[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