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Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series

Jan W. Kantelhardt1,2, Stephan A. Zschiegner1, Eva Koscielny-Bunde3,1, Armin Bunde1,Shlomo Havlin4,1, and H. Eugene Stanley2

1 Institut fur Theoretische Physik III, Justus-Liebig-Universitat, D-35392 Giessen, Germany2 Center for Polymer Studies and Department of Physics, Boston University, Boston MA 02215, USA3 Potsdam Institute for Climate Impact Research, P. O. Box 60 12 03, D-14412 Potsdam, Germany

4 Department of Physics and Gonda-Goldschmied Medical Diagnostics Research Center, Bar-Ilan University, Ramat-Gan 52900, Israel

We develop a method for the multifractal characterization of nonstationary time series, which is based ona generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to thestandard partition function-based multifractal formalism, and prove that both approaches are equivalent forstationary signals with compact support. By analyzing several examples we show that the new method canreliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA resultsfor original series to those for shuffled series we can distinguish multifractality due to long-range correlationsfrom multifractality due to a broad probability density function. We also compare our results with the wavelettransform modulus maxima (WTMM) method, and show that the results are equivalent.

PACS numbers: 05.40.-a, 05.45.Tp

I. INTRODUCTION

In recent years the detrended fluctuation analysis (DFA)method [1] has become a widely-used technique for the de-termination of (mono-) fractal scaling properties and the de-tection of long-range correlations in noisy, nonstationary timeseries [2–5]. It has successfully been applied to diverse fieldssuch as DNA sequences [1,6], heart rate dynamics [7,8], neu-ron spiking [9], human gait [10], long-time weather records[11], cloud structure [12], geology [13], ethnology [14], eco-nomics time series [15], and solid state physics [16]. Onereason to employ the DFA method is to avoid spurious detec-tion of correlations that are artifacts of nonstationarities in thetime series.

Many records do not exhibit a simple monofractal scalingbehavior, which can be accounted for by a single scaling ex-ponent. In some cases, there exist crossover (time-) scaless× separating regimes with different scaling exponents [3,4],e. g. long-range correlations on small scaless ≪ s× and an-other type of correlations or uncorrelated behavior on largerscaless ≫ s×. In other cases, the scaling behavior is morecomplicated, and different scaling exponents are requiredfordifferent parts of the series [5]. This occurs, e. g., when thescaling behavior in the first half of the series differs from thescaling behavior in the second half. In even more compli-cated cases, such different scaling behavior can be observedfor many interwoven fractal subsets of the time series. In thiscase a multitude of scaling exponents is required for a full de-scription of the scaling behavior, and a multifractal analysismust be applied.

In general, two different types of multifractality in time se-ries can be distinguished: (i) Multifractality due to a broadprobability density function for the values of the time series.In this case the multifractality cannot be removed by shuf-fling the series. (ii) Multifractality due to different long-range(time-) correlations of the small and large fluctuations. Inthiscase the probability density function of the values can be a

regular distribution with finite moments, e. g. a Gaussiandistribution. The corresponding shuffled series will exhibitnon-multifractal scaling, since all long-range correlations aredestroyed by the shuffling procedure. If both kinds of mul-tifractality are present, the shuffled series will show weakermultifractality than the original series.

The simplest type of multifractal analysis is based upon thestandard partition function multifractal formalism, which hasbeen developed for the multifractal characterization of nor-malized, stationary measures [17–20]. Unfortunately, thisstandard formalism does not give correct results for nonsta-tionary time series that are affected by trends or that cannotbe normalized. Thus, in the early 1990s an improved multi-fractal formalism has been developed, the wavelet transformmodulus maxima (WTMM) method [21], which is based onwavelet analysis and involves tracing the maxima lines in thecontinuous wavelet transform over all scales. Here, we pro-pose an alternative approach based on a generalization of theDFA method. This multifractal DFA (MF-DFA) does not re-quire the modulus maxima procedure, and hence does not in-volve more effort in programming than the conventional DFA.

The paper is organized as follows: In Section II we de-scribe the MF-DFA method in detail and show that the scal-ing exponents determined via the MF-DFA method are iden-tical to those obtained by the standard multifractal formalismbased on partition functions. In Section III we introduce sev-eral multifractal models, where the scaling exponents can becalculated exactly, and compare these analytical results withthe numerical results obtained by MF-DFA. In Section IV, weshow how the comparison of the MF-DFA results for originalseries to the MF-DFA results for shuffled series can be used todetermine the type of multifractality in the series. In SectionV, we compare the results of the MF-DFA with those obtainedby the WTMM method for nonstationary series and discussthe performance of both methods for multifractal time seriesanalysis.

1

II. MULTIFRACTAL DFA

A. Description of the method

The generalized multifractal DFA (MF-DFA) procedureconsists of five steps. The first three steps are essentially iden-tical to the conventional DFA procedure (see e. g. [1–5]). Letus suppose thatxk is a series of lengthN , and that this seriesis of compact support, i.e.xk = 0 for an insignificant fractionof the values only.• Step 1: Determine the “profile”

Y (i) ≡i

∑

k=1

[xk − 〈x〉] , i = 1, . . . , N. (1)

Subtraction of the mean〈x〉 is not compulsory, since it wouldbe eliminated by the later detrending in the third step.• Step 2: Divide the profileY (i) into Ns ≡ int(N/s) non-overlapping segments of equal lengths. Since the lengthN ofthe series is often not a multiple of the considered time scales,a short part at the end of the profile may remain. In order not todisregard this part of the series, the same procedure is repeatedstarting from the opposite end. Thereby,2Ns segments areobtained altogether.• Step 3: Calculate the local trend for each of the2Ns seg-ments by a least-square fit of the series. Then determine thevariance

F 2(s, ν) ≡1

s

s∑

i=1

{Y [(ν − 1)s+ i] − yν(i)}2 (2)

for each segmentν, ν = 1, . . . , Ns and

F 2(s, ν) ≡1

s

s∑

i=1

{Y [N − (ν −Ns)s+ i] − yν(i)}2 (3)

for ν = Ns+1, . . . , 2Ns. Here,yν(i) is the fitting polynomialin segmentν. Linear, quadratic, cubic, or higher order poly-nomials can be used in the fitting procedure (conventionallycalled DFA1, DFA2, DFA3,. . .) [1,8]. Since the detrendingof the time series is done by the subtraction of the polynomialfits from the profile, different order DFA differ in their capa-bility of eliminating trends in the series. In (MF-)DFAm [mthorder (MF-)DFA] trends of orderm in the profile (or, equiv-alently, of orderm − 1 in the original series) are eliminated.Thus a comparison of the results for different orders of DFAallows one to estimate the type of the polynomial trend in thetime series [3,4].• Step 4: Average over all segments to obtain theqth orderfluctuation function

Fq(s) ≡

{

1

2Ns

2Ns∑

ν=1

[

F 2(s, ν)]q/2

}1/q

, (4)

where, in general, the index variableq can take any real valueexcept zero [22]. Forq = 2, the standard DFA procedure is

retrieved. We are interested in how the generalizedq depen-dent fluctuation functionsFq(s) depend on the time scalesfor different values ofq. Hence, we must repeat steps 2 to 4for several time scaless. It is apparent thatFq(s) will increasewith increasings. Of course,Fq(s) depends on the DFA orderm. By construction,Fq(s) is only defined fors ≥ m+ 2.• Step 5: Determine the scaling behavior of the fluctuationfunctions by analyzing log-log plotsFq(s) versuss for eachvalue ofq. Several examples of this procedure will be shownin Section III. If the seriesxi are long-range power-law corre-lated,Fq(s) increases, for large values ofs, as a power-law,

Fq(s) ∼ sh(q). (5)

In general, the exponenth(q) may depend onq. For stationarytime series,h(2) is identical to the well-known Hurst expo-nentH (see e. g. [17]). Thus, we will call the functionh(q)generalized Hurst exponent.

For monofractal time series with compact support,h(q) isindependent ofq, since the scaling behavior of the variancesF 2(s, ν) is identical for all segmentsν, and the averaging pro-cedure in Eq. (4) will give just this identical scaling behaviorfor all values ofq. Only if small and large fluctuations scaledifferently, there will be a significant dependence ofh(q) onq:If we consider positive values ofq, the segmentsν with largevarianceF 2

s (ν) (i. e. large deviations from the correspondingfit) will dominate the averageFq(s). Thus, for positive valuesof q, h(q) describes the scaling behavior of the segments withlarge fluctuations. Usually the large fluctuations are character-ized by a smaller scaling exponenth(q) for multifractal series[23]. On the contrary, for negative values ofq, the segmentsν with small varianceF 2

s (ν) will dominate the averageFq(s).Hence, for negative values ofq, h(q) describes the scalingbehavior of the segments with small fluctuations, which areusually characterized by a larger scaling exponent.

However, the MF-DFA method can only determineposi-tivegeneralized Hurst exponentsh(q), and it already becomesinaccurate for strongly anti-correlated signals whenh(q) isclose to zero. In such cases, a modified (MF-)DFA techniquehas to be used. The most simple way to analyze such datais to integrate the time series before the MF-DFA procedure.Hence, we replace thesinglesummation in Eq. (1), which isdescribing the profile from the original dataxk, by adoublesummation,

Y (i) ≡

i∑

k=1

[Y (k) − 〈Y 〉] . (6)

Following the MF-DFA procedure as described above, we ob-tain generalized fluctuation functionsFq(s) described by ascaling law as in Eq. (5), but with larger exponentsh(q) =h(q) + 1,

Fq(s) ∼ sh(q) = sh(q)+1. (7)

Thus, the scaling behavior can be accurately determined evenfor h(q) which are smaller than zero (but larger than−1) for

2

some values ofq. We note thatFq(s)/s corresponds toFq(s)in Eq. (5). If we do not subtract the average values in each stepof the summation in Eq. (6), this summation leads to quadratictrends in the profileY (i). In this case we must employ at leastthe second order MF-DFA to eliminate these artificial trends.

B. Relation to standard multifractal analysis

For stationary, normalized records with compact supportthe multifractal scaling exponentsh(q) defined in Eq. (5) aredirectly related, as shown below, to the scaling exponentsτ(q)defined by the standard partition function-based multifractalformalism.

Suppose that the seriesxk of lengthN is a stationary, nor-malized sequence. Then the detrending procedure in step 3of the MF-DFA method is not required, since no trend has tobe eliminated. Thus, the DFA can be replaced by the stan-dard Fluctuation Analysis (FA), which is identical to the DFAexcept for a simplified definition of the variance for each seg-mentν, ν = 1, . . . , Ns, in step 3 [see Eq. (2)]:

F 2FA(s, ν) ≡ [Y (νs) − Y ((ν − 1)s)]2. (8)

Inserting this simplified definition into Eq. (4) and usingEq. (5), we obtain

{

1

2Ns

2Ns∑

ν=1

|Y (νs) − Y ((ν − 1)s)|q

}1/q

∼ sh(q). (9)

For simplicity we can assume that the lengthN of the seriesis an integer multiple of the scales, obtainingNs = N/s andtherefore

N/s∑

ν=1

|Y (νs) − Y ((ν − 1)s)|q ∼ sqh(q)−1. (10)

This already corresponds to the multifractal formalism usede. g. in [18,20]. In fact, a hierarchy of exponentsHq similarto ourh(q) has been introduced based on Eq. (10) in [18].

In order to relate also to the standard textbook box countingformalism [17,19], we employ the definition of the profile inEq. (1). It is evident that the termY (νs) − Y ((ν − 1)s) inEq. (10) is identical to the sum of the numbersxk within eachsegmentν of sizes. This sum is known as the box probabilityps(ν) in the standard multifractal formalism for normalizedseriesxk,

ps(ν) ≡

νs∑

k=(ν−1)s+1

xk = Y (νs) − Y ((ν − 1)s). (11)

The scaling exponentτ(q) is usually defined via the partitionfunctionZq(s),

Zq(s) ≡

N/s∑

ν=1

|ps(ν)|q ∼ sτ(q), (12)

whereq is a real parameter as in the MF-DFA above. Some-timesτ(q) is defined with opposite sign (see e. g. [17]).

Using Eq. (11) we see that Eq. (12) is identical to Eq. (10),and obtain analytically the relation between the two sets ofmultifractal scaling exponents,

τ(q) = qh(q) − 1. (13)

Thus, we have shown thath(q) defined in Eq. (5) for the MF-DFA is directly related to the classical multifractal scaling ex-ponentsτ(q). Note thath(q) is different from the generalizedmultifractal dimensions

D(q) ≡τ(q)

q − 1=qh(q) − 1

q − 1, (14)

that are used instead ofτ(q) in some papers. Whileh(q) isindependent ofq for a monofractal time series with compactsupport,D(q) depends onq in that case. Our assumption ofcompact support of the seriesxk can be directly observed inEq. (14), since the fractal dimension of the support isD(0) ≡−τ(0) = 1.

Another way to characterize a multifractal series is the sin-gularity spectrumf(α), that is related toτ(q) via a Legendretransform [17,19],

α = τ ′(q) and f(α) = qα− τ(q). (15)

Here,α is the singularity strength or Holder exponent, whilef(α) denotes the dimension of the subset of the series that ischaracterized byα. Using Eq. (13), we can directly relateαandf(α) to h(q),

α = h(q) + qh′(q) and f(α) = q[α− h(q)] + 1. (16)

III. FOUR ILLUSTRATIVE EXAMPLES

A. Example 1: monofractal uncorrelated and long-rangecorrelated series

As a first example we apply the MF-DFA method tomonofractal series with compact support, for which the gen-eralized Hurst exponenth(q) is expected to be independent ofq,

h(q) = H and τ(q) = qH − 1. (17)

Such series have been discussed in the context of conventionalDFA in several studies before, see e.g. [3–5]. Long-range cor-related random numbers are usually generated by the Fouriertransform method, see e. g. [17,24]. Using this method we cangenerate long-range anti-correlated (0 < H < 0.5), uncorre-lated (H = 0.5), or (positively) long-range correlated (0.5 <H < 1) series. The latter are characterized by a power-law de-cay of the autocorrelation functionC(s) ≡ 〈xk xk+s〉 ∼ s−γ

for large scaless with γ = 2 − 2H if the series is stationary.

3

Alternatively, all stationary long-range correlated series canbe characterized by the power-law decay of their power spec-tra, S(f) ∼ f−β with frequencyf andβ = 2H − 1. NotethatH corresponds to the Hurst exponent of the integratedtime series here.

Figure 1 shows the generalized fluctuation functionsFq(s)for all three types of monofractal series (H = 0.75, 0.5, 0.25)and severalq values. On large scaless, we observe theexpected power-law scaling behavior according to Eq. (5),which corresponds to straight lines in the log-log plot. InFig. 1(d), the scaling exponentsh(q) determined from theslopes of these straight lines are shown versusq. Althougha slightq dependence is observable, the values ofh(q) are al-ways very close to theH of the generated series that has beenanalyzed. The degree of theq dependence observed for thismonofractal series allows to estimate the usual fluctuationofh(q) to be expected for monofractal series in general.

Next, we analyze multifractal series for whichτ(q) can becalculated exactly, and compare the numerical results withtheexpected scaling behavior.

B. Example 2: binomial multifractal series

In the binomial multifractal model [17–19], a series ofN =2nmax numbersk with k = 1, . . . , N is defined by

xk = an(k−1)(1 − a)nmax−n(k−1), (18)

where0.5 < a < 1 is a parameter andn(k) is the number ofdigits equal to 1 in the binary representation of the indexk,e. g.n(13) = 3, since 13 corresponds to binary 1101.

The scaling exponentsτ(q) can be calculated straightfor-wardly. According to Eqs. (11) and (18) the box probabilityp2s(ν) in theνth segment of size2s is given by

p2s(ν) = ps(2ν − 1) + ps(2ν)

= [(1 − a)/a+ 1]ps(2ν) = ps(2ν)/a.

Thus, according to Eqs. (12) and (18),

Zq(s) =

N/s∑

ν=1

[ps(ν)]q =

N/2s∑

ν=1

[ps(2ν − 1)]q + [ps(2ν)]q

=

[

(1 − a)q

aq+ 1

] N/2s∑

ν=1

[ps(2ν)]q

= [(1 − a)q + aq]

N/2s∑

ν=1

[p2s(ν)]q = [(1 − a)q + aq] Zq(2s)

and according to Eqs. (12) and (13),

τ(q) =− ln[aq + (1 − a)q]

ln(2), (19)

h(q) =1

q−

ln[aq + (1 − a)q]

q ln(2). (20)

Note thatτ(0) = −1 as required. There is a strong non-linear dependence ofτ(q) uponq, indicating multifractality.The same information is comprised in theq dependence ofh(q). The asymptotic values areh(q) → − ln(a)/ ln(2) forq → +∞ andh(q) → − ln(1− a)/ ln(2) for q → −∞. Theycorrespond to the scaling behavior of the largest and weakestfluctuations, respectively. Note thath(q) becomes indepen-dent ofq in the asymptotic limit, whileτ(q) approaches linearq dependences.

Figure 2 shows the MF-DFA fluctuation functionsFq(s)for the binomial multifractal model witha = 0.75. The re-sults for MF-DFA1 and MF-DFA4 are compared in parts (a)and (b). Fig. 2(c) shows the corresponding slopesh(q) forthree values ofa together with the exact results obtained fromEq. (20). The numerical results are in good agreement withEq. (20), showing that the MF-DFA correctly detects the mul-tifractal scaling exponents. Figures 2(d) and (e) show the cor-responding exponentsτ(q) = qh(q) − 1 [see Eq. (13)] andthe correspondingf(α) spectrum calculated fromh(q) usingthe modified Legendre transform (16). Both are also in goodagreement with Eq. (19). We have also checked that the re-sults for the binomial multifractal model remain unchangedifthe double summation technique [see Eq. (6)] is applied. Weobtain slopesh(q) = h(q) + 1 as expected in Eq. (7). Notethat there is no need to use this modification, except ifh(q) isclose to zero or has negative values.

C. Example 3: dyadic random cascade model with log-Poissondistribution

For another independent test of the MF-DFA, we employ analgorithm based on random cascades on wavelet dyadic treesproposed in [25] (see also [26]). This algorithm builds a ran-dom multifractal series by specifying its discrete waveletco-efficientscn,m, defined recursively,

c1,1 = 1, cn,2m−1 = Wcn−1,m, cn,2m = Wcn−1,m,

where n = 2, . . . , nmax (with N = 2nmax) and m =1, . . . , 2n−2. The values ofW are taken from a log-Poissondistribution,|W | = exp(P ln δ + γ), whereP is Poisson dis-tributed with〈P 〉 = λ. There are three independent parame-ters,λ, δ, andγ. Inverse wavelet transform is applied to createthe multifractal random seriesxk once the wavelet coefficientscn,m are known,

xk =

nmax∑

n=1

2n−1

∑

m=1

cn,mψn,m(k), (21)

whereψn,m(k) is a set of wavelets forming an orthonor-mal wavelet basis. Here, we employ the Haar wavelets,ψn,m(k) ≡ 2(n−nmax−1)/2ψ[2n−nmax−1k −m] with ψ(x) ≡1 for 0 < x ≤ 0.5,ψ(x) ≡ −1 for 0.5 < x ≤ 1 andψ(x) ≡ 0otherwise. For this model the multifractal scaling exponentsare given by [25]

4

τ(q) =λ(1 − δq) − γq

ln 2− 1, (22)

h(q) = [λ(1 − δq) − γq]/(q ln 2). (23)

Figure 3 shows the MF-DFA fluctuation functionsFq(s)for the dyadic random cascade model. The numerically deter-mined slopesh(q) for three sets of parameters are comparedwith the exact results obtained from Eq. (23) and the goodagreement shows that the MF-DFA correctly detects the mul-tifractal scaling exponents. Large deviations occur only forvery small moments (q < −10), indicating that the range ofqvalues should not exceed−10.

D. Example 4: uncorrelated multifractal series with power-lawdistribution function

The examples discussed in the previous three subsectionswere based on series involving long-range correlations. Inthepresent example we want to apply the MF-DFA method toan uncorrelated series, that nevertheless exhibits multifractalscaling behavior due to the broad distribution of its values. Wedenote byP (x) the probability density function of the valuesxk in the series. The distributionP (x) does not affect themultifractality of a series on large scaless, if all moments

〈|x|q〉 ≡

∫ ∞

−∞

|x|qP (x) dx (24)

are finite. Here we choose a (normalized) power-law proba-bility distribution function,

P (x) = αx−(α+1) for 1 ≤ x <∞ with α > 0 (25)

andP (x) = 0 for x < 1, where already the second momentdiverges ifα ≤ 2. In this case, the series exhibits multifractalscaling behavior on all scales. Note, that Eq. (25) becomesidentical to a Levy distribution of classα for large values ofx. The parameterα is not related to the Holder exponentα inEq. (15). The scaling properties of random walks with Levydistributed steps (Levy flights and Levy walks) have been an-alyzed in [27–29]. The multifractal nature of Levy processeshas been investigated in [30,31].

In order to derive the multifractal spectrum, let us considers uncorrelated random numbersrk, k = 1, . . . , s, distributedhomogeneously in the interval[0, 1]. Obviously, the typicalvalue of the minimum of the numbers,rmin(s) ≡ mins

k=1rk,will be rmin(s) = 1/s. It can be easily shown that the num-bersrk are transformed into numbersxk distributed accord-ing to the power-law probability distribution function (25) byrk → xk = r

−1/αk . Thus, the typical value of the maximum

of thexk will be xmax(s) ≡ maxsk=1xk = [rmin(s)]

−1/α =s1/α.

If α ≤ 2, the fluctuations of the profileY (i) [Eq. (1)] andthe corresponding DFA varianceF 2(s, ν) [Eq. (2)] will bedominated by the square of the largest valuex2

max(s) = s2/α

in the segment ofs numbers, since the second moment of the

distribution (25) diverges. Now the whole series consists ofNs ≡ int(N/s) segments of lengths and not just of onesegment. For some segmentsν, [F 2(s, ν)]1/2 is larger thanits typical valuexmax(s) = s1/α, since the maximum withinthe whole series of lengthN is xmax(N) = N1/α. In or-der to calculateFq(s) [Eq. (4)], we need to take into accountthe whole distributionPs(y) of the valuesy ≡ [F 2(s, ν)]1/2.Since each of the maxima in theNs segments correspondsto an actual numberxk and thesexk are random numbersfrom the power-law distribution (25), it becomes obvious, thatthe distribution of the maxima will have the same form, i. e.Ps(y) ∼ P (x = y) for largey. Small values ofy are ex-cluded because of the maximum procedure, but the largexk

values are very likely to be identical to the maxima of thecorresponding segments. Since the smallest maxima for seg-ments of lengths are of the order ofxmax(s) = s1/α, thelower cutoff for Ps(y) must be proportional tos1/α. Fromthe normalization condition

∫ ∞

As1/α Ps(y) dy = 1 (with anunimportant prefactorA < 1) we get

Ps(y) = Aααsy−(α+1). (26)

Now Fq(s) [Eq. (4)] can be calculated by integration fromthe minimum valueAs1/α of y ≡ [F 2(s, ν)]1/2 to the maxi-mum valueN1/α. Fors≪ N we obtain

Fq(s) ∼

[

∫ N1/α

As1/α

yqPs(y) dy

]1/q

∼∣

∣

∣AαsN q/α−1 −Aqsq/α

∣

∣

∣

1/q

∼

{

s1/q (q > α)s1/α (q < α)

.

Comparing with Eq. (5), we finally get

h(q) ∼

{

1/q (q > α)1/α (q ≤ α)

. (27)

Note thatτ(q) follows a linearq dependence,τ(q) = q/α− 1for q < α, while it is equal to zero forq > α accordingto Eq. (13). Hence, the series of uncorrelated power-law dis-tributed values has rather bi-fractal [31] instead of multifractalproperties. Sinceh(2) = 1/2 holds exactly for all values ofα, it is not possible to recognize the multifractality due to thebroad power-law distribution of the values if only the conven-tional DFA is applied. The second moment shows just theuncorrelated behavior of the values. In a very recent preprint[29] this behavior has been interpreted as a failure of the DFAand corresponding non-detrending methods for series with abroad distribution, and another method to determine the ex-ponent1/α has been proposed. We believe that a multifractaldescription with more than one exponent is required to charac-terize this kind of series, and thus any method calculating justone exponent will be insufficient for a full characterization.

Figure 4(a) shows the MF-DFA3 fluctuation functions forseries of independent random numbersxk ∈ [1,∞) dis-tributed according to Eq. (25) withα = 1. Since the scalingexponentsh(q) become very close to zero asymptotically for

5

large positive values ofq according to Eq. (27), we must usethe modified MF-DFA technique involving the double sum asdescribed in the last paragraph of Subsection II.A. Hence, forthis technical reason,Fq(s)/s is calculated instead ofFq(s).The corresponding slopesh(q) − 1 are identical toh(q), seeEq. (7). In Fig. 4(b) the slopesh(q) for series withα = 0.5,1.0, and 2.0 are compared with the theoretical result Eq. (27),and nice agreement is observed.

IV. COMPARISON OF THE MULTIFRACTALITY FORORIGINAL AND SHUFFLED SERIES

A. Distinguishing the two types of multifractality

As already mentioned in the introduction, two differenttypes of multifractality in time series can be distinguished.Both of them require a multitude of scaling exponents forsmall and large fluctuations. (i) Multifractality of a time se-ries can be due to a broad probability density function for thevalues of the time series, and (ii) multifractality can alsobedue to different long-range correlations for small and largefluctuations. The example discussed in Subsection III.D, theuncorrelated multifractal series with a power-law probabilitydensity function, is of type (i), while the examples discussedin Subsections III.A – III.C are of type (ii), where the prob-ability density function of the values is a regular distributionwith finite moments [32].

Now, we would like to distinguish between these two typesof multifractality. The most easy way to do so is by analyzingalso the corresponding randomly shuffled series. In the shuf-fling procedure the values are put into random order, and thusall correlations are destroyed. Hence the shuffled series frommultifractals of type (ii) will exhibit simple random behavior,hshuf(q) = 0.5, i. e. non-multifractal scaling like in Fig. 1(b).For multifractals of type (i), on the contrary, the originalh(q)dependence is not changed,h(q) = hshuf(q), since the mul-tifractality is due to the probability density, which is notaf-fected by the shuffling procedure. If both kinds of multifrac-tality are present in a given series, the shuffled series willshowweaker multifractality than the original one.

The effect of the shuffling procedure is illustrated inFig. 5(a), where the MF-DFA2 fluctuation functionsF shuf

−10 (s)andF shuf

10 (s) are shown for shuffled series for three of themultifractal examples taken from the previous section. Ran-dom behavior,hshuf(q) = 0.5, is observed for the series thatwere long-range correlated or generated from the dyadic ran-dom cascade model before the shuffling procedure [upper fourcurves in Fig. 5(a)]. In contrast, we observe the original multi-fractal scaling for the shuffled multifractal series with power-law probability density functionP (x) ∼ x−2 [lower twocurves in Fig. 5(a)]. Thehshuf(q) dependences are shown inFig. 6, which can be compared with the corresponding slopesshown in Figs. 1(d), 3(b), and 4(b). Thus, the fluctuationanalysis of the shuffled series,F shuf

q (s), directly indicates the

presence of type (i) multifractality, which is due to a broadprobability distribution, by deviations fromhshuf(q) = 0.5.

Now we want to determine directly the magnitude of the (ii)multifractality, which is due to correlations. For that purposewe compare the fluctuation function for the original series,Fq(s), with the result for the corresponding shuffled series,F shuf

q (s). Differences between these two fluctuation func-tions directly indicate the presence of correlations in theorig-inal series. These differences can be observed best in a plotof the ratioFq(s)/F

shufq (s) versuss [33]. Since the anoma-

lous scaling due to a broad probability density affectsFq(s)andF shuf

q (s) in the same way, only multifractality due to cor-relations will be observed inFq(s)/F

shufq (s). This is illus-

trated in Fig. 5(b) for the same three multifractal examplesasin Fig. 5(a). In order not to have increased statistical errors inthe results when considering the ratioFq(s)/F

shufq (s) instead

of Fq(s) itself, F shufq (s) can be calculated by averaging over

a large number of randomly shuffled series generated from thesame original series.

The scaling behavior of the ratio is

Fq(s)/Fshufq (s) ∼ sh(q)−hshuf(q) = shcor(q). (28)

Note thath(q) = hshuf(q) + hcor(q). If only distributionmultifractality [type (i)] is present,h(q) = hshuf(q) dependson q and hcor(q) = 0. On the other hand, deviations ofhcor(q) from zero indicate the presence of correlations, anda q dependence ofhcor(q) indicates correlation multifractal-ity [type (ii)]. If only correlation multifractality is present,hshuf(q) = 0.5 andh(q) = 0.5 + hcor(q). If both, distribu-tion multifractality and correlation multifractality arepresent,both,hshuf(q) andhcor(q) depend onq.

B. Significance of the results

In Figs. 1-6 we have shown the results of the MF-DFA forsingle configurations of long time series. Now we addressthe significance and accuracy of the MF-DFA results for shortseries. How much do the numerically determined exponentsh(q) vary from one configuration (sample series) to the next,and how close are the average values to the theoretical val-ues? In other words, how large are the statistical and system-atical deviations of exponents practically determined by theMF-DFA for finite series? These questions are particularlyimportant for short series, where the statistics is poor. Ifthevalues ofh(q) are determined inaccurately, the multifractalproperties will be reported inaccurately or even false conclu-sions on multifractal behavior might be drawn for monofractalseries.

To address the significance and accuracy of the MF-DFAresults we generate, for each of the three examples consideredalready in Fig. 5, 100 series of lengthN = 213 = 8192 andcalculateh(−10), h(+10), hshuf(−10), andhshuf(+10) foreach of these series. The corresponding histograms are shownin Fig. 7. For the long-range power-law correlated series with

6

H = 0.75 we find the following mean values and standarddeviations of the generalized Hurst exponents:

h(−10) = 0.80 ± 0.03, hshuf(−10) = 0.56 ± 0.02,

h(+10) = 0.72 ± 0.04, hshuf(+10) = 0.48 ± 0.02.

The mean values for the original series are rather close to, butnot identical to the theoretical valueH = 0.75. The meanvalue for q = −10 is about two standard deviations largerthan 0.75, while the value forq = +10 is slightly smaller.These deviations, though, certainlycannotindicate multifrac-tality, since we analyzed monofractal series. Instead, they aredue to the finite, random series, where parts of the series haveslightly larger and slightly smaller scaling exponent justbystatistical fluctuations. By considering negative values of qwe focus on the parts with small fluctuations, which are usu-ally described by a larger scaling exponent [23]. For positivevalues ofq we focus on the parts with large fluctuations usu-ally described by a smaller value ofh. Thus for short recordswe always expect a slight difference betweenh(−10) andh(+10) even if the series are monofractal. If this differenceis weak, one has to be very careful with conclusions aboutmultifractality. Practically it is always wise to compare withgenerated monofractal series with otherwise similar propertiesbefore drawing conclusions regarding the multifractalityof atime series. In addition to the statistical fluctuations of thehvalues, the averageh(−10) is usually determined slightly toolarge, whileh(+10) is slightly too small. The same behav-ior is obtained if the WTMM method is used instead of theMF-DFA, as we will show in Subsection V.C.

The same kind of difference is also observed for the aver-agehshuf(−10) andhshuf(+10) values. After all correlationshave been destroyed by the shuffling,hshuf = 0.5 is expectedsince the probability density is Gaussian with all finite mo-ments. The deviations fromhshuf = 0.5 we observe for thefinite random series are characteristic for monofractal seriesof this length (N = 8192). Only for the second moment weobtainhshuf(2) = 0.5 exactly if a sufficient number of seriesis considered.

For multifractal series generated from the dyadic randomcascade model, Fig. 7(c,d) shows the histograms of the scal-ing exponentsh(−10), h(+10),hshuf(−10), andhshuf(+10).Their averages and standard deviations,

h(−10) = 0.69 ± 0.04, hshuf(−10) = 0.57 ± 0.02,

h(+10) = 0.54 ± 0.02, hshuf(+10) = 0.48 ± 0.02.

have to be compared with the theoretical values from Eq. (23),h(−10) = 0.743 andh(+10) = 0.567. Surprisingly, themeanh(−10) is smaller than the theoretical value in this ex-ample, but for the meanh(+10) the deviation is similar tothe deviation observed for the monofractal data in the pre-vious example. Again, similar results are obtained with theWTMM method. For the shuffled series, the mean general-ized Hurst exponents are practically identical to those fortheshuffled monofractal series [the averagehshuf(−10) is larger

by half the standard deviation], and both are evidently consis-tent with monofractal uncorrelated behavior,h(q) = 0.5, asdiscussed above. Hence, the series from the dyadic randomcascade model show no signs of distribution multifractalityand are characterized by correlation multifractality only.

The histograms of the scaling exponents for our last exam-ple, the power-law distributed random numbers withP (x) ∼x−2, are shown in Fig. 7(e,f). The corresponding mean valuesand standard deviations,

h(−10) = 1.24 ± 0.09, hshuf(−10) = 1.26 ± 0.09,

h(+10) = 0.11 ± 0.03, hshuf(+10) = 0.11 ± 0.04.

show obviously no differences between original and shuffledseries as expected for uncorrelated series. This indicatesthatthe multifractality is due to the broad probability densityfunc-tion only. The values have to be compared withh(−10) = 1andh(+10) = 0.1 from Eq. (27). As usual, the average valueof h(−10) is too large because we analyzed short series.

V. COMPARISON TO THE WAVELET TRANSFORMMODULUS MAXIMA METHOD

A. Brief description of the wavelet transform modulus maximamethod

The wavelet transform modulus maxima (WTMM) method[21] is a well-known method to investigate the multifractalscaling properties of fractal and self-affine objects in thepres-ence of nonstationarities. It is an application of the wavelettransform with continuous basis functions. One defines thewavelet-transform of a seriesxk of lengthN by

W (n, s) =1

s

N∑

k=1

xk ψ[(k − n)/s]. (29)

Note that in this case the seriesxk are analyzed directly in-stead of the profileY (i) defined in Eq. (1). Here, the functionψ(x) is the analyzing wavelet ands is, as above, the scaleparameter. The wavelet is chosen orthogonal to the possi-ble trend. If the trend can be represented by a polynomial,a good choice forψ(x) is them-th derivative of a Gaussian,ψ(m)(x) = dm(e−x2/2)/dxm. This way, the transform elimi-nates trends up to(m− 1)th order.

Now, instead of averaging over all values ofW (n, s), oneaverages, within the modulo-maxima method, only the lo-cal maxima of|W (n, s)|. First, one determines for a givenscales, the positionsni of the local maxima of|W (n, s)|as function ofn, so that|W (ni − 1, s)| < |W (ni, s)| ≥|W (ni + 1, s)| for i = 1, . . . , imax. Then one sums up theqth power of these maxima,

Z(q, s) =

imax∑

i=1

|W (ni, s)|q. (30)

7

The reason for this maxima procedure is that the absolutewavelet coefficients|W (n, s)| can become arbitrarily small.The analyzing waveletψ(x) must always have positive valuesfor somex and negative values for otherx, since it has to beorthogonal to possible constant trends. Hence there are alwayspositive and negative terms in the sum (29), and these termsmight cancel. If that happens,|W (n, s)| can become closeto zero. Since such small terms would spoil the calculationof negative moments in Eq. (30), they have to be eliminatedby the maxima procedure. In the MF-DFA, the calculation ofthe variancesF 2(s, ν) in Eq. (2), i. e. the deviations from thefits, involves only positive terms under the summation. Thevariances cannot become arbitrarily small, and hence no max-imum procedure is required for series with compact support.

In addition, the MF-DFA variances will always increase ifthe segment lengths is increased, because the fit will alwaysbe worse for a longer segment. In the WTMM method, incontrast, the absolute wavelet coefficients|W (n, s)| need notincrease with increasing scales, even if only the local maximaare considered. The values|W (n, s)| might become smallerfor increasings since just more (positive and negative) termsare included in the summation (29), and these might canceleven better. Thus, an additional supremum procedure has beenintroduced in the WTMM method in order to keep the depen-dence ofZ(q, s) on s monotonous: If, for a given scales, amaximum at a certain positionni happens to be smaller than amaximum atn′

i ≈ ni for a lower scales′ < s, thenW (ni, s)is replaced byW (n′

i, s′) in Eq. (30). There is no need for such

a supremum procedure in the MF-DFA.Often, scaling behavior is observed forZ(q, s), and scal-

ing exponentsτ (q) can be defined that describe howZ(q, s)scales withs,

Z(q, s) ∼ sτ(q). (31)

The exponentsτ(q) characterize the multifractal properties ofthe series under investigation, and theoretically they areiden-tical to theτ(q) defined in Eq. (12) [21] and related toh(q) inEq. (13).

B. Examples for series with nonstationarities

Since the WTMM method has been developed to analyzemultifractal series with nonstationarities, such as trends orspikes, we will compare its performance with the performanceof the MF-DFA for such nonstationary series. In Fig. 8 theMF-DFA fluctuation functionFq(s) and its scaling behaviorare compared with the rescaled WTMM partition sumZ(q, s)for the binomial multifractal described in Subsection III.B.To test the detrending capability of both methods, we haveadded linear as well as quadratic trends to the generated mul-tifractal series. The trends are removed by both methods, ifa sufficiently high order of detrending is employed. The de-viations from the theoretical values of the scaling exponentsh(q) [given by Eq. (20)] are of similar size for the MF-DFA

and the WTMM method. Thus, the detrending capability andthe accuracy of both methods is equivalent.

We also obtain similar results for a monofractal long-rangecorrelated series with additional spikes (outliers) that consistof large random numbers and replace a small fraction of theoriginal series in randomly chosen positions. The spikes leadto multifractality on small scaless, while the series remainsmonofractal on large scales. Thus, the effects of the spikesare eliminated neither by the WTMM method nor by the MF-DFA, but both methods again give rather equivalent results.

C. Significance of the results

The last problem we address is a comparison of the signifi-cance of the results obtained by the MF-DFA and the WTMMmethod. The significance of the MF-DFA results has alreadybeen discussed in detail in Subsection IV.B. Here we willcompare the significance of both methods for short and longseries.

We begin with the significance of the results for randomseries involving neither correlations nor a broad distribution[as in Fig. 1(b)]. Fig. 9 shows the distribution of the mul-tifractal Hurst exponentsh(−10) andh(+10) calculated bythe MF-DFA as well as by the WTMM using the relationh(q) = [τ (q) + 1]/q based on Eq. (13). Similar to the resultspresented in Fig. 7, we have analyzed 100 generated series ofuncorrelated random numbers. In addition, we compare theresults for the (relatively short) series lengthN = 213 = 8192and forN = 216 = 65532. Ideally, both,h(−10) andh(+10), should be equal to the Hurst exponent of the uncorre-lated monofractal series,H = 0.5. The histograms show thatsimilar deviations as well as remarkable fluctuations of theexponents occur for both methods, as discussed in SubsectionIV.B for the MF-DFA. We find the following mean values andstandard deviations,

h(−10) =

0.55 ± 0.03 for MF-DFA (N = 8k)0.52 ± 0.02 for MF-DFA (N = 64k)0.58 ± 0.05 for WTMM (N = 8k)0.56 ± 0.03 for WTMM (N = 64k)

and h(+10) =

0.49 ± 0.03 for MF-DFA (N = 8k)0.49 ± 0.02 for MF-DFA (N = 64k)0.46 ± 0.04 for WTMM (N = 8k)0.48 ± 0.02 for WTMM (N = 64k)

.

As already discussed in Subsection IV.B, the deviations of theaverageh(q) values fromH = 0.5 do not indicate multifrac-tality. For the WTMM method and short series, one has to bevery careful in order not to draw false conclusions from resultslike h(−10) = 0.58 andh(+10) = 0.46. The correspondingresults of the MF-DFA are closer to the theoretical value.

Figure 10 shows the distribution of the multifractal scalingexponentsh(−10) andh(+10) calculated for generated mul-tifractal series from the binomial model witha = 0.75 de-scribed in Subsection III.B. Like for Fig. 9, 100 generated se-ries have been analyzed for each of the histograms. Now, the

8

differences between the distributions ofh(−10) andh(+10)are much larger, indicating multifractality. We find

h(−10) =

1.88 ± 0.06 for MF-DFA (N = 8k)1.89 ± 0.03 for MF-DFA (N = 64k)1.86 ± 0.05 for WTMM (N = 8k)1.89 ± 0.02 for WTMM (N = 64k)

and h(+10) =

0.50 ± 0.02 for MF-DFA (N = 8k)0.51 ± 0.01 for MF-DFA (N = 64k)0.46 ± 0.01 for WTMM (N = 8k)0.47 ± 0.01 for WTMM (N = 64k)

.

These values must be compared with the theoretical valuesh(−10) = 1.90 andh(+10) = 0.515 from Eq. (20). Again,the MF-DFA results turn out to be slightly more significantthan the WTMM results. The MF-DFA seems to have slightadvantages for negativeq values and short series, but in theother cases the results of the two methods are rather equiva-lent. Besides that, the main advantage of the MF-DFA methodcompared with the WTMM method lies in the simplicity ofthe MF-DFA method.

VI. CONCLUSION

We have generalized the DFA, widely recognized as amethod to analyze the (mono-) fractal scaling properties ofnonstationary time series. The MF-DFA method allows a re-liable multifractal characterization of multifractal nonstation-ary time series. The implementation of the new method is notmore difficult than that of the conventional DFA, since justone additional step, aq dependent averaging procedure, is re-quired. We have shown for stationary signals that the gen-eralized (multifractal) scaling exponenth(q) for series withcompact support is directly related to the exponentτ(q) ofthe standard partition function-based multifractal formalism.Further, we have shown in several examples that the MF-DFAmethod can reliably determine the multifractal scaling behav-ior of the time series, similar to the WTMM method which isa more complicated procedure for this purpose. For short se-ries and negative moments, the significance of the results forthe MF-DFA seems to be slightly better than for the WTMMmethod.

Contrary to the WTMM method, the MF-DFA method asdescribed in Subsection II.A requires series of compact sup-port, because the averaging procedure in Eq. (4) will onlywork if F 2(s, ν) > 0 for all segmentsν. Although mosttime series will fulfill this prerequisite, it can be overcome bya modification of the MF-DFA technique in order to analyzedata with fractal support: We restrict the sum in Eq. (4) to thelocal maxima, i. e. to those termsF 2(s, ν) that are larger thanthe termsF 2(s, ν − 1) andF 2(s, ν + 1) for the neighboringsegments. By this restriction all termsF 2(s, ν) that are zeroor very close to zero will be disregarded, and series with frac-tal support can be analyzed. The procedure reminds slightlyof the modulus maxima procedure in the WTMM method (see

Subsection V.A). There is no need, though, to employ a con-tinuously sliding window or to calculate the supremum overall lower scales for the MF-DFA, since the variancesF 2(s, ν),which are determined by the deviations from the fit, will al-ways increase when the segment sizes is increased. In themaxima MF-DFA procedure the generalized Hurst exponenth(q) defined in Eq. (5) will depend onq and even divergefor q → 0 for monofractal series with non-compact support.Thus, it is more appropriate to consider the scaling exponentτ(q), calculating

∑

F 2(s,ν−1)<F 2(s,ν)≥F 2(s,ν+1)

[F 2(s, ν)]q/2 ∼ sτ(q). (32)

This extended MF-DFA procedure will also be applicable fordata with fractal support.

In a later work we will apply the MF-DFA method to arange of physiological and meteorological data.

Acknowledgements

We would like to thank Yosef Ashkenazy for useful discus-sions and the Deutsche Akademischer Austauschdienst, theDeutsche Forschungsgemeinschaft, the German Israeli Foun-dation, and the Minerva Foundation for financial support.

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[11] E. Koscielny-Bunde, A. Bunde, S. Havlin, H.E. Roman, Y.Gol-dreich, and H.-J. Schellnhuber, Phys. Rev. Lett.81, 729 (1998).K. Ivanova and M. Ausloos, Physica A274, 349 (1999); P.Talkner and R.O. Weber, Phys. Rev. E62, 150 (2000).

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[22] The value ofh(0), which corresponds to the limith(q) for q →

0 for time series with compact support, cannot be determineddirectly using the averaging procedure in Eq. (4) because ofthediverging exponent. Instead, a logarithmic averaging procedurehas to be employed,

F0(s) ≡ exp

{

1

4Ns

2Ns∑

ν=1

ln[

F 2(s, ν)]

}

∼ sh(0). (33)

Note thath(0) cannot be defined for time series with fractalsupport, whereh(q) diverges forq → 0.

[23] For the maximum scales = N the fluctuation functionFq(s)is independent ofq, since the sum in Eq. (4) runs over only twoidentical segments (Ns ≡ [N/s] = 1). For smaller scaless ≪

N the averaging procedure runs over several segments, and theaverage valueFq(s) will be dominated by theF 2(s, ν) fromthe segments with small (large) fluctuations ifq < 0 (q > 0).Thus, fors ≪ N , Fq(s) with q < 0 will be smaller thanFq(s)with q > 0, while both become equal fors = N . Hence, if weassume an homogeneous scaling behavior ofFq(s) followingEq. (5), the slopeh(q) in a log-log plot ofFq(s) with q < 0

versuss must be larger than the corresponding slope forFq(s)with q > 0. Thus,h(q) for q < 0 will usually be larger thanh(q) for q > 0.

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[26] Y. Ashkenazy, S. Havlin. P. Ch. Ivanov, C. K. Peng, V. Schulte-Frohlinde, and H. E. Stanley, cond-mat/0111396 (unpublished).

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[29] N. Scafetta and P. Grigolini, cond-mat/0202008 (unpublished).[30] S. Jaffard, Probab. Theory Rel.114, 207 (1999).[31] H. Nakao, Phys. Lett. A266, 282 (2000).[32] The example of the binomial multifractal series (Subsection

III.B) can also show multifractality due to a broad probabil-ity density function for the valuesxk. If the parametera ischosen to be very close to one or if very long series are con-sidered, corresponding to large values ofnmax, the minimumvalue in the series,(1 − a)nmax , will be very small comparedwith the maximum valueanmax [see Eq. (18)]. In this case thelog-binomial probability density function will become broad,approaching a log-normal form. Since the scaling behavior ofuncorrelated log-normal distributed series corresponds to themultifractal scaling behavior observed in the example of uncor-related power-law distributed series withα = 2 (see SubsectionIII.D), distribution multifractality [type (i)] will occur in addi-tion to the correlation multifractality [type (ii)]. For the serieswith a = 0.75 andN = 8192 (nmax = 13) considered in Sub-section III.B, we observe only type (ii) multifractality causedby long-range correlations.

[33] The ratioFq(s)/F shufq (s) can also be used to eliminate sys-

tematic deviations from the expected power-law scaling behav-ior that occur at very small scaless < 10 especially for smallvalues ofq (see Figs. 1-3). A similar procedure has alreadybeen introduced for the conventional DFA recently (see Sub-section 3.1 of [3]). Since the deviations are systematic fortheMF-DFA method, they occur in both,Fq(s) andF shuf

q (s), andthey should cancel in the ratio.

10

10-4

10-2

(b)

Fq(

s )

0.250.500.751.00 (d)

h (q )

q

10-5

10-3

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Fq(

s )

101

102

103

104

10-3

10-1

(c)

Fq(

s )

s

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10 (e)

ττ (q )

FIG. 1. The MF-DFA fluctuation functionsFq(s) are shownversus the scales in log-log plots for (a) long-range correlatedmonofractal series withH = 0.75, (b) uncorrelated random serieswith H = 0.5 (white noise), and (c) long-range anti-correlated se-ries withH = 0.25. The different symbols correspond to the differ-ent values of the exponentq in the generalized averaging procedure,q = −10 (2), −2 (◦), −0.2 (△), +0.2 (▽), +2 (3), andq = +10(+). MF-DFA2 has been employed, and the curves have been shiftedby multiple factors of4 for clarity. The straight dashed lines havethe corresponding slopesH and are shown for comparison. Part (d)shows theq dependence of the asymptotic scaling exponenth(q) de-termined by fits in the regime200 < s < 5000 for H = 0.25 (2),0.5 (◦), and0.75 (△). The very weak dependence onq is consis-tent with monofractal scaling. In part (e) theq dependence ofτ (q),τ (q) = qh(q) − 1, is shown.

10-8

10-6

10-4

10-2 (a)

q = 10 q = 2 q = 0.2 q = -0.2 q = -2 q = -10

Fq(

s )

101

102

103

104

10-9

10-7

10-5

10-3 (b)

Fq(

s )

s

01234 (c)

a = 0.90, MF-DFA1 a = 0.75, MF-DFA1 a = 0.75, MF-DFA4 a = 0.60, MF-DFA1

h (q )

q-20 -10 0 10 20-60

-30

0 (d)

ττ (q )

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0 (e)

f (αα)

αα

FIG. 2. The MF-DFA fluctuation functionsFq(s) are shown ver-sus the scales in log-log plots for the binomial multifractal modelwith a = 0.75 (a) for MF-DFA1 and (b) for MF-DFA4. The symbolsare the same as for Fig. 1. The straight dashed lines have the corre-sponding theoretical slopesh(−10) = 1.90 andh(+10) = 0.515and are shown for comparison. In part (c) theq dependence of thegeneralized Hurst exponenth(q) determined by fits in the regime50 < s < 500 is shown for MF-DFA1 anda = 0.9 (△), a = 0.75(◦), anda = 0.6 (2), as well as for MF-DFA4 anda = 0.75 (▽).Parts (d) and (e) show the corresponding exponentsτ (q) and the cor-responding singularity spectrumf(α) for a = 0.75 determined bythe modified Legendre transform (16), respectively. The lines are thetheoretical values obtained from Eq. (20).

11

101

102

103

104

10-5

10-3

10-1

(a)

Fq(

s )

s

0.6

0.8

1.0

1.2

ττ (q )

(b)

λ = 1, δ = 0.90, γ = -δ4/2

λ = ln 2, δ = 0.90, γ = -δ4/2

λ = 1, δ = 0.93, γ = -2δ4/5h (

q )

-20 -10 0 10 20-20

-10

0

10 (c)

q

FIG. 3. (a) The MF-DFA2 fluctuation functionsFq(s) are shownversus the scales in a log-log plot for the dyadic random cascademodel with log-Poisson distribution with parametersλ = 1, δ = 0.9,and γ = −δ4/2. The symbols are the same as for Fig. 1. Thedashed straight lines have the theoretical slopesh(−10) = 0.743and h(+10) = 0.567 and are shown for comparison. (b) Theqdependence of the generalized Hurst exponenth(q) determined byfits is shown for MF-DFA2 and different parameters (see legend).The lines are the theoretical values obtained from Eq. (23).In part(c) τ (q) = qh(q) − 1 is shown.

101

102

103

10410

-3

10-1

101

103

(a)

Fq(

s )

s

0

1

2 α = 0.5

α = 1.0

α = 2.0

(b)

h (q )

q-20 -10 0 10 20-40-30-20-10

0 (c)

ττ(q

)

FIG. 4. (a) The modified and rescaled MF-DFA3 fluctuation func-tions Fq(s)/s[∼= Fq(s)] are shown versus the scales in a log-logplot for a series of independent numbers with a power-law probabil-ity density distributionP (x) ∼ x−(α+1) with α = 1. The symbolsare the same as for Fig. 1. The straight dashed lines have the cor-responding theoretical slopesh(−10) = 1 andh(+10) = 0.1 andare shown for comparison. (b) Theq dependence of the generalizedHurst exponenth(q) = h(q)− 1 determined by fits on large scalessis shown for MF-DFA3 andα = 0.5 (2), 1.0 (◦), and2.0 (△). Thelines are the theoretical values obtained from Eq. (27). In part (c) thecorrespondingτ (q) is shown. The broad distribution of the valuesleads to multifractality (bi-fractality) in all three cases.

101

102

103

10410

-6

10-4

10-2

100

s

(b)

F

±10(

s) /

F±1

0shuf( s

)

s

101

102

103

10410

-12

10-7

10-2

103

(a)

F±1

0shuf( s

)

FIG. 5. (a) The MF-DFA fluctuation functionsF shuf−10 (s) (open

symbols) andF shuf10 (s) (filled symbols) are shown versus the scales

in a log-log plot for randomly shuffled series of long-range correlatedseries withH = 0.75 (2), for the dyadic random cascade model withlog-Poisson distribution with parametersλ = ln 1, δ = 0.9, andγ = −δ4/2 (◦), and for power-law distributed random numbersxk

with P (x) ∼ x−2 (△). The correlations and the multifractality aredestroyed by the shuffling procedure for the first two series,but forthe broadly distributed random numbers the multifractality remains.The dashed line has the slopeH = 0.5 and is shown for comparison.(b) The ratios of the MF-DFA2 fluctuation functionsFq(s) of theoriginal series and the MF-DFA2 fluctuation functionsF shuf

q (s) ofthe randomly shuffled series are shown versuss for the same mod-els as in (a), correlated series (▽), dyadic random cascade model(3), and power-law distributed random numbers (+ for q = −10,× for q = +10). The deviations from the slopehcor = 0 indicatelong-range correlations.

12

0.5

1.0(a)

h shuf( q

)

q-20 -10 0 10 20

-20

-10

0

10 (b)

ττ shuf( q

)

FIG. 6. (a) Theq dependence of the slopeshshuf(q) of the samemodels as in Fig. 5(a). The lines indicate the theoretical values:H = 0.5 for shuffled data with narrow distribution, andh(q) fromEq. (27) for the series of numbers with a power-law probability den-sity distribution. The symbols are the same as in Fig. 5. Part(b)showsτshuf(q) = qhshuf(q) − 1.

0

20

40

60(a)

original shuffled

# of

ser

ies

(b)

0.4 0.5 0.6 0.7 0.80

20

40

60(c)

# of

ser

ies

0.4 0.5 0.6 0.7 0.8

(d)

1.0 1.1 1.2 1.3 1.4 1.50

20

40

60(e)

# of

ser

ies

h(−−10), hshuf

(−−10)0.0 0.1 0.2 0.3 0.4

(f)

h(+10), hshuf

(+10)

FIG. 7. (a) Histograms of the generalized Hurst exponentsh(−10) (black bars) andhshuf(−10) (grey bars) for 100 gener-ated monofractal series withH = 0.75. The exponents havebeen fitted to MF-DFA2 fluctuation functions in the scaling range400 < s < 2000. (b) Same as (a), but forh(+10) andhshuf(+10). (c,d) Same as (a,b), but for the dyadic random cas-cade model with log-Poisson distribution and parametersλ = ln 1,δ = 0.9, andγ = −δ4/2. The corresponding theoretical values areh(−10) = 0.743 andh(+10) = 0.567 from Eq. (23) for the originalseries andhshuf = 0.5 for the shuffled series. From the histogramof h(+10) it would be hard to draw any conclusions regarding mul-tifractality. (e,f) Same as (a,b), but for power-law distributed randomnumbers with the distributionP (x) ∼ x−2. The corresponding the-oretical values from Eq. (27) areh(−10) = 1 andh(+10) = 0.1for the original and the shuffled series. The length of all series isL = 8192. The figure shows that correlations and multifractality dueto correlations (a-d) are eliminated by the shuffling procedure, whilemultifractality due to a broad distribution (e,f) remains.It further al-lows to estimate the statistical fluctuations in the scalingexponentsh(q) determined by the MF-DFA for monofractal (a,b), correlationmultifractal (c,d) and distribution multifractal (e,f) series.

10-9

10-7

10-5

10-3 (a)

q = 10 q = 2 q = 0.2 q = -0.2 q = -2 q = -10

Fq(

s )

101

102

103

-6-4-202 (b)

log 10

(s Z

(q,s

))/q

s

0.5

1.0

1.5

2.0

2.5 (c)

linear trend, MF-DFA2 linear trend, WTMM quadratic trend, MF-DFA2 quadratic trend, WTMM quadratic trend, MF-DFA3h (

q )

q-20 -10 0 10 20

-40

-20

0(d)

ττ(q

)

13

FIG. 8. (a) The MF-DFA2 fluctuation functionsFq(s) are shownversus the scales in log-log plots for the binomial multifractal modelwith a = 0.75 and an additional linear trendxk → xk+k/500L. (b)The scaled WTMM partition functions[sZ(q, s)]1/q are shown forthe same series and the same values ofq. The symbols are the sameas for Fig. 1. (c) Theq dependence of the generalized Hurst expo-nenth(q) for the generated series with linear trend for the MF-DFA2(2) and the second order WTMM (◦) methods. Corresponding re-sults for a binomial multifractal with an additional quadratic trendare also included for MF-DFA2 (△) and second order WTMM (▽)methods. The quadratic trend causes deviations from the line indicat-ing the theoretical values [obtained from Eq. (20)], which disappearif MF-DFA3 is employed (×). The values ofh(q) have been deter-mined by fits toFq(s) andZ(q, s) in the regime50 < s < 2000.The relationh(q) = [τ(q) + 1]/q from Eq. (13) has been used toconvert the exponentτ(q) from Eq. (31) intoh(q). (d) Theq depen-dence ofτ (q).

0

20

40

60 (a)

MF-DFA WTMM

# of

ser

ies

(b)

0.5 0.6 0.70

20

40

60 (c)

# of

ser

ies

h(−−10)0.4 0.5 0.6

(d)

h(+10)

FIG. 9. (a) Histograms of the generalized Hurst exponentsh(−10) for 100 random uncorrelated series withH = 0.5. The ex-ponents have been fitted to MF-DFA2 fluctuation functionsF

−10(s)(black bars) in the scaling range40 < s < 2000 and to WTMMresultsZ(−10, s) (grey bars) in the scaling range5 < s < 250. Thelength of the series isL = 8192. The relationh(q) = [τ (q) + 1]/qfrom Eq. (13) has been used to convert the exponentτ(q) fromEq. (31) intoh(q). (b) Same as (a), but forh(+10). (c,d) Sameas (a,b), but for longer series (L = 65536), where statistical fluc-tuations are reduced. The figure shows that the MF-DFA seems togive slightly more reliable results than the WTMM method forshortseries and negative moments (q = −10), see (a). In the other cases,the performance of both methods is similar.

0

20

40

60

80

100(a) MF-DFA

WTMM

# of

ser

ies

(b)

1.7 1.8 1.9 2.00

20

40

60

80

100(c)

# of

ser

ies

h(−−10)0.4 0.5 0.6

(d)

h(+10)

FIG. 10. Same as Fig. 9 for the binomial model witha = 0.75.The theoretical values of the generalized Hurst exponents areh(−10) = 1.90 and h(+10) = 0.515 according to Eq. (20).The figure shows that our findings regarding the performance of theMF-DFA and WTMM methods for uncorrelated monofractal seriesin Fig. 9 also hold for multifractal series.

14