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1、SpatialPoissonProcessesTheSpatialPoissonProcessConsideraspatialconfigurationofpointsintheplane:Notation:LetSbeasubsetofR2.(R,R2,R3,…)LetAbethefamilyofsubsetsofS.Forlet
2、A
3、denotethesizeofA.(length,area,volume,…)LetN(A)=thenumberofpointsinthesetA.(AssumeSisnormalizedtohavevolume1.)
4、ThenisahomogeneousPoissonpointprocesswithintensityif:Foreveryfinitecollection{A1,A2,…,An}ofdisjointsubsetsofS,N(A1),N(A2),…,N(A3)areindependent.ForeachAlternatively,aspatialPoissonprocesssatisfiesthefollowingaxioms:IfA1,A2,…,Anaredisjointregions,thenN(A1),N(A2),…,N(An)areindepen
5、dentrv’sandN(A1UA2U…UAn)=N(A1)+N(A2)+…+N(An)TheprobabilitydistributionofN(A)dependsonthesetAonlythroughit’ssize
6、A
7、.ThereexistsasuchthatThereisprobabilityzeroofpointsoverlapping:Iftheseaxiomsaresatisfied,wehave:fork=0,1,2,…ConsiderasubsetAofS:Thereare3pointsinA…howaretheydistribu
8、tedinA?AExpectauniformdistribution…Infact,forany,wehaveProof:So,weknowthat,fork=0,1,…,n:ie:N(B)
9、N(A)=n~bin(n,
10、B
11、/
12、A
13、)Generalization:ForapartitionA1,A2,…,AmofA:forn1+n2+…+nm=n.(Multinomialdistribution)SimulatingaspatialPoissonpatternwithintensityoverarectangularregionS=[a,b]x[c,d
14、].simulateaPoisson()numberofpoints(perhapsbyfindingthesmallestnumberNsuchthat)scatterthatnumberofpointsuniformlyoverS(foreachpoint,drawU1,U2,indepunif(0,1)’sandplaceitat((b-a)U1+a),(d-c)U2+c)Consideratwo-dimensionalPoissonprocessofparticlesintheplanewithintensityparameter.Let’sd
15、eterminethe(random)distanceDbetweenaparticleanditsnearestneighbor.Forx>0,So,forx>0.In3-Dwecouldshowthat:Example:SpatialPatternsinStatisticalEcologyConsiderawideexpanseofopengroundofauniformcharacter(suchasthemuddybedofarecentlydrainedlake).Thenumberofwind-dispersedseedsoccurring
16、inanyparticular“quadrat”onthissurfaceiswellmodeledbyaPoissonrandomvariable.ThereasonthistendstobetrueisduetothebinomialapproximationtothePoissondistributionwhichwillholdiftherearemanyseedswithanextremelysmallchanceoffallingintothequadrat.Supposenowthattheprobabilitythataseedgerm
17、inatesispandthattheyarenotsufficientlypackedtog
