空间泊松点过程

SpatialPoissonProcessesTheSpatialPoissonProcessConsideraspatialconfigurationofpointsintheplane:Notation:LetSbeasubsetofR2.(R,R2,R3,…)LetAbethefamilyofsubsetsofS.Forlet|A|denotethesizeofA.(length,area,volume,…),AALetN(A)=thenumberofpointsinthesetA.(AssumeSisnormalizedtohavevolume1.)ThenisahomogeneousPoissonpointprocesswithintensityif:Foreveryfinitecollection{A1,A2,…,An}ofdisjointsubsetsofS,N(A1),N(A2),…,N(A3)areindependent.AA{N(A)}0Foreach,AA.|)A|Poisson(~N(A)Alternatively,aspatialPoissonprocesssatisfiesthefollowingaxioms:i.IfA1,A2,…,Anaredisjointregions,thenN(A1),N(A2),…,N(An)areindependentrv’sandN(A1UA2U…UAn)=N(A1)+N(A2)+…+N(An)ii.TheprobabilitydistributionofN(A)dependsonthesetAonlythroughit’ssize|A|.iii.Thereexistsasuchthat0|)Ao(||A|1)P(N(A)iv.Thereisprobabilityzeroofpointsoverlapping:11)P(N(A)1)P(N(A)lim0|A|Iftheseaxiomsaresatisfied,wehave:fork=0,1,2,…k!|)A|(ek)P(N(A)k|A|-ConsiderasubsetAofS:Thereare3pointsinA…howaretheydistributedinA?AExpectauniformdistribution…Infact,forany,wehaveProof:AB|A|||B1)N(A)|1P(N(B)1))P(N(A1)N(A)1,P(N(B)1)N(A)|1P(N(B)1))P(N(A1))BN(A1,P(N(B)C|A|-|BA|-|B|-e|A|ee||BC|A|||BSo,weknowthat,fork=0,1,…,n:k-nk|A|||B-1|A|||Bknn)N(A)|kP(N(B)ie:N(B)|N(A)=n~bin(n,|B|/|A|)Generalization:ForapartitionA1,A2,…,AmofA:n)N(A)|n)N(A,...,n)N(A,n)P(N(Amm2211m21nmn2n1m21|A||A||A||A||A||A|!n!n!nn!forn1+n2+…+nm=n.(Multinomialdistribution)SimulatingaspatialPoissonpatternwithintensityoverarectangularregionS=[a,b]x[c,d].simulateaPoisson()numberofpoints1N1i-ieU(perhapsbyfindingthesmallestnumberNsuchthat)scatterthatnumberofpointsuniformlyoverS(foreachpoint,drawU1,U2,indepunif(0,1)’sandplaceitat((b-a)U1+a),(d-c)U2+c)Consideratwo-dimensionalPoissonprocessofparticlesintheplanewithintensityparameter.Let’sdeterminethe(random)distanceDbetweenaparticleanditsnearestneighbor.Forx0,x)P(D(x)FDx)P(D-1centereddiskinparticlesotherP(no-1)xareawithparticletheat22x-e-1So,forx0.2x-DDex2(x)Fdxd(x)fIn3-Dwecouldshowthat:3x-De-1(x)F343x-2DDex4(x)Fdxd(x)f34Example:SpatialPatternsinStatisticalEcologyConsiderawideexpanseofopengroundofauniformcharacter(suchasthemuddybedofarecentlydrainedlake).Thenumberofwind-dispersedseedsoccurringinanyparticular“quadrat”onthissurfaceiswellmodeledbyaPoissonrandomvariable.ThereasonthistendstobetrueisduetothebinomialapproximationtothePoissondistributionwhichwillholdiftherearemanyseedswithanextremelysmallchanceoffallingintothequadrat.Supposenowthattheprobabilitythataseedgerminatesispandthattheyarenotsufficientlypackedtogethertointeractatthisstage.Question:Whatisthedistributionofthenumberofgerminatedseeds?Answer:ThisisathinnedPoissonprocess….pwithrate(acceptprobabilityis)pSo,thesurvivingseedscontinuetobedistributed“atrandom”.SimulationProblem:Type1andtype2seedswillgerminatewithprobabilitiesp1andp2,respectively.Type1plantswillproduceKoffshootplantsonrunnersrandomlyspacedaroundtheplantwhereK~geom(p).(P(K=0)=p)Twotypesofseedsarerandomlydispersedonaone-acrefieldaccordingtotwoindependentPoissonprocesseswithintensities.and21Supposethattheone-acrefieldisevenlydividedinto10x10quadrats.Assumethatthenumberofoffshootplantsthatfallintoaquadratdifferentfromtheirparentplantsisnegligible.Aparticularinsectpopulationcanonlybesupportedifatleast75%ofthequadratscontainatleast35plants.21andUsingp=0.9,p1=0.7,andp2=0.8,explorethevaluesofthatwillgivetheinsectpopulationa95%chanceofsurviving.Usethehugelysimplifyingassumptionthatthereisnotimecomponenttothisprocess(and,inparticular,thatoffshootplantsdonothavefurtheroffshoots)Keepinmindthatwedon’treallyhavetokeeptrackofwheretheindividualplantsare,onlythenumberineachquadrat..piiNotethatwedon’thavetoconsidergerminationoftheplantsasasecondstepafterthearrivaloftheseeds–insteadconsiderathinnedPoissonnumberofplantsofTypeiwithrateTipsonsimulatingthis:Ratherthandrawinguniformlydistributedlocationsfortheseeds,wecansimulatethenumbersforeachquadratseparately(andignorelocations)usingthefactthateachquadratwillcontainPoisson()germinatingseeds./100piiItwouldbeniceifwecouldfurthermodifythePoissonnumberofseedsforType1.Wecan,atleast,simplifythegenerationofoffshootplants,dealingwithallplantsinaparticularquadrattogetherbyaddinganegative-binomialnumberofplantstoeachquadrat.Howtodealwithoffshootplants…