Applications of Discrete-time Markov Chains and Poisson Processes to Air Pollution Modeling and Studies PDF
Preview Applications of Discrete-time Markov Chains and Poisson Processes to Air Pollution Modeling and Studies
SpringerBriefs in Mathematics SeriesEditors KrishnaswamiAlladi NicolaBellomo MicheleBenzi TatsienLi MatthiasNeufang OtmarScherzer DierkSchleicher BenjaminSteinberg VladasSidoravicius YuriTschinkel LoringW.Tu GeorgeYin PingZhang SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. Forfurthervolumes: http://www.springer.com/series/10030 Eliane Regina Rodrigues • Jorge Alberto Achcar Applications of Discrete-time Markov Chains and Poisson Processes to Air Pollution Modeling and Studies 123 ElianeReginaRodrigues JorgeAlbertoAchcar AreadelaInvestigacio´nCient´ıfica UniversidadedeSa˜oPaulo InstitutodeMatema´ticas Av.Bandeirantes3900 UniversidadNacionalAuto´nomadeMe´xico SaoPaulo,Brazil MexicoCity,DF04510 Mexico ISSN2191-8198 ISSN2191-8201(electronic) ISBN978-1-4614-4644-6 ISBN978-1-4614-4645-3(eBook) DOI10.1007/978-1-4614-4645-3 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012944741 MathematicsSubjectClassification(2010):60J10,60J20,60G55,62F15,65C05,60K15,60K20,62N05, 92F99 ©ElianeReginaRodrigues2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To myparentsJoa˜oandThereza, and mysistersJaneandMoˆnica. Muitoobrigadapor tudo. ERR To myparentsAlbertoandMadalena (in memoriam). JAA Acknowledgements OurthankstoAct.GuadalupeTzintzunandDr.Adria´nA.Ferna´ndez-Bremauntzfor providingtheozonedatafromthemonitoringstationsoftheMetropolitanAreaof MexicoCityandalsoinformationabouttheenvironmentalregulationsinMexico. Thankyou,GiselaOrt´ız-Rodr´ıguezandRomualdaH.Pe´rez-Mun˜oz,forallowing ustousethematerialfromyourMaster’sDissertationandUndergraduateFinalYear Report,respectively. We also thank two anonymous reviewers for their comments and suggestions regardingthepresentationofthiswork. AspecialthankstoDonnaChernyk,theAssociateEditorofSpringerBriefs,for contactingusaboutsubmittingaBriefsproposal. We also thank the publishers of the works we have used here for giving us permissiontodoso. ThankyoutoC.andJ.forkeepingcompanyduringthewritinghours. During the preparation of this work partial financial support was provided by the project PAPIIT-IN104110-3 of the Direccio´n General de Apoyo al Personal Acade´micooftheUniversidadNacionalAuto´nomadeMe´xico,Mexico. vii Contents 1 Introduction .................................................................. 1 2 MarkovChainModels....................................................... 11 2.1 Introduction............................................................. 11 2.2 DescriptionoftheMathematicalModel............................... 12 2.3 BayesianFormulation.................................................. 14 2.4 ApplicationtoOzoneAirPollution ................................... 17 3 PoissonModelsandTheirApplicationtoOzoneData .................. 25 3.1 Introduction............................................................. 25 3.2 HomogeneousPoissonModels........................................ 26 3.3 Non-homogeneousPoissonModels................................... 40 3.4 ModelswiththePresenceofChange-Points.......................... 54 4 ModelingtheTimeBetweenOzoneExceedances ........................ 65 4.1 Introduction............................................................. 65 4.2 TheMathematicalModels ............................................. 66 4.3 AnApplicationtoOzoneData......................................... 69 5 SomeCountingProcessesandOzoneAirPollution...................... 79 5.1 Introduction............................................................. 79 5.2 DescriptionoftheIndependentandBivariateModels................ 80 5.3 ACopulaModel........................................................ 85 6 Comments..................................................................... 91 References......................................................................... 93 Appendix:ProgramCode....................................................... 97 A.1 RCode fortheNon-homogeneousPoissonModels withNoChange-Points................................................. 97 A.1.1 WeibullRateFunction......................................... 97 ix x Contents A.1.2 GeneralizedGoel–OkumotoRateFunction.................. 98 A.1.3 Musa–OkumotoRateFunction ............................... 100 A.2 WinBugsCode.......................................................... 102 A.2.1 WinBugs Code for the Non-homogeneous ModelswithOneChange-Point .............................. 102 A.2.2 WinBugsCodefortheTimesBetweenExceedances........ 102 Index............................................................................... 105
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