空间泊松点过程

SpatialPoissonProcesses,TheSpatialPoissonProcess,Consideraspatialconfigurationofpointsintheplane:,Notation:,LetSbeasubsetofR2.(R,R2,R3,),LetAbethefamilyofsubsetsofS.,Forlet|A|denotethesizeofA.(length,area,volume,),LetN(A)=thenumberofpointsinthesetA.,(AssumeSisnormalizedtohavevolume1.),ThenisahomogeneousPoissonpointprocesswithintensityif:,ForeveryfinitecollectionA1,A2,AnofdisjointsubsetsofS,N(A1),N(A2),N(A3)areindependent.,Foreach,Alternatively,aspatialPoissonprocesssatisfiesthefollowingaxioms:,TheprobabilitydistributionofN(A)dependsonthesetAonlythroughitssize|A|.,Thereexistsasuchthat,Thereisprobabilityzeroofpointsoverlapping:,Iftheseaxiomsaresatisfied,wehave:,fork=0,1,2,ConsiderasubsetAofS:,Thereare3pointsinAhowaretheydistributedinA?,A,Expectauniformdistribution,Infact,forany,wehave,Proof:,So,weknowthat,fork=0,1,n:,ie:N(B)|N(A)=nbin(n,|B|/|A|),Generalization:,ForapartitionA1,A2,AmofA:,forn1+n2+nm=n.,(Multinomialdistribution),SimulatingaspatialPoissonpatternwithintensityoverarectangularregionS=a,bxc,d.,scatterthatnumberofpointsuniformlyoverS,(foreachpoint,drawU1,U2,indepunif(0,1)sandplaceitat(b-a)U1+a),(d-c)U2+c),Consideratwo-dimensionalPoissonprocessofparticlesintheplanewithintensityparameter.,Letsdeterminethe(random)distanceDbetweenaparticleanditsnearestneighbor.,Forx0,So,forx0.,In3-Dwecouldshowthat:,Example:SpatialPatternsinStatisticalEcology,Considerawideexpanseofopengroundofauniformcharacter(suchasthemuddybedofarecentlydrainedlake).,Thenumberofwind-dispersedseedsoccurringinanyparticular“quadrat”onthissurfaceiswellmodeledbyaPoissonrandomvariable.,ThereasonthistendstobetrueisduetothebinomialapproximationtothePoissondistributionwhichwillholdiftherearemanyseedswithanextremelysmallchanceoffallingintothequadrat.,Supposenowthattheprobabilitythataseedgerminatesispandthattheyarenotsufficientlypackedtogethertointeractatthisstage.,Question:Whatisthedistributionofthenumberofgerminatedseeds?,(acceptprobabilityis),So,thesurvivingseedscontinuetobedistributed“atrandom”.,SimulationProblem:,Type1andtype2seedswillgerminatewithprobabilitiesp1andp2,respectively.,Type1plantswillproduceKoffshootplantsonrunnersrandomlyspacedaroundtheplantwhereKgeom(p).(P(K=0)=p),Twotypesofseedsarerandomlydispersedonaone-acrefieldaccordingtotwoindependentPoissonprocesseswithintensities,Supposethattheone-acrefieldisevenlydividedinto10 x10quadrats.,Assumethatthenumberofoffshootplantsthatfallintoaquadratdifferentfromtheirparentplantsisnegligible.,Aparticularinsectpopulationcanonlybesupportedifatleast75%ofthequadratscontainatleast35plants.,Usingp=0.9,p1=0.7,andp2=0.8,explorethevaluesofthatwillgivetheinsectpopulationa95%chanceofsurviving.,Usethehugelysimplifyingassumptionthatthereisnotimecomponenttothisprocess(and,inparticular,thatoffshootplantsdonothavefurtheroffshoots),Keepinmindthatwedontreallyhavetokeeptrackofwheretheindividualplantsare,onlythenumberineachquadrat.,NotethatwedonthavetoconsidergerminationoftheplantsasasecondstepafterthearrivaloftheseedsinsteadconsiderathinnedPoissonnumberofplantsofTypeiwithrate,Tipsonsimulatingthis:,Ratherthandrawinguniformlydistributedlocationsfortheseeds,wecansimulatethenumbersforeachquadratseparately(andignorelocations)usingthefactthateachquadratwillcontainPoisson()germinatingseeds.,ItwouldbeniceifwecouldfurthermodifythePoissonnumberofseedsforType1.,Wecan,atl