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

WaterScience

ScienceDirect

journalhomepage:www.elsevier.com/locate/wsj

Improving

accuracy

of

downscaling

rainfall

by

combining

predictions

of

different

statistical

downscale

models

Yassin

Z.

Osman

,

Mawada

E.

Abdellatif

a_{Engineering,SportandSciencesAcademicGroup,UniversityofBolton,DeaneRoad,BoltonBL35AB,UK} b_{SchooloftheBuiltEnvironment,LiverpoolJohnMooresUniversity,ByromStreet,LiverpoolL33AF,UK}

Received26March2016;receivedinrevisedform4October2016;accepted13October2016 Availableonline19November2016

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.

©2016NationalWaterResearchCenter.ProductionandhostingbyElsevierB.V.ThisisanopenaccessarticleundertheCC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

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

1. Introduction

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

∗_{Correspondingauthor.}

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

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

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).

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

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

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.

Inputparameters Unstandardizedcoefficients Standardizedcoefficients t Sig.

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.

Inputparameters Unstandardizedcoefficients Standardizedcoefficients t Sig.

B Std.error Beta

(Constant) 2.852 0.175 16.317 0.00000

Laggedmeansealevel −1.836 0.159 −0.290 −11.538 0.00000

Relativehumidityat500hp 0.589 0.070 0.137 8.459 0.00000

Surfacevorticity 0.822 0.099 0.168 8.336 0.00000

Surfacespecifichumidity 2.081 0.255 0.387 8.176 0.00000

Temp −3.321 0.395 −0.412 −8.417 0.00000

Geopotentialheightat850hp 1.222 0.203 0.199 6.021 0.00000

Surfacerelativehumidity −0.837 0.161 −0.205 −5.186 0.00000

Airflowstrengthat500hp −0.941 0.153 −0.201 −4.172 0.00000

Table1e

StepwiseregressioncoefficientsandsignificanceforeachpredictorinAutumnat5%significancelevel.

Inputparameters Unstandardizedcoefficients Standardizedcoefficients t Sig.

B Std.error Beta

(Constant) 1.996 0.072 27.681 0.00000

Laggedmeansealevel −1.219 0.105 −0.306 −11.554 0.00000

Relativehumidityat500hp 0.537 0.066 0.136 8.099 0.00000

Surfacevorticity 0.770 0.083 0.195 9.232 0.00000

Airflowstrengthat500hp 0.479 0.061 0.122 7.845 0.00000

Geopotentialheightat850hp 0.665 0.134 0.163 4.971 0.00000

Temp −2.030 0.352 −0.359 −5.761 0.00000

Surfacespecifichumidity 1.517 0.299 0.326 5.067 0.00000

Surfacerelativehumidity −0.460 0.145 −0.088 −3.166 0.00156

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

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

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

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

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

Fig.4.Annualrainfalltotalofthefivemodelscomparedwithobservedrainfall.

.Z.

Osman,

M.E.

Abdellatif

/

W

ater

Science

30

(2016)

61–75

71

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

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.

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

CANNmodelproducesrainfallmuchsimilartotheobservedoneintermsofdataspreadingaroundthemean.Itisan additionalevidencethatcombiningpredictionsfromdifferentdowscalemodelscanproducebetterresults.

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.

References

Abrahart,R.J.,See,L.M.,2002.Multi-modeldatafusionforriverflowforecasting:anevaluationofsixalternativemethodsbasedontwocontrasting catchments.Hydrol.EarthSyst.Sci.6(4),655–670.

Ajami,N.K.,Duan,Q.,Gao,X.,Sorooshian,S.,2006.Multimodelcombinationtechniquesforanalysisofhydrologicalsimulations:applicationto DistributedModelIntercomparisonProjectresults.J.Hydrometeorol.7,755–768.

Beuchat,X.,Schaefli,B.,Soutter,M.,Mermoud,A.,2012.Arobustframeworkforprobabilisticprecipitationsdownscalingfromanensembleof climatepredictionsappliedtoSwitzerland.J.Geophys.Res.117,D03115.

Busuioc,A.,Tomozeiu,R.,Cacciamani,C.,2008.Statisticaldownscalingmodelbased oncanonicalcorrelationanalysisforwinterextreme precipitationeventsintheEmilia-Romaniaregion.Int.J.Climatol.28,449–464.

Chandler,R.E.,Wheater,H.S.,2002.Analysisofrainfallvariabilityusinggeneralizedlinearmodels:acasestudyfromthewestofIreland.Water Resour.Res.38,1192,http://dx.doi.org/10.1029/2001WR000906.

Clement,R.L.,1989.Combiningforecasts:areviewandannotatedbibliography.Int.J.Forecast.5,559–583.

Coulibaly,P.,Hache,M.,Fortin,V.,Bobee,B.,2005.Improvingdailyreservoirinflowforecastswithmodelcombination.ASCEJ.Hydrol.Eng.10 (2),91–99.

Fealy,R.,Sweeney,J.,2007.StatisticaldownscalingofprecipitationforselectionofsitesinIrelandemployingageneralisedlinearmodelling approach.Int.J.Climatol.27,2083–2094.

Fowler,H.J.,Blenkinsop,S.,Tebaldi,C.,2007.Linkingclimatechangemodellingtoimpactsstudies:recentadvancesindownscalingtechniques forhydrologicalmodelling.Int.J.Climatol.27,1547–1578.

French,M.N.,Krajewski,W.F.,Cuykendall,R.R.,1992.Rainfallforecastinginspaceandtimeusinganeuralnetwork.J.Hydrol.137,1–31.

Goubanova,K.,Echevin,V.,Dewitte,B.,Codron,F.,Takahashi,K.,Terray,P.,Vrac,M.,2010.Statisticaldownscalingofsea-surfacewindoverthe Peru-Chileupwellingregion:diagnosingtheimpactofclimatechangefromtheIPSL-CM4model.Clim.Dyn.36(7–8),1365–1378.

Hammerstorm,D.,1993.Workingwithneuralnetworks.IEEESpectr.,46–53.

Harpham,C.,Wilby,R.L.,2005.Multi-sitedownscalingofheavydailyprecipitationoccurrenceandamounts.J.Hydrol.312,235–255.

Hashmi,M.Z.,Shamseldin,Y.A.,Melville,B.W.,2012.Statisticallydownscaledprobabilisticmulti-modelensembleprojectionsofprecipitation changeinawatershed.Hydrol.ProcessJ.,http://dx.doi.org/10.1002/hyp.8413.

Hassan,Z.,Harun,S.,2012.ApplicationofstatisticaldownscalingmodelforlongleadrainfallpredictioninKuraurivercatchmentofMalaysia. Malays.J.CivilEng.24(1),1–12.

Hibon,Michele,Evgeniou,Theodoros,2005.Tocombineornottocombine:selectingamongforecastsandtheircombinations.Int.J.Forecast.21, 15–24.

Huth,R.,1999.StatisticaldownscalingincentralEurope:evaluationofmethodsandpotentialpredictors.Clim.Res.J.13,91–101.

Laplace,P.S.,1818.DeuxièmesupplémentàlaThéorieAnalytiquedesProbabilités.Courcier.

Muluye, G.Y., 2012. Comparison of statistical methods for downscaling daily precipitation. J. Hydroinform., http://dx.doi.org/10.2166/ hydro.2012.197.

Salameh,T.,Drobinski,P.,Vrac,M.,Naveau,P.,2009.StatisticaldownscalingofnearsurfacewindfieldovercomplexterraininsouthernFrance. Meteorol.Atmos.Phys.103,253–265.

See,L.,Openshaw,S.,2000.Ahybridmulti-modelapproachtoriverlevelforecasting.Hydrol.Sci.J.45(4),523–536.

Shamseldin,A.Y.,O’Connor,K.M.,Liang,G.C.,1997.Methodsforcombiningtheoutputsofdifferentrainfall-runoffmodels.J.Hydrol.197, 203–229.

Shamseldin,A.Y.,1997.Applicationofaneuralnetworktechniquetorainfall-runoffmodelling.J.Hydrol.199,272–294.

Viney,N.R.,Bormann,H.,Breuer,L.,Bronstert,A.,Croke,B.F.W.,Frede,H.G.,Hubrechts,T.,Huisman,L.,Jakeman,J.A.,Kite,A.J.,Lanini, G.A.,Leavesley,J.,Lettenmaier,G.,Lindstrom,D.P.,Seibert,G.,Sivapalan,J.,Willems,M.P.,2009.Assessingtheimpactoflandusechange onhydrologybyensemblemodeling(LUCHEM)II:ensemblecombinationsandpredictions.Adv.WaterResour.32,147–158.

Vrac, M.,Naveau,P., 2007.Stochastic downscalingof precipitation:from dryevents toheavy rainfalls.Water Resour. Res.43,W07402,

http://dx.doi.org/10.1029/2006WR005308.

Wilby,R.L.,Hay,L.E.,Leavesley,G.H.,1999.AcomparisonofdownscaledandrawGCMoutput:implicationsforclimatechangescenariosinthe SanJuanRiverbasin,Colorado.J.Hydrol.225,67–91.

Wilks,D.S.,Wilby,R.L.,1999.Theweathergenerationgame:areviewofstochasticweathermodels.Prog.Phys.Geogr.23,329–357.

Wilby,R.L.,Dawso,C.W.,Barrow,E.M.,2002.SDSM–Adecisionsupporttoolfortheassessmentofregionalclimatechangeimpacts.Environ. Model.Softw.17,147–159.

Wilby,R.L.,Charles,S.P.,Zorita,E.,Timbal,B.,Whetton,P.,Mearns,L.O.,2004.GuidelineforUseofClimateScenariosDevelopmentfrom StatisticalDownscalingMethods.EnvironmentalAgency,KingCollegeLondon,CSIROLandandWater,GKSS,BureauofMeteorology, CSIROAtmosphericResearchandNationalCentreforAtmosphericResearch.

Wilby,R.L.,Dawson,C.W.,2013.StatisticalDownscalingModel(SDSM),UserManual.Version5.1.1.

Wilks,D.,1995.StatisticalMethodsinAtmosphericSciences.AcademicPressLimited,London.

Willmott,C.J.,Rowe,C.M.,Mintz,Y.,1985.Climatologyoftheterrestrialseasonalwatercycle.J.Climatol.5,589–606.

Xiong,L.,Shamseldin,A.Y.,O’Connor,K.M.,2001.Anon-linearcombinationoftheforecastsofrainfall-runoffmodelsbythefirst-order Takagi-Sugenofuzzysystem.J.Hydrol.245(1–4),196–217.

Yadav,D.,Naresh,R.,Sharma,V.,2010.StreamflowforecastingusingLevenberg–Marquardtalgorithmapproach.IJWREE3(1),30–40.