.Basedontheintroducedconcepts,Sec.V describes the proposed topological fusion scheme. We consider theproblemofclassifierfusionwithmissinginformationthat can be solved with our approach in Sec. VI.Sec. VII provides thediscussionofourresultsandfuturedirectionsofour topological approach. II. BOOLEAN AND TOPOLOGICAL EVENT SPACES Throughoutthispaper,weconsiderfinitesets.Atopological space, or just a topology, is an ordered pair (u, J), where u is aset(whereeachelementiscalledapoint),andJ isa collectionofsubsetsof u ,suchthat e J, u e J,and J is closedunderfiniteintersectionsandarbitraryunionsofits elements[12].TheelementsofJarecalledopensets. Suppose A e J. Theset-theoreticalcomplementofthesubset A is -A = u\A;anysubset -A thatisthecomplementofan open set is called a closed set. The collection of closed sets is also closed under unions and intersections. It is possible for a set to be closed and open at the same time (clopen), or neither. Theextremecasesoftopologicalspacesarethediscrete topology,whichconsistsofallsubsetsofu,thatisthe powerset 2H,andtheindiscretetopologyconsistingofonly two elements: and u. Designating u tobethesetofelementaryevents,orthe groundset,wemodelthesetofallpossibleeventsbythe topology J. The discrete topology corresponds to the Boolean event space that is used by the classical probability. The events in Boolean space are closed under the operations ofsetintersection,union,andcomplementation.Theyforma Booleanalgebraofevents,whichdirectlycorrespondsto classicalpropositionallogic.Thetopologicalspaceisclosed under union and intersection,but not under complementation. Itcorrespondstointuitionisticlogic[13].Thus,topological modelingallowscapturingthefeaturesofintuitionisticlogic, withtherelaxationofthelawofexcludedmiddleasits centerpiece.Evidencenegatingacertaineventdoesnot excludethepossibilityofsomeadditionalthirdevent occurring.Inadditiontocapturingintuitionisticlogic, 18th International Conference on Information FusionWashington, DC - July 6-9, 2015978-0-9964527-1-72015 ISIF 2092topological spaces are also suitable for probabilistic reasoning. Inordertoconsiderprobabilitiesontopologicaleventspaces weneedtobrieflydiscusstheconnectionbetweentopology and lattice theory. III.LATTICE THEORY AND TOPOLOGICAL SPACES A comprehensive treatment of Lattice Theory can be found in [14] and here we only provide the basic definitions. A partial orderonasetP isabinaryrelation,denotedby thatis reflexive, antisymmetric, and transitive. Sets with partial order arecalledposets.Givenasubset S L P,anelement x e P is anupperboundof S if (vs e S), s x.Thelowerboundis defined analogously. The least upper bound is an element x eP such that x is an upper bound of S , and for any other upper boundy ofS,x y.Thegreatestlowerboundisdefined analogously.Givenanytwoelements,x, y e P,wedenotetheirleast upperboundanddually,greatestlowerbound,by x v y and x A y. These operations are respectively called join and meet. Apartiallyorderedset P suchthatjoinandmeetexistfor all pairs of its elements is called a lattice. The lattice is called complete if meet and join exist for any subset of P.Afinitelatticealwayshastheleastelement,orbottom, denoted by 0, and the greatest element (top), denoted by 1. An importantoperationonalatticeisthatofcomplementation.For any element oe P, its complement is an element o'e P, such that oiA o = u and oiv o = 1. Note that for an arbitrary elementofP,itscomplementmaynotexist,oritmayhave multiple complements.Afinitelatticeis often visualized asa Hasse diagram [14], for example see Fig.1.Alatticeiscalleddistributiveifforanylatticeelements o, b, cthedistributivelawo A (b v c) = (o A b) v (o A c) is satisfied.Inadistributivelattice,itisknownthatanelement can have at most one complement. Latticeofsetsisalatticeformedbyafamilyofsubsetsof thepowersetofu ,P L 2H,withthebinaryrelation corresponding to set inclusion: for all A, Be u, A B =A LB andthemeetandjoinoperationscorrespondtotheset theoreticalintersectionandunion.Thetopelementisthe setu,andthebottomelementistheemptyset.Lattice complementationoperationisjusttheset-theoretic complementation: Ai= u\A. Finite distributivelatticecanbeviewed asa latticeofsets, andthisisexpressedbyBirkhoffsrepresentationtheorem [15]. A special kind of distributive lattice is a Boolean lattice, where each element has a unique complement. Sincenoteveryelementofadistributivelatticehasa complement,aweakernotion,calledpseudo-complement,is defined.Foralatticeelement oe P,itspseudo-complement isanelemento-e P,suchthat o-A o = u ando- isthe largestsuchelementtodoso(i.e.,meet o to0).Itturnsout thateveryelementofafinitedistributivelatticehasaunique pseudo-complement;thoughitmaynotbetruethato-v o =1. Fromthedefinitionsisitintuitivelyclearthatatopology can be thought of as a distributive lattice of sets. In fact, there isadirectcorrespondencebetweentheoperationsofclosure andinteriorofasetendowedwithatopologyandthe operationofpseudo-complementation.Intermsofevent modeling,thepseudo-complementcorrespondsto intuitionistic negation, or refutation [7]. Latticetheorypointofviewisusefulduetotherecent interestinlatticebelieffunctions.Wewilldiscussnextthe theoryofbelieffunctionsondifferenttypesoflattice,and relate them to topological modeling.IV.BELIEF FUNCTIONS AND PROBABILITIES ON A LATTICE Dempster-Shafertheoryofbelieffunctions[1]assumesa finite set u of possible answers to a question, referred to as the frameofdiscernment. Theeventscorrespondtosubsetsof u, withtheiruncertaintyquantifiedbythebasicprobability assignment,orbpa(alsoreferredtoasbasicbelief assignment,orbba),whichisafunction m: 2H- |u,1] such that m() = u, (1) and m(A)ALH= 1. (2) Thekeyideahereisthatnotallsubsetsof u encodeinferred information; somearebasic events and can beassigned basic (elementary)probabilitymass.Thosesubsetsthatweassign non-zerom()arecalledfocalelements.Thisstructureis calledabodyofevidence. Basedonourdiscussionoflattice theory, thebody of evidenceis defined on thepowerset of u, or equivalently, on a Boolean lattice. A belief function Bcl: 2H- |u,1] is computed from the bpa as follows: Bcl(A) = m(B)BLA, (3) whereA L u.Thebpacanberecoveredfromthebelief function using the following formula, m(A) = (-1)|A\B|Bcl(B)BLA. (4) Thusthereisaone-to-onecorrespondencebetweenm and Bcl. The belief functions satisfy the following properties: Bcl() = u, Bcl(u) = 1, Bcl __An=1_ (-1)|I|+1Bcl __AeI_.Ic{1,2..n],I= (5) Thelast property shows that belief functions arenon-additive eveninthecaseoftwodisjoinedsubsets A and B: Bcl(A UB) Bcl(A) +Bcl(B) ,incontrasttotheadditivityof probability functions.2093 If,andonlyif,thefocalelementsofbpaare1-element subsets(singletons)of u,does Bcl becomeequivalentand reduces to a probability function. Thus, a classical probability functionisacaseofabelieffunctionwithbpaassignedto singletonelementsoftheBooleanlattice.Inthiscase,the function Bcl becomes additive. Giventwobodiesofevidencewithbpas m1and m2 that need to becombined or fused, theconjunctiveconsensusis given by the following formula: m1r2(A) =m1(B)m2(C)BrC=A. (6) The result of (6) needs to be normalized in order to satisfy (2).Dividingm1r2 bythetotalconflictingmass,Kachieves normalization.K is defined in (7). K =m1(B)m2(C)BrC= (7) Formulas(8)and(9)correspondtotheDempstersruleof combination. The un-normalized version (6) of the Dempsters rule is used in Transferable Belief Models (TBM) [16]. Other combinationruleshavebeenproposedaswell[2],[17].The rule (6) can easily be generalized for more than two bpas. Since Bcl is not additive in general, it is usually transformed intoaprobabilityfunctionusingthepignistictransformation [16].Rationaldecisionsaremadebasedonpignistic probabilities, or betting changes, defined as follows, Bctm(o) = m(A)|A|, vo e u.{A|ALH,ueA] (8)

Itturnsoutthatthebelieffunctionframeworkcanbe generalizedtothelatticesetting[18].Foranyposet (P, ), andafunction: P - R,theMbiustransform[19]isthe function m: P - R such that (x) = m(y)