Table Of Contentepl draft
Multifractal Height Cross-Correlation Analysis: A New Method
for Analyzing Long-Range Cross-Correlations
Ladislav Kristoufek1,2
1 Institute of Economic Studies, Charles University, Opletalova 26, Prague, CZ-110 00, Czech Republic
2 Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod Vodarenskou vezi
4, Prague, CZ-182 08, Czech Republic
2
1 PACS 89.75.-k–Complex systems
0 PACS 05.45.Tp–Time series analysis in nonlinear dynamics
2 PACS 89.65.Gh–Econophysics
n
Abstract – We introduce a new method for detection of long-range cross-correlations and mul-
a
tifractality – multifractal height cross-correlation analysis (MF-HXA) – based on scaling of qth
J
order covariances. MF-HXA is a bivariate generalization of the height-height correlation analysis
9
of Barabasi & Vicsek [Barabasi, A.L., Vicsek, T.: Multifractality of self-affine fractals, Physical
1
Review A 44(4), 1991]. The method can be used to analyze long-range cross-correlations and
] multifractality between two simultaneously recorded series. We illustrate a power of the method
T on both simulated and real-world time series.
S
.
n
i
f
-
q
The research of long-range dependence and multifrac-
[
tality has been growing significantly in recent years with
2 applicationtoawiderangeofdisciplines[1–10]. Recently, ν T(cid:88)∗/ν q q
vthe examination of long-range cross-correlations has be- Kxy,q(τ)= T∗ |∆τXtYt|2 ≡(cid:104)|∆τXtYt|2(cid:105) (1)
3 t=1
7come of interest as it provides additional information
4about the examined processes. Carbone [11] generalized where time interval τ generally ranges between ν =
3the detrending moving average (DMA) method for higher τmin,...,τmax. ScalingrelationshipbetweenKxy,q(τ)and
1.dimensions. Podobnik & Stanley [12] adjusted the de- the generalized bivariate Hurst exponent Hxy(q) is ob-
0trended fluctuation analysis for two time series and intro- tained as
2duced the detrended cross-correlation analysis (DCCA).
1Zhou [13] further generalized the method and introduced Kxy,q(τ)∝τqHxy(q). (2)
:
vthemultifractaldetrendedcross-correlationanalysis(MF- For q = 2, the method can be used for the detection
XiDXA). Jiang & Zhou [14] then implemented moving aver- of long-range cross-correlations solely and we call it the
age filtering to MF-DXA algorithm creating MF-X-DMA. height cross-correlation analysis (HXA). Obviously, MF-
r
aIn this paper, we introduce two new methods for an HXA reduces to the height-height correlation analysis of
analysisoflong-rangecross-correlations–themultifractal Barabasi et al. [15] for X =Y . Note that it makes sense
t t
height cross-correlation analysis (MF-HXA) and its spe- to analyze the scaling according to Eq. 2 only for de-
cial case of the height cross-correlation analysis (HXA). trended series X and Y and only for q > 0 [5]. A type
t t
To analyze long-range cross-correlations, we generalize of detrending can generally take various forms – polyno-
the q-th order height-height correlation function for two mial, moving averages and other filtering methods – and
simultaneously recorded series. Let us consider two series is applied for each time resolution ν separately.
X and Y with time resolution ν and t = ν,2ν,...,ν(cid:98)T(cid:99), The bivariate Hurst exponent 0<H (2)<1 has simi-
t t ν xy
where (cid:98)(cid:99) is a lower integer sign. For better legibility, larpropertiesandinterpretationasaunivariateHurstex-
we denote T∗ = ν(cid:98)T(cid:99), which varies with ν, and we ponent. ForH (2)>0.5,theseriesarecross-persistentso
ν xy
write the τ-order difference as ∆ X ≡ X −X and that a positive (a negative) value of ∆X ∆Y is more sta-
τ t t+τ t t t
∆ X Y ≡ ∆ X ∆ Y . Height-height covariance function tisticallyprobabletobefollowedbyanotherpositive(neg-
τ t t τ t τ t
is then defined as ative)valueof∆X ∆Y . ConverselyforH (2)<0.5,
t+1 t+1 xy
p-1
Ladislav Kristoufek1,2
theseriesarecross-antipersistentsothatapositive(aneg- α(q)= Hx(q)+Hy(q) ⇒H (q)= Hx(q)+Hy(q)
ative)valueof∆X ∆Y ismorestatisticallyprobabletobe 2 xy 2
t t
(10)
followed by a negative (a positive) value of ∆X ∆Y .
t+1 t+1
Weshowthattheserelationshipsareindeedtrueforar-
Notethateventwopairwiseuncorrelatedprocessescanbe
cross-persistent1. tificially generated processes later in the text. Therefore,
we need to distinguish between two types of long-range
The expected values of the bivariate Hurst exponents
cross-correlations: (i)long-rangecross-correlationscaused
have been partly discussed in [12–14]. It has been shown
by long-range dependence of the separate series, and (ii)
that
long-range cross-correlations caused by scaling of covari-
H (q)= Hx(q)+Hy(q) (3) ances between |∆τXt|q2 and |∆τYt|q2.
xy 2 In order to test validity of the method, we present re-
sults for several artificial series. In the analysis, we ap-
for all q > 0 for pairwise uncorrelated and correlated
ply MF-HXA with changing τ = 5,...,100 and fixed
processes. We present some new insights into this rela- max
τ =1. Inturn,weobtainthe99%jackknifeconfidence
tion. To better understand the behavior of the bivariate min
intervals under an assumption of a normally distributed
Hurstexponent, weuseastandardmultifractalformalism
Hurstexponentwithanunknownvariance. Theestimated
[16]. Consider processes X and Y are multifractal with
t t
Hurst exponent is then taken as a mean of the exponents
generalized Hurst exponents H (q) and H (q) so that
x y
based on the various τ . This way, we can comment on
max
theresultswithstatisticalpower[5]. Intheprocedure,we
(cid:104)|∆ X |q(cid:105)∝τqHx(q) (4)
τ t
apply filtering of a constant trend. We now turn to the
(cid:104)|∆ Y |q(cid:105)∝τqHy(q). (5) artificial processes.
τ t
First, we start with the Mandelbrot’s binomial multi-
In the same way, we can write the joint scaling of two
fractal (MBM) measures [17,18]. Let m > 0, m > 0
series (compare with Eq. 1) as 0 1
and m +m = 1 and let us work on interval [0,1]. In
0 1
(cid:104)|∆τXtYt|q2(cid:105)∝τqHxy(q). (6) ttherevafilrsst[0s,t1a]gaen,dth[e1,1m],aswshoefn1thiesrediivsidtehde mintaosstwmo sinubtihne-
2 2 0
first subinterval and the mass m in the second one. In
Usingthedefinitionofcovariance, theleftpartofEq. 6 1
the following stage, each subinterval is again halved and
can be rewritten as
its mass is divided between the smaller subintervals in a
q q q q q
(cid:104)|∆τXtYt|2(cid:105)=(cid:104)|∆τXt|2(cid:105)(cid:104)|∆τXt|2(cid:105)+cov(|∆τXt|2,|∆τYt|2). ratio m : m . After k stages, we obtain a series of 2k
(7) 0 1
values. Note that the values are deterministic as there
From Eqs. 4 and 5, the first part of the right-hand side
is no noise added in the simplest version of the method.
of Eq. 7 implies
For an interval [z,z + 2−k], the value µ has a value of
(cid:104)|∆τXt|q2(cid:105)(cid:104)|∆τYt|q2(cid:105)∝τqHx(q)+2Hy(q) (8) µ[z,z+2−k]=mk0ϕ0mk1ϕ1, where ϕ0 and ϕ1 stand for the
relative frequencies of numbers 0 and 1 in a binary devel-
which corresponds to Eq. 3. Therefore, the crucial part opment of 2kz, respectively. We construct two series with
of long-range cross-correlations and multifractality is the m0 = 0.3,0.4 and k = 16. Results are presented in Fig.
scaling of covariances between |∆τXt|q2 and |∆τYt|q2 with 1a,showingthatthebivariateHurstexponentHxy(q)does
varying τ. Consider now a scaling exponent α(q) and a not deviate significantly from the average value of Hx(q)
scaling relationship and Hy(q) even though the analyzed series are strongly
correlated.
cov(|∆τXt|q2,|∆τYt|q2)∝τqα(q). (9) Second,weapplyMF-HXAonARFIMAprocesseswith
correlatednoiseterms. ARFIMA(0,d,0)processisdefined
Thisleadsustothreesimpleimplications. Ifcovariances as x = (cid:80)∞ a (d)x +ε where 0 < d < 0.5 is a free
t i=1 i t−i t
do not scale with τ, then Eq. 3 holds. If the covariances parameter, relatedtoHurstexponentasH =d+0.5, and
scale with τ, the other two are as follows: a (d) = dΓ(i−d)/(Γ(1−d)Γ(1+i)). We simulate long-
i
rangedependentseriesoflength104. Todescribeinfluence
H (q)+H (q) H (q)+H (q)
α(q)(cid:54)= x y ⇒H (q)(cid:54)= x y ofthecorrelationsonMF-HXAestimates,wegeneratese-
2 xy 2
ries with correlated noise ε ∼ N(0,1) and five cases are
t
1For example, let us have pairwise uncorrelated processes Xt investigated – correlation coefficients for the noise terms
and Yt following fractional Gaussian noise with Hx(2) = 0.9 and are set to 1, 0.5, 0, -0.5 and -1. The results are shown
Hy(2) = 0.7 and thus Hxy(2) = 0.8. Even though the two pro- in Fig. 1b-f. The estimates of H (q) are not signifi-
xy
cesses are independently generated (and thus uncorrelated), they
are cross-persistent. If Xt > 0 and Yt > 0, then it is statistically cantly different from Hx(q)+2Hy(q) for any q or any corre-
moreprobable(basedonpersistenceoftheseparateprocesses)that lation coefficient value. This result is in hand with the
also Xt+1 >0 and Yt+1 >0 than otherwise. Therefore, if the pro- resultsshownin[14]–pairwisecorrelationshavenoeffect
cessesmovedtogetherinperiodt,itisstatisticallymorelikelythat
on the H estimation.
theywillmovetogetherinperiodt+1aswell(andviceversa),i.e. xy
theprocessesarecross-persistent. Third, we analyze the behavior of two-component
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Multifractal Height Cross-Correlation Analysis
ARFIMA processes [19]. For parameters d and d , the Commodities database). Even though the real-world se-
1 2
two-component ARFIMA(d ,d ) processes X and Y are ries are of the same length order as the simulated pro-
1 2 t t
described by the following set of equations: cesses, which scale even up to τ =100 and q =10, K (q)
τ
usuallydoesnotscaleforτ >20andq >3fordailyfinan-
Xt =[Wxt+(1−W)yt]+εt cial data [5]. Also, we apply linear filtering according to
Yt =[(1−W)xt+Wyt]+νt [4]. The generalized Hurst exponents are then estimated
xt =(cid:80)∞i=1ai(d1)Xt−i by varying τmax between 5 and 20 for 0.1≤q ≤3.
y =(cid:80)∞ a (d )Y
t i=1 i 2 t−i For the stock indices, we analyze the series of vol-
ume and volatility for the longest available datasets –
Here, W is a free parameter (0.5 ≤ W ≤ 1) controlling
from 11.10.1984 to 26.4.2011 for NASDAQ (6,693 ob-
a strength of coupling between X and Y , and ε ,ν ∼
t t t t servations) and from 3.1.1950 to 26.4.2011 for S&P500
N(0,1) are noise terms. Note that for W = 1, we obtain
(15,428 observations). We take absolute returns, defined
two decoupled ARFIMA processes, whereas for W < 1,
as|logP −logP |whereP isastockindexclosingprice,
the two processes have long memory of the process itself t t−1 t
as a measure of volatility. Volume series are transformed
as well as of the other one. In our simulations, we con-
as a relative deviation from a moving average of traded
sider d = d = 0.3 with W = 0.5,0.75 (practically, the
1 2 volume in approximately past two trading years (500 ob-
case W = 1 has been investigated in the previous para-
servations) to control for an exponential increase of the
graph). The results are shown in Fig. 1g,h. For both
traded volume in past decades (Fig. 3a,c). The estimated
W = 0.75 and W = 0.5, we notice deviations of H (q)
xy generalized Hurst exponents are shown in Fig. 3b,d. For
from Hx(q)+Hy(q) starting already at q = 0.1. The devi-
2 both stock indices, the trading volume and volatility are
ations are statistically insignificant for lower moments q
strongly persistent as well as cross-persistent. Neverthe-
(due to rather short series, T = 104), but become statis-
less, the bivariate Hurst exponent H (q) does not differ
xy
tically significant for higher moments (for q > 1.3 when
significantlyfromtheaverageofH (q)andH (q),i.e. the
x y
W =0.5 and for q >2.5 when W =0.75). The effect gets
cross-persistence of the series is mainly due to the persis-
strongerwithlowerW. Indeed, theseareexpectedresults
tence of the separate processes and the fact that the pro-
astheconstructionofthetwo-componentARFIMAmixes
cesses are correlated (Fig. 4e,f). The scaling of K (τ)
xy,q
the long memory of the separate processes together.
is very stable up to τ = 20 and for all examined qs (Fig.
In Fig. 2, we present the results based on separation
4a,b). Theresultsareinhandwith[12]whofoundweaker
in Eq. 7, i.e. scaling of separate processes and scaling
persistence of the process of traded volume. However, the
of covariances of |∆τXt|q2 and |∆τYt|q2. For illustrational definitions of traded volume differ from our study.
purposes, we show only the case q = 2. For MBM (Fig.
For the WTI spot and futures prices, we cover a period
2a),thescalingofcovariancesisslightlylowerthantheav-
from 2.1.1986 to 26.4.2011 (6,348 observations) and ana-
erage of Hurst exponents, yet remains well between them.
lyze the logarithmic returns r = log(P )−log(P ) and
This is reflected in the fact that the estimated H (2) is t t t−1
xy thevolatilityagainintheformofabsolutereturns. There-
not equal to the average of estimated H (2) and H (2)
x y sults are shown in Fig. 3e,f. For both returns and volatil-
but is rather close to the lower confidence interval (Fig.
ity,theestimatesofthegeneralizedHurstexponentsprac-
1a),yetthedeviationisstillinsignificant. InFig. 2b,four
ticallyoverlapforallq. Ononehand,thereturnsshowno
cases of correlated ARFIMA processes are illustrated. All
signs of long-range correlations or cross-correlations. On
four processes show α(2) ≈ 0.7, which perfectly fits the
the other hand, the volatility of separate processes show
expectations. We can see that the covariances are higher
strong persistence as well as cross-persistence. Moreover,
for highly correlated series than the less correlated series,
the generalized Hurst exponents vary only slightly with
butthescalingrelationremainsthesameforall. Thecase
q and are not even monotonically declining as expected
of uncorrelated ARFIMA processes exhibits no scaling of
formultifractalprocesses,suggestingthattheprocessesof
covariances (as these vary around zero) and is thus not
volatilityaremonofractal. Yetagain,thecross-persistence
shown. In Figs. 2c,d, the two-component ARFIMA pro-
of the series is mainly due to the persistence of the sepa-
cessesareillustrated. Here, thedifferencebetweenscaling
rate processes and high correlation between the processes
of covariances and the pair of K (τ) and K (τ) is re-
x,2 y,2 (Fig. 4g,h) as H (q) does not significantly deviate from
markable for both W =0.75 and W =0.5. The scaling of xy
Hx(q)+Hy(q). The scaling of K (τ) shows different be-
covariances is expectedly stronger for W =0.5. These re- 2 xy,q
havior for returns and volatilities. As for volatility, the
sults perfectly support the calculations presented in Eqs.
scalingisverystableuptoτ =20andq =3. Oncontrary,
4 – 10.
the scaling for returns becomes less stable with growing q
To show potential use of the method, we study differ-
(Fig. 3c,d).
ent real-world financial series, which we consider the out-
puts of the complex systems – daily volatility and vol- Inconclusion,weintroducethenewmethodforananal-
ume series of NASDAQ and S&P500 stock indices (fi- ysis of long-range cross-correlations and multifractality
nance.yahoo.comdatabase),anddailyreturnsandvolatil- – the multifractal height cross-correlation analysis. The
ityofspotandfuturespricesofWTICrudeOil(NYMEX scaling of covariances of the absolute values of the series
p-3
Ladislav Kristoufek1,2
givesadditionalinformationaboutdynamicsoftwosimul- [19] Podobnik B., Horvatic D., Lam Ng A., Stanley H.
taneously recorded series and can cause divergence of the and Ivanov P., Physica A , 387 (2008) 3954.
bivariate Hurst exponent from the average of the sepa-
rate univariate Hurst exponents. A utility of the method
has been shown on several artificial series as well as the
real-world time series. We argue that even though major-
ity of the analyzed series are cross-persistent, such cross-
persistence is mainly caused by persistence of the sepa-
rate processes and the fact that the series are correlated.
The scaling of covariances of the absolute values of the
examined processes is with good agreement with this re-
sult. A larger study comparing bias and efficiency of MF-
HXAcomparedtotheothermethodsanalyzinglong-range
cross-correlations(MF-X-DFAandMF-X-DMA)shallfol-
low.
∗∗∗
The support from the Czech Science Foundation under
Grants402/09/H045and402/09/0965andfromtheGrant
Agency of the Charles University (GAUK) under project
118310 are gratefully acknowledged. The author would
also like to thank J. Barunik and L. Vacha for helpful
comments and discussions.
REFERENCES
[1] Shiogai Y., Stefanovska A. and McClintock P.,
Physics Reports , 488 (2010) 51.
[2] Liao F., Garrison D. and Jan Y.-K., Microvascular
Research , 80(1) (2010) 44.
[3] Zuo R., Cheng Q. and Xia Q., Journal of Geochemical
Exploration , 102 (2009) 37.
[4] Di Matteo T., Aste T. and Dacorogna M., Journal
of Banking and Finance , 29 (2005) 827.
[5] Di Matteo T., Quantitative Finance , 7 (2007) 21.
[6] Chen C.-C., Lee Y.-T. and Chang Y.-F., Physica A ,
387 (2008) 4643.
[7] RehmanS.,Chaos,SolitonsandFractals,39(2009)499.
[8] HayakawaM., HattoriK., NickolaenkoA.andRa-
binowicz L., Physics and Chemistry of the Earth , 29
(2004) 379.
[9] Barunik J. and Kristoufek L., Physica A , 389(18)
(2010) 3844.
[10] KristoufekL.,Chaos,SolitonsandFractals,43(2010)
68.
[11] Carbone A., Physical Review E , 76 (2007) 056703.
[12] Podobnik B. and Stanley H., Physical Review Letters
, 100 (2008) 084102.
[13] Zhou W.-X., Physical Review E , 77 (2008) 066211.
[14] Jiang Z.-Q. and Zhou W.-X., Physical Review E ,
(2011) Forthcoming.
[15] BarabasiA.-L.,SzepfalusyP.andVicsekT.,Physica
A , 178 (1991) 17.
[16] CalvetL.andFisherA.,Multifractalvolatility: theory,
forecasting, and pricing (Academic Press) 2008.
[17] Meneveau C. and Sreenivasan K., Physical Review
Letters , 59 (1987) 1424.
[18] Mandelbrot B., Fisher A. and Calvet L., Cowles
Foundation Discussion Paper , 1164 (1997) .
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Multifractal Height Cross-Correlation Analysis
(a) (b) (c) (d)
(e) (f) (g) (h)
Fig. 1: (a) Binomial multifractal measures. GeneralizedHurstexponents(y-axis)dependentonmomentsq∈[0.1,10](x-axis)withstep
of 0.1. Hx(q) for MBM with m0 = 0.3 (bold black line) varies only weakly with q compared to Hy(q) for MBM with m0 = 0.4 (bold
dashed black line). Hxy(q) (bold red line) is not significantly different from the average of Hx and Hy (dotted black line) for all q (the
99% jackknife confidence intervals around Hxy(q) in gray). (b) - (f) ARFIMA(0,d,0) processes with correlated noise. Generalized Hurst
exponents (y-axis) dependent on moments q ∈ [0.1,10] (x-axis) with step of 0.1. Hx(q) for ARFIMA(0,d,0) with H = 0.8 (bold black
line) and Hy(q) for ARFIMA(0,d,0) with H =0.6 (bold dashed black line). The rest of the notation and parameters setting holds from
(a). Figs. (b) – (f) show ARFIMA(0,d,0) processes with correlated noise with correlation coefficients ρ = 1, ρ = 0.5, ρ = 0, ρ = −0.5
and ρ=−1, respectively. There is no significant deviation of Hxy(q) from Hx(q)+2Hy(q) for all q and for all examined correlations. (g) -
(h) Two component ARFIMA(d1,d2) processes. Generalized Hurst exponents (y-axis) dependent on moments q ∈[0.1,10] (x-axis) with
step of 0.1. Here, we use two component ARFIMA(d1,d2) processes with d1 = d2 = 0.3 and varying W. For W = 0.75 (g), Hxy(q) is
significantlyhigherthan Hx(q)+2Hy(q) forhighermoments(q>2.5). ForW =0.5(h),thedeviationofHxy(q)from Hx(q)+2Hy(q) ishigher
thanforcase(g)andthestatisticallysignificantdeviationfromtheaveragestartsatlowermoments(q>1.3).
(a) (b) (c) (d)
Fig.2: ScalingofKx,2(τ),Ky,2(τ)andcovariancesbetween|∆τXt|and|∆τYt|. ThescalingisshownforMBM(a),correlatedARFIMA
processes (b) and two-component ARFIMA processes (c,d). For MBM, we observe slight divergence of α(2) from the average of Hx(2)
andHy(2),whichremainsinsignificant(seeFig. 1a). CorrelatedARFIMAprocessesshowpracticallyperfectfittheexpectedα(2)of0.7
(ARFIMA processes with d=0.1 and d=0.3). Two-component ARFIMA processes exhibit remarkable deviation of α from the average
of Hurst exponents. Note that the fits (dashed black lines) and slopes are estimated on the whole sample from τmin =1 to τmax =20.
TheresultsareinhandwithexpectationsbasedonEqs. 4–10andinagreementwithFig. 1.
p-5
Ladislav Kristoufek1,2
(a) (b) (c)
(d) (e) (f)
Fig. 3: (a), (b) NASDAQ. (a) Evolution of logarithmic prices (red, left y-axis) and transformed traded volumes (black, right y-axis) in
between11.10.1984and26.4.2011. (b)GeneralizedHurstexponents(y-axis)dependentonmomentsq∈[0.1,3](x-axis)withstepof0.1.
Hx(q) for NASDAQ volatility (bold black line) and Hy(q) for NASDAQ traded volume (bold dashed black line) both vary with q while
stronger variation is present for volume. Hxy(q) (bold red line) is not statistically different from the average (dotted line) of Hx(q) and
Hy(q)foranyq. (c), (d) S&P500. Thetimeperiodcoveredrangesfrom3.1.1950to26.4.2011. Samenotationandestimationparameters
setting hold here. Hxy again does not differ from the average of the univariate Hurst exponents. (e) WTI crude oil spot and futures
pricesreturns. Samenotationholds,Hx representsthedynamicsofspotreturnsandHy forfuturesreturns. Thereisagainnosignificant
deviation of Hxy(q) from Hx(q)+2Hy(q). (f) WTI crude oil spot and futures prices volatility. The notation holds. Generalized Hurst
exponentspracticallyoverlayforbothseriesaswellasforthejointdynamics.
(a) (b) (c) (d)
(e) (f) (g) (h)
Fig. 4: (a) – (d) Scaling of Kxy,q for NASDAQ, SPX and WTI. The scaling functions are presented by black lines, where the cases of
q = 1,2,3 are in bold. For these three cases, the best fits are illustrated (dashed lines). The scaling is very stable for NASDAQ, SPX
and WTI volatility up to τ = 20 for all examined q so that the fits are almost undistinguishable from the scaling functions. For WTI
returns, the scaling is less stable with increasing q. For the analyzed series, it implies that scaling is better for higher values of Hurst
exponents. (e) – (h) Scaling of Kx,q, Ky,q and covariances between |∆τXt| and |∆τYt| for NASDAQ, S&P500 and WTI. Best linear
fitsarerepresentedbydashedlinesandestimatedslopesarenoted. Forillustrationalpurposes,weshowonlythecaseq=2. Thescaling
exponents α(2) are approximately equal to the average of estimated Hurst exponents. This implies that eventual cross-persistence (for
casesofNASDAQ,S&P500andWTIvolatility)ismajorlycausedbypersistenceoftheseparateprocessesandthefactthattheprocesses
arepairwisecorrelated. NotethatthedifferencesofestimatesfromFig3arecausedbythefactthathere,weestimatetheexponentsfor
τ betweenτmin=1andτmax=20,whileforFig. 3,weusethejackknifeestimates.
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