Forecasting Financial Time Series with Support Vector Machines.

ForecastingFinancialTimeSerieswithSupportVectorMachinesBasedonDynamicKernels

JohannesMager

InstituteofComputerArchitecturesUniversityofPassau,GermanyEmail:mager@ m.uni-passau.de

UlrichPaasche

NeuralResearchCenterMunichGmbH

Munich,Germany

Email:ulrich.paasche@nrcm.de

BernhardSick

InstituteofComputerArchitecturesUniversityofPassau,GermanyEmail:sick@ m.uni-passau.de

Abstract—Thetechnicalanalysisof nancialtimeseriesandinparticularthepredictionoffuturedevelopmentsisachallengingproblemthathasbeenaddressedbymanyresearchersandpractitionersduetothepossiblepro t.Weprovideaforecastingtechniquebasedonacertainmachinelearningparadigm,namelysupportvectormachines(SVM).SVMgainedmoreandmoreimportanceforpracticalapplicationsinthepastyearsastheyhaveexcellentgeneralizationabilitiesduetotheprincipleofstructuralriskminimization.However,standardkernelfunctionsforSVMarenotabletocomparetimeseriesofvariablelengthappropriately,i.e.,whenweassumethatthesetimeseriesmustbescaledinanon-linearway.Therefore,weusethedynamictimewarping(DTW)techniqueasakernelfunction.Wedemonstratefortwo nancialtimeseries(FDAXandFGBLfutures)thatexcellentresultscanbeobtainedwiththisapproach.

I.INTRODUCTION

Thepredictionoffuturestockmarketdevelopmentsisaproblemthathasbeenattractingtheattentionofbothpracti-tionersandresearchersformanydecades.Itcaneasilybeseenthattherearecertainrecurringpatternsinthehistoryofmarketprices,andtherearevariousapproachesforclassifyingthem[1],[2].Butamuchhardertaskistorecognizesuchpatternsintheconstantlyevolving nancialmarketsearlyandwithsuf cientreliability.Evenworse:Itstillisheavilydisputed,whetherchartpatternsallowforapredictionofcertainfutureeventsatall.

Inthisarticle,weproposeamachinelearningtechniqueforforecasting nancialtimeseries,whichreliesonthepopulartechniqueofsupportvectormachines(SVM).Usingalargehistoricalsetofreal-world nancialtimeseries,weexaminetheperformanceofdifferentvariantsandparametersettings.Tofurtherincreasethepredictionaccuracyontimeseries,standardkernelsoftheSVMarereplacedbyspecialdynamickernelfunctions,whichareadaptedforanalyzingtemporaldata.Wewillshowthattheutilizationofthesekernelsresultsinasigni cantlybetteraccuracyanditbecomespossibletooutperformthemarket’soveralldevelopment.

Withtheintegrationofthistechniqueintoaframeworkfortechnicalanalysis,Investox,itisalsopossibletoevaluatetheperformanceusingavirtualtradingagentonhistoricaldataandusethesystemon“live”datafeeds.

Thearticleisorganizedasfollows:InSectionII,weprovideashortinsightintotheprinciplesoftechnicalanalysisand nancialmarketdataanddiscusssomerelatedwork.In

SectionIII,SVManddynamickernelfunctionsareintroduced.SectionIVfollowswiththeexperiments:We rstexplaintherationalebehindtheconstructeddatasetsandsetouttheutilizederrormeasures.Thereafter,theresultsofourexperimentsaredocumented.Finally,SectionVsummarizesthemajorinsightandgivesanoutlooktofutureresearch.

II.FINANCIALANALYSIS

A.PrinciplesofTechnicalAnalysis

Theanalysisof nancialmarketscanbedividedintotwobig elds:Whereasfundamentalanalysistriestoanalyzealleconomicfactorsofacompanyoramarketinordertocalcu-latethetruevalueofacommercialpaper,technicalanalystsassumethatallimportantinformationforthepaper’sfuturedevelopmentisalreadycontainedinitspastbehavior[3].Therefore,futuremovementscanbeanticipatedbythoroughlyanalyzingthestock’shistoryanditsinherentpatterns[4].Whiletheprinciplesofsomeofthetechniquesutilizedforatechnicalanalysisdatebacktothe18thcentury,theirvalidityhaspermanentlybeendisputed.Mostpopularly,theef cientmarkethypothesis[5]statesthat nancialmarketsareinformationallyef cient,and,therefore,allpastinformationisalreadycontainedineachstock’slastvalue.Asaresult,itisclaimedthattechniquesforatechnicalanalysiscannotperformbetterthanarandomwalkonthechartortheoveralldevelopmentofthemarket.Despiteallobjections,itstillwasnotpossibletoprooftheinvalidityoftechnicalanalysis,anditstechniquesaregainingpopularityamongboth,investorsandresearchers.

B.CharacteristicsofFinancialMarketData

The nancialinstrumentsusedforourworkaretwofu-tures,derivativeinstrumentstradedattheEuropeanderivativesexchangeEurex[6].Afuturescontractgivestheholdertheobligationtobuy(longposition)orsell(shortposition)aspeci edunderlyingassetatadistinctdateinthefutureandatapre-speci edprice.Thisdualitygivesthetraderthepossibilitytobene tfromrisingaswellasfromfallingmarketprices[7].

Aseverytransactioninamarketvariestheratioofsupplyanddemand,marketpricescanchangeinverysmallandir-regularintervals.Tofacilitateanalysis,thedataiscompressedintointervalsofacertainsize.Consequently,itispossibleto

notonlyidentifyasinglepriceforeachinterval,buttoextractadditionalinformation:Theopen,high,low,andclosepricesforthisinterval,namedOHLC-data(seeFig.

1).

Fig.1.Ontheleftsideweseethemarketrateofacertainequityduringoneday.Ontherightside,thesedatahavebeencompressedanddepictedusingtheso-calledcandlesticklayout:Theupperandlowershadowsmarktheday’shighestandlowesttradedprices,whereasthebodyofthecandlespansfromtheopentothecloseprice.Thecolorofthebodyillustratestheequity’sdevelopmentduringtheday:Ifthepricewentup,thebodyiswhiteandblackotherwise.

C.RelatedResearchintheFieldofTechnicalAnalysisOverthelast15years,therehasbeenavastamountofscienti cinvestigationstousingmachinelearningmethodsfortechnicalanalysis.

[8],forexample,useabackpropagationneuralnetwork(multilayerperceptron)withonehiddenlayertopredictthedailyclosepricesofthestockindexS&P500,andcomparetheresultstoanARIMA-model.Asaresult,theyshowthatalthoughtheneuralnetworkhasahighertolerancetomarket uctuations,itsoutputistoovolatiletoindicatelong-termtrends.Abettersuitedapproachisdescribedin[9],whichutilizesrecurrentElmanneuralnetworks[10]forforecastingforeignexchangeprices.Itiscombinedwithamechanismtoautomaticallychooseandoptimizethenetwork’sparameters.Asaresult,itishighlightedthattheforecastsdonotdifferasmuchbetweendifferentmodelsasbetweendifferentinputdata.Foronlytwooutof veexchangerates(JPY/USDandGBP/USD),reliablepredictionsarepossible,whereasfortheotherrates,thepredictionaccuracyissimilartoanaiveforecast.

[11]usesamodi edSVMmodelforregressionwiththe(static)Gaussiankernel.ByadjustingtheregularizationconstantCwithaweightfunction,recenterrorsaremoreheavilypenalizedthandistanterrors,thusincreasingthein uenceofthemostrecentstockprices.Inadditiontothat,[12]addsasimilarweightfunctiontothethresholdε,whichlimitsthetoleranceofVapnik’sε-insensitiveerrorfunction[13].Thisapproachhelpstofurtherreducethecomplexityofthebuiltmodelandthenumberofsupportvectors.Furtheremphasizingtheneedofthoroughdatapreparation,[14]usessupportvectorclassi cationcombinedwithavarietyofdifferentpre-processingmethods.Asakernelfunction,

theyusethepolynomialkernelinadditiontotheGaussiankernelfunction,paredtoabackpropagationnetwork,theGaussianversionheavilyincreasesthemeasuredpredictionaccuracy.

Althoughallthesearticleswereabletopresentsomesuccessintheirexperiments,themajor awisobvious:Withastatickernelfunctionitisonlypossibletoincorporateacertain(limited)amountofinformationaboutthechart’shistory.Theinherenttemporalstructureofthedatacannotbeanalyzedap-propriately,leadingtorelativelypoorandunstablepredictionresults.

III.SUPPORTVECTORMACHINESWITHDYNAMIC

KERNELFUNCTIONSA.FundamentalsofSupportVectorMachines

Inthisarticle,cost-sensitivesupportvectormachines(C-SVM)andν-SVMareusedtoclassifythetimeseriesusingcharacteristicattributesextractedfromthetimeseriesasinputs.Basically,SVMuseahyperplanetoseparatetwoclasses[15]–[18].Forclassi cationproblemsthatcannotbelinearlyseparatedintheinputspace,SVM ndasolutionusinganon-linearmappingfromtheoriginalinputspaceintoahigh-dimensionalso-calledfeaturespace,whereanoptimallyseparatinghyperplaneissearched.Thosehyperplanesarecalledoptimalthathaveamaximalmargin,wheremarginmeanstheminimaldistancefromtheseparatinghyperplanetotheclosest(mapped)datapoints(so-calledsupportvectors).Thetransformationisusuallyrealizedbynonlinearkernelfunctions.C-SVMandν-SVMbothallow,butalsominimizemisclassi cation.

Comparedtothepopulararti cialneuralnetworks,SVMhaveseveralkeyadvantages:Bydescribingtheproblemasaconvexquadraticoptimizationproblem,theyareensuredtoconvergetoauniqueglobaloptimuminsteadofonlyapossiblylocaloptimum.Additionally,byminimizingthestructuralriskofmisclassi cation,SVMarefarlessvulnerabletoover tting,oneofthemajordrawbacksofstandardneuralnetworks.B.RelatedWorkintheFieldofDynamicKernelFunctionsAnoverviewandcomparisonofmethodsfortimeseriesclassi cationwithSVMcanbefoundin[19]or[20],forinstance.OnecommonmethodforclassifyingtimeserieswithSVMistouseoneofthedefaultstatickernels(i.e.,poly-nomialorGaussian).Forspeakerveri cation[21],phoneticclassi cation[22],orinstrumentclassi cation[23]thishassuccessfullybeendone.Abigdisadvantageofthisapproachisthatstatickernelsareunabletodealwithdataofdifferentlength.Therefore,itisnecessarytore-samplethetimeseriestoacommonlength,ortoextracta xednumberoffeaturesbeforestatickernelscanbeapplied.Itisobviousthatthere-samplingorthereductiontosomeextractedfeaturesinducesalossofinformationandisnotverywellsuitedtodealwithtimeseriesofvariablelength,wherealinearfunctionforre-scalingisnotapplicable.Amoresophisticatedapproachistousemethodsthatdirectlycomparethedatapointsoftwotimeseriesinamore exibleway,forexamplewith

tangentdistance[24],timealignment[25]–[27],ordynamictimewarpingkernels[28].Alsoprobabilisticmodels,suchasHMM8hiddenMarkovmodels)andGMM(Gaussianmixturemodels),thataretrainedonthetimeseriesdata,canbeusedincombinationwithSVM.Theso-calledFisher-kernelshavebeenwidelyused,e.g.,forspeechrecognition[29],[30],speakeridenti cation[31]–[33],orwebaudioclassi cation[34].[35],[36]usedanothersimilaritymeasureonGMM,theKullback-Leiblerdivergence,forspeakeridenti cationandveri cation.

Altogether,wecanstatethatdynamickernelfunctions[20]incorporatetemporalinformationdirectlyintoansupportvectormachine’skernelanduseitforcalculatingthesimilaritybetweendifferentinputtimeseries.Therefore,itbecomespos-sibletoalsodetectsimilaritiesbetweenmisalignedsequencesoravaryingfrequencyofthecontainedpatterns.C.DynamicTimeWarpingasKernelFunctionforSVMInourwork,weusedakernelbasedonthedynamictimewarping(DTW)method,whichhaspreviouslybeenutilizedforhandwritingandspeechrecognitionin[27],[28].Wealsorelyonownresultsdescribedin

[37].

Fig.2.ExamplefortheresultsobtainedwithDTW:Thecorrespondenceofpointsoftwosimilartimeseries(oneisdrawnwithaconstantoffsethere)isindicatedbyconnectinglines.

TheDTWkerneltakestwoinputtimeseriesandcalculatestheirsimilaritybydetermininganoptimalso-calledwarpingpathconsistingofpairsoftheirrespectivepoints.Eachpointofoneseriesisassignedtooneormorepointsoftheotherseries,obeyingthreeconstraints:

The rstandthelastpointsofbothseriesareassignedtoeachother.

Allassignmentsrespecttheseries’temporalorder.

Everypointofbothseriesbelongstoatleastoneassign-ment.

Thewarpingpathwiththeminimumsumofdistancesinitsassignmentswillbechosenastheoptimalwarpingpath.Otherdynamickernels,suchasthelongestcommonsubse-quence(LCSS)kernelwepresentedandinvestigatedin[37]followasimilarapproach.

IV.TESTSANDEXPERIMENTS

A.PreparationsandDataSetConstruction

Forourwork,weusedtheSVMroutinesfromthesoftwarepackageLibSVM[38].Theimplementationofthedynamickernelfunctionsfollows[37].

Tocomparetheforecastingaccuracyofthedifferentmodels,avarietyofdifferentmeasuresareusedintheliterature.How-ever,[39]and[40]showthatallofthepopularmeasuresareeithernotinvarianttoscalingorcontainunde nedintervals.Therefore,weusedthemeanabsolutescalederror(MASE)asproposedby[40],whichscalesthemeasurederrorusingthemeanabsoluteerrorofanaiveforecast(alsocalledrandomwalk).Thisforecastingtechniquesimplyassumesthattheresultforthenextpatternequalsthepreviousresult.

IfYtdenotestheobservationattimet∈{1,...,n}andFtistheforecast,wecallet=Yt Fttheforecasterror.Themeanabsolutescalederrorisde nedasthearithmeticmeanoftheforecasterrorsscaledbytheaverageerrorofarandomwalk:

MASE=mean

et |Y (1)i Yi 1| .

i=2Consequently,aMASEsmallerthan1.0indicatesthattheforecastingmethodperformsbetterthananaiveforecast.Appliedtothedomainoftechnicalanalysis,wecanseethatconstantMASEvaluessmallerthan1.0contradicttheef cientmarkettheory.Additionally,wespeci edthehitrateHITSofallforecasts,whichsimplyisthepercentageofcorrectlypredictedtrendsinthechart:HITS=

|{Fi|(Yi Yi 1)·(Fi Fi 1)>0,i=1,...,n}|

n

.

(2)

Fig.3.Inthediagram,weseehowthehistoryoftheFDAXwasdividedintosixdifferent,overlappingseriesofasizeof1000dayseach.Thelast250valuesofeachpart(approximatelyoneyear)wasusedtocalculatethepredictionaccuracyofthedevelopedsystemonthisspeci ctimeseries.Asaresult,amaximumnumberof750valueswasusedfortraining.

Forourexperiments,wedecidedtousetwopopularfutures:TheFDAXfutureonthestockindexDAX,andtheFGBLfutureonGermangovernmentbonds.Asallfuturespriceshaveapre-de nedenddateand,therefore,containperiodicbehaviorandpointsofdiscontinuity,thedatawasmanuallyadjusted.Tominimizetheimpactoftemporaryanomalies,we

decidedtoverifyourresultspiecewiseontheentirehistoryofthetwocharts,bydividingthemintoatotalof20differenttimeseriesofdailyvalues(seeFig.3).Forallexperiments,adailycompressionofthedatawasused.B.ExperimentSetupandResults

Theoverallorganizationoftheconductedexperimentswasmadeupofseveralparts:Firstofall,weexaminedtheperformanceofseveraldifferentinputandoutputseries.Wethencompareddifferentkernelfunctionsanddeterminedtheirbestparametersettings.Inthefollowingstep,differentvariantsoftheSVMtechniquewerecompared.Finally,weinvestigatedoptimalsettingsforthetotalamountandthelengthoftheinputseriesusedfortrainingandprediction.

Asoutputdata,itisalwayspossibletotrytopredicttheactualclosepriceofthenextday.Forusingthepredictioninatradingsystem,itismoreinterestingtopredictanupcomingtrend.ThiscanbedoneusingtherateofchangeROCnforagivenperiodnonatimeseriesY:

ROCn(Yt)=100·

Yt Yt n

Y.

(3)

t n

Earlyexperimentsshowedthatthetheforecastingaccuracycanbeconsiderablyincreasedusingthispre-processingfunc-tion.

Weconductedextensivetests,whereweexaminedmanydifferentinputtimeseriesandtheirperformanceinconjunctionwiththeoutputseries.Thebestresultswereachievedusingamultidimensionalinputvectorconsistingofseveralratesofchangewithdifferentperiods.ThisvectorincorporatesthetimeseriesROC1,ROC2,ROC3,ROC5,andROC8,andwillbedenotedROC5inthefollowing.Asaresult,thedifferentvaluesateachtimeexpress,bywhichratiothecurrentpricediffersfromadistinctpriceinthepast.TheresultsofourtestsaresetoutinTableI.

TABLEI

THEVALUESSHOWTHEPREDICTIONACCURACYOFAν-SUPPORTVECTORREGRESSIONSYSTEMUSINGTHEDYNAMICTIMEWARPINGKERNELFORDIFFERENTINPUTANDOUTPUTSERIES:WHILETHEOUTPUTSERIESROC2ANDROC5DESCRIBEROCOUTPUTSWITHDIFFERENTPERIODS,CLOSE–OPENDENOTESTHEDEVIATIONBETWEENADAY’S

OPENANDCLOSEPRICES.INCONTRASTTOTHEONE-DIMENSIONAL

INPUTSERIES

CLOSE,OHLC4ANDROC5AREMULTI-DIMENSIONAL

INPUTS,BUILTOFTHEDAY’SFOUROHLCVALUESORDIFFERENTRATES

OFCHANGE.

THELASTROWSHOWSTHEPERFORMANCEOFTHENAIVE

FORECASTINGMETHOD.ASTHEERRORMEASUREMASEISSCALEDBYTHEERROROFNAIVEFORECAST,ITALWAYSRESULTSINTHEVALUE1.

Output→ROC2

ROC5

Close–Open↓InputMASEHITSMASEHITSMASEHITSClose0.95350.49901.51310.48650.64390.4958OHLC40.96130.50011.52990.48910.64540.4924ROC50.77560.76041.09410.82630.53640.7382naive

1.0000

0.6727

1.0000

0.8027

1.0000

0.4829

Inasecondstep,wecomparedtheperformanceofSVMwithdifferentkernelfunctions.Fortheseexperiments,threedifferentdynamickernelfunctionstakenfrom[41]wereused:Thedynamictimewarpingkernel(DTW)aswellasthelongestcommonsubsequencekernelswithglobal(LCSS-global)aswellaslocalscaling(LCSSlocal).Asaresult,theDTW-kernelwasnotonlyconsiderablyfasterthanitsopponents.Duringthewholetraining,theLCSSkernelswerenotonceabletooutperformthepredictionaccuracyoftheDTWkernelonthetestdata(seeFig.4).Additionally,theLCSSkernelsappearedtorelyonspeci cattributes(features),whereastheDTWkernelshowedgoodresultsforalldatasets.ForthechoiceoftheSVMtype,weconductedclassi cationandregressionexperiments:Apartfromtheε-SVR(supportvectorregression)[42]andtheν-SVR[43],wemeasuredtheperformancefortheC-SVC(supportvectorclassi cation[44]andtheν-SVC[43].Insteadoftryingtopredictactualvalues,thesetechniquesweretrainedtoclassifythedataintotwocategories:oneforexpectedincreasing(rising),theanotheroneforexpecteddecreasing(falling)trends.Asaresult,wesawthatthepredictionresultsoftheν-SVRsigni cantlyoutperformedallothervariants,regardingbothMASE(forregressiontypes)andthehitrate,withtheε-SVRformulationrankingsecond.

Finally,weconductedsomeexperimentsinwhichwevariedthetotalamountandthelengthoftheinputtimeseriesoftheSVM.Con rmingtheobservationof[45],anincreaseintheamountofinputinformationdoesnotnecessarilyincreasethepredictionaccuracy.Instead,wecanseeinFig.4thatasmalleramountofcurrentinformationsigni cantlyimprovesthepredictionaccuracycomparedtoalargebacklogofhis-toricalinformation.C.MajorFindings

Fortheinputinformation,westatedthatthesheeramountofhistoricaldatadoesnotnecessarilyproducebetterresults.Instead,themainfocusshouldlieonthoroughpre-processingroutinestocapturetemporalpatternsofdifferentscale.Inthisregard,theappliedtechniqueofcreatingamulti-dimensionalvectorwithratesofchangeofdifferentmagnitudeworkedexceptionallywell.

Apartfromthat,ourresultsclearlyshowthehighabilityofSVMwithdynamickernelfunctionsintheareaof nancialtimeseriesforecasting.TheDTWkernelwasabletoproduceahitrateofupto70%overthewholehistoryofbothexaminedderivatives,comparedtoahitrateofonly47%forthenaiveforecast.Thisisevenmorerelevantasthehitratedirectlycorrelatestotheinputofcommonalgorithmictradingsystemsystems,triggeringactionswitheachtrendshift.

V.CONCLUSIONANDOUTLOOK

Inthisarticle,ashortintroductionintothe eldoftechnicalanalysisof nancialtimeserieshasbeengiven,andtheapplicationofSVMwithdynamickernelfunctionsinthisdomainhasbeenexamined.Aswedescribed,thedevelopedtechniquehasahighabilitytopredictfuturepricemovements

Fig.4.Thesegraphsshowsthedependenceofthepredictiononthesizeofthetimeslotusedeachtimeforpredictionandtraining:Theusedkernelfunctionsarefromlefttoright:DTW,LCSSglobal,andLCSSlocal.Theroundmarksinthediagramdenotetheresultswithatrainingsetof75periods,whereassquareandtriangularmarksshowtheresultsfor150and300periods.

ingdynamickernelfunctions,itispossibletouseawholerangeoftheprecedingseriesandanalyzeitasawholewiththeSVM’skernel.Aswecouldshow,thisapproachsigni cantlyincreasesthepredictionaccuracyandreliablyperformsbetterthanastandardnaiveforecast.

Forreal-worldexperimentsandapplicationsofthedevel-opedsystem,aninterfacetothetechnicalanalysissoftwareInvestox[46]wascreated(seeFig.5).Usingthisapplication,itbecomesnotonlypossibletoverifytheresultsonhistoricaldatausingavirtualbroker,butalsotoapplythesystemdirectlytocurrentdatainputsinaconstantlyevolvingmarketenvironment(seealso[47]).

Inourfutureresearch,theperformanceofthedevelopedsystemwillbeexaminedindifferenttradingconstellations.Contrarytotheworkonend-of-daydata,theperformanceofthetechniqueisalsohighenoughtouseitintheareaofintra-dayforecasting.Thisinvolvespredictionsinintervalsofonlyseveralminutes,ifnotjustinseconds’intervals.Inthisenvironmentofhighuncertaintyandconstanttrendshift,verydifferentrequirementsmayapply.Ontheotherhand,itisalsopossibletonotonlyusetheinputofonepre-processedtimeseries,buttocombinedifferentmarketpricesforpredictingacertainvalue.Thiskindofinter-marketanalysismayhavethepotentialtodetect uctuationsinaspeci cpriceandprematurelyratetheresultingin uenceonthetargetvalue.

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