空间泊松点过程PPT课件.ppt

SpatialPoissonProcesses 1 TheSpatialPoissonProcess Consideraspatialconfigurationofpointsintheplane 2 Notation LetSbeasubsetofR2 R R2 R3 LetAbethefamilyofsubsetsofS Forlet A denotethesizeofA length area volume LetN A thenumberofpointsinthesetA AssumeSisnormalizedtohavevolume1 3 ThenisahomogeneousPoissonpointprocesswithintensityif Foreveryfinitecollection A1 A2 An ofdisjointsubsetsofS N A1 N A2 N A3 areindependent Foreach 4 Alternatively aspatialPoissonprocesssatisfiesthefollowingaxioms TheprobabilitydistributionofN A dependsonthesetAonlythroughit ssize A 5 Thereexistsasuchthat Thereisprobabilityzeroofpointsoverlapping 6 Iftheseaxiomsaresatisfied wehave fork 0 1 2 7 ConsiderasubsetAofS Thereare3pointsinA howaretheydistributedinA A Expectauniformdistribution 8 Infact forany wehave Proof 9 So weknowthat fork 0 1 n ie N B N A n bin n B A 10 2020 1 8 11 Generalization ForapartitionA1 A2 AmofA forn1 n2 nm n Multinomialdistribution 12 SimulatingaspatialPoissonpatternwithintensityoverarectangularregionS a b x c d scatterthatnumberofpointsuniformlyoverS foreachpoint drawU1 U2 indepunif 0 1 sandplaceitat b a U1 a d c U2 c 13 Consideratwo dimensionalPoissonprocessofparticlesintheplanewithintensityparameter Let sdeterminethe random distanceDbetweenaparticleanditsnearestneighbor Forx 0 14 So forx 0 In3 Dwecouldshowthat 15 Example SpatialPatternsinStatisticalEcology Considerawideexpanseofopengroundofauniformcharacter suchasthemuddybedofarecentlydrainedlake Thenumberofwind dispersedseedsoccurringinanyparticular quadrat onthissurfaceiswellmodeledbyaPoissonrandomvariable ThereasonthistendstobetrueisduetothebinomialapproximationtothePoissondistributionwhichwillholdiftherearemanyseedswithanextremelysmallchanceoffallingintothequadrat 16 Supposenowthattheprobabilitythataseedgerminatesispandthattheyarenotsufficientlypackedtogethertointeractatthisstage Question Whatisthedistributionofthenumberofgerminatedseeds acceptprobabilityis So thesurvivingseedscontinuetobedistributed atrandom 17 SimulationProblem Type1andtype2seedswillgerminatewithprobabilitiesp1andp2 respectively Type1plantswillproduceKoffshootplantsonrunnersrandomlyspacedaroundtheplantwhereK geom p P K 0 p Twotypesofseedsarerandomlydispersedonaone acrefieldaccordingtotwoindependentPoissonprocesseswithintensities Supposethattheone acrefieldisevenlydividedinto10 x10quadrats 18 Assumethatthenumberofoffshootplantsthatfallintoaquadratdifferentfromtheirparentplantsisnegligible Aparticularinsectpopulationcanonlybesupportedifatleast75 ofthequadratscontainatleast35plants Usingp 0 9 p1 0 7 andp2 0 8 explorethevaluesofthatwillgivetheinsectpopulationa95 chanceofsurviving Usethehugelysimplifyingassumptionthatthereisnotimecomponenttothisprocess and inparticular thatoffshootplantsdonothavefurtheroffshoots 19 Keepinmindthatwedon treallyhavetokeeptrackofwheretheindividualplantsare onlythenumberineachquadrat Notethatwedon thavetoconsidergerminationoftheplantsasasecondstepafterthearrivaloftheseeds insteadconsiderathinnedPoissonnumberofplantsofTypeiwithrate Tipsonsimulatingthis Ratherthandrawinguniformlydistributedlocationsfortheseeds wecansimulatethenumbersforeachquadratseparately andignorelocations usingthefactthateachquadratwillcontainPoisson germinatingseeds 20 ItwouldbeniceifwecouldfurthermodifythePoissonnumberofseedsforType1 Wecan atleast simplif