PHYSICAL REVIEW E 76, 056703 共2007兲
Algorithm to estimate the Hurst exponent of high-dimensional fractals
Anna Carbone
Physics Department, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
共Received 19 June 2007; published 13 November 2007兲
We propose an algorithm to estimate the Hurst exponent of high-dimensional fractals, based on a generalized
high-dimensional variance around a moving average low-pass filter. As working examples, we consider rough
surfaces generated by the random midpoint displacement and by the Cholesky-Levinson factorization algorithms. The surrogate surfaces have Hurst exponents ranging from 0.1 to 0.9 with step 0.1, and different sizes.
The computational efficiency and the accuracy of the algorithm are also discussed.
DOI: 10.1103/PhysRevE.76.056703
PACS number共s兲: 05.10.⫺a, 05.40.⫺a, 05.45.Df, 68.35.Ct
I. INTRODUCTION
The scaling properties of random curves and surfaces can
be quantified in terms of the Hurst exponent H, a parameter
defined in the framework of the fractional Brownian walks
introduced in 关1兴. A fractional Brownian function f共r兲 : Rd
2
→ R, is characterized by a variance ␴H
,
␴H2 = 具关f共r + ␭兲 − f共r兲兴2典 ⬀ 储␭储␣
with
␣ = 2H,
with r = 共x1 , x2 , . . . , xd兲, ␭ = 共␭1 , ␭2 , . . . , ␭d兲,
= 冑␭21 + ␭22 + ¯ + ␭2d; a power spectrum SH,
S H ⬀ 储 ␻ 储 −␤
with
␤ = d + 2H,
and
共1兲
储␭储
共2兲
with ␻ = 共␻1 , ␻2 , . . . , ␻d兲 the angular frequency, 储␻储
= 冑␻21 + ␻22 + ¯ + ␻2d; and a number of objects NH of characteristic size ⑀ needed to cover the fractal,
NH ⬀ ⑀−D
with
D = d + 1 − H,
共3兲
D being the fractal dimension of f共r兲. The Hurst exponent
ranges from 0 to 1, taking the values H = 0.5, H ⬎ 0.5, and
H ⬍ 0.5, respectively for uncorrelated, correlated, and anticorrelated Brownian functions.
The application of fractal concepts, through the estimate
of H, has been proven useful in a variety of fields. For example, in d = 1, heartbeat intervals of healthy and sick hearts
are discriminated on the basis of the value of H 关2,3兴; the
stage of financial market development is related to the correlation degree of return and volatility series 关4兴; coding and
noncoding regions of genomic sequences have different correlation degrees 关5兴; and climate models are validated by
analyzing long-term correlation in atmospheric and oceanographic series 关6,7兴. In d 艌 2 fractal measures are used to
model and quantify stress induced morphological transformation 关8兴; isotropic and anisotropic fracture surfaces
关9–13兴; static friction between materials dominated by hard
core interactions 关14兴; diffusion 关15,16兴 and transport 关17,18兴
in porous and composite materials; mass fractal features in
wet or dried gels 关19兴 and in physiological organs 共e.g., lung兲
关20兴; hydrophobicity of surfaces with hierarchic structure undergoing natural selection mechanism 关21兴 and solubility of
nanoparticles 关22兴; and digital elevation models 关23兴 and
slope fits of planetary surfaces 关24兴.
A number of fractal quantification methods—based on
Eqs. 共1兲–共3兲 or on variants of these relationships—such as
rescaled range analysis 共R / S兲, detrended fluctuation analysis
1539-3755/2007/76共5兲/056703共7兲
共DFA兲, detrending moving average analysis 共DMA兲, and
spectral analysis, have been thus proposed to accomplish accurate and fast estimates of H in order to investigate correlations at different scales in d = 1. A comparatively small
number of methods able to capture spatial correlations, operating in d 艌 2, have been proposed so far 关25–32兴. This
work is addressed to develop an algorithm to estimate the
Hurst exponent of high-dimensional fractals and thus is intended to capture scaling and correlation properties over
space. The proposed method is based on a generalized highdimensional variance of the fractional Brownian function
around a moving average. In Sec. II, we report the relationships holding for fractals with arbitrary dimension. It is argued that the implementation can be carried out in directed
or isotropic mode. We show that the detrending moving average 共DMA兲 method 关30–32兴 is recovered for d = 1. In Sec.
III, the feasibility of the technique is proven by implementing the algorithm on rough surfaces—with different size N1
⫻ N2 and Hurst exponent H—generated by the random midpoint displacement 共RMD兲 and by the Cholesky-Levinson
factorization 共CLF兲 methods 关33,34兴. The generalized variance is estimated over subarrays n1 ⫻ n2 with different size
共“scales”兲 and then averaged over the whole fractal domain
N1 ⫻ N2. This feature reduces the bias effects due to nonstationarity with an overall increase of accuracy—compared to
the two-point correlation function, whose average is calculated over all the fractal. Furthermore—compared to the twopoint correlation function, whose implementation is carried
out along one-dimensional lines 共e.g., for the fracture problem, the two-point correlation functions are measured along
the crack propagation direction and the perpendicular one兲,
the present technique is carried out over d-dimensional structures 共e.g., squares in d = 2兲. In Sec. IV, we discuss the accuracy and range of applicability of the method.
II. METHOD
In order to implement the algorithm, the generalized vari2
is introduced as follows:
ance ␴DMA
N −m
2
␴DMA
=
N −m
N −m
2
2
d
d
1 1 1
¯ 兺 关f共i1,i2, . . . ,id兲
兺
兺
N i1=n1−m1 i2=n2−m2
id=nd−md
− f̃ n1,n2,. . .,nd共i1,i2, . . . ,id兲兴2 ,
共4兲
where f共i1 , i2 , . . . , id兲 = f共i兲 is a fractional Brownian function
056703-1
©2007 The American Physical Society
PHYSICAL REVIEW E 76, 056703 共2007兲
ANNA CARBONE
defined over a discrete d-dimensional domain, with maximum sizes N1 , N2 , . . . , Nd. It is i1 = 1 , 2 , . . . , N1, i2
= 1 , 2 , . . . , N2 , . . ., id = 1 , 2 , . . . , Nd. n = 共n1 , n2 , . . . , nd兲 defines
the subarrays ␯d of the fractal domain with maximum values
n1 max = max兵n1其, n2 max = max兵n2其 , . . .. nd max = max兵nd其; m1
= int共n1␪1兲, m2 = int共n2␪2兲 , . . ., md = int共nd␪d兲 and ␪1, ␪2, . . .␪d
are parameters ranging from 0 to 1; N = 共N1 − n1 max兲共N2
− n2 max兲 ¯ 共Nd − nd max兲. The function f̃ n1,n2,. . .,nd共i1 , i2 , . . . , id兲
= f̃ is given by
f̃ n1,n2,. . .,nd共i1,i2, . . . ,id兲
1
=
n 1n 2 ¯ n d
N −m
n1−1−m1 n2−1−m2
兺 兺
k =−m k =−m
1
1
2
2
␴DMA
=
¯
2
nd−1−md
⫻
兺
k =−m
d
will be further clarified in Sec. III, where the algorithm is
implemented over fractal surfaces with different sizes. For
2
each subarray ␯d, the corresponding value of ␴DMA
is calculated and finally plotted on log-log axes.
To elucidate the way the algorithm works, in the following we consider its implementation for d = 1 and d = 2. The
case d = 1 reduces to the detrending moving average 共DMA兲
method already used for long-range correlated time series
关30–32兴.
One-dimensional case. By posing d = 1 in Eq. 共4兲, one
obtains
f共i1 − k1,i2 − k2, . . . ,id − kd兲,
共5兲
d
which is an average of f calculated over the subarrays ␯d.
Equations 共4兲 and 共5兲 are defined for any value of
n1 , n2 , . . . , nd and for any shape of the subarrays, however, it
is preferable to choose subarrays with n1 = n2 = ¯ · = nd to
avoid spurious directionality in the results. The generalized
2
varies as 共冑n21 + n22 + ¯ + n2d兲2H as a consevariance ␴DMA
quence of the property 共1兲 of the fractional Brownian functions.
Upon variation of the parameters ␪1 , ␪2 , . . . , ␪d in the
range 关0,1兴, the indexes i1 , i2 , . . . , id and k1 , k2 , . . . , kd of the
sums in Eqs. 共4兲 and 共5兲 are accordingly set within ␯d. In
particular, 共i1 , i2 , . . . , id兲 coincides, respectively, with 共a兲 one
of the vertices of ␯d for ␪1 = ␪2 = ¯ = ␪d = 0 and ␪1 = ␪2 = ¯
= ␪d = 1 or 共b兲 the center of ␯d for ␪1 = ␪2 = ¯ = ␪d = 1 / 2. It is
worthy of note that the choice ␪1 = ␪2 = ¯ = ␪d = 1 / 2 corresponds to the isotropic implementation of the algorithm,
while ␪1 = ␪2 = ¯ = ␪d = 0 and ␪1 = ␪2 = ¯ = ␪d = 1 correspond
to the directed implementation. For example, in d = 2, the
isotropic implementation implies that the variance defined by
Eq. 共4兲 is referred to a moving average f̃ calculated over
squares n1 ⫻ n2 whose center is 共i1 , i2兲. Conversely, the directed implementation implies that the function f̃ is calculated over squares n1 ⫻ n2 with one of the vertices in 共i1 , i2兲.
The directed mode is of interest to estimate H in fractals with
preferential growth direction, e.g., biological tissues 共lung兲,
epitaxial layers, and crack propagation in fracture 共anisotropic fractals兲. If the fractal is isotropic and the accuracy is
a priority, the parameters ␪1 , ␪2 , . . . , ␪d should be preferably
taken equal to 1 / 2 to achieve the most precise estimate of H.
The dependence of the algorithm on ␪ for d = 1 has been
discussed in 关32兴.
In order to calculate the Hurst exponent, the algorithm is
implemented through the following steps. The moving average f̃ is calculated for different subarrays ␯d, by varying
n1 , n2 , . . . , nd from 2 to the maximum values
n1 max , n2 max , . . . , nd max. The values n1 max , n2 max , . . . , nd max
depend on the maximum size of the fractal domain. In order
to minimize the saturation effects due to finite size, it should
be n1 max Ⰶ N1; n2 max Ⰶ N2 ; . . . ; nd max Ⰶ Nd. These constraints
1
1
1
兺 关f共i1兲 − f̃ n1共i1兲兴2 ,
N1 − n1 max i1=n1−m1
共6兲
where N1 is the length of the sequence, n1 is the sliding
window, and n1 max = max兵n1其 Ⰶ N1. The quantity m1
= int共n1␪1兲 is the integer part of n1␪1 and ␪1 is a parameter
ranging from 0 to 1. The relationship 共6兲 defines a generalized variance of the sequence f共i1兲 with respect to the function f̃ n1共i1兲 as follows:
f̃ n1共i1兲 =
1
n1
n1−1−m1
兺
f共i1 − k1兲,
共7兲
k1=−m1
which is the moving average of f共i1兲 over each sliding window of length n1. The moving average f̃ n1共i1兲 is calculated
for different values of the window n1, ranging from 2 to the
2
is then calculated
maximum value n1 max. The variance ␴DMA
according to Eq. 共6兲 and plotted as a function of n1 on loglog axes. The plot is a straight line, as expected for a power2
on n1 as follows:
law dependence of ␴DMA
2
␴DMA
⬃ n2H
1 .
共8兲
Equation 共8兲 allows one to estimate the scaling exponent H
of the series f共i1兲. Upon variation of the parameter ␪1 in the
range 关0,1兴, the index k1 in f̃ n1共i1兲 is accordingly set within
the window n1. In particular, ␪1 = 0 corresponds to the average f n1共i1兲 over all the points to the left of i1 within the
window n1; ␪1 = 1 corresponds to the average f n1共i1兲 over all
the points to the right of i1 within the window n1; ␪1 = 21
corresponds to the average f n1共i1兲 with the reference point in
the center of the window n1.
Two-dimensional case. For d = 2, the generalized variance
defined by Eq. 共4兲 writes
2
␴DMA
=
1
共N1 − n1 max兲共N2 − n2 max兲
N1−m1
⫻
兺
N2−m2
兺
i1=n1−m1 i2=n2−m2
with f̃ n1,n2共i1 , i2兲 given by
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关f共i1,i2兲 − f̃ n1,n2共i1,i2兲兴2 ,
共9兲
ALGORITHM TO ESTIMATE THE HURST EXPONENT OF …
1
f̃ n1,n2共i1,i2兲 =
n 1n 2
PHYSICAL REVIEW E 76, 056703 共2007兲
n1−1−m1 n2−1−m2
兺
兺
f共i1 − k1,i2 − k2兲. 共10兲
k1=−m1 k2=−m2
The average f̃ is calculated over subarrays with different size
n1 ⫻ n2. The next step is the calculation of the difference
f共i1 , i2兲 − f̃ n1,n2共i1 , i2兲 for each subarray n1 ⫻ n2. A log-log plot
2
,
of ␴DMA
2
␴DMA
⬃ 关冑n21 + n22兴2H ⬃ sH
共11兲
s = n21 + n22,
yields a straight line with slope H.
as a function of
Depending upon the values of the parameters ␪1 and ␪2,
entering the quantities m1 = int共n1␪1兲 and m2 = int共n2␪2兲 in
Eqs. 共9兲 and 共10兲, the position of 共k1 , k2兲 and 共i1 , i2兲 can be
varied within each subarray. 共i1 , i2兲 coincides with a vertex of
the subarray if 共i兲 ␪1 = 0, ␪2 = 0; 共ii兲 ␪1 = 0, ␪2 = 1; 共iii兲 ␪1 = 1,
␪2 = 0; and 共iv兲 ␪1 = 1, ␪2 = 1 共directed implementation兲. The
choice ␪1 = ␪2 = 1 / 2 corresponds to take the point 共i1 , i2兲 coinciding with the center of each subarray n1 ⫻ n2 共isotropic
implementation兲 关41兴.
III. RESULTS
In order to test feasibility and robustness of the proposed
method, synthetic rough surfaces with assigned Hurst exponents have been generated by the random midpoint displacement 共RMD兲 and by the Cholesky-Levinson factorization
共CLF兲 algorithms 关33,34兴. The widespread use of the RMD
algorithm is based on the trade-off of its fast, simple, and
efficient implementation to the limited accuracy, especially
for H Ⰶ 0.5 and H Ⰷ 0.5. Conversely, the Cholesky-Levinson
factorization algorithm is one of the most accurate techniques to generate 1D and 2D fractional Brownian functions,
at the expense of a more complex implementation structure
关42兴.
2
as a function of s are
In Fig. 1, the log-log plots of ␴DMA
shown for the synthetic fractal surfaces generated by the
RMD 共circles兲 and by the CLF method 共squares兲. The surfaces have Hurst exponents Hin ranging from 0.1 to 0.9 with
step 0.1. The domain sizes are, respectively, N1 ⫻ N2 = 256
⫻ 256 共a兲, N1 ⫻ N2 = 1024⫻ 1024 共b兲, and N1 ⫻ N2 = 4096
⫻ 4096 共c兲. The dashed lines show the behavior that should
be exhibited by variances varying exactly as sHin over the
2
as a function s are
entire range of scales. The plots of ␴DMA
in good agreement with the behavior expected on the basis of
Eq. 共11兲. The quality of the fits is higher for the surfaces
generated by the CLF method, confirming that the RMD algorithm synthesizes less accurate fractals. By comparing the
results of the simulation 共symbols兲 to the straight lines corresponding to full linearity over the whole range 共dashed兲,
deviations from the full linearity can be observed especially
for the small surfaces at the extremes of the scale. A plot of
the slopes for the fractal surfaces generated by the CLF algorithm is shown in Fig. 2 for different sizes of the fractal
domain. A detailed discussion of the origin of the deviations
at low and large scales is reported in the Sec. IV.
Finally, we show three examples of digital images currently mapped to fractal surfaces with reference to the color
intensity, i.e., to the level of red, green, and blue 共RGB兲. The
2
FIG. 1. 共Color online兲 Log-log plot of ␴DMA
for fractal surfaces,
respectively, with size N1 ⫻ N2 = 256⫻ 256 共a兲, N1 ⫻ N2 = 1024
⫻ 1024 共b兲, and N1 ⫻ N2 = 4096⫻ 4096 共c兲. The data refer to fractal
surfaces generated by the RMD 共black circles兲 and by the CLF
共blue squares兲 methods. The Hurst exponent Hin—input of the
RMD and CLF algorithms—varies from 0.1 to 0.9 with step 0.1.
The results correspond to the isotropic implementation, i.e., with
the parameters ␪1 = ␪2 = 1 / 2 in the Eq. 共9兲. The dashed lines represent the behavior expected for full linearity, i.e., the log-log plot of
curves varying as sHin. It is worthy of note that the CLF results are
much closer to the full-linear behavior compared to the RMD ones.
Hurst exponents estimated by the proposed method are, respectively, H = 0.1 共a兲, H = 0.5 共b兲, and H = 0.9 共c兲 for the
images in Fig. 3.
IV. DISCUSSION
The proposed algorithm is characterized by short execution time and ease of implementation. By considering the
case d = 2, the function f̃ n1,n2共i1 , i2兲 is indeed simply obtained
by summing the values of f共i1 , i2兲 over each subarray n1
⫻ n2. Then the sum is updated at each step by adding the last
and discarding the first row 共column兲 of each sliding array
n1 ⫻ n2. For higher dimensions, the sum is updated at each
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PHYSICAL REVIEW E 76, 056703 共2007兲
ANNA CARBONE
FIG. 2. 共Color online兲 Plot of the values of H obtained by linear
fit of the data shown in Figs. 1共a兲–1共c兲 obtained by implementing
the proposed algorithm on the fractal surfaces generated by the
Cholesky-Levinson factorization method. The dashed line represents the ideal behavior: H = Hin.
step by adding and discarding a d − 1-dimensional set of each
array n1 ⫻ n2 ⫻ ¯ ⫻ nd. The algorithm does not use arbitrary
parameters, the computation simply relying on averages of f.
We will now argue on the origin of the deviations at small
and at large scales.
FIG. 3. 共Color online兲. Cloudy sky images, respectively, with
Hurst exponent H = 0.1 共a兲, H = 0.5 共b兲, and H = 0.9 共c兲. Such heterogeneous surfaces can be represented as fractals by mapping their
color intensity in terms of RGB content.
FIG. 4.
HT共␻1 , ␻2兲.
共Color online兲 Plot of the transfer function
Deviations occurring at large scales. The deviations from
the linearity at large scales, leading to the saturation of the
2
␴DMA
values, are due to finite size effects. The small surfaces
do not contain enough data to make the evaluation of the
scaling law over the subarrays statistically meaningful. By
comparing the data in Figs. 1共a兲–1共c兲, one can note that the
saturation effect reduces upon increasing the size N1 ⫻ N2 of
the fractal surface. The finite size effects become negligible
when the conditions n1 max Ⰶ N1; n2 max Ⰶ N2 ; . . . ; nd max Ⰶ Nd
are fulfilled.
Deviations occurring at small scales. The deviations occurring at low scales are related to the departure of the lowpass filter from the ideality. This problem also occurs with
one-dimensional fractals 共time series兲 resulting in the quite
generally reported overestimation of H in anticorrelated signals and underestimation of H in correlated signals 关35–39兴.
We will discuss the origin of these deviations by means of
the filter transfer function HT共␻兲 关40兴. The algorithm is
based on a generalized variance of the function f with respect to f̃. The function f̃ is the output of a low-pass filter
driven by f, with impulse response a box-car function. In the
Appendix, the transfer function HT共␻兲 of f̃ is explicitly calculated and shown in Fig. 4 for d = 2. For an ideal low-pass
filter, the transfer function should be one or zero, respectively, at frequencies lower or higher than the cutoff frequency. However, in real low-pass filters, at frequencies
lower than the cutoff frequency, all the components of the
signal suffer some attenuation but ␻ = 0. The cutoff frequencies of HT共␻兲 are ␻i = ␲ / ␶i, i.e., the first zeroes of the functions sin ␻i␶i / ␻i␶i in the Eq. 共A4兲. Moreover, in real filters,
at frequencies higher than ␲ / ␶i, due to the presence of the
sidelobes, components of the signals lying in the bands
共␲ / ␶i , 2␲ / ␶i兲 ; 共2␲ / ␶i , 3␲ / ␶i兲 ; . . .., are not fully filtered out.
As a result, the function f̃ contains 共a兲 less components with
frequency lower than ␻i = ␲ / ␶i and 共b兲 more components
with frequency higher than ␻i = ␲ / ␶i compared to what
would be expected with an ideal low-pass filter. The lack of
low-frequency components depends on the central lobe,
while the excess of high-frequency components depends on
the sidelobes. The excess of high-frequency components results in the decrease of the difference f − f̃, i.e., in the de2
and, finally, in the increase of the slope of the
crease of ␴DMA
log-log plot. Conversely, the lack of low-frequency compo-
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ALGORITHM TO ESTIMATE THE HURST EXPONENT OF …
PHYSICAL REVIEW E 76, 056703 共2007兲
TABLE I. Slopes HI, HII, HIII, and relative errors ⌬HI, ⌬HII, ⌬HIII of the curves plotted in Fig. 1共b兲
共squares兲 with N1 ⫻ N2 = 1024⫻ 1024. The slopes have been calculated by linear fit, respectively, over the
ranges 10艋 s 艋 100 共HI兲, 10艋 s 艋 1000 共HII兲, and 10艋 s 艋 10000 共HIII兲.
Hin
HI
⌬HI
HII
⌬HII
HIII
⌬HIII
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.1346
0.2272
0.3233
0.4205
0.5178
0.6171
0.7185
0.8207
0.9253
+3.46⫻ 10−1
+1.36⫻ 10−1
+7.77⫻ 10−2
+5.12⫻ 10−2
+3.56⫻ 10−2
+2.85⫻ 10−2
+2.64⫻ 10−2
+2.58⫻ 10−2
+2.81⫻ 10−2
0.1073
0.2050
0.2995
0.3970
0.4973
0.5973
0.6956
0.7999
0.8999
+7.30⫻ 10−2
+2.50⫻ 10−2
−1.67⫻ 10−3
−7.50⫻ 10−3
−5.40⫻ 10−3
−4.50⫻ 10−3
−6.29⫻ 10−3
−1.25⫻ 10−4
−1.11⫻ 10−4
0.0718
0.1700
0.2716
0.3691
0.4752
0.5617
0.6770
0.7659
0.8679
−2.822⫻ 10−1
−1.500⫻ 10−1
−9.467⫻ 10−2
−7.725⫻ 10−2
−4.960⫻ 10−2
−6.383⫻ 10−2
−3.286⫻ 10−2
−4.263⫻ 10−2
−3.567⫻ 10−2
nents results in the increase of the difference f − f̃, i.e., in the
2
and, finally, in the decrease of the slope of
increase of ␴DMA
the log-log plot. The two effects are more relevant with
smaller values of the scales, when the filter nonideality is
greater. Moreover, as one can deduce from Eqs. 共2兲 and 共A6兲,
the effect of the sidelobes dominates in high-frequency rich
fractals with H ⬍ 0.5, while the effect of the central lobe is
dominant in low-frequency rich fractals with H ⬎ 0.5.
In order to gain further insight in the above theoretical
arguments, we report in Table I the slopes HI, HII, and HIII of
the curves 共squares兲 plotted in Fig. 1共b兲 over different
ranges. The slopes have been calculated by linear fit, respectively, over the ranges 10艋 s 艋 100 共HI兲, 10艋 s 艋 1000 共HII兲,
and 10艋 s 艋 10000 共HIII兲. The relative errors ⌬H = 共H
− Hin兲 / Hin are given, respectively, in the third, fifth, and seventh columns. The slope HI is greater than the expected value
Hin. The slope HII is overestimated for H = 0.1 and H = 0.2
and underestimated for H ⬎ 0.2. The slope HIII is underestimated since the effects of the finite size of the fractal domain
play a dominant role.
We address the question if the artifacts due to the filter
nonideality described above might be corrected somehow. In
the remaining of this section, we will thus consider the use of
windows whose general effect is to increase the width of the
central lobe while reducing those of the sidelobes of the
function HT共␻兲 共a detailed description of these methods can
be found in 关40兴兲. By restricting our discussion to the present
technique, the correction is performed by using the following
variant of the relationship 共5兲:
f̃ n쐓 ,n
1 2,. . .,nd
values are summed together and weighted by ␣. In Fig. 5, we
2
/ sHin obtained by implementing the alshow the ratio ␴DMA
gorithm, respectively, with the function f̃ 共solid lines兲 and f̃ 쐓
共dashed lines兲 in the range 10艋 s 艋 100. To avoid the overlap
in the scaled variance obtained for different values of Hin, the
2
␴DMA
/ sHin has been shifted. The plots are shown in logarithmic scales to allow the correction by direct comparison with
2
/ sHin is
the data plotted in log scales in Fig. 1. The ratio ␴DMA
noticeably closer to a constant when the implementation is
performed with f̃ 쐓, with a corresponding reduction of two
orders of magnitude in the relative error ⌬HI of the slope HI.
V. CONCLUSION
We have put forward an algorithm to estimate the Hurst
exponent of fractals with arbitrary dimension, based on the
2
defined by Eq.
high-dimensional generalized variance ␴DMA
共4兲.
The methods currently used to estimate the Hurst exponent of high-dimensional fractals are based on: 共i兲 1-d twopoint correlation and structure functions operated along dif-
共i1,i2, . . . ,id兲 = 共1 − ␣兲f n1,n2,. . .,nd共i1,i2, . . . ,id兲
+ ␣ f̃ n1,n2,. . .,nd共i1 − 1,i2 − 1, . . . ,id − 1兲,
共12兲
where ␣ = n1n2 , . . . , nd / 关共n1 + 1兲共n2 + 1兲 ¯ 共nd + 1兲兴. Equation
共12兲 reduces for d = 1 to the exponentially weighted moving
average 共EWMA兲. In practice, the difference between Eq. 共5兲
and Eq. 共12兲 is that the function f̃ 쐓 places more importance to
the data around the point i1 , i2 , . . . , id. This is achieved by
assigning to the function a weight 共1 − ␣兲, while all the other
2
FIG. 5. Plot of the function ␴DMA
/ sHin, respectively, with f̃ defined by the Eq. 共4兲 共solid lines兲 and f̃ 쐓 defined by the Eq. 共12兲
共dashed lines兲. The data refer to fractal surfaces generated by the
CLF algorithm with N1 ⫻ N2 = 1024⫻ 1024. It can be noted that the
deviations from the constant behavior at small scales are reduced
with f̃ 쐓 implying a corresponding reduction of the relative error ⌬HI
in the slope HI.
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PHYSICAL REVIEW E 76, 056703 共2007兲
ANNA CARBONE
␪1 = ␪2 = , . . . , = ␪d = 0. Equation 共A1兲 can be rewritten as a
convolution integral as follows:
ferent directions, and 共ii兲 high-d Fourier and wavelet
transforms 关9–13,29兴. The advantage of the methods 共i兲 is the
ease of implementation. Their drawback is the limited accuracy due to biases and nonstationarities, being these functions are calculated over the entire fractal domain. The methods 共ii兲 are more accurate, however, their implementation is
complicated especially for data set with limited extension.
2
is scaled, meaning that it is
The generalized variance ␴DMA
calculated over subarrays of the whole fractal domain by
means of the function f̃. The scales are set by the size of the
subarrays n1 ⫻ n2 ⫻ ¯ . ⫻ nd. Therefore, the proposed
method exhibits at the same time 共a兲 ease of implementation,
being based on a variancelike approach and 共b兲 high accuracy, being calculated over scaled subarrays rather than on
the whole fractal domain.
A further important feature of the proposed algorithm is
that it can be implemented “isotropically” or in “directed”
mode to accomplish estimates of H in fractals having preferential growth direction, e.g., biological tissues, epitaxial
layers, or in crack propagation in fracture. The isotropic
implementation is obtained by taking ␪1 = ␪2 = ¯ = ␪d = 1 / 2 in
Eq. 共4兲. This choice implies that the reference point
共i1 , i2 , . . . , id兲 of the moving average lies in the center of each
subarray n1 ⫻ n2 ⫻ ¯ ⫻ nd and thus f̃ is calculated by summing the values of f around 共i1 , i2 , . . . , id兲. Conversely, to
implement the algorithm in a preferential direction 共directed
implementation兲, the reference point must be coincident with
one of the extremes of the segment n1, or with one of the
vertices of the square grid n1 ⫻ n2 or of the d-dimensional
array n1 ⫻ n2 ⫻ ¯ ⫻ nd. The directed implementation can be
performed by choosing, for example, ␪1 = ␪2 = ¯ = ␪d = 0.
Further generalizations of the proposed method can be
envisaged for application to the analysis of time-dependent
spatial correlations in d 艌 2.
that is thus d times the function sin ␻i␶i / ␻i␶i.
The Fourier transform F̃ of the function f̃ can be obtained
by means of the following relationship:
APPENDIX: TRANSFER FUNCTION OF f̃
F̃共␻1, ␻2, . . . , ␻d兲 = HT共␻1, ␻2, . . . , ␻d兲F共␻1, ␻2, . . . ␻d兲,
The function f̃, defined by Eq. 共5兲, corresponds to the
discrete form of the integral as follows:
1
f̃共x1,x2, . . . ,xd兲 =
␶ 1␶ 2 ¯ ␶ d
⫻
冕
x2
x2−␶2
冕
x1
x1−␶1
dx2⬘ ¯
dx1⬘
冕
xd
xd−␶d
dxd⬘ f共x1⬘,x2⬘, . . . ,xd⬘兲,
共A1兲
f̃共x1,x2, . . . ,xd兲 =
1
␶ 1␶ 2 ¯ ␶ d
⫻
冕
冕
dx*1U
−⬁
⬁
dx*dU
−⬁
−
⬁
冉冊
x*d
␶d
冉 冊冕
x*1
␶1
⬁
dx*2U
−⬁
冉冊
x*2
¯
␶2
f共x1 − x*1,x2 − x*2, . . . ,xd
x*d兲,
共A2兲
with the convolution kernels given by the boxcar function as
follows:
U共x*i /␶i兲 =
再
1 for 0 ⬍ x*/␶i ⬍ 1
0 elsewhere.
冎
The transfer function can be calculated as follows:
H T共 ␻ 1, ␻ 2, . . . , ␻ d兲 =
1
␶ 1␶ 2 ¯ ␶ d
⫻
冕
␶d
0
冕 冕
␶1
␶2
dx1
0
0
dx2 ¯
dxd exp关− i2␲共␻1x1 + ␻2x2 + ¯
+ ␻dxd兲兴
共A3兲
that can be written as
d
H T共 ␻ 1, ␻ 2, . . . , ␻ d兲 = 兿
i=1
sin ␻i␶i
␻ i␶ i
共A4兲
共A5兲
where F共␻1 , ␻2 , . . . , ␻d兲 is the Fourier transform of the function f共x1 , x2 , . . . , xd兲. The power spectrum S̃ of the function f̃
is given by
S̃共␻1, ␻2, . . . , ␻d兲 = 兩HT共␻1, ␻2, . . . , ␻d兲兩2S共␻1, ␻2, . . . , ␻d兲,
共A6兲
where for the sake of simplicity we have considered the case
where S共␻1 , ␻2 , . . . , ␻d兲 is the power spectrum of the function f共x1 , x2 , . . . , xd兲.
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关41兴 The source and executable files of the proposed algorithm can
be downloaded at www.polito.it/noiselab/utilities
关42兴 We use the CLF algorithm included in the package
FRACLAB downloadable at http://www.irccyn.ec-nantes.fr/
hebergement/FracLab/
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