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Notes_7,GEOS585A,Spring201517DetrendingTrendinatimeseriesisaslow,gradualchangeinsomepropertyoftheseriesoverthewholeintervalunderinvestigation.Trendissometimeslooselydefinedasalongtermchangeinthemean(Figure7.1),butcanalsorefertochangeinotherstatisticalproperties.Forexample,tree-ringseriesofmeasuredringwidthfrequentlyhaveatrendinvarianceaswellasmean(Figure7.2).Traditionally,seasonalorperiodiccomponents,andirregularfluctuations,andthevariouspartswerestudiedseparately.Modernanalysistechniquesfrequentlytreattheserieswithoutsuchroutinedecomposition,butseparateconsiderationoftrendisstilloftenrequired.Detrendingisthestatisticalormathematicaloperationofremovingtrendfromtheseries.Detrendingisoftenappliedtoremoveafeaturethoughttodistortorobscuretherelationshipsofinterest.Inclimatology,forexample,atemperaturetrendduetourbanwarmingmightobscurearelationshipbetweencloudinessandairtemperature.Detrendingisalsosometimesusedasapreprocessingsteptopreparetimeseriesforanalysisbymethodsthatassumestationarity.Manyalternativemethodsareavailablefordetrending.Simplelineartrendinmeancanberemovedbysubtractingaleast-squares-fitstraightline.Morecomplicatedtrendsmightrequiredifferentprocedures.Forexample,thecubicsmoothingsplineiscommonlyusedindendrochronologytofitandremovering-widthtrendthatmightnotbelinear,ornotevenmonotonicallyincreasingordecreasingovertime.Instudyingandremovingtrend,itisimportanttounderstandtheeffectofdetrendingonthespectralpropertiesofthetimeseries.Thiseffectcanbesummarizedbythefrequencyresponseofthedetrendingfunction.7.1IdentifyingtrendIdentificationoftrendinatimeseriesissubjectivebecausetrendcannotbeunequivocallydistinguishedfromlowfrequencyfluctuations.Whatlooksliketrendinashortsegmentofatimeseriessegmentoftenprovestobealow-frequencyfluctuation–perhapspartofacycle--inthelongerseries.Byextension,wecanviewtheentireobservedtimeseriesasasegmentofanunknowninfinitelylongseries,andcannotbesurethatanobservedchangeinmeanoverthatsegmentisnotpartofsomelow-frequencyfluctuationimpartedbyastationaryprocess.Sometimesknowledgeofthephysicalsystemhelpsinidentifyingtrend.Forexample,adecreaseofringwidthofatreewithtimeisexpectedpartlyongeometricalgrounds:theannualincrementofwoodisbeinglaiddownonanever-increasingcircumference.Ifthevolumeofwoodproducedannuallylevelsoffasthetreeages,ringwidthwouldstillbeexpectedtodecline.Figure7.2.Trendinmeanandvariance.RingwidthsfromaDouglas-firtreeinJemezMountains,NewMexico,1785-2007.Bothmeanringwidthandvarianceofringwidthdeclinewithageoftree.Figure7.1.Trendinmean.AstrongtrenddominatestheDecemberatmosphericCO2concentrationatMaunaLoa,Hawaii,1958-2007.Source:“agecurve”inringwidthcanbecomputedassumingthecross-sectionalareaofwoodaddedeachyearisconstant(Figure7.3).Suchaconceptualmodelwasusedindendrochronologyasjustificationfor“modifiednegativeexponential”detrending(Fritts1976).Ifaphysicalbasisislacking,weneedtorelyonstatisticalmethodstoquantifytrend.Statisticalmethodscanhelpdistinguishtrendfromothervariations.Asimplestatisticaltechniqueofidentifyinglineartrendistoregresstheobservedtimeseriesagainsttimeandtesttheestimatedslopecoefficientoftheregressionequationforsignificance(Haan2002).Thenullhypothesisisthattheslopecoefficientiszero(nolineartrend)andthealternativehypothesisisthattheslopediffersfromzero.At-testappliedtotheestimatedslopecoefficientwillindicaterejectionoracceptanceofthenullhypothesis.Thisapproachcanbeextendedtomultiplelinearregressionfortrendsinmeanmorecomplexthansimplelineartrend(Haan2002).Nonparametrictestsarealsoavailableforidentifyingtrend.TheMann-Kendalltestisonesuchtestcommonlyusedinclimatologyandhydrology(Salas1993).Thefrequencydomainisparticularlyusefulhere.GrangerandHatanaka(1964p.130)givesomeinsightintospectralinterpretationoftrend.Theyconcludethatweareunabletodifferentiatebetweenatruetrendandaverylowfrequencyfluctuation,andgivethefollowingadvice:Ithasbeenfoundusefulbytheauthortoconsideras“trend”inasampleofsizenallfrequencieslessthan1(2)nasthesewillallbemonotonicincreasingifthephaseiszero,butitmustbeemphasizedthatthisisanarbitraryrule.Itmayalsobenotedthatitisimpossibletotestwhetheraseriesisstationaryornot,givenonlyafinitesampleasanyapparenttrendinmeancouldarisefromanextremelylowfrequencyfluctuation.Ifweapplytheabovereasoningtoa500-yeartree-ringseries,wewouldsaythatvariationswithperiodlongerthantwicethesamplesize,or1000years,shouldberegardedastrend.Inanotherpaper,Granger(1966)defines‘trendinmean’ascomprisingallfrequencycomponentswhosewavelengthexceedsthelengthoftheobservedtimeseries.Cooketal.(1990)refertoGranger’s(1966)“trendinmean”conceptingivingsuggestionsfordetrendingtree-ringdata:Giventheabovedefinitionoftrendinmean,anotherobjectivecriterionforselectingtheoptimalfrequencyresponseofadigitalfilterisasfollows.Selecta50%frequency-responsecutoffinyearsforthefilterthatequalssomelargepercentageoftheserieslength,n.Thisisthe%ncriteriondescribedinCook(1985).TheresultsofCook(1985)suggestthatthepercentagei