arXiv:1005.0877v2 [q-fin.ST] 8 Jun
2010PRE/MFDMADetrendingmovingaveragealgorithmformultifractalsGao-FengGu1,
2andWei-Xing Zhou1, 2, 3, 4, 5, 1School of Business, East
ChinaUniversityof ScienceandTechnology, Shanghai 200237,
China2ResearchCenterforEconophysics, EastChinaUniversityof
ScienceandTechnology, Shanghai 200237, China3School of Science,
East ChinaUniversityof ScienceandTechnology, Shanghai 200237,
China4EngineeringResearchCenterof Process Systems
Engineering(Ministryof Education),East ChinaUniversityof Science
andTechnology, Shanghai 200237, China5Research Center onFictitious
Economics &DataScience,Chinese Academy of Sciences, Beijing
100080, China(Dated: June9,2010)The detrendingmovingaverage (DMA)
algorithmis
awidelyusedtechniquetoquantifythelong-termcorrelationsof
non-stationarytimeseriesandthelong-rangecorrelationsof fractal
sur-faces,which contains a parameterdetermining the positionof the
detrending window. We developmultifractal
detrendingmovingaverage(MFDMA)algorithmsfortheanalysisof
one-dimensionalmultifractal measuresandhigher-dimensional
multifractals, whichisageneralizationof theDMAmethod. The
performance of the one-dimensional and two-dimensional MFDMA
methods is investi-gated using synthetic multifractalmeasures with
analytical solutionsfor backward (= 0), centered( =
0.5),andforward(= 1) detrending windows. Wend that
theestimatedmultifractalscalingexponent(q) and the
singularityspectrumf() areingoodagreement withthe
theoreticalvalues.Inaddition, thebackwardMFDMAmethodhasthebest
performance, whichprovidesthemostaccurateestimates of
thescalingexponents withlowest error bars,
whilethecenteredMFDMAmethodhastheworseperformance.
ItisfoundthatthebackwardMFDMAalgorithmalsoout-performsthemultifractaldetrendeductuationanalysis(MFDFA).Theone-dimensionalbackwardMFDMAmethodisappliedtoanalyzingthetimeseriesof
Shanghai StockExchangeCompositeIndexanditsmultifractal
natureisconrmed.PACSnumbers: 05.45.Df,05.40.-a, 05.10.-a,89.75.DaI.
INTRODUCTIONFractals andmultifractals are ubiquitous
innaturalandsocial sciences[13]. Therearealargenumberofmethods
developed to characterize the properties of
frac-talsandmultifractals. TheclassicmethodistheHurstanalysis or
rescaledrange analysis (R/S) for time se-ries [4, 5] andfractal
surfaces [6]. Thewavelet
trans-formmodulemaxima(WTMM)methodisamorepow-erful tool
toaddressthemultifractality[711], evenforhigh-dimensional
multifractal measures inthe elds
ofimagetechnologyandthree-dimensionalturbulence[1216].
Anotherpopularmethodisthedetrendeductua-tionanalysis (DFA),
whichhas theadvantagesof easyimplementation and robust estimation
even for short sig-nals [1719]. The DFA method was originally
invented
tostudythelong-rangedependenceincodingandnoncod-ingDNAnucleotidessequence[20]
andthenappliedtotimeseriesinvariouselds[2124]. TheDFAalgorithmwas
extended to analyze the multifractal time series,which is termed as
multifractal detrended uctuationanalysis(MFDFA)[25].
TheseDFAandMFDFAmeth-ods were also generalizedto analyze
high-dimensionalfractalsandmultifractals[26].Amorerecentmethodisbasedonthemovingaver-age(MA)
or mobileaveragetechnique[27], [email protected]
proposedbyVandewalle andAusloos toestimatethe Hurst exponent of
self-anity signals [28] and
furtherdevelopedtothedetrendingmovingaverage(DMA)byconsidering the
second-order dierence between the orig-inal signal and its moving
average function [29]. Becausethe DMA method can be easily
implemented to estimatethecorrelationpropertiesof
non-stationaryserieswith-out any assumption, it is widely applied
to the analysis ofreal-worldtimeseries[3037]
andsyntheticsignals[3840]. Recently, Carbone extendedthe
one-dimensionalDMA method to higher dimensions to estimate the
Hurstexponents of higher-dimensional fractals [41,42].
Exten-sivenumerical experimentsunveil thattheperformanceof theDMA
method are comparable to the DFA methodwith slightly dierent
priorities under dierent situations[39,43].Inthis paper,
weextendthe DMAmethodtomul-tifractal
detrendingmovingaverage(MFDMA), whichis designed to analyze
multifractal time series andmultifractal surfaces. Further
extensions to higher-dimensional versions are straightforward. The
perfor-mance of the MFDMAalgorithms is investigated us-ingsynthetic
multifractal measures withknownmulti-fractalproperties.
WealsocomparetheperformanceofMFDMAwithMFDFA,andndthatMFDMAissupe-riortoMFDFAformultifractalanalysis.Thepaperisorganizedasfollows.
InSec. II, wede-scribe the algorithmof one-dimensional
MFDMAandshowtheresultsof numerical simulations. Wealsoap-plythe
one-dimensional MFDMAtoanalyzethe time2seriesof intradayShanghai
StockExchangeCompositeIndex(SSEC). InSec. III,
wedescribethealgorithmoftwo-dimensional MFDMAandreporttheresultsof
nu-mericalsimulationsaswell. WediscussandconcludeinSec.IV.II.
ONE-DIMENSIONALMULTIFRACTALDETRENDINGMOVINGAVERAGEANALYSISA.
AlgorithmStep1. Consideratimeseriesx(t), t =1, 2, ,
N.Weconstructthesequenceofcumulativesumsy(t) =t
i=1x(i), t = 1, 2, , N. (1)Step2.
Calculatethemovingaveragefunctiony(t)inamovingwindow[35], y(t)
=1n(n1)(1)
k=(n1)y(t k), (2)where n isthewindow size,x isthelargest integer
notgreater thanx, xis thesmallest integer not smallerthanx,
andisthepositionparameterwiththevaluevaryingintherange[0, 1].
Hence, themovingaveragefunctionconsiders (n1)(1) datapoints
inthepast and(n 1)points inthefuture. Weconsiderthreespecial
casesinthis paper. Therstcase=0refers to thebackward moving average
[39],in which themovingaveragefunctiony(t) is calculatedover all
thepast n 1datapoints of thesignal. Thesecondcase = 0.5 corresponds
tothecentered moving average [39],wherey(t) contains half past and
half future informationineachwindow.
Thethirdcase=1iscalledthefor-wardmovingaverage,wherey(t)considersthetrendofn
1datapointsinthefuture.Step 3. Detrend the signal series by
removing the mov-ing average functiony(i) from y(i), and obtain the
resid-ualsequence(i)through(i) = y(i) y(i), (3)wheren (n 1)iN (n
1).Step 4. The residual series (i) is dividedinto Nndisjoint
segments with the same size n, where Nn=N/n 1. Eachsegment canbe
denotedbyvsuchthatv(i)=(l + i)for1in, wherel =(v
1)n.Theroot-mean-squarefunctionFv(n)withthesegmentsizencanbecalculatedbyF2v(n)
=1nn
i=12v(i). (4)Step 5. The qth order overall uctuation
functionFq(n)isdeterminedasfollows,Fq(n) =
1NnNn
v=1Fqv(n)
1q, (5)whereqcantakeanyrealvalueexceptforq= 0. Whenq=
0,wehaveln[F0(n)] =1NnNn
v=1ln[Fv(n)], (6)accordingtoLH ospitalsrule.Step6.
Varyingthevaluesof segmentsizen,
wecandeterminethepower-lawrelationbetweenthefunctionFq(n)andthesizescalen,whichreadsFq(n)
nh(q).
(7)Accordingtothestandardmultifractalformalism,themultifractal
scalingexponent(q)canbeusedtochar-acterizethemultifractalnature,whichreads(q)
= qh(q) Df, (8)whereDfisthefractaldimensionofthegeometricsup-port
of the multifractal measure [25]. For time
seriesanalysis,wehaveDf=1. Ifthescalingexponentfunc-tion (q) is a
nonlinear function of q, the signal hasmultifractal nature. It is
easytoobtainthe singular-ity strength function (q)and
themultifractal spectrumf()viatheLegendretransform [44]
(q) = d(q)/dqf(q) = q (q). (9)B. NumericalexperimentsIn the
numerical experiments, we generate
one-dimensionalmultifractalmeasuretoinvestigatetheper-formanceofMFDMA,which
iscompared withMFDFA.Weapplythep-model,
amultiplicativecascadingpro-cess, tosynthesizethemultifractal
measure[45]. Start-ingfromameasureuniformlydistributedonanin-terval
[0, 1]. In the rst step, the measure is
redis-tributedontheinterval,1,1= p1tothersthalfand1,2= p2= (1p1) to
the second half. One partitionsit into two sub-lines with the same
length. In the (k+1)-th step,themeasure k,ion each of
the2klinesegmentsisredistributedintotwoparts,wherek+1,2i1=
k,ip1andk+1,2i=k,ip2. Werepeat theprocedurefor 14times and nally
generate the one-dimensional multi-fractal measurewiththelength214.
Inthispaper, wepresent the results whenthe parameters are
p1=0.3andp2= 0.7. Theresultsforotherparameters
arequal-itativelythesame.31011021031010105100nFq(n)
(a)MFDMA,=0MFDMA,=0.5MFDMA,=1MFDFA4 3 2 1 0 1 2 3 48642024q(q)
(b)MFDMA,=0MFDMA,=0.5MFDMA,=1MFDFA4 3 2 1 0 1 2 3
40.40.200.20.4q(q) (c)MFDMA,=0MFDMA,=0.5MFDMA,=1MFDFA0.4 0.6 0.8 1
1.2 1.4 1.600.20.40.60.81f() (d)MFDMA,=0MFDMA,=0.5MFDMA,=1MFDFAFIG.
1. (Color online)Multifractal analysisof theone-dimensional
multifractal binomial
measureusingthethreeMFDMAalgorithmsandtheMFDFAapproach.
(a)Power-lawdependenceoftheuctuationfunctionsFq(n)withrespecttothescalenforq=4,
q=0, andq=4. Thestraightlinesarethebestpower-lawtstothedata.
Theresultshavebeentranslatedverticallyforbettervisibility.
(b)Multifractal
massexponents(q)obtainedfromtheMFDMAandMFDFAmethodswiththetheoretical
curveshownasasolidline.
(c)Dierences(q)betweentheestimatedmassexponentsandtheirtheoreticalvaluesforthefouralgorithms.
(d)Multifractal
spectraf()withrespecttothesingularitystrengthforthefourmethods.Thecontinuouscurveisthetheoreticalmultifractalspectrum.We
calculate the uctuation function Fq(n) of the syn-thetical
multifractal measure using the MFDMAandMFDFAmethods,
andpresenttheuctuationfunctionFq(n)inFig.1(a).
WendthatthefunctionFq(n)wellscales with the scale size n. Using the
least squares ttingmethod,
weobtaintheslopesh(q)forMFDMA(=0,=0.5and=1)andMFDFArespectively,whichareillustratedinTableI.
ItisfoundthattheerrorbarsofthethreeMFDMAalgorithmsareall
smallerthantheMFDFA method,which implies that it is easier to
deter-minethescalingrangesfortheMFDMAalgorithms. Inmostcases,
thealgorithmsunderestimatetheh(q)
val-uesandthebackwardMFDMAapproach givesthebestestimates. Thereis
aninterestingfeatureinFig.
1(a)showingevidentlog-periodicoscillationsintheMFDFAFq(n) curves,
which isintrinsic for themultifractal bino-mialmeasure[46].We plot
the multifractal scalingexponents (q) ob-tainedfromMFDMA(=0,
=0.5and=1)andTABLEI. TheMFDMAexponents h(q) for q =4, -2, 0,2,
and4of theone-dimensional syntheticmultifractal
mea-surewiththeparameters p1 =0.3andp2 =0.7usingtheMFDMA(=0,
=0.5and=1)andMFDFAmethods.Thenumbers intheparentheses
arethestandarderrors
oftheregressioncoecientestimatesusingthet-testatthe5%signicancelevel.qMFDMAMFDFA
Analytic = 0 = 0.5 = 1-4 1.505(4) 1.401(12) 1.496(2) 1.490(17)
1.499-2 1.354(3) 1.249(8) 1.337(4) 1.326(9) 1.3590 1.114(4)
1.022(5) 1.096(5) 1.074(6) 1.1262 0.874(6) 0.788(3) 0.859(5)
0.804(11) 0.8934 0.749(9) 0.667(4) 0.736(6) 0.670(15)
0.753MFDFAinFig.1(b). Thetheoreticalformulaof(q)ofthemultifractal
measuregeneratedbythep-model dis-4101102103104103102101100101nFq(n)
(a)q=4q=2q=0q=2q=40.3 0.4 0.5 0.6 0.7 0.80.50.60.70.80.91f()
(b)Real dataShuffled data4 2 0 2 44202q(q)FIG. 2.
(Coloronline)Multifractal analysisof the5-minreturntimeseriesof
theSSECindexusingthebackwardMFDMAmethod. (a) Power-lawdependenceof
theuctuationfunctions F(n) withrespect tothescale n. Thesolidlines
aretheleast-squareststothedata. Theresultscorrespondingtoq=2, q=0,
q=2andq=4havebeentranslatedverticallyforclarity. (b)Multifractal
spectraf()of therawreturnseriesof SSECanditsshuedseries. Inset:
Multifractal
scalingexponents(q)asafunctionofq.cussedabovecanbeexpressed
by[44]th(q) = ln(pq1 + pq2)ln 2, (10)which has been illustrated in
Fig. 1(b) as well. In order toquantitatively evaluate the
performance of MFDMA andMFDFA, we calculate the relative estimation
errors of thenumerical values of (q) in reference to the
correspondingtheoreticalvaluesth(q)(q) = (q) th(q),
(11)whichareshowninFig. 1(c). When0
