空间泊松点过程

空间泊松点过程TheSpatialPoissonProcessConsideraspatialconfigurationofpointsintheplane:Notation:

LetSbeasubsetofR2.(R,

R2,

R3,…)

LetAbethefamilyofsubsetsofS.

Forlet|A|denotethesizeofA. (length,area,volume,…)

LetN(A)=thenumberofpointsinthesetA.(AssumeSisnormalizedtohavevolume1.)ThenisahomogeneousPoissonpointprocesswithintensityif:

Foreveryfinitecollection{A1,A2,…,An}ofdisjointsubsetsofS,N(A1),N(A2),…,N(A3)areindependent.

ForeachAlternatively,aspatialPoissonprocesssatisfiesthefollowingaxioms:IfA1,A2,…,Anaredisjointregions,thenN(A1),N(A2),…,N(An)areindependentrv’sandN(A1UA2U…UAn)=N(A1)+N(A2)+…+N(An)TheprobabilitydistributionofN(A)dependsonthesetAonlythroughit’ssize|A|.ThereexistsasuchthatThereisprobabilityzeroofpointsoverlapping:大家有疑问的，可以询问和交流可以互相讨论下，但要小声点Iftheseaxiomsaresatisfied,wehave:fork=0,1,2,…ConsiderasubsetAofS:Thereare3pointsinA…howaretheydistributedinA?AExpectauniformdistribution…Infact,forany,wehaveProof:So,weknowthat,fork=0,1,…,n:ie:N(B)|N(A)=n~bin(n,|B|/|A|)Generalization:ForapartitionA1,A2,…,AmofA:forn1+n2+…+nm=n.(Multinomialdistribution)SimulatingaspatialPoissonpatternwithintensity overarectangularregionS=[a,b]x[c,d].

simulateaPoisson()numberofpoints(perhapsbyfindingthesmallestnumberNsuchthat)

scatterthatnumberofpointsuniformlyoverS(foreachpoint,drawU1,U2,indepunif(0,1)’sandplaceitat((b-a)U1+a),(d-c)U2+c)Consideratwo-dimensionalPoissonprocessofparticlesintheplanewithintensityparameter.Let’sdeterminethe(random)distanceDbetweenaparticleanditsnearestneighbor.Forx>0,So,forx>0.In3-Dwecouldshowthat:Example:SpatialPatternsinStatisticalEcologyConsiderawideexpanseofopengroundofauniformcharacter(suchasthemuddybedofarecentlydrainedlake).Thenumberofwind-dispersedseedsoccurringinanyparticular“quadrat”onthissurfaceiswellmodeledbyaPoissonrandomvariable.ThereasonthistendstobetrueisduetothebinomialapproximationtothePoissondistributionwhichwillholdiftherearemanyseedswithanextremelysmallchanceoffallingintothequadrat.Supposenowthattheprobabilitythataseedgerminatesispandthattheyarenotsufficientlypackedtogethertointeractatthisstage.Question:Whatisthedistributionofthenumberofgerminatedseeds?Answer:ThisisathinnedPoissonprocess…

withrate(acceptprobabilityis)So,thesurvivingseedscontinuetobedistributed“atrandom”.SimulationProblem:

Type1andtype2seedswillgerminatewithprobabilitiesp1andp2,respectively.

Type1plantswillproduceKoffshootplantsonrunnersrandomlyspacedaroundtheplantwhereK~geom(p).(P(K=0)=p)

Twotypesofseedsarerandomlydispersedonaone-acrefieldaccordingtotwoindependentPoissonprocesseswithintensities

Supposethattheone-acrefieldisevenlydividedinto10x10quadrats.

Assumethatthenumberofoffshootplantsthatfallintoaquadratdifferentfromtheirparentplantsisnegligible.

Aparticularinsectpopulationcanonlybesupportedifatleast75%ofthequadratscontainatleast35plants.

Usingp=0.9,p1=0.7,andp2=0.8,explorethevaluesofthatwillgivetheinsectpopulationa95%chance