SpatialPoissonProcesses空間泊松點(diǎn)過(guò)程TheSpatialPoissonProcessConsideraspatialconfigurationofpointsintheplane:空間泊松點(diǎn)過(guò)程N(yùn)otation:

LetSbeasubsetofR2.(R,

R2,

R3,…)

LetAbethefamilyofsubsetsofS.

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

LetN(A)=thenumberofpointsinthesetA.(AssumeSisnormalizedtohavevolume1.)空間泊松點(diǎn)過(guò)程ThenisahomogeneousPoissonpointprocesswithintensityif:

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

Foreach空間泊松點(diǎn)過(guò)程Alternatively,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|.空間泊松點(diǎn)過(guò)程ThereexistsasuchthatThereisprobabilityzeroofpointsoverlapping:空間泊松點(diǎn)過(guò)程Iftheseaxiomsaresatisfied,wehave:fork=0,1,2,…空間泊松點(diǎn)過(guò)程ConsiderasubsetAofS:Thereare3pointsinA…h(huán)owaretheydistributedinA?A

Expectauniformdistribution…空間泊松點(diǎn)過(guò)程Infact,forany,wehaveProof:空間泊松點(diǎn)過(guò)程So,weknowthat,fork=0,1,…,n:ie:N(B)|N(A)=n~bin(n,|B|/|A|)空間泊松點(diǎn)過(guò)程Generalization:ForapartitionA1,A2,…,AmofA:forn1+n2+…+nm=n.(Multinomialdistribution)空間泊松點(diǎn)過(guò)程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)空間泊松點(diǎn)過(guò)程Consideratwo-dimensionalPoissonprocessofparticlesintheplanewithintensityparameter.Let’sdeterminethe(random)distanceDbetweenaparticleanditsnearestneighbor.Forx>0,空間泊松點(diǎn)過(guò)程So,forx>0.In3-Dwecouldshowthat:空間泊松點(diǎn)過(guò)程Example:SpatialPatternsinStatisticalEcologyConsiderawideexpanseofopengroundofauniformcharacter(suchasthemuddybedofarecentlydrainedlake).Thenumberofwind-dispersedseedsoccurringinanyparticular“quadrat”onthissurfaceiswellmodeledbyaPoissonrandomvariable.ThereasonthistendstobetrueisduetothebinomialapproximationtothePoissondistributionwhichwillholdiftherearemanyseedswithanextremelysmallchanceoffallingintothequadrat.空間泊松點(diǎn)過(guò)程Supposenowthattheprobabilitythataseedgerminatesispandthattheyarenotsufficientlypackedtogethertointeractatthisstage.Question:Whatisthedistributionofthenumberofgerminatedseeds?Answer:ThisisathinnedPoissonprocess…

withrate(acceptprobabilityis)So,thesurvivingseedscontinuetobedistributed“atrandom”.空間泊松點(diǎn)過(guò)程SimulationProblem:

Type1andtype2seedswillgerminatewithprobabilitiesp1andp2,respectively.

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

Twotypesofseedsarerandomlydispersedonaone-acrefieldaccordingtotwoindependentPoissonprocesseswithintensities

Supposethattheone-acrefieldisevenlydividedinto10x10quadrats.空間泊松點(diǎn)過(guò)程

Assumethatthenumberofoffshootplantsthatfallintoaquadratdifferentfromtheirparentplantsisnegligible.

Aparticularinsectpopulationcanonlybesupportedifatleast75%ofthequadratscontainatleast35plants.

Usingp=0.9,p1=0.7,andp2=0.8,explorethevaluesofthatwillgivetheinsectpopulationa95%chanceofsurviving.

Usethehugelysimplifyingassumptionthatthereisnotimecomponenttothisprocess(and,inparticular,thatoffshootplantsdonothavefurtheroffshoots)空間泊松點(diǎn)過(guò)程

Keepinmindthatwedon’treallyhavetokeeptrackofwheretheindividualplantsare,onlythenumberineachquadrat.

Notethatwedon’thavetoconsidergerminationoftheplantsasasecondstepafterthearrivaloftheseeds–insteadconsiderathinnedPoissonnumberofplantsofTypeiwithrateTipsonsimulatingthis:

Ratherthandrawinguniformlydistributedlocationsfortheseeds,wecansimulatethenumbersforeachquadratseparately(andignorelocations)usingthefactthateachquadratwillcontainPoisson()germinatingseeds.空間泊松點(diǎn)過(guò)程

ItwouldbeniceifwecouldfurthermodifythePoissonnumberofseedsforType