Statistical Modelling of the Relationship Between

arXiv:physics/0701173v1[physics.ao-ph]15Jan2007StatisticalModellingoftheRelationshipBetweenMainDevelopmentRegionSeaSurfaceTemperatureandLandfallingAtlanticBasinHurricaneNumbersRomanBinter(RMSandLSE)StephenJewson(RMS)∗ShreeKhare(RMS)February2,2008AbstractWearebuildingahurricanenumberpredictionschemethatrelies,inpart,onstatisticalmodellingoftheempiricalrelationshipbetweenAtlanticseasurfacetemperaturesandlandfallinghurricanenumbers.Wetestoutanumberofsimplestatisticalmodelsforthatrelationship,usingdatafrom1900to2005anddatafrom1950to2005,andforbothallhurricanenumbersandintensehurricanenumbers.Theresultsareverydifferentfromthecorrespondinganalysisforbasinhurricanenumbers.1IntroductionWeareinterestedindevelopingpracticalmethodsforthepredictionofthedistributionofthenumberofhurricanesthatmightmakelandfallintheUSoverthenext5years.Onepossiblewaytomakesuchpredictionsisviaa2-stepmethodthatinvolvespredictingmaindevelopmentregion(MDR)seasurfacetemperature(SST),andthenpredictinglandfallinghurricanenumbersasafunctionoftheMDRSST.ThefirststepofpredictingMDRSSThasbeenconsideredinMeagherandJewson(2006)andLaeppleetal.(2006).Thispaperinvestigatesthesecondstep,andconsidersstatisticalrelationsthatonemightusetomodeltherelationshipbetweenMDRSSTandthenumberoflandfalls.Thispapercloselyfollowsanearlierpaper(Binteretal.,2006)inwhichwemodelledtherelationshipbetweenMDRSSTandthenumberofhurricanesinthebasin.Thedataandthemodelsweuse,andtheformatoftheresultswepresent,arealltakenfromthatpaper.Readersshouldrefertothatpaperforfurtherdetailsincludingashortdiscussion,withreferences,givinganoverviewofthephysicalrelationshipbetweenSSTandhurricanes.Therestofthisarticleproceedsasfollows:insection2weshowtheresultsfromourtestsonthelandfallinghurricanedata,andinsection3wediscusswhatwehavefound.2ResultsWenowpresentresultsfromourcomparisonsoftheabilityofvariousstatisticalmodelstorepresenttherelationshipbetweenMDRSSTandlandfallinghurricanenumbers.Firstweconsidermodelsforthetotalnumberoflandfallinghurricanes,fortheperiods1900-2005and1950-2005,andthenweconsidermodelsforintenselandfallinghurricanenumbersforthesametwoperiods.2.1Allhurricanes,1900-2005Thefirstresultswepresentarebasedonalllandfallinghurricanes,anddatafrom1900to2005.Thescatterplotshowninfigure1showsthenumberofhurricanesversustheSSTduringthisperiod.ThepictureisdramaticallydifferentfromwhatwesawwhenweconsideredtherelationshipbetweenSSTandthenumberofhurricanesinthebasininBinteretal.(2006).Inthatcasetherewasaclearpositiverelationship.Inthiscase,thereis,atleastprimafacie,norelationshipatall.Table1showsthatthelinearcorrelationis0.16andtherankcorrelationis0.12.∗Correspondenceemail:[e-mail]2showsthescorecomparisonsforthesixmodelsforthisdataset,andtable4showsthep-valuesforthepairwisecomparisonofthesemodels.Thebestmodel,intermsofout-of-sampleRMSEperformance,istheexponentialnegativebinomialmodel,butthescoreisonlyslightlybetterthanthetrivialflatpoissonmodelwithhasnorelationshipbetweenlandfallingnumbersandSST.Thepoint-wisecomparisonsshowsthatthenon-trivialmodelsarenotstatisticallydistinguishablefromthetrivialmodel.Asfarasthelog-likelihoodscoresintable2areconcerned,wefindthattheflatpoissonmodeldefeatsthelinearanddampedlinearnormalmodelsinastatisticallysignificantway.Theremainingmodelsdonotbeattheflat-linemodelinastatisticallysignificantway.Intable3weseethattheslopeparametersofallthenon-trivialmodelsarenotsignificantlydifferentfromzero.Inotherwords,basedontheseparameterestimatesandstandarderrors,wecertainlycouldn’trejectanull-hypothesisthatthereisnorelationatallbetweenMDRSSTandthenumberoflandfallinghurricanes.Thedampingparameterinthedampedlineartrendmodelismuchlessthan1(0.73),inresponsetotheweak(orperhapsnon-existent)signalthatwearetryingtoidentify.ThelinearrelationshipsbetweenSSTandhurricanenumbers,forwhattheyareworth,giveasensitivityofbetween0.64and0.69hurricanesperdegree.Insummary:wedon’tseeanyindicationofarelationshipbetweenMDRandSSTandlandfallinghurricanenumbers.2.2Allhurricanes,1950-2005GiventhelackofasignificantrelationshipbetweenSSTandhurricanenumbersonthedatafrom1900to2005,itisinterestingtoseeifwecanfindoneusingthemorerecentdata.Ontheonehand,themorerecentdataismoreaccurate,andsoitmightbemorelikelywecandetectarelationship.Ontheotherhand,usinglessdatawillmakeitevenhardertoestimatetheparametersofthemodels.Table6showsthatallthenon-trivialmodelsdefeattheflatpoissonmodel.Theresultsforlinearnormal,dampedlinearnormalandlinearpoissonarestatisticallysignificant.Asfarasthelog-likelihoodscoresareconcerned,thelinearnormalanddampedlinearnormalmodelsaredefeatedbytheflatpoissonmodelinastatisticallysignificantway.Thelinearpoissonmodeldefeatsthetrivialmodelinastatisticallysignificantway(forRMSE).However,basedontheseresults,it’shardtoconcludedefinitivelythatthelinearpoissonmodelisbest.Forinstance,inthepairwisecomparisonsforRMSEandlog-likelihood,theexponentialpoissonmodelalsodefeatsthelinearpoissonmodelinastatisticallysignificantway.Onceagaintheslopeparametersinthenon-trivialmodelsareallindistinguishablefromzero.Thelinearrelat