Improving accuracy of downscaling rainfall by combining predictions of different statistical downscale models

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ScienceDirect

WaterSciencexxx(2016)xxx–xxx journalhomepage:www.elsevier.com/locate/wsj

Improving

accuracy

of

downscaling

rainfall

by

combining

predictions

of

different

statistical

downscale

models

Yassin

Z. Osman

a

,

Mawada

E. Abdellatif

b,∗

a_Engineering,_Sport_and_Sciences_Academic_Group,_University_of_Bolton,_Deane_Road,_Bolton_BL3_5AB,_UK b_School_of_the_Built_Environment,_Liverpool_John_Moores_University,_Byrom_Street,_Liverpool_L3_3AF,_UK

Received26March2016;receivedinrevisedform4October2016;accepted13October2016

Abstract

Aflexibleframeworkofmulti-modelofthreestatisticaldownscalingapproacheswasestablishedinwhichpredictionsfromthese modelswereusedasinputstoArtificialNeuralNetwork(ANN).TraditionalANN,SimpleAverageMethod(SAM),andcombining models(SDSM,MultipleLinearRegressions(MLR),GeneralizedLinearModel(GLM))wereappliedtoastudiedsitein North-westernEngland.Modelperformancecriteriaofeachoftheprimaryandcombiningmodelswereevaluated.Theobtainedresults indicatethatdifferentdownscalingmethodscangaindiverseusefulnessandweaknessinsimulatingvariousrainfallcharacteristics underdifferentcircumstances.ThecombiningANNmodelshowedmoreadaptabilityby acquiringbetteroverallperformance, whileGLM,MLRandshowedcomparableresultsandtheSDSMrevealsrelativelylessaccurateresultsinmodellingmostofthe rainfallamount.FurthermoretraditionalANNhasbeentestedandshowedpoorperformanceinreproducingtheobservedrainfall comparedwithabovemethods.Theresultsalsoshowthatthesuperiorityofthecombiningapproachmodeloverthesinglemodels ispromisingtobeimplementedtoimprovedownscalingrainfallatasinglesite.

Keywords: Downscaling;SDSM;GLM;MLR;Combiningmodel;Neuralnetworks

1. Introduction

Oneofthemajorfindingsofforecastingorpredictionresearchoverthelastquartercenturyhasbeenthatgreater predictiveaccuracycanoftenbeachievedbycombiningforecastsorpredictionsfromdifferentmethodsorsources. The combiningapproachgenerallyadvocatesthesynchronoususeof the forecastor predictionfromanumberof forecastingorpredictingmodelstoproduceanoverallcombinedorintegratedforecastorpredictionwhichcanbeused asanalternative tothatproducedbyasinglemodel.Thebasichypothesismade inthecombiningapproachisthat

∗_{Corresponding}_author.

E-mailaddress:[email protected](M.E.Abdellatif). PeerreviewunderresponsibilityofNationalWaterResearchCenter.

http://dx.doi.org/10.1016/j.wsj.2016.10.002

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differentmodelscapturedifferentaspectsofthedataandhencethecombinationoftheseaspectswouldproducebetter variableestimatesthanthoseproducedbyanyoneoftheindividualmodelsinvolvedinthecombination.Combination canbeaprocessasstraightforwardastakingasimpleaverageofthedifferentforecasts,inwhichcasetheconstituent forecastsareallweightedequally.Other,moresophisticatedtechniquesareavailabletoo,suchastryingtoestimatethe optimalweightsthatshouldbeattachedtotheindividualforecasts,sothatthosethatarelikelytobethemostaccurate receiveagreater weightintheaveragingprocess.Withinthiscontextthecurrentstudyhascomebyproposingthe useofdownscaledrainfallpredictedbydifferentdownscalingmodelsandcombiningthemusingsimpleaverageand artificialneuralnetworkmethods.

Useofcombiningapproachindifferentfieldsofforecastingiswelldocumentedandgoesbacktothe17thcentury.

Laplace(1818)hasstated,whencombiningresultsoftwoforecasts,that“Incombiningtheresultsofthesetwomethods,

onecanobtainaresultwhoseprobabilitylawoferrorwillbemorerapidlydecreasing”.Sincethendifferentmethods havebeen developedtofind‘optimal’combinationsof forecasts.Bothsimulationandempirical studieshavebeen carriedouttotestthemodelsandBayesianinterpretationshavealsobeenpresented.Theresultshavebeenvirtually unanimous:combiningmultipleforecastsleadstoincreasedforecastaccuracy.Thishasbeentheresultwhetherthe forecastsarejudgmentalorstatistical,econometricorextrapolation.Clement(1989)andHibonandEvgeniou(2005)

haveproducedanexcellentreviewforthemethodsusedincombiningforecastsandmoreinformationcanbefound inthenamedreferencesandwouldnotberepeatedhere.However,themethodsusedinthefieldofhydrologywillbe brieflydiscussedhere.

Combiningforecastsofdifferentrainfall-runoffmodelswasfirstusedbyShamseldin(1997)andShamseldinetal.

(1997).Thishasthenbeenfollowedbyseveralstudieswhichhavedealtwithmulti-modelcombinationofhydrological

models(e.g.Xiongetal.,2001;AbrahartandSee,2002;Coulibalyetal.,2005;Ajamietal.,2006;Vineyetal.,2009). Asthe natureof thecombinationfunctionisunknownandnotheory existstoanalyticallyderivethecombination functionfromahydrologicalorphysicalpointofview,thepreviousstudieshaveusedempiricaldata-drivenmodelling toderivethecombinationfunctionandsuchuseisveryappropriate.Inalltheaforementionedstudiesbothlinearand non-linearcombinationfunctionshavebeenused(e.g.linearregression,neuralnetworkandfuzzylogic)toproduce multi-modelriverflows.Applicationresultsfromthesestudieshavedemonstratedthepotential capabilitiesof the multi-modelcombinationapproachinimprovingtheaccuracyandreliabilityofhydrologicalmodellingresultsand havelaid thefoundationfor furtheruse of thisapproachinrainfall-runoffmodelling(Shamseldin, 1997;Seeand

Openshaw,2000;Xiongetal.,2001;Coulibalyetal.,2005).Thesuccessstoriesofusingthecombiningapproachin

thefieldofrainfall-runoffforecastinghaveformedgreatmotivationsforthecurrentstudy.

Rainfallisoneofthemostdifficultelementsofthehydrologicalcycletoforecast(Frenchetal.,1992)andgreat uncertaintiesstillaffecttheperformancesofbothstochasticanddeterministicrainfallpredictionmodels.Interesting perspectivesforthefuturerainfallareofferedbynumericalglobalcirculationmodels(GCMs),however,upuntilnow, theyunfortunatelydonotseemabletoprovideaccuraterainfallforecastsatthetemporalandspatialresolutionrequired bymanyhydrologicapplications(Brath,1999).To overcometheproblemofspatialandtemporaldatalimitations, rainfalldisaggregationandsimulationapproach(particularlyforclimatechangestudies)forthecoarseresolutionof theGCMsisgenerallyused.Thisapproachisgenerallyreferredtoasdownscaling.

Duringthelasttwodecades,extensiveresearchhasbeenconductedondownscalingmethodsandtheirapplications. Numeroustechniquesandmethodshavebeenproposedandusedwhichcanbebroadlydividedintostatisticaland dynamicalmethods.Statisticaldownscalingisthemostwidelyusedmethodindownscalingclimatevariablesfrom GCMs.Itrelateslarge-scaleclimatevariables(predictors)toregionalandlocalvariables(predictands).Thelarge-scale outputsoftheGCMsimulationarethenfedintothisstatisticalmodeltoestimatethecorrespondinglocalandregional climatecharacteristics(Wilbyetal.,2004).

Differenttechniquesforstatisticaldownscaling(SD)havebeenemployedincludingregression-basedtechniques, weatherpatternclassificationandweathergenerators.ThefundamentalassumptionusedintheSDtechniquesisthat thederivedrelationshipsbetweentheobservedpredictors(climatevariables)andpredictand(i.e.rainfall)willremain constantunderconditionsofclimatechangeandthattherelationshipsaretime-invariant(Yarnaletal.,2001;Fowler etal.,2007).OneoftheprimaryadvantagesofSDtechniquesisthattheyarecomputationallyinexpensiveandthus canbeeasilyappliedtooutputsfromdifferentGCMexperiments(Wilbyetal.,2004).

Severalregression-basedtechniqueshavebeenusedtodownscaletheprecipitationwithdifferentcapabilitiesfor eachmethodincludinglinearandnon-linearregression.Beuchatetal.(2012)andFealyandSweeney(2007)used GeneralizedLinearModels(GLMs)whichperformwellinreproducinghistoricalrainfallstatisticsinSwitzerlandand

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Ireland,respectively.Muluye(2012)employedthehybrid(SDSM),ANN,andnearestneighbour-basedapproaches (KNN)inCanadawithgreaterskillsforANNmodels.AnotherstudycarriedbyHassanandHarun(2012)showsthat theSDSMmodelcanbewellacceptableinregardtoitsperformanceindownscalingofthedailyandannualrainfall inMalaysia.Resultsfromthreedownscalingmethods(multiplelinearregressions,multiplenon-linearregression,and stochasticweathergenerator)weresuccessfullyusedbyHashmietal.(2012)inclimateimpactstudyandtheoutcome isencouraginganyfutureattemptsforcombiningtheresultsofmultiplestatisticaldownscalingmethods.Moreover manyotherstudiesusedlinearregression(Busuiocetal.,2008;Goubanovaetal.,2010),nonparametricregression basedonsplines,generalizedadditivemodels(VracandNaveau,2007;Salamehetal.,2009)indownscalingrainfall forclimatechangeimpactandadaptationsstudiesandobtainedgoodresults.

Inthepresentstudy,themulti-modelcombiningapproachhasbeenappliedtotheareaofdownscalingrainfallfrom theoutputsofGCM.Themainobjectiveistoimprovedownscalingrainfallpredictionbycombiningpredictionsfrom differentstatisticaldownscalingmodels.ThemodelsusedtodownscalerainfallinthestudiedsitearetheMultiple LinearRegression (MLR),theGeneralized LinearModel(GLM),theSDSM(Wilbyetal., 2002).The combining modelsusedaretheSimpleAverageMethod(SAM)andtheArtificialNeuralNetwork(ANN)whichbothcompared withtraditionalANN(nocombinationapproach).

2. Studyareaanddata

TheCrewedrainageareainNorthWestofEngland(NW),Fig.1,isselectedforthisstudy.Theexposureofthe NWregiontowesterlymaritimeairmassesandthepresenceofextensiveareasofhighgroundmeanthattheregion isconsideredasoneofthewettestplacesintheUK,withaverageannualrainfallover3200mm.Therainfallinthe CreweareaisrecordedatWorelstonStation(WR).

Twoprincipaldatasetswereemployedduringthecalibrationandvalidationofthedailyrainfalldownscalemodels. Firstly,theobserveddailyrainfalldataset,collectedfromWRstationwasobtained fromtheEnvironmentAgency forEnglandandWales,fortheperiod1961–1990.Secondly,thelarge-scaleobservedclimaticpredictorsdatasetwas obtainedfromtheNationalCentreforEnvironmentPredictions(NCEP/NCAR).Originallyatresolutionof2.50×2.50 degrees,thisdatawasre-griddedtoconfirmwithoutputoftheHadCM3GCMthathasgridresolutionof2.50×3.750. ThusInverseDistanceWeighted(IDW)method(Willmottetal.,1985)ofinterpolationwasappliedpriortotheuseof GCMoutputinprediction.IDWinterpolationexplicitlyimplementstheassumptionthatthingsthatareclosetoone

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anotheraremorealikethanthosethatarefartherapart.Topredictavalueforanyunmeasuredlocation,IDWwilluse themeasuredvaluessurroundingthepredictionlocation.Thosemeasuredvaluesclosesttothepredictionlocationwill havemoreinfluenceonthepredictedvaluethanthosefartheraway.Thus,IDWassumesthateachmeasuredpointhas alocalinfluencethatdiminisheswithdistance.Itweightsthepointsclosertothepredictionlocationgreaterthanthose fartheraway,hencethenameinversedistanceweighted.

Thetwosets ofdatawereneeded tobuild therainfalldownscalemodelof thedrainage area.The relationships betweenlarge-scaleatmosphericdataandlocalvariablesareimportantforsimulatingfuturerainfallconditionedby climateprojections.Thereare26large-scaleclimatevariablesusedinthisstudywhichwereassessedfortheirinfluence onrainfallattheWRstationforwinter,spring,summerandautumnseasons.Thesevariableswereusedforthepurpose ofconstructingtherainfallmodel(observed).Observeddailypredictorvariableshavebeennormalizedwithrespect totheir1961–1990meansandstandarddeviations.

3. Methodology

Themethodologyfollowedinthisstudyconsistsofthreesteps.Thefirststepwasscreeningforpredictorsofrainfall inthestudiedsiteusingNCEPlarge-scaleclimaticvariables.Thesecondstepinvolvedbuildingofthreeseasonalrainfall modelsusingtheSDSDM,MLRandGLMdownscalemodelswiththedatainperiod1961–1975usedforcalibration andthatinperiod1976–1990usedforvalidation.Thethirdstepinvolvedcombiningofthepredictionsobtainedfrom thethreedownscalemodelsusingSimpleAverageMethod(SAM)andArtificialNeuralNetworkcombiningmethod (ANN).Belowisabriefdescriptionforeachofthesestepsandthemodelsused.

3.1. Screeningforpredictors

Selectionofappropriatepredictorsisthemostimportantstepinrainfalldownscalingexercise.Itwouldgenerally notbeusefultoincludeallpotential predictorsinafinalmodel.Thisisbecausethe predictorvariablesarealmost alwaysmutuallycorrelated,sothatthefullsetofpotentialpredictorscontainsredundantinformation(Wilks,1995).

Thepredictors–rainfallrelationsinthisresearchareformedbasedoncorrelationcoefficientsbetweenthem.Stepwise regressionisappliedinthepresentstudyforselectionofpredictorsfromtheNCEPclimaticdataasithasbeenshown asapowerfulmethodbymanypreviousstudies(e.g.Huth,1999;HarphamandWilby,2005).Thepoolofpredictors usedweredailyvaluesof 26variablescomprisingsurfacepressure,temperature andhumidityaswellasupperair measuresofwindspeedanddirection,vorticity,divergence,humidity,temperatureandgeo-potentialheight.

Thescreeningprocesshasyieldedthemostpowerfulandparsimoniousseasonalmodelpossibleconsistingofeight predictorspresentedinTable1.Inordertoremoveanyinconsistenciesassociatedwiththepresenceofsmallrainfall values,athresholdof0.3mmwasappliedtothedataasrainfallvalueslessthanthisthresholdareconsideredtobedry daysandrepresentedwithzero.Thoseequaltoorgreaterthanthethresholdwereconsideredwetdays.

3.2. Downscalemodels 3.2.1. SDSM

SDSMusesahybridstochasticweathergeneratorandmulti-linearregressionmethodtosimulatelocalprecipitation ateachstationconditionalonregionalcirculationandatmosphericmoisturepredictors(HarphamandWilby,2005). Thus,ithastheabilitytocapturetheinter-annualvariabilitybetterthanotherstatisticaldownscalingapproaches,e.g. weathergenerators,weathertyping(Wilbyetal.,2002).SDSMisacombinationofastochasticweather generator

Table1a

Stepwiseregressionmodelsperformancesummaryforthefourseasons.

R R2 _F _Sig.

JFD 0.516 0.267 167.720 0.000

MAM 0.448 0.200 117.807 0.000

JJA 0.454 0.206 139.226 0.0000

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Table1b

StepwiseregressioncoefficientsandsignificanceforeachpredictorinWinterat5%significancelevel.

Inputparameters Unstandardizedcoefficients Standardizedcoefficients t Sig.

B Std.error Beta

(Constant) 1.694 0.195 8.670 0.00000

Laggedmeansealevel −1.074 0.074 −0.406 −14.545 0.00000

Surfacespecifichumidity 2.443 0.491 0.466 4.976 0.00000

Airflowstrengthat500hp 0.503 0.048 0.167 10.464 0.00000

Surfacevorticity 0.689 0.061 0.225 11.370 0.00000

Geopotentialheightat850hp 0.753 0.090 0.268 8.381 0.00000

Relativehumidityat500hp 0.330 0.052 0.109 6.300 0.00000

Temp −2.406 0.526 −0.391 −4.571 0.00001

Surfacerelativehumidity −0.345 0.115 −0.065 −2.998 0.00273

Table1c

StepwiseregressioncoefficientsandsignificanceforeachpredictorinSpringat5%significance.

B Std.error Beta (Constant) 2.100 0.073 28.599 0.00000 ncepmslpeu+1 −0.920 0.095 −0.259 −9.719 0.00000 ncepr500eu 0.563 0.057 0.169 9.875 0.00000 ncepp zeu 0.683 0.071 0.199 9.663 0.00000 ncepp850eu 0.620 0.119 0.171 5.232 0.00000 ncepp5feu 0.173 0.054 0.048 3.184 0.00147 ncepshumeu 1.919 0.286 0.368 6.716 0.00000 nceptempeu −1.720 0.283 −0.342 −6.077 0.00000 nceprhumeu −0.575 0.124 −0.165 −4.640 0.00000 Table1d

StepwiseregressioncoefficientsandsignificanceforeachpredictorinSummerat5%significancelevel.

B Std.error Beta

(Constant) 2.852 0.175 16.317 0.00000

Temp −3.321 0.395 −0.412 −8.417 0.00000

Airflowstrengthat500hp −0.941 0.153 −0.201 −4.172 0.00000

approachandatransferfunctionmodel(Wilbyetal.,2002)needingtwotypesofdailydata.Thefirsttypecorresponds tolocalpredictandsofinterest(e.g.temperature,precipitation,etc.)andthesecondtypecorrespondstothedataof large-scalepredictors(NCEPandGCM)ofagridboxclosesttothestudyarea.Correlationandpartialcorrelation analysisisperformedinSDSMbetweenthepredictandofinterestandpredictorstoselectasetofpredictorsmost relevantforthesiteinquestion(Wilbyetal.,2002).AlthoughSDSMhasitsownfunctionforpredictorsscreening, howeverthisfunctionwasnotusedinthepresentstudy.

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Table1e

StepwiseregressioncoefficientsandsignificanceforeachpredictorinAutumnat5%significancelevel.

B Std.error Beta

(Constant) 1.996 0.072 27.681 0.00000

Airflowstrengthat500hp 0.479 0.061 0.122 7.845 0.00000

Temp −2.030 0.352 −0.359 −5.761 0.00000

SDSMmodelsrainfallintwosteps,thefirststepisdevelopingofanoccurrencerainfallmodelusingthescreened predictorsasdescribedinEq.(1)(Wilbyetal.,1999):

Oi =α0+ n j=1

αjpji (1)

Thesecondstepisrainfallamountmodelwhichusesthesamescreenedpredictorsinaregressionmodelasdescribed

inEq.(2)(Wilbyetal.,1999):

Ri_SDSM=β0+ n j=1

βjpji+ei (2)

whereOiistheconditionalprobabilityofdailyrainfalloccurrenceondayi,Ri_SDSMaredailyrainfallamounts,pjiare predictors,nisnumberofpredictors,αandβaremodel parametersestimatedbydualsimplexalgorithmandei is modellingerror.

TheversionofSDSMusedinthisstudyisversion5.1.1whichisfreelydownloadedfromthesoftwarewebsite.A fulldescriptionofthesoftware,itsvariousfunctionsandmathematicalformulationcanbefoundintheUserManual

(WilbyandDawson,2013).

3.2.2. MLR

MultipleLinearRegression(MLR)isoneofthemostwidelyusedformsof regression.In thepresentstudythe rainfallismodelledinMLRasamountonly(nooccurrenceprocess)bysolvingalinearmodeloftheform:

Ri_MLR=β0+ n j=1

βjpji+εi (3)

whereεi∼N(0,σ2)isaGaussianerrortermwithvarianceσ2,allothersymbolsinEq.(3)havethesamemeaningas inEqs.(1)and(2).

TheMLRmodeldevelopedinthisstudywasprogrammedinMATLABandusedmethodofmaximumlikelihood toestimatemodelparameters.ThemainproblemwithMLRmodelisthatittriestomodeltheconditionalmean,which isnotbestsuitedforpredictingextremes.However,thefocusinthisstudyispredictionofrainfalltimeseriesnotonly theextremesandhenceuseofMLRisjustified.

3.2.3. GLM

GeneralizedLinearModel(GLM)belongs tolinearregressionfamily,butdiffersfromtheMLRbyassuminga distributionotherthanthenormal distributionforthe responsevariable Ri inEq.(3).TheGLM formused inthe

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RSDSM RMLR RGLM Input layer . . . Hidden laye . . . rs Output layer CANN

Fig.2.CombinedANNstructure.

presentstudyrelatestheresponsevariable(Ri),whosedistributionhasavectormeanμ=(μ1,...,μm)tooneormore covariates(p)viatherelationships(FealyandSweeney,2007):

μ=E(R) (4) g(μ)=ν (5) Ri_GLM=ν=β0+ n j=1 βjpji (6)

Aloglinkfunction,g(μ),andgammadistributionwereemployedforthepurposesofmodellingtherainfalland

β aremodelparameters.Whilethemixedexponentialdistributionhasbeenfoundtoprovideabetterfittorainfall amounts(WilksandWilby,1999)therelationshipbetweenthemeanandvarianceforthisdistributionmakesitdifficult toincorporateintoaGLM.Nonetheless,thegammadistributionGLMhasbeenfoundtobeagoodfittoprecipitation amountsinanumberofregions(ChandlerandWheater,2002)andhenceusedhere.

TheGLMmodeldevelopedinthisstudywasprogrammedinMATLABandusedmethodofmaximumlikelihood toestimatemodelparameters.

3.3. Combiningmodels 3.3.1. SAM

TheSimpleAverageMethod(SAM)takesthearithmeticaverageoftheforecastorpredictionobtainedfromthe threedownscalemodels,treatingtheforecastsorpredictionofeachmodelofhavingthesameweightinthecombined forecast.Thiscanbeexpressedasfollows:

Ri_SAM=1 3{R i SDSM+R i MLR+R i GLM} (7)

TheSAMisanaïveforecastcombinationmethod,whichcanworkverywellwhentheconstituentmodelshave practicallythesamelevelofperformance;itismoresensibletouseitpurelyasabaselineagainstwhichtheresultsof moresophisticatedcombinationmethodscanbecompared.

3.3.2. ANN

TheArtificialNeuralNetworkmethodusestheforecastorpredictionobtainedfromthethreedownscalemodelsas inputstoaMulti-LayerfeedForwardArtificialNeuralNetwork(MLF-ANN)model.ThestructureoftheMLF-ANN modelusedinthisstudyconsistsofaninputlayer,anoutputlayerandtwo“hidden”layerslocatedbetweentheinput andtheoutputlayersasshowninFig.2.Eachneuronofaparticularlayerhasconnectionpathwaystoalltheneurons inthefollowingadjacentlayer,butnonetothoseofitsownlayerortothoseofthepreviouslayer(ifany).Likewise, nodesinnon-adjacentlayersareunconnected.Intheoutputlayer,thereisonlyoneneuron,forthesingleoutput.The numberofneuronsintheinputandoutputlayersisdeterminedbythenumberofelementsintheexternalinputarray andoutputarrayofthenetwork,respectively.Thenumberofneuronsinthehiddenlayersisdeterminedbytrialand

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Table2a

StructureofcombiningANNmodelintermsofnumberofhiddenneuronsforeightinputsandoneoutputs.

Season Hiddenlayerneurons Totalno.ofmodelparameters(connections/biases) Totalno.ofdatapointsintrainingset

JFD 25,25 901 1353

MAM 20,15 511 1380

JJA 25,25 901 1380

SON 20,20 601 1365

Table2b

StructureoftraditonalANNmodelintermsofnumberofhiddenneuronsforeightinputsandoneoutputs.

Season Hiddenlayerneurons Totalno.ofmodelparameters(connections/biases) Totalno.ofdatapointsintrainingset

JFD 7 71 1335

MAM 13 131 1380

JJA 16 161 1380

SON 10 91 1365

error(Hammerstorm,1993)forthebestperformingmodelaspresentedinTable2.Thefinaloutput,RANN,fromthe networkshowninFig.2isobtainedbythefollowingequation:

RANN =fout ⎛ ⎝n k θkfhidden2 ⎧ ⎨ ⎩ m j βjkfhidden1 ₃ i Riαij+α0j +β0j ⎫ ⎬ ⎭ +θ0 ⎞ ⎠ (8) where

Ri:theinputtothenetworkfromtheprimarydownscalemodels. αij:theconnectionweightsbetweennodesoftheinputandhiddenlayer. βjk:theconnectionweightsbetweennodesofhiddenlayer1andhiddenlayer2. θk:theconnectionweightsbetweennodesofhiddenlayer2andouterlayer.

α0j,β0jandθ0areneuronthresholds(orbaseflow)inhidden1,hidden2andoutputlayers,respectively. mandnarenumbersofneuronsinhiddenlayer1andhiddenlayer2.

fhidden1,fhidden2andfoutarethelogistic,logisticandidentitytransfersfunctionsforhiddenlayer1,hiddenlayer2and outputlayer,respectively.

Theweightsandthresholdvaluesconstitutetheparametersofthenetwork,whichareusuallyestimatedbycalibrating (ortraining)ofthenetwork.Thisisusuallyachievedbyminimizingthesumofthesquaresofthedifferencesbetweenthe networkoutputseries(RANN),andthecorrespondingobservedrainfall,Robs,usingnonlinearoptimizationalgorithms.In thepresentresearch,thefasterback-propagationalgorithmofLevenberg–Marquardt(Yadavetal.,2010)ofMATLAB 7.11wasused,whichwasdesignedtospeedupthetrainingprocess.

Moreover,traditionalANNwithsamepredictorsinputsasinthethreemodelsSDSM,MLRandGLMhasbeen appliedandtrainedwithsametrainingalgorithmofCANN.Thiswillinvestigateitsowncapabilitiescomparedwith theothermethodsandthecombiningapproaches.

4. Resultsanddiscussion

The methodologydescribed above was sequentially followed. Observed dailyrainfall timeseries obtained for Worleston(WR)stationfor30yearswasusedtogetherwiththelarge-scaleobservedclimaticvariablesofNCEPdata; correspondtotheNorthWalesGrid(NW)ofHadCMGCM,toestablishseasonalpredictant–predictorsrelationship inthestudiedsite.Usingthemethodofstepwiseregression,theappropriatepredictorsfordailyrainfallinthisstation arepresentedinTable1.Thepredictorswerefoundtobethesameforallfourseasons.

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Stepwiselinearregressionisamethodofregressingmultiplevariableswhilesimultaneouslyremovingthosethat aren’timportantwhichhasbeendonethroughSPSS.Stepwiseregressionessentiallydoesmultipleregressionanumber oftimes,eachtimeremovingtheweakestcorrelatedvariable.Attheendthevariablesthatexplainthedistributionbest willbeleft.ResultsinTable1ashowR(correlation),R2(coefficientofdetermination),F-valueandsignificancelevel ofthatF-value.TheF-valueisstatisticallysignificantwithtypicallyp<.05,thissignifiesthatthemodelsdidagood jobofpredictingtheoutcomevariableandthatthereisasignificantrelationshipbetweenthesetofpredictorsandthe dependentvariable(rainfall).

AftertheevaluationoftheF-valueandR2,itisimportanttoevaluatetheregressionbetacoefficients:unstandardized andstandardized. Thebetacoefficientscanbe negativeorpositive, haveat-valueandsignificance of thatt-value

associatedwithit.Ifthebetacoefficientisnotstatisticallysignificant(i.e.,thet-valueisnotsignificant),nostatistical significancecanbeinterpretedfromthatpredictor.Iftheregressionbetacoefficientispositive,theinterpretationis thatforevery1-unitincreaseinthepredictorvariable,thedependentvariablewillincreasebytheunstandardizedbeta coefficientvalue.ResultsforthemodelcoefficientsandtheirsignificanthavebeenpresentedinTables1b–1ewhich showSig.figuresbelow0.05.Thentheselectedpredictorswerethenusedtobuildprimaryrainfalldownscalemodels usingeachofthemodellingmethodsdescribedinSection3.2.

Theobserveddailyrainfalldataset(1961–1990)withcorrespondingselectedpredictorshavebeendividedintotwo setscomprisingcalibration(period1961–1975)andverification(period1976–1990)sets.Alloftheprimarymodels werecalibratedandverifiedusingthesamecalibrationandvalidationperiods.HavingbuiltthethreeSDSM,MLR andGLMprimarydownscalemodelsfor eachseason,thecombiningSAMandCANNseasonalmodelswerebuilt usingoutputsfromtheseprimarymodelsasinputs.TheSAMmodelwasbuiltassimplearithmeticaverageofthethree primarydownscalemodels,whereastheANNmodelwasdevelopedusingthenetworkstructureshowninFig.2.Two typesofactivationfunctionshavebeenused,thelog-sigmoidforthehiddenlayerandlineartransferfunctioninthe outputlayer.AppropriatenumbersofneuronsineachofthetwohiddenlayersofthenetworkarepresentedinTable2a

foreachseasonalmodel.Thetwohiddenlayershavebeenusedaftermanytrialsbecauseonelayerfailedtogivebest fitforthedataandhenceitleadstoalesseraccuratemodel.Moreover,samepredictorsthatusedinthethreestatistical downscalemodelsweredirectlyappliedtoANNwhichresultsinnetworkstructuresinTable2b(traditionalANN). Performanceestimatessuchascross-validationbysplittingthedataintocalibrationandvalidationdatasethasbeen usedinbothtraditionalANNandCANNmodelswith90%ofthedatawereselectedrandomlyforcalibrationand10% forvalidationinMatlab.

CapabilitiesofCANNandtraditionaloneintermsofgeneralizationandavoidingoverfittinghavebeenalso inves-tigatedbycomparingthenumberofmodelsparameters(connectionsweightsandbiases)withnumberofdatapoints setthatusedtotrainthenetwork.Tables2aand2bshowthattheANNsusedlessparametersthanthedatausedwhich increasetheconfidenceofusingtheANNsforsimulationandprediction.Furthermoretheearlystoppingapproachhas beenusedtopreventoverfittingandimprovethegeneralizationduringthetrainingsotherunautomaticallywillstop thetrainingifANNexperienceanyoverfitting.

The efficiencyandabilityof eachprimary andcombiningmodeltopredict rainfallamountthatbest matchthe observedrainfallareexpressedhereintermsof correlationcoefficient(R)androotmeansquareerror(RMSE)and

0 20 40 60 80 100 120 140

Jan Feb March April May Jun July Aug Sep Oct Nov Dec

Rai

nf

all

(mm)

Observed SDSM MLR Possion CANN CSAM ANN

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presentedinTable 3.Table 3 showsthat, withoutexception,the ANN combiningmodel (CANN) producesdaily rainfallestimatesthatpossessahighercorrelationcoefficient(R)withtheobservedrainfallandalowerRMSEthan thecorrelationcoefficientvaluesassociatedwiththeprimarymodelsandeventheSAMcombiningmodel(CSAM). TheSAMcombiningapproachisrelativelyunskilfulcomparedtotheCANNapproachandeventotheotherthree methodswhiletraditionalANNshowedleastskilfulcomparedtoall.Amongtheprimarydownscalemodels,theGLM andMLRprimarymodelsperformbetterthantheSDSMinestimatingdailyrainfallatthestation.

0 200 400 600 800 1000 1200 1400 1600 19 61 19 63 19 65 19 67 19 69 19 71 19 73 19 75 19 77 19 79 19 81 19 83 19 85 19 87 19 89 Annual R ai nf all (mm )

Observed SDSM MLR GLM CANN CSAM ANN

Fig.4.Annualrainfalltotalofthefivemodelscomparedwithobservedrainfall.

-5 0 5 10 15 20 25 30 35 40 45 Rai nf all (mm) (a) -10 0 10 20 30 40 50 60 70 80 Rai nf all (mm) -10 0 10 20 30 40 50 60 70 Rai nf all (mm) -10 0 10 20 30 40 50 60 Rai nf all (mm) (b) (c) (d)

Fig.5.DailyBoxplotoftheseasonalrainfall(a)winter,(b)spring,(c)summerand(d)autumnforthedifferentmodellingmethodsshowingthe statistics:maximum,upperquantile,mean,lowerquantileandminimum.

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this article in press as: Osman, Y .Z., Abdellatif, M.E., Impro ving accurac y of do wnscaling rainf all by combining of dif ferent statistical do wnscale models. W ater Sci. (2016), http://dx.doi.or g/10.1016/j.wsj.2016.10.002

AR

TICLE IN PRESS

of P ages 15 Y .Z. Osman, M.E. Abdellatif / W ater Science xxx (2016) xxx–xxx 11 Table3

Modelsstatisticsandefficiencyforcalibrationandvalidationperiods.

Model Winter Spring Summer Autumn

Mean STD Skewness R RMSE Mean STD Skewness R RMSE Mean STD Skewness R RMSE Mean STD Skewness R RMSE

OBSER 1.91 3.35 3.27 – – 1.72 3.39 3.96 – – 1.87 4.25 4.60 – – 2.13 3.96 3.36 – – SDSM 2.42 4.20 2.88 0.33 4.45 2.35 4.40 3.77 0.23 4.93 2.56 5.51 3.76 0.25 6.09 3.10 5.37 2.90 0.25 5.90 MLR 2.01 1.69 0.61 0.53 2.84 1.86 1.56 0.68 0.24 3.37 2.15 1.99 1.00 0.45 3.80 2.32 1.92 0.69 0.48 3.48 GLM 1.90 2.47 3.29 0.55 2.88 1.74 1.93 3.15 0.21 3.53 1.97 2.92 6.61 0.45 3.94 2.23 2.56 3.42 0.44 3.65 CANN 2.00 2.33 2.77 0.62 2.63 1.83 1.73 2.12 0.36 3.16 2.00 2.88 3.68 0.55 3.55 2.31 2.58 2.20 0.57 3.26 CSAM 2.11 2.44 2.13 0.50 3.01 1.98 2.06 2.13 0.29 3.43 2.23 2.98 3.12 0.40 4.11 2.55 2.84 2.08 0.40 3.87 ANN 3.60 5.06 1.90 0.21 6.28 2.67 3.92 1.88 −0.11 5.29 2.05 3.33 2.42 −0.09 5.46 3.25 4.61 1.69 −0.11 6.26

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Fig.6.Averagewet(a)anddry(b)spelllengthforthefourseasonsduringcalibrationandverificationperiod1961–1990.

InadditiontothestatisticsandefficiencyresultspresentedinTable3,fivemorediagnostictestsareperformedon thethreeprimary,twocombiningdownscalemodelsandtraditionalANNtoensuretheirsuitabilityfordownscaling futurerainfall inthe studysite. ThesearedemonstratedinFigs. 3–7for calibrationandvalidationdataset. Fig.3

showscomparisonplotsoftheaveragemonthlyrainfallamountbetweentheobservedandrainfallsimulatedbythe fivemodelsforthewholeperiod1961–1990.Theplotsdemonstrateagooddegreeofagreementbetweentheobserved andsimulatedaveragemonthlyrainfallsbythecombiningANNmodel.Itcanclearlybededucedfromtheseplotsthat thecombiningANNmodelisabletoreproducethemonthlyrainfallandthereforeitisanimprovementovertheother primarydownscalemodels.

Fig.4showstheinter-annualvariabilityforrainfallinthestudiedsite,betweentheobservedandsimulatedseriesfor thewholeperiod1961–1990.Thetotalyearlyvalueswouldappeartohavebeenadequatelycapturedbythecombining ANNmodelbetterthantheotherthreeprimarymodels,thecombiningSAMmodelandtraditionalANN.Therefore theseresults,togetherwiththoseinFig.3,demonstratethatthecombiningANN(CANN)modelismorereliablein reproducingtheobservedrainfallwhichisanimportantrequirementwhenassessingclimateimpactsonhydrological systems.

InFig.5theDailyBoxplotsofthesimulatedseasonalrainfallcharacteristicsisrepresentedassmallverticalbars showingsomestatisticsofrainfalseriessimulatedbydifferentdownscalemodels.Theplottedstatisticsaremeasures ofhowmuchspreadthereisaroundtheaverage;withclosenessofthesimulatedstatisticvaluetothecorresponding observedstatisitcvalueindicatesgoodrepresentationfortheobservedspreadofthedata.ByviewingtheDailyBox plotsoftheseasonalrainfallinFig.5,theerrorbarscorrespondtotheCANNmodelappearmuchclosertotheobserved onesforallseasons.ThosecorrespondtotheMLRmodelrankthelowest.Thisisaclearindicationthatthecombining

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Fig.7.Probabilitydensityfunctionofdailyextremesrainfallfor(a)observed,(b)SDSM,(c)MLR,(d)GLM,(e)combiningANN,(f)combining SAMand(g)traditionalANNduringcalibrationverificationperiod1961–1990.

CANNmodelproducesrainfallmuchsimilartotheobservedoneintermsofdataspreadingaroundthemean.Itisan additionalevidencethatcombiningpredictionsfromdifferentdowscalemodelscanproducebetterresults.

Fig.6aandb show annualaverage dryand wetspells,resepctively, yielded bydifferent downscalemodels in comparisontotheobservedone.Athresoldof0.3mmisusedtodistinguishadrydayfromawetone.InFig.6athe averageseasonaldryspellscomputedfromtherainfallsimulatedbytheSDSM,MLR,GLM,CSAMandtraditional ANNmodelsarelowerthantheobservedones,whereastheaverageseasonaldryspellscomputedfromthecombining ANNmodelismuchclosertotheobserservedone.Conversly,inFig.6b,theaveragewetspellcomputedfromthe raifallsimulatedbytheMLR,GLM,traditionalANNandCSAMmodelsaresignificantlyoverestimatedtheobserved ones,whereastheSDSMandthecombiningANNproducedanaverageseasonalwetdaysreasonablymatchingthe observedones.TheclosenessoftheaverageseasonaldryandwetspellsproducedbythecombiningANNmodelto theobservedones,isanotherdesireablepropertyneededindownscalingmodelresultswhenusedinclimateimpact modelling.

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Thelastdiagonistictestforexamingtheperformanceofthedifferentprimarymodelsandthecombiningonesisthe probabilitydesnsityfunctions,pdf,forthesimulatedannualmaximumrainfallserie(AM)obtainedfromeachmodel.

Fig.7a–fshowsshapesofthepdfproducedtheAMseriesbyeachmodel.Concerningtheshapeofthedistribution,it canbeobservedthatthepdfoftheobservedrainfallskewsslightlytotheleftwhilethosefromtheSDSMandGLM skewtotheleftmorethantheobservedonewhileCSAMskewtoleftwithlessdegreecomparedtoobservedone.The

pdfoftheMLRandtraditionalANNmodeltendstoresemblethenormaldistribution,whereasthatofthecombining ANNskewsslightlytotheleftsimilartothepdfofobservedrainfall.Similarly,theboundariesoftheextremes(along thex-axis)aredifferentfordifferentdownscalemodels,withthoseoftheobservedandthecombiningANNaremuch closertoeachother.Theanalogyinskewnessofthepdfshapeandclosenessintheextremesboundariesbetweenthe observedandthecombiningANNsuggestthatthevariabilityofrainfallproducedbythecombiningANNismuch similartothoseoftheobservedones.

5. Conclusions

Threestatisticaldownscalingprimarymodelshavebeencomparedwithtwocombiningmodelsintermsoftheir abilityto downscaledailyrainfall overa selectedsite inNorthwest England.Daily observedrainfalldata for the period1961–1990,togetherwiththeobservedNCEPdata,wasusedtocalibrate andverify themodels.Anumber ofdiagnostictestsorparameterswasused tomeasuretheabilityandperformanceofeachprimary andcombining modeltodownscalethedailyrainfall.ThestatisticalresultsshowedthecombiningANNmodelsperformedbetterin downscalingseasonalrainfallmuchclosertotheobservedseriesthantheprimarySDSM,GLMandMLRmodels, traditionalANNandthecombiningSAMmodel.Theotherdiagnostictestsofmonthlyaveragerainfall,theinter-annual variability,theDailyBoxplots,theannualaveragedry/wetspellsandtheprobabilitydensityfunctionplots,which areimportantrequirementsforassessingclimatechangeimpact,haveallrevealedthatthecombiningANN model generallyperformsbetterinreproducingtheinter-annualvariabilityandmagnitudeoftherainfallincomparisontothe otherprimaryandcombiningmodels.WhileSDSMshowmuchcloserperformancetoCANNintermofreproducing wetanddryspelllengthhowever itissignificantlyoverestimatetheannualvariability,averagemonthly,anddaily statisticsoftherainfall.ThismeansthatSDSMcanbeabletosimulatetheoccurrenceproperlybutnottheamount unlikeCANNwhichisgoodforboth.

Overall,thispaperhighlightstheimportanceofacknowledginglimitationsandadvantagesofdifferentstatistical downscalingmethods,andalsoimpliesthatthereisaroomforimprovementsbycombiningthesemodels.Theresults obtainedforthisstudiedcatchmentarepromisingas wellas encouragingandcanbeextendedtomultiplesiteand regionsinthefuture.

Conﬂictsofinterest

Thereisnoconflictofinterestforthisresearch. Acknowledgments

TheauthorswouldliketoacknowledgethehelpreceivedfromtheEnvironmentAgencyforEnglandandWalesand theMetOfficeUKinformofdatareceivedanddiscussiontakenplaceduringtheperiodofthisstudy.

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