filtering and smoothing techniques-presenting time series analysis

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1、Chapter4Filteringandsmoothingtechniques4.1Presentingthechallenge4.1.1DescribingtheproblemDynamicalsystemsarecharacterisedbytwotypesofnoise,wherethefirstoneiscalledobservationaloradditivenoise,andthesecondoneiscalleddynamicalnoise.Intheformer,thesystemisunaffectedbythisno

2、ise,insteadthenoiseisameasurementproblem.Theobserverhastroublepreciselymeasuringtheoutputofthesystem,leadingtorecordedvalueswithaddednoiseincrement.Thisadditivenoiseisexternaltotheprocess.Inthelatter,thesysteminterpretsthenoisyoutputasaninput,leadingtodynamicalnoisebeca

3、usethenoiseinvadesthesystem.Dynamicalnoisebeinginherenttofinancialtimeseries,wearenowgoingtosummarisesomeofthetoolsproposedinstatisticalanalysisandsignalprocessingtofilteritout.Infinancialtimeseriesanalysis,thetrendisthecomponentcontainingtheglobalchange,whilethelocalchang

4、esarerepresentedbynoise.Ingeneral,thetrendischaracterisedbyasmoothfunctionrepresentinglong-termmovement.Hence,trendsshouldexhibitslowchanges,whilenoiseisassumedtobehighlyvolatile.Trendfiltering(TF)attemptsatdifferentiatingmeaningfulinformationfromexogenousnoise.Thesepara

5、tionbetweentrendandnoiseliesatthecoreofmodernstatisticsandtimeseriesanalysis.TFisgenerallyusedtoanalysethepastbytransforminganynoisysignalintoasmootherone.Itcanalsobeusedasapredictivetool,butitcannotbeperformedonanytimeseries.Forinstance,trendfollowingpredictionssuppose

6、thatthelastobservedtrendinfluencesfuturereturns,butthetrendmaynotpersistinthefuture.Aphysicalprocesscanbedescribedeitherinthetimedomain,bythevaluesofsomequantityhasafunctionoftimeh(t),orinthefrequencydomainwheretheprocessisspecifiedbygivingitsamplitudeHasafunctionoffreque

7、ncyf,thatis,H(f)with1

8、filteringtoeliminatenoiseatloworhighfrequencies,orrequiringabandpassfilteriftheinterestingpartofthesignalliesonl