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