On the scaling ranges of detrended fluctuation analysis for long-memory correlated short series of data

Transcription

1 On the scaling ranges of detrended flctation analysis for long-memory correlated short series of data arxiv:126.17v1 [physics.data-an] 5 Jn 212 Darisz Grech (1) and Zygmnt Mazr (2) (1) Institte of Theoretical Physics, University of Wroc law, Pl. M.Borna 9, P-5-24 Wroc law, Poland (2) Institte of Experimental Physics, University of Wroc law, Pl. M.Borna 9, P-5-24 Wroc law, Poland Abstract We examine the scaling regime for the detrended flctation analysis (DFA) - the most poplar method sed to detect the presence of long memory in data and the fractal strctre of time series. First, the scaling range for DFA is stdied for ncorrelated data as a fnction of length of time series and regression line coefficient R 2 at varios confidence levels. Next, an analysis of artificial short series with long memory is performed. In both cases the scaling range is fond to change linearly both with and R 2. We show how this dependence can be generalized to a simple nified model describing the relation = (,R 2,H) where H (1/2 H 1) stands for the Hrst exponent of long range atocorrelated data. Or findings shold be sefl in all applications of DFA techniqe, particlarly for instantaneos (local) DFA where enormos nmber of short time series has to be examined at once, withot possibility for preliminary check of the scaling range of each series separately. Keywords: scaling range, detrended flctation analysis, Hrst exponent, power laws, time series, long memory, econophysics, complex systems PACS: 5.45.Tp, 5.4.-a, a, Da, Gh, n dgrech@ift.ni.wroc.pl 1

2 1 Introdction and description of the method. Detrended flctation analysis (DFA)[1, 2, 3] is now considered the main tool in searching for fractal [4, 5, 6], mltifractal [7, 8] and long memory effects in ordered data. There is more than one thosand articles pblished on DFA and its applications so far. The detrended techniqe has been widely applied to varios topics, jst to mention: genetics (see e.g. [2, 9, 1, 11], meteorology (see e.g. [12, 13, 14]), cardiac dynamics (see e.g. [15, 16]), astrophysics (see e.g. [17]), finances (see e.g. [18, 19, 2, 21, 22, 23, 24]) and many others. The indisptable advantage of DFA over other available methods searching for the Hrst exponent H [25, 26] in series of data, like the rescaled range method (R/S) [7, 25, 26, 27], is that DFA is shown to be resistant to some extent to non-stationarities in time series [28]. We will not describe the DFA techniqe in details here, for it is done in many other pblications (see e.g. [29, 3, 31, 32]). Instead, we will focs mainly on the isses which are relevant for the so called scaling range being the goal of this article. Briefly, the DFA method contains the following steps: (i) the time series x(t) (t = 1, 2,..., ) of data (random walk) is divided into non-overlapping boxes (time windows) of length τ each, (ii) the linear trend 1 is fond within each box and then sbtracted from the signal giving so called detrended signal ˆx(t), (iii) the mean-sqare flctation F 2 (τ) of the detrended signal is calclated in each box and then F 2 (τ) is averaged over all boxes of size τ, (iv) the procedre is repeated for all box sizes τ (1 < τ < ). One expects that the power law F 2 (τ) box τ 2H (1) is flfilled for stationary signal 2 where. box is the expectation vale - here, the average taken over all boxes of size τ. The latter eqation allows to make the linear fit in loglog scale to extract the vale of H exponent necessary in varios applications. One can also look alternatively at the above relationship as a link between the variance of the detrended random walk ˆx(t) and its dration time t, i.e. ˆx 2 (t) t 2H what reflects the precise definition of Hrst exponent in stochastic processes. The H exponent clearly indicates the randomness natre of this process. One deals with ncorrelated steps in data series if H = 1/2, once for other vales of H these steps are respectively anticorrelated ( < H < 1/2) or atocorrelated with (positive) long memory (1/2 H 1). The edge part of time series is sally not covered by any box. Some athors sggest to overcome this difficlty performing DFA in two opposite directions in time series, i.e. according to increasing and then according to decreasing time arrow (see e.g. [33]). The average of mean-sqare flctations from sch divisions is then taken for evalation of time series properties. We proposed another soltion in Refs.[34, 35]. If the remaining part of time series has the length τ/2 < τ, we cover it by an additional box of size τ partly 1 the sbtracted trend can also be mimicked by nonlinear polynomial fnction of order k in so called DFA-k schemes - we will not discs this isse in details here 2 this property holds also for non-stationary, positively atocorrelated (H > 1/2) time series [28] 2

3 overlapping the preceding data. If < τ/2, we do not take into accont the part of data contained in. Sch recipe is particlarly sefl in the local version of DFA [18, 34, 35, 36, 37, 38, 39], where the time arrow is important. Throghot this article we will apply the latter approach. If time series are infinitely long, the formla in Eq.(1) holds for all τ s. However, in practise we always deal with finite, and sometimes with rather short time series. Particlarly, it is a case for the mentioned already instantaneos or local DFA analysis, where one wants to find a dynamics of fractal properties changing in time and (or) their time dependent long memory in data. Covering the data series with boxes, we are finally stck with sitation that for small nmber of boxes covering the time series (for large τ ), the scaling is not revealed in Eq.(1) de to small statistics we deal with. In other words, we are allowed in this case to take τ only within some range τ min τ τ max called the scaling range. One expects within this range sfficiently good performance of the power law, ths leading to H exponent extraction via linear fit. Bt what does this sfficiently good performance exactly mean? In most research activities athors end p with τ max 1/4, where is the total length of considered data. Is it still good or already too large scaling range? This problem is somehow circmvented in papers bt it does have impact on the final reslts. The aim of this and other forthcoming article [4] is to confront this isse. Or approach will be different than the one pblished in [41, 42, 43]. The goal is to find qalitative and qantitative dependence between the scaling range τ max and main parameters of time series like its length, level of long memory described by the Hrst exponent H, and the goodness of linear fit indced by the form of Eq.(1) in log-log scale. The latter one is sally measred by the R 2 regression line coefficient. All this can be done at desired confidence level (C) indicating the minimal ratio of time series flfilling the fnctional dependence = (,R 2,H). We are going to find this relation below. Throghot this paper we assmed that τ min = 8 becase below this threshold a significant lack of scaling in DFA is observed de to emergence of artificial atocorrelations associated with too short brsts of data in τ boxes. We start with analysis of ncorrelated data in the next section and proceed with long memory correlated time series in section 3. Section 4 tries to obtain an nified formla for scaling range vs and R 2 for all H 1/2. Althogh the presented considerations are done exclsively for DFA method, they can be easy extended to other detrended methods introdced in literatre, in particlar to those based on moving averages [44, 45, 46, 47]. The latter analysis is left to another pblication [4]. 2 DFA scaling ranges for ncorrelated data The starting point for the entire search is the statistical analysis of an ensemble of artificially generated time series with a given length. For this ensemble we find the percentage rate of series which are below the specified level of regression line fit parameter R 2. This rate will obviosly depend on the maximm size of the box τ max. The larger τ max, the 3

4 percentage rate of series not matching the assmed criterion for R 2 will be also larger. Fig.1. illstrates this fact for two specified series lengths = 1 3 and = of ncorrelated data increments (H = 1/2) drawn from the normalized Gassian distribtion. The rejection rate, i.e the percentage rate of series not matching the assmed criterion for R 2 is shown there for different R 2 vales. We took two particlar vales of rejection rate in frther analysis: 2.5% and 5.%, connected with confidence levels C = 97.5% and C = 95.% respectively. All data have been gathered nmerically on a set of artificially generated time series of length between for the above-stated confidence levels. The τ max vale corresponding to reqired C and for given R 2 is identified with the scaling range and referred to exactly as. Introdcing for convenience a new parameter = 1 R 2, we may search for a () dependence for 2 1 4, for different vales of and for selected C s. The reslts are presented in a series of graphs in Figs. 2, 3 and reveal a very good linear relationship between the scaling range profile and the length of ncorrelated data 3 : (,) = A()+B() (2) The fnctional dependence of coefficients A() and B() on has to be frther specified from the regression line fit of the above eqation. The latter procedre yields to the vales of A and B estimated for the spread of parameters and gathered in Fig.4. We see from these graphs that the dependence of A() is again linear for both cases of C = 97.5% and C = 95%, while the vale of B varies very weakly with, what legitimates s to accept B() = b = const. Ultimately, the foregoing considerations lead to the following simple formla describing the fll scaling range dependence on and : (,) = (a+a )+b (3) with some nknown constants a, a and b to be fitted. We made the fit for Eq.(3) reqiring minimization of mean absolte error (MAE) and simltaneosly, minimization of the maximal relative error (ME) for each of the fitting points 4 ( i, j ). The MAE denoted as MAE () is nderstood as MAE () = 1/N (ij) ij ( exp ij (,) ij (,))/ ij (,) (4) where ij (,) ( i, j ) is taken from Eq.(3) for the particlar choice = i and = j, while exp ij (, ) is the respective vale simlated nmerically for given ensemble of time series, and N (ij) conts different (ij) pairs. 3 obviosly (,) Z, so in fact only the integer part of RHS of Eq.(2) shold be taken for determination of (,) 4 we considered j = (1+j) where j = 1,2,...,9 and i covering the range from = p to = as indicated on plots 4

5 Similarly ME marked below as ME is simply defined as ME = max (ij) ( exp ij (,) ij (,)) / ij (,)) (5) Note that some pairs ( i, j ) are not permitted by the specific C demand 5. It is seen already in Fig.1. These points are therefore absent in Figs 2-3, 5-6. The fitting procedre led to the vales of parameters in eqation Eq.(3) gathered in Table 1. The exemplary reslts of scaling ranges for the wide spread of and vales are presented graphically in Figs.5,6. Whenever (,) comes ot negative in the fond fitting patterns for the particlar length of the series, one shold interpret this as a lack of scaling range at the given confidence level C for the reqired vale of regression line coefficient R 2 within DFA. 3 DFA scaling range for long-memory correlated data The analysis presented in the previos section can be extended to time series manifesting long memory. The series with.5 < H <.9 are of particlar interest since they correspond to long-range atocorrelated data one often meets in practice in varios areas. To constrct sch signals we sed Forier filtering method (FFM) [48]. The level of atocorrelations in this approach was directly modlated by the choice of atocorrelation fnction C(δt) which satisfies for stationary series with long memory the known power law [49]: C(s) x(t+s) x(t) H(2H 1)s 2H 2 (6) where x(t) = x(t+1) x(t), (t = 1,2,..., 1) are increments of discrete time series, s is the time-lag between observations, H is the Hrst exponent [25, 26], and the average is taken over all data in series. We start with similar analysis as the one shown in Fig.1 for ncorrelated data. Fig.7 presents an example of plot made for the ensemble of atocorrelated signals of length = 1 3 with H =.7. The percentage rate of rejected time series not satisfying the assmed goodness R 2 of DFA fit is shown there for several distinct R 2 as a fnction of maximal boxsize τ max. The otcomeof sch analysis forarangeofsimlated datalengths and for varios Hrst exponents can be collected in nmber of plots as in Figs.8a,9a for (), and in Fig.8b,9b for () dependence. To make the figre readable and de to lack of space, only plots for =.2 and = 1 3 are shown. The relations for other vales look qalitatively the same. We shold not be srprised, taking into accont the reslts of the previos chapter, that these relationships are again linear. Ths the formla in Eq.(2) is more general and coefficients A() and B() are linear fnction of also for series with memory. The latter relationships are drawn in details for H =.6,.7,.8 5 we fond that following pairs: ( < 3, 1 ),( < 18, 2 ),( < 15, 3 ),( < 1, 4 ), ( < 1, 5 ), ( < 8, 6 ), ( < 8, 7 ), ( < 6, 8 ) do not match the C = 97.5% reqirement, and: ( < 24, 1 ),( < 18, 2 ),( < 12, 3 ),( < 1, 4 ),( < 8, 5 ),( < 8, 6 ),( < 6, 7 ) do not match the C = 95% demand 5

6 H \C a 97.5% a 97.5% b 97.5% MAE 97.5% ME 97.5% a 95% a 95% b 95% 95% MAE 95% ME H = % 5.1% % 5.1% H = % 5.2% % 5.3% H = % 4.6% % 4.6% H = % 4.8% % 6.% Table 1: Reslts of the best fit for coefficients in Eq.(3) fond for series with varios atocorrelation level measred by H exponent and for chosen two confidence levels: 97.5% and 95%. The accracy of fitted parameters are respectivel: a = ±1 3, a = ±1 5, b =. in Fig.1. In particlar, we notice from Fig. 1b the similar behavior of B() coefficient for atocorrelated data as it has been observed in the previos section for ncorrelated signals, i.e. B() remains almost constant as a fnction of. Moreover, its dependence on H is also negligible. Ths, the formla postlated in Eq.(3) applies also for atocorrelated data with a, a and b coefficients to be fitted independently for each H. We did sch a fit for series with long memory, assming the same criterions for MAE and ME as previosly. The reslts are collected in Table 1 for two different confidence levels and are shown graphically in Figs These figres generalize plots shown for H =.5 in Figs.5,6. The extremely good linear relationship of Eq.(3) is kept for atocorrelated signals p to = 1 4. Only for highly atocorrelated series (H >.8) or very long ones ( 1 4 ) we noticed some slight departre from the linear dependence 6. 4 Towards nified model of scaling ranges Finally, we shold investigate if there exists a nified formla with the minimal nmber of free parameters, able to describe all scaling ranges of both ncorrelated and atocorrelated data. So far we know that Eq.(3) with parameters fitted according to Table 1 describes verywell (,)dependenceforgivenh. Wesholddiscssthentheformofrelationships a(h), a (H), and b(h) in the relation (,,H) = (a(h)+a (H))+b(H) (7) ooking at the bottom panels of Figs.4, 1 one perceives immediately that the assmption b(h) = const can be jstified. Similarly, we may easily notice from data collected in Table 1 that a (H)/(a(H)) O(1 1 ). It means that the component a(h) gives the leading contribtion to the linear factor a(h)+a (H) in Eq. (7) for each vale of H and therefore, one shold focs mainly on a(h) dependence depicted in Fig.17. The latter relationship also appears to be linear, which allows to represent Eq.(7) in its simplest nified form containing the smallest nmber of for free parameters (α,β,α,γ) as follows: 6 the predicted scaling ranges from Eq.(3) were nevertheless lower in these cases than the ones coming from the direct simlation 6

7 α 97.5% β 97.5% α 97.5% γ 97.5% MAE 97.5% 95% ME % 5.9% α 95% β 95% α 95% γ 95% 95% MAE 95% ME % 5.7% Table2: ResltsofthebestfitforcoefficientsinnifiedformlainEq.(8). Fitwasdonefor all data coming from investigated series, separately for two chosen confidence levels: 97.5% and 95%. The accracy of fitted parameters have been estimated: α = β = 1 3, δα = 1 5, γ =. (,,H) = ((αh +β)+α )+γ (8) Demanding minimization of MAE and ME dring fitting procedre of the proposal given in Eq.(8) to all data points exp ( i, j,h) indicated in previos sections, we arrive with the best fit reslts for these free parameters as shown in Table 2. The obtained nified formla can be particlarly sefl while doing interpolation to arbitrary atocorrelation levels 1/2 < H < 1. In fact the fit based on Eq.(8) is of the same qality as the one prodced by Eq.(3) (see Table 1 and 2 to compare MAE and ME errors). The difference between two fitting methods is so negligible that it cannot be noticed graphically. Therefore the fitting lines shown in series of Figs describe eqally well the nified model based on data from Table 2 and the local fit based on data from Table 1. We may also easy conclde from Eq.(8) that the average relative change in the scaling range δ(δh)/(h) de to the small change δh in Hrst exponent is given as δ(δh, ) (H, ) αh (9) and varies from 3% (at R 2 =.99) to 1% (at R 2 =.97) for any change δh =.1 in the investigated signal. 5 Discssion and Conclsions In this stdy we searched for the scaling range properties of the most sbstantial power law between flctationsofdetrended randomwalkf 2 (τ)andthelengthofthetimewindowτ in which sch flctations are measred. This power law proposed within DFA techniqe gives s an important information abot the natre of randomness in stochastic process via link between the scaling exponent H and the atocorrelation exponent between steps of random walk. Therefore, the precise knowledge of scaling range dependence on any other involved parameters is a sbstantial task and has an impact on the final otcomes of DFA power law qoted in Eq.(1). We did or simlations on the ensemble of short and medim-length time series with We varied also their atocorrelation 7

8 properties in order to reflect properties of real random walk signals mostly existing in natre. First, it has been fond that for ncorrelated process, the scaling range of DFA power law is the perfect linear fnction of data length and the goodness of linear fit to power law formla in Eq.(1). Moreover, this linear relationship extends also to time series with long memory. The niform shape of (, ) dependence for different memory levels in data, rises the qestion if one nified simple formla describing dependence of scaling range on all parameters in a game, i.e. (,,H) exists. We fond sch a formla, and showed that it fits data obtained from nmerical simlations no worse than patterns previosly fond in this article for (, ) dependence at separate vales of H. The nified formla contains only for free parameters, which were calclated with high precision and are presented in Table 2. We showed also that scaling range grows with a long memory level present in time series on the average of 3 1% for every δh =.1 (see Eq.(9)). A rather slight increase in the scaling range for the series with memory in comparison with the array of ncorrelated data may entitle s to simplify the scaling range for the series with long memory, sing a model for ncorrelated data, i.e. with H = 1/2. The presented reslts can be considered therefore as the lower limit for the DFA scaling range profile. The relations we fond strike with their simplicity and make a sefl recipe how to determine the scaling ranges, especially for short time series wherever one needs to consider very large data sets arranged in shorter sbseries. In particlar, these reslts can be sed in search for evolving (time-dependent) local Hrst exponent in large amont of moving time windows. The extension of this approach to other techniqes of flctation analysis (FA) can also be done [4]. References [1] C.-K. Peng, S. Havlin, H. E. Stanley, and A.. Goldberger, Chaos 5, 82 (1995). [2] C.-K. Peng, S. V. Bldyrev, S. Havlin, M. Simons, H. E. Stanley,and A.. Goldberger, Phys.Rev.E 49, 1685 (1994). [3] A. Bnde, S. Havlin, J. W. Kantelhardt, T. Penzel, J. H. Peter, and K. Voigt, Phys. Rev. ett. 85, 3736 (2). [4] J. Feder, Fractals (Plenm, New York, 1988). [5] H.-O. Peitgen, H. Jrgens, D. Sape, Chaos and fractals (Springer, Berlin, 24). [6] D. Sornette, Critical phenomena in natral sciences (Springer, Berlin, 24). [7] B. B. Mandelbrot, Mltifractals and 1/f noise: wild self-affinity in physics, selected works ( ) (Springer, Berlin 1999). 8

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11 rejection rate (%) (a) τ max rejection rate (%) (b) τ max Figre 1: Percentage rate (%) of rejected time series with scaling range not providing indicated goodness of regression line fit R 2 in DFA procedre. Reslts are based on the ensemble of time series and are drawn as a fnction of maximal box size τ max for two lengths of ncorrelated data: (a) = 1 3 and (b) =

12 H =.5 C = 97.5% (a) =.2 =.3 =.4 = H =.5 C = 97.5% = 1 = 3 = 6 = (b) Figre 2: Dependence between scaling range as in Fig.1 and: (a) time series length or (b) the goodness of linear fit = 1 R 2. Examples of (, = fixed) and (, = fixed) relations for varios and vales are shown at 97.5% confidence level. 12

13 H =.5 C = 95.% (a) (b) =.2 =.3 =.4 = H =.5 C = 95.% = 1 = 3 = 6 = Figre 3: Same as in Fig.2 bt at 95% confidence level. 13

14 ,4,35 H =.5,3,25 A,2,15,1 97.5% 95.%,5 (a),,1,2,3,4,5,6,,1,2,3,4,5,6-2 H =.5 B % 95.% -1 (b) -12 Figre 4: (a) Dependence between fitted coefficient A in Eq.(2) and the goodness of DFA fit = 1 R 2 for ncorrelated data (H =.5) shown for two distinct confidence levels: C = 97.5%, C = 95.%. (b) Same for B coefficient in Eq.(2). 14

15 (a) C = 97.5% C = 97.5% = 1 = 3 = 6 = (b) Figre 5: Best fit reslts of the relationship sggested in Eq.(3) at C = 97.5% level. Continos lines represent the fit of Eq.(3) to data (marked as points). (a) Reslts shown for(, = fixed) atr 2 = 1 =.96,.97,.98,.985. (b)reslts for( = fixed,) shown for chosen lengths of ncorrelated data = 1 3, 3 1 3, 6 1 3, 1 4. Parameters of fits are gathered in Table 1. 15

16 C = 95.% (a) C = 95.% = 1 = 3 = 6 = 1 (b) Figre 6: Same as in Fig.5 bt at C = 95% level. 16

17 rejection rate (%) τ max Figre 7: Percentage rate (%) of rejected time series as a fnction of maximal box size τ max. Exemplary reslts arebased ontheensemble of5 1 4 timeseries foratocorrelated signal of length = 1 3 with H =.7. 17

18 H =.6 H =.7 H =.8 (a) 2 =.2 C = 97.5% = 1, C = 97.5% H =.6 H =.7 H =.8 (b) Figre 8: Dependence between scaling range and: (a) time series length or (b) the goodness of linear fit = 1 R 2 for atocorrelated signals. The atocorrelation level is indicated by Hrst exponent H. Plots are shown only for = 1 3 and =.2 (R 2 =.98)bttherelations(,)fortheremaining valesofandlookqalitatively very similar (not shown). The linear relationship (,) is seen at C = 97.5% level. 18

19 H =.6 H =.7 H =.8 (a) =.2 C = 95.% = 1, C = 95.% (b) H =.6 H =.7 H =.8 Figre 9: Same as in Fig.8 bt at C = 95% 19

20 .4.35 C = 97.5%.3.25 A H =.6 H =.7 H = (a) C = 97.5% B H =.6 H =.7 H =.8-12 (b) Figre 1: (a) Dependence between coefficient A in Eq.(2) and the goodness of DFA fit = 1 R 2 for atocorrelated data (H =.6,H =.7,H =.8). Confidence level C = 97.5% is considered bt C = 95% looks similarly (not shown). (b) Same for B coefficient in Eq.(2). 2

21 C = 97.5% 35 C = 97.5% 3 =.15 3 =.2 25 H = H =.6 H =.7 H = H = H =.6 H =.7 H = C = 97.5% 8 C = 97.5% 7 =.3 7 = H =.5 H =.6 H =.7 H = H =.5 H =.6 H =.7 H = Figre 11: Best fit reslts of Eq.(7) fond for simlated series of atocorrelated data at C = 97.5% level and shown for 3. Continos lines represent fit of Eq.(7) to data points marked as dots for chosen. The cases of other vales (not shown de to lack of space) look identically. Parameters of the fit are gathered in Table 2. 21

22 5 5 C = 95.% C = 95.% 4 =.15 4 = H =.5 H =.6 H =.7 H = H =.5 H =.6 H =.7 H = C = 95.% C = 95.% =.3 = H =.5 H =.5 2 H =.6 H =.7 2 H =.6 H =.7 H = H = Figre 12: Same as in Fig.11 for C =.95%. 22

23 C = 97.5% = H =.5 H =.6 H =.7 H = C = 97.5% = H =.5 H =.6 H =.7 H = Figre 13: Same dependence as in Fig.11 shown for longer series of data ( 1 4 ). Only cases for =.15 and =.3 are presented de to lack of space. Plots for other R 2 coefficients look qalitatively the same. 23

24 C = 95.% = H =.5 H =.6 H =.7 H = C = 95.% = H =.5 H =.6 H =.7 H = Figre 14: Same as in Fig.13 for C = 95%. 24

25 3 1 C = 97.5% 9 C = 97.5% 25 = 1 8 = H =.5 H =.6 H =.7 H = H =.5 H =.6 H =.7 H = C = 97.5% = 6 3 C = 97.5% = H =.5 15 H =.5 H =.6 1 H =.6 5 H =.7 H =.8 5 H =.7 H = Figre 15: Best fit reslts of Eq.(7) fond for simlated series of atocorrelated data at C = 97.5% as in Fig.11 and shown as the fnction of. The cases of several lengths of data are shown. Other vales (not shown de to lack of space) show also linear dependence on. Parameters of the fits are gathered in Table 2. 25

26 C = 95.% = C = 95.% = H =.5 6 H =.5 1 H =.6 4 H =.6 5 H =.7 H =.8 2 H =.7 H = C = 95.% = 6 4 C = 95.% = H =.5 H =.6 H =.7 H = H =.5 H =.6 H =.7 H = Figre 16: Same as in Fig.15, bt for C = 95%. 26

27 9 C = 97.5% 8 C = 95.% a H Figre 17: Dependence between the fitted a(h) parameter (see Eq.(7)) and the atocorrelation level in data expressed by Hrst exponent H. 27