new measure of multifractality and its application in finances

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1、NewmeasureofmultifractalityanditsapplicationinfinancesDariuszGrech∗andGrzegorzPamula†InstituteofTheoreticalPhysicsPl.M.Borna9,UniversityofWroclaw,PL-50-204Wroclaw,PolandAbstractWeprovideanalternativemethodforanalysisofmultifractalpropertiesoftimeseries.Thenewapproa

2、chtakesintoaccountthebehaviorofthewholemultifractalprofileofthegeneralizedHurstexponenth(q)forallmomentordersq,notlimitedonlytotheedgevaluesofh(q)describinginMFDFAscalingpropertiesofsmallestandlargestfluctuationsinsignal.Themeaningofthisnewmeasureisclarifiedanditspro

3、pertiesareinvestigatedforsyntheticmultifractaldataaswellasforrealsignalsfromstockmarkets.Weshowthattheproposednewmeasureisfreeofproblemsonecanmeetinrealnonstationarysignals,whilesearchingtheirmultifractalsignatures.Keywords:multifractality,spuriousmultifractality,

4、finitesizeeffects,multifractaldetrendedanalysis,multifractalbias,scaling,timeseriesanalysis,newmultifractalmeasure,generalizedHurstexponentPACS:05.45.Tp,89.75.Da,05.40.-a,89.65.Gh,89.75.-k1IntroductionThemultifractaldetrendedfluctuationanalysis(MFDFA)[1]becameinlastt

5、enyearsthemajorarXiv:1309.5466v1[q-fin.ST]21Sep2013techniqueusedforstudyingthemultifractalpropertiesincomplexsystemsandintimeseries.Morethen500papersdiscussesMFDFAissuesinvarietyofproblemsrelatedwithcomplexity(see,e.g.,[2]–[18]).ThecentralroleinMFDFAisplayedbytheq

6、-thmomentoffluctuationfunction(q∈R)definedas[1]∗dgrech@ift.uni.wroc.pl†gpamula@ift.uni.wroc.pl1()1/q1X2N2q/2F(q,τ)=[Fˆ(τ,k)](1)2Nk=1forq6=0,and()1X2N2F0(τ)=expln[Fˆ(τ,k)](2)4Nk=1forq=0,where1XτFˆ2(τ,k)=x−P(j)2(3)(k−1)τ+jkτj=1andxj(j=1,...,Nτ)aredatainseries,τisthes

7、izeofwindowboxinwhichdetrendingisperformed,whilePk(j)isthepolynomialtrendsubtractedforj-thdataink-thwindowbox(k=1,...,N).Theobservedscalinglawh(q)F(q,τ)∼τ(4)iscrucialwithinMFDFAtoestimatethemultifractalpropertiesofasignalgivenbytheprofileofgeneralizedHurstexponents

8、h(q).Thestrengthofmultifractalitypresentindataisusuallystudiedasaspread∆h∆h≡h(−Q)−h(Q)(5)ofsomeedgevaluesofgeneralizedHurstexponenth∓calculatedatthefixed