Forecasting Financial Time Series with Support Vector Machin(3)

[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

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