空间泊松点过程.ppt

SpatialPoissonProcesses TheSpatialPoissonProcess Consideraspatialconfigurationofpointsintheplane 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 Foreach Alternatively aspatialPoissonprocesssatisfiesthefollowingaxioms TheprobabilitydistributionofN A dependsonthesetAonlythroughit ssize A Thereexistsasuchthat Thereisprobabilityzeroofpointsoverlapping Iftheseaxiomsaresatisfied wehave fork 0 1 2 ConsiderasubsetAofS Thereare3pointsinA howaretheydistributedinA A Expectauniformdistribution Infact forany wehave Proof 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 SimulatingaspatialPoissonpatternwithintensityoverarectangularregionS a b x c d 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 SpatialPatternsinStatisticalEcology Considerawideexpanseofopengroundofauniformcharacter suchasthemuddybedofarecentlydrainedlake Thenumberofwind dispersedseedsoccurringinanyparticular quadrat onthissurfaceiswellmodeledbyaPoissonrandomvariable ThereasonthistendstobetrueisduetothebinomialapproximationtothePoissondistributionwhichwillholdiftherearemanyseedswithanextremelysmallchanceoffallingintothequadrat Supposenowthattheprobabilitythataseedgerminatesispandthattheyarenotsufficientlypackedtogethertointeractatthisstage Question Whatisthedistributionofthenumberofgerminatedseeds acceptprobabilityis So thesurvivingseedscontinuetobedistributed atrandom SimulationProblem Type1andtype2seedswillgerminatewithprobabilitiesp1andp2 respectively Type1plantswillproduceKoffshootplantsonrunnersrandomlyspacedaroundtheplantwhereK geom 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 Keepinmindthatwedon treallyhavetokeeptrackofwheretheindividualplantsare onlythenumberineachquadrat Notethatwedon thavetoconsidergerminationoftheplantsasasecondstepafterthearrivaloftheseeds insteadconsiderathinnedPoissonnumberofplantsofTypeiwithrate Tipsonsimulatingthis Ratherthandrawinguniformlydistributedlocationsfortheseeds wecansimulatethenumbersforeachquadratseparately andignorelocations usingthefactthateachquadratwillcontainPoisson germinatingseeds ItwouldbeniceifwecouldfurthermodifythePoissonnumberofseedsforType1 Wecan atle