PHYSICS OF PLASMAS
VOLUME 9, NUMBER 4
APRIL 2002
Investigation of rescaled range analysis, the Hurst exponent, and long-time
correlations in plasma turbulence
M. Gilmorea)
Electrical Engineering Department, University of California, Los Angeles, California 90095
C. X. Yu
Department of Modern Physics, University of Science and Technology of China, Hefei 230026,
People’s Republic of China
T. L. Rhodes and W. A. Peebles
Electrical Engineering Department, University of California, Los Angeles, California 90095
共Received 12 October 2001; accepted 18 January 2002兲
A detailed investigation of rescaled range (R/S) analysis to search for long-time correlations 共via
the Hurst exponent, H) in plasma turbulence is presented. In order to elucidate important issues
related to R/S analysis, structure functions 共SFs兲, one of several techniques available for calculating
H, are also applied, and comparisons between the two methods are made. Time records of both
simulated data and fluctuation reflectometry data from the DIII-D tokamak 关J. L. Luxon and L. G.
Davis, Fusion Technol. 8, 441 共1985兲兴 are analyzed. It is found that the R/S method can be used to
accurately determine H, provided a long enough data record is used, and that H is an indicator of
persistence in the data. In addition, subtleties of the correct application of both methods are
discussed, and potential advantages of SFs are pointed out. © 2002 American Institute of Physics.
关DOI: 10.1063/1.1459707兴
I. INTRODUCTION
the relation of H to signal persistence. For comparison, we
also present calculations of H based on structure functions8
共one of a number of methods to calculate H, including
scaled-windowed variance,9 dispersional analysis,10 etc.兲,
which serves to elucidate issues of applying the R/S method.
It is found that the R/S method can be used to accurately
determine H in plasma fluctuation data provided a long
enough data record is used, and that H is an indicator of
persistence. In practice this means that H must be evaluated
at time lags greater than 5–10 local autocorrelation times,
␶ L . It is also shown that the structure function 共SF兲 method
can be used to evaluate H as well, and that lags greater than
1⫻ ␶ L are sufficient. The disparity between R/S and SF
methods is a result of fundamental differences between the
two algorithms, and helps to make clear the issues related to
the correct application of the R/S method.
Recent models based on self-organized criticality
共SOC兲1 as well as models based on standard turbulence2
have predicted the existence of long-time correlations, or
persistence, in the turbulent fluctuations of magnetically confined plasmas. It is proposed that these long-time correlations, manifested as long-range tails in the autocorrelation
functions of the fluctuations, may have a significant effect on
the turbulence driven cross-field transport.
In order to search for such persistence in time records of
plasma fluctuation measurements, the rescaled range (R/S)
analysis has been applied, for example, by Carreras et al.3
Following the original work of Hurst,4 the R/S method was
used to calculate the scaling exponent 共Hurst exponent兲, H,
to give a quantitative measure of the persistence of a signal.5
0.5⬍H⬍1 was taken to indicate persistence, while H⫽0.5
indicates an uncorrelated process. 0⬍H⬍0.5 indicates anticorrelation.
Recently, however, two articles appeared6,7 questioning
the use of the Hurst parameter, determined via R/S analysis,
to indicate persistence in plasma fluctuations. These articles
used the R/S method to calculate the Hurst exponent for
Langmuir probe data acquired in the edge of the Tore Supra
tokamak. The authors suggested that the R/S method is not
appropriate for evaluating persistence in fusion devices, and
that the information given by H is contained entirely in the
short-range correlations of the system.
In this article, we present a detailed investigation of the
rescaled range analysis to determine the Hurst exponent, and
II. CALCULATION OF THE HURST EXPONENT AND
DATA DESCRIPTION
There are two classes of definitions of self-similarity,
which are based on two different stochastic processes. The
standard definition is that a process Y ⫽ 兵 Y (t),t⭓0 其 is selfsimilar with self-similarity 共Hurst兲 parameter H苸(0,1) if it
satisfies
d
Y 共 ␭t 兲 ⫽ ␭ H Y 共 t 兲 ,
d
where ⫽ means that the two random functions have the same
finite joint distribution function for any constant ␭⬎0. A
process Y satisfying Eq. 共1兲 is nonstationary, but its increment process X⫽ 兵 X(t)⫽Y (t⫹1)⫺Y (t),t⭓0 其 can be 共and
a兲
Electronic mail: gilmore@ee.ucla.edu
1070-664X/2002/9(4)/1312/6/$19.00
共1兲
1312
© 2002 American Institute of Physics
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Phys. Plasmas, Vol. 9, No. 4, April 2002
Investigation of rescaled range analysis . . .
is typically assumed to be兲 stationary. The canonical example
of such a process is fractional Brownian motion 共fBm兲,
which can be regarded as a generalization of the well-known
Brownian motion which has H⫽1/2. Although the power
spectral density 共PSD兲 is not defined for a nonstationary selfsimilar process such as fBm, it has been shown that a timeaveraged PSD, S( ␻ ), of such a process can be obtained by
means of a time-frequency analysis to give11
S 共 ␻ 兲 ⫽c s 兩 ␻ 兩 ⫺ ␤ ,
1⬍ ␤ ⫽2H⫹1⬍3.
共2兲
A second definition of self-similarity involves stationary
processes. In this case, a stationary process X⫽ 兵 X(t),t⬎0 其
is self-similar with self-similarity parameter H if there exists
a real number ␣ 苸(0,2) and a constant c ␳ such that
lim ␶ 共 ␶ 兲 ⫽c ␳ ␶ ⫺ ␣ ,
␶ →⬁
0⬍ ␣ ⫽2⫺2H⬍2,
共3兲
where ␳ ( ␶ ) is the normalized autocorrelation function
共ACF兲. Then, the PSD of such a process exhibits the powerlaw scaling
lim S 共 ␻ 兲 ⫽c s 兩 ␻ 兩 ⫺ ␤ ,
␻ →⬁
⫺1⬍ ␤ ⫽2H⫺1⬍1,
共4兲
where c s ⫽ ␴ 2 ␲ ⫺1 c ␳ ⌫(2H⫺1)sin( ␲ (1⫺H)), and ⌫( ) is
the gamma function.12 A typical example of such a process is
fractional Gaussian noise 共fGn兲, which is the increment process of fBm.
It should be noted that the scaling exponents of the PSDs
for these two self-similar processes 共fBm and fGn兲 relate to
H differently as ␤ ⫽2H⫹1 共fBm兲 and ␤ ⫽2H⫺1 共fGn兲.
This means that the processes with fBm and fGn statistics do
not share the same correlation structure 共only the process
with fGn statistics and increments of the process with fBm
statistics share the same correlation structure兲, even if they
share the same value of H. Therefore, to avoid ambiguity, it
is necessary to specify not only the H value for a self-similar
process, but also the statistics of the process.
In the R/S method, a self-similar scaling
R
lim 共 ␶ 兲 ⫽C ␶ H ,
␶ →⬁ S
共5兲
is sought, where R and S denote the range and standard deviation, respectively, of the cumulative-summed 共integrated兲
time series W k , ␶ denotes a time lag, and C is a constant. For
a time record segment of length n which is a subset of the
total data record of length N (n⭐N), X⫽ 兵 X t :t
⫽1,2, . . . ,n 其 , W k is given by W k ⫽X 1 ⫹X 2 ⫹¯⫹X k
⫺kX̄(n), where X̄ is the mean. The ratio R/S, or rescaled
range, for a given lag n is then given by
R 共 n 兲 max兵 0,W 1 ,W 2 , . . . ,W n 其 ⫺min兵 0,W 1 ,W 2 , . . . ,W n 其
⫽
.
S共 n 兲
S共 n 兲
共6兲
It should be noted that the scaling, Eq. 共5兲, is defined for
the case of fBm statistics. Therefore, the first step in the
process is to integrate the data to generate the surrogate time
record W k . Thus, the R/S method is applicable to raw data
X k having fGn statistics. Alternately, raw data having fBm
1313
statistics could be analyzed by omitting the integration step
关so that X i replaces W i in 共6兲兴. In analysis of real data, this is
an extremely important point.
Practically, one computes R/S for a number of lags n up
to the total time record length N. A log–log plot of R/S vs n
then gives H from the slope, as evident from Eq. 共5兲. If the
data record has mixed statistics, this will be manifested in
different regions of the R/S vs lag plot. The scaling region
constituted by data having fBm statistics will have a constant
scaling exponent generally with a value close to 1. Only the
scaling region constituted by data having fGn statistics will
have a scaling exponent H⬍1. For example, the common
case of a turbulence time record characterized by an autocorrelation with a Gaussian central peak of width ␶ L 共local
decorrelation time兲, and an extended algebraic tail shows a
two slope R/S function separated by a breakpoint. The time
lag of the breakpoint is related to ␶ L . The scaling region
with time lags larger than ␶ L is the ‘‘self-similarity range,’’
over which H should be determined, while the scaling range
with time lags smaller than ␶ L is the local turbulence decorrelation range, wherein the process is nonstationary and
tends to give a biased estimate of H near 1.
In the structure function method, a self-similar scaling of
the qth-order structure function, S q ( ␶ ),
S q 共 ␶ 兲 ⫽C q ␶ qH 共 q 兲
共7兲
is sought, where
S q 共 ␶ 兲 ⫽ 具 兩 W 共 t 1 ⫹ ␶ 兲 ⫺W 共 t i 兲 兩 q 典 ,
共8兲
i denotes the ith data point, and 具 典 denotes the ensemble
average. This method is discussed in detail by Davis et al.8
Here, H is determined practically, by plotting S q ( ␶ ) vs ␶ on a
log–log plot, and taking the slope, which is equal to qH.
Given the caveats above, Hurst exponents were calculated for both numerically generated 共simulated兲 data and
fluctuation reflectometer data from the DIII-D tokamak. The
numerical data were generated by an exact frequency domain
method introduced by Davies and Harte,13 and discussed in
greater detail by Percival.14 This method utilizes the fact that
correlation function is a set of normally distributed variables,
X i , with fGn statistics is given by
␳ 共 i 兲 ⫽ 21 兩 i⫹1 兩 22H ⫹ 21 兩 i⫹1 兩 2H ⫺ 兩 i 兩 2H .
共9兲
As discussed by Percival, after fast Fourier transforming
␳ (i), the power spectrum can be operated on and combined
with independent, identically distributed Gaussian random
variables to obtain randomized spectral amplitudes, then reverse transformed to generate a time series with fGn statistics and arbitrary, chosen values of H. Time series with fractional Brownian motion statistics can then be constructed by
integrating the fGn series.
Reflectometer data were acquired from a set of O-mode,
fixed frequency homodyne systems located on the outboard
midplane of DIII-D.15 Data presented here were digitized at
800 ksamples/s, and typically cover many hundreds of milliseconds of the discharge. Local autocorrelation times were
on the order of 20 ␮s.
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1314
Phys. Plasmas, Vol. 9, No. 4, April 2002
FIG. 1. R/S vs time lag of 100 000 point simulated data record with H
⫽0.7 for scrambled and unscrambled data.
Gilmore et al.
FIG. 3. Average of q⫽0.5,1.0,1.5, . . . ,5.0 structure functions vs time lag of
simulated data record with H⫽0.7 for scrambled and unscrambled data.
Structure functions are raised to the 1/q power, so the slope gives H directly.
Slope of unscrambled record⫽0.699. Slopes of scrambled records ⬃0.5 for
N⬎⌬N.
III. ANALYSIS OF DATA
In order to demonstrate the relationship between a high
value of the Hurst parameter, H, and long-time correlation,
the test proposed by the authors of Ref. 6 has been applied to
a 100 000 point simulated time series of fGn data with H
⫽0.7. The data have been broken into blocks of length ⌬N
⫽10, 50, 100, and 500 points, and the blocks scrambled
randomly. The scrambled and unscrambled data were then
used to calculate R/S versus time lag, ␶, as well as the autocorrelation function, calculated by the method proposed by
Carreras et al.16
The results for the R/S and autocorrelation functions are
shown in Figs. 1 and 2, respectively. In Fig. 1 it can be seen
that there are clear changes in the slopes of the R/S curves
with breakpoints, N break , at about five times the block length,
⌬N. In each case of randomized data, H⫽0.49– 0.51 is obtained by calculating the slopes of the R/S function in the lag
range larger than N break . In contrast, the unscrambled data
FIG. 2. Normalized autocorrelation function of 100 000 point simulation
data record, with H⫽0.7, scrambled and unscrambled.
gives H⫽0.70 within this scaling range. This is consistent
with the absence of long-time correlation in the case of
scrambled data, as expected.
As shown in Fig. 2, the extended tail present in the autocorrelation function of the unscrambled data is removed
from the scrambled data cases, and correlation disappears at
time lags greater than approximately the block length ⌬N,
implying that scales larger than ⌬N have been decorrelated
by scrambling. These results clearly demonstrate that the
high value of H is closely related to long-time correlations,
not local correlation.
It might be assumed that the breakpoint in the R/S plots
of the scrambled data should occur at the block size, ⌬N,
since a random scrambling of such blocks removes any correlations on scales larger than ⌬N 共cf. Fig. 2兲. In fact, if
structure function analysis is used, this is precisely what is
found. For example, the average of the qth-order structure
functions, S q (n), q⫽0.5,1.0,1.5, . . . ,5, is plotted versus lag
n in Fig. 3, again for the original and scrambled data record
of Figs. 1 and 2. Note that in this plot, each S q ( ␶ ) is raised to
the 1/q power, so that the slope gives H directly. It can be
seen that the breakpoint in the slope occurs at the block size,
⌬N, in each case, as expected. Again, the Hurst exponents
calculated over the range above the breakpoints are H
⫽0.49– 0.51 for the scrambled data, which is consistent with
the results estimated by the R/S analysis.
Naturally, the question arises as to why the R/S plot
gives a breakpoint at several times ⌬N, while the SF breaks
at ⌬N. In order to understand this behavior, the R/S and SF
algorithms need to be examined in detail. Both algorithms
for estimating H share a common structure: subdividing the
data into nonoverlapping or overlapping bins, calculating
single lag statistics 共scaling at a given lag ␶兲, then calculating
transcale statistics 共slope vs lag兲. But the details of the algorithms are different. For R/S analysis, it can be seen from
Eq. 共5兲 that in order to obtain the single scale statistics 共i.e.,
the scaling at one lag or bin size, n), the ‘‘local statistics’’
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Phys. Plasmas, Vol. 9, No. 4, April 2002
Investigation of rescaled range analysis . . .
1315
FIG. 5. Breakpoint lag normalized to ⌬N vs Hurst exponent from R/S
共solid line兲 and SF 共dashed line兲 calculations from simulated data for various
values of ⌬N, ⌬N⫽10 共circles兲, ⌬N⫽50 共squares兲, and ⌬N⫽100 共triangles兲.
FIG. 4. Diagrammatic representation of the effect of random block scrambling on R/S and SF computations. Data, W k , is a 128 point record 共detrended兲 with H⫽0.8. 共a兲 Unscrambled data and the case of lag, n⫽64. 共b兲
Scrambled data where blocks ii, iii, and iv have been exchanged and ⌬N
⫽n/2⫽32.
must first be estimated by calculating the range, R, of cumulate deviation from the mean within a single bin of size n.
Then the ‘‘bin-based statistic’’ 共i.e., the scaling at a given bin
size, n, or lag, ␶ ), is estimated by calculating the arithmetic
mean of the R/S values for all bins of a particular scale size,
n. Because the range is a measure of the ‘‘memory’’ or correlation between successive time lags, it will also contain
some information on correlations at scales greater than n. In
the case of scrambled data, the R/S value within a single bin
will be almost the same as that of the unscrambled data if the
scale n is so small as to contain only a few blocks of size
⌬N, since there will be little randomizing of the blocks.
However, if the scale, n, is long enough to contain a large
number of blocks, the long-time correlation in the R/S value
will be removed by a large amount of randomizing within a
single bin of scale n. Moreover, it should be noted that Eq.
共5兲 holds as ␶ →⬁, because of the asymptotic property of the
R/S estimation.3,5 Therefore, in cases of multiple scaling exponents, H, such as in the case of the scrambled data, there
must exist a transition range of several times ⌬N, which
links the two scaling regions.
In contrast, the SF method calculates the relative increment of a function over a scale n instead of the ‘‘memory’’ or
correlation between successive time lags within a single bin,
and uses these in estimating the single scale statistics. The
moments for bins of a particular scale, n, calculated from the
single scale statistics, are essentially the qth order generalized correlation function, and only contain information about
correlation at that scale. In addition, Eq. 共7兲 holds for all ␶
because of the exact scaling property of SFs.11 Therefore, in
the case of scrambled data, the structure function of any
scale greater than ⌬N should be decorrelated by the randomization of blocks.
This difference between the R/S and SF algorithms is
illustrated in Fig. 4. Here, a 128 point fBm data record
共imagined to be part of a much larger total record兲 with H
⫽0.8 is plotted, and the case of lag n⫽64 is considered.
Figure 4共a兲 plots the unscrambled data. For the first bin,
points 1– 64, the range is given by the difference of the
points shown as dots, R⫽W max⫺W min , while the qth order
SF is computed using points A and B 关cf. Eqs. 共6兲 and 共8兲兴. In
Fig. 4共b兲, the data record has been scrambled in blocks of
size ⌬N⫽32, and blocks ii, iii, and iv have been rearranged
as shown. When R/S is computed for the first lag bin 共points
1– 64兲 the range, R, is again given by the difference of the
same points, W max⫺W min . The ratio R/S is therefore unchanged in spite of scrambling. The points W max and W min
will only end up in the same lag bin after scrambling with
some probability less than one, of course, but it can be seen
that on average it will take a random scrambling with ⌬N
significantly less than the bin size, n, for the ‘‘memory’’ of
R/S values to be completely erased. In contrast, the
scrambled SF is computed from points A and C, as shown, so
that a direct measure of the correlation between these points
is computed, and the ‘‘local’’ behavior within block i does
not affect the calculation.
If this is the case, then one might expect that the position
of the breakpoint in the R/S analysis would depend on the
intensity of long-time dependence present in the original
data. This is, in fact, what is seen. Figure 5 plots the breakpoint, relative to ⌬N, for simulated data with H varying
from 0.6 to 0.9. As can be seen, the breakpoint occurs later in
lag for higher values of H for R/S calculations, but is unchanged in the SF case.
Additionally, the same behavior is found in real plasma
fluctuation data, and is therefore not an artifact of the numerically generated data. An example is shown in Figs. 6
and 7, where autocorrelation functions and R/S of reflectometry data from the DIII-D tokamak are plotted. The local
1/e decorrelation time of the turbulence is ␶ L ⬇20 ␮s. In Fig.
7 the unscrambled data can be seen to have one breakpoint at
approximately 30 ␮s 共a somewhat longer lag than ␶ L ), and a
second at approximately 3000– 4000 ␮s. We interpret these
breakpoints as separating the local-correlation, mesoscale,
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1316
Gilmore et al.
Phys. Plasmas, Vol. 9, No. 4, April 2002
FIG. 6. Normalized autocorrelation function of fluctuation reflectometry
data from the DIII-D tokamak for scrambled and unscrambled data. 40 GHz
O-mode homodyne reflectometer channel, shot 103498, 1050–1080 ms.
and global-scale ranges. The process in the range below the
first breakpoint is nonstationary, and the H exponent in this
range is always biased to H⬃1. The region between the two
breakpoints is the scaling range that we seek. Fitting this
range yields H⬇0.77.
It can be seen that the scrambled data exhibit breakpoints at lags several times greater than the scrambling time,
exactly as for the simulated data. Above the breakpoints, H
⬇0.5, and fits become increasingly less accurate as the
breakpoint increases in lag. SF analysis shows a consistent
breakpoint at a lag equal to the scrambling time, just as for
the simulated data cases 共e.g., Fig. 3兲.
Returning to the R/S plot, Fig. 7, it can be seen that if
one wishes to determine H for lag times in the so-called
‘‘mesoscale’’—i.e., the range of scales greater than the local
events 共local correlation兲, but less than ‘‘global’’ time scales
FIG. 7. R/S vs time lag of fluctuation reflectometry data from the DIII-D
tokamak for scrambled and unscrambled data. 40 GHz O-mode homodyne
reflectometer channel, shot 103498, 1050–1080 ms.
characteristic of the entire process—then the R/S slope
should be determined for times greater than the breakpoint
lag. Since, in the R/S case, the breakpoint occurs at several
times ␶ L , the lower bound on the fitting range should always
be several times ␶ L . In addition, there needs to be a sufficient range of lags, relative to ␶ L , over which to perform the
fit. When we adhere to these guidelines, we are able to correctly reproduce any numerically simulated value of H we
choose. In the SF method, of course, a lower bound lag of
⌬N is acceptable, since the breakpoint occurs at ⌬N when
SFs are used.
In Fig. 3 of Ref. 6, it appears that this fitting is at issue.
It seems that in performing linear fits to determine H, lags
less than the breakpoint were included, giving erroneous values of H. Furthermore, the total length of the data record was
only around 100␶ L , as discussed previously, which can lead
to errors since estimates of R/S are degraded as n approaches
the total record length, N, as pointed out by van Milligan
et al.17
When the same relative amount of data (100␶ L ) as in
Ref. 6 is taken for the data of Fig. 1, and the scaling fit is
performed in a range down to ␶ L , the trend of ‘‘H’’ vs ⌬N
shown in Fig. 4 of Ref. 6 is reproduced. That is, there is an
apparent saturation of H with ⌬N. The reason is simply because the effects of local correlations are being included by
encompassing lags below the breakpoint.
IV. CONCLUSIONS
In conclusion, we find the following. The rescaled range
method allows the accurate determination of the Hurst exponent of time records that have been constructed numerically
with a selected value of H. The self-similarity scaling, H,
must be determined within the appropriate range of time lags
that excludes both local correlations and global scaling
events. Furthermore, there must be a sufficient scaling range
to give an accurate fit within this range. In practice, this
means that there must be a long enough data record relative
to the local autocorrelation time, ␶ L . For example, given
that the R/S breakpoint occurs at ␶ ⭐10␶ L , a time record of
at least 1000␶ L would yield a scaling region of at least two
decades 共assuming that 1000␶ L is less than any global scaling time兲.
When the data are shuffled into blocks of length ⌬N, a
breakpoint in the R/S scaling occurs at 5 – 10⌬N. The position of the breakpoint depends on the level of persistence, as
shown in Fig. 5. This seems to support the statement by van
Milligan et al.17 that ‘‘This fact that the position of the ‘break
point’ is proportional to ⌬N is a strong indication of the
existence of long-range correlations . . . .’’ The breakpoint
occurrence at lags of 5 – 10⌬N, rather than ⌬N in the R/S
method has to do with the local scaling behavior that exists
at short lags, as discussed previously. The structure function
method, in contrast, yields a breakpoint at ⌬N.
Finally, the same behavior is found when applying rescaled range and structure function analysis to real fluctuation
data from reflectometry on the DIII-D tokamak. We therefore
conclude that the Hurst parameter, derived from R/S analysis
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Phys. Plasmas, Vol. 9, No. 4, April 2002
共or SFs兲, is indicative of persistence in plasma fluctuation
data sets.
ACKNOWLEDGMENTS
This work was supported by the U.S. National Science
Foundation under Grant No. 0078372, and the U.S. DOE
under Grant No. DE-FG03-99ER54527.
1
D. E. Newman, B. A. Carreras, P. H. Diamond, and T. S. Hahm, Phys.
Plasmas 3, 1858 共1996兲.
2
J. A. Krommes, Phys. Plasmas 7, 1752 共2000兲.
3
B. A. Carreras, B. Ph. van Milligen, M. A. Pedrosa et al., Phys. Plasmas 5,
3632 共1998兲.
4
H. E. Hurst, Trans. Am. Soc. Civ. Eng. 116, 770 共1951兲.
5
B. B. Mandelbrot and J. R. Willis, Water Resour. Res. 4, 909 共1968兲.
6
G. Wang, G. Antar, and P. Devynck, Phys. Plasmas 7, 1181 共2000兲.
7
G. Antar, P. Devynck, and G. D. Wang, Phys. Plasmas 7, 5269 共2000兲.
Investigation of rescaled range analysis . . .
1317
8
A. Davis, A. Marshak, W. Wiscombe, and R. Calahan, J. Geophys. Res.
99, 8055 共1994兲.
9
M. J. Cannon, D. B. Percival, D. C. Caccia, G. M. Raymond, and J. B.
Bassingthwiaghte, Physica A 241, 606 共1997兲.
10
J. B. Bassingthwaighte and G. M. Raymond, Ann. Biomed. Eng. 23, 491
共1995兲.
11
P. Flanrin, IEEE Trans. Inf. Theory 35, 197 共1989兲.
12
J. Beren, Statistics for Long-Memory Processes 共Chapman and Hall, New
York, 1994兲, pp. 41– 45.
13
R. B. Davies and D. S. Harte, Biometrika 74, 95 共1987兲.
14
D. B. Percival, Comput. Sci. Statistics 24, 534 共1992兲.
15
E. J. Doyle, T. Lehecka, N. C. Luhmann, Jr., W. A. Peebles, R. Philipona,
K. H. Burrell, R. J. Groebner, H. Matsumoto, and T. H. Osborne, Rev. Sci.
Instrum. 61, 3016 共1990兲.
16
B. A. Carreras, D. E. Newman, B. P. van Milligen, and C. Hidalgo, Phys.
Plasmas 6, 485 共1999兲.
17
B. Ph. van Milligan, C. Hidalgo, and B. A. Carreras, Phys. Plasmas 7,
5267 共2000兲.
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