Evaluation of two numerical methods to measure the Hausdorff dimension of the fractional Brownian motion ∗ Sergei M. Prigarin † Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch of Russian Academy of Sciences Klaus Hahn, and Gerhard Winkler § Institute of Biomathematics and Biometry GSF-National Research Centre for Environment and Health

‡

¶

Abstract The aim of the paper is to study by Monte Carlo simulation statistical properties of two numerical methods (the extended counting method and the variance counting method) developed to estimate the Hausdorff dimension of a time series and applied to the fractional Brownian motion. Key words: fractal set, Hausdorff dimension, extended counting method, variance counting method, generalized Wiener process, fractional Brownian motion. Mathematical Subject Classification: 28A80, 62M10, 65C05.

1

Introduction

Fractals and multifractals are used as mathematical models of different physical objects and social phenomena in various fields of knowledge: biology, ∗

Supported by Russian Presidential Program ”Leading scientific schools” (HIII4774.2006.1) † e-mail: [email protected] ‡ pr. Lavrentieva 6, Novosibirsk, 630090, Russia § e-mail: [email protected], [email protected] ¶ Ingolst¨ adter Landstraße 1, 85764 Neuherberg/M¨ unchen, Germany

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geophysics, financial mathematics, computer science, and others. Let us specify only a few of many research directions were the theory of fractals occupies a significant part: plasma physics, turbulence, porous media, biological systems, fractal antennas, image compression, computer graphics, data traffic, financial markets, functional magnetic resonance imaging, etc. Theoretical concepts such as deterministic and random fractals, multifractal singular measures, fractal dimension and fractional analysis play an important role in the present-day development of mathematics and physics (see, for example, [2, 3, 4, 5, 7, 8, 9, 15]). A significant research problem is the numerical analysis of fractal and multifractal structures, and, in particular, the estimation of the fractal dimension of time series. In spite of the variety of developed numerical methods the problem of their validity remains topical. In our paper we study by the Monte Carlo method properties of two numerical methods designed to measure the Hausdorff dimension of a time series. The first method is a modification of the extended counting method originally destined for arbitrary sets in multidimensional spaces and adapted here for arbitrary time series (see Section 2). The second method is the variance counting method which is appropriate for Gaussian random processes with stationary increments (see Section 3). As testing time series we used realizations of the fractional Brownian motion with Hausdorff dimension in the interval [1, 2). Numerical modeling of the fractional Brownian motion is a non-trivial problem that we shall not discuss here. An essential point is that in Monte Carlo simulation we used a numerical method that is exact in a probabilistic sense (see Section 4). The results of the Monte Carlo simulation and the analysis of the results are presented in Section 5.

2

The extended counting method

The extended counting method [13, 14] was proposed as an alternative to the box counting method to compute the Hausdorff dimension of a compact set in a finite dimensional space. According to the extended counting method the fractal dimension of a set A ⊂ R2 is estimated in the following way. 1. Let us fix a point (x0 , y0 ) ∈ R2 and represent the space R2 as a sum (1) of disjoint “atomic” squares Sij of a small size d1 : 2

R =

+∞ X

(1)

Sij ,

(1)

Sij = [x0 + id1 , x0 + (i + 1)d1 ) × [y0 + jd1 , y0 + (j + 1)d1 ).

i,j=−∞

We mark the atomic squares which contain points of the set A. 2

2. Let us fix a natural number M and consider a set of (not necessarily disjoint) “exploratory” squares (2)

Skn = [x0 + kd1 , x0 + kd1 + d2 ) × [y0 + nd1 , y0 + nd1 + d2 ), where d2 = M d1 , and k, n are integers. Every such an exploratory square (2) (1) Skn consists of M 2 atomic squares Sij of which some may contain points of (2) the set A. For a fixed exploratory square Skn we denote by Nkn the number (1) (2) of atomic squares Sij ⊂ Skn which contain points from A. 3. We find Nmax = maxkn Nkn and compute the value xdim[d1 , d2 ](A) =

log Nmax log Nmax = . log d2 − log d1 log M

(1)

For the proper parameters d1 , d2 this number xdim[d1 , d2 ](A) will be interpreted as an estimator of the fractal dimension of the set A (for details see [13]). To apply the extended counting method it is necessary to fix the parameters: the origin of the fine grid (x0 , y0 ), the size of the atomic squares d1 , and the coefficient M that defines the size of the exploratory squares, d2 = M d1 . Further, we shall call parameter M an exploratory factor. The main point of the extended counting method can be formulated in the following way. The box counting method is applied for many subsets of the fractal set and the maximum of the subsets’ dimensions is taken as the fractal dimension of the set. On the other hand, the box counting method, that is used for the subsets, is extremely simplified (the box counting regression line is built only on the basis of 2 points).

2.1

Realization of the extended counting method for a time series

To apply the extended counting method to a (finite) time series x(ih), i = 1, . . . , N , defined on a grid with step h it is necessary (i) to define the parameters of the method and (ii) to mark the atomic squares which are assumed to be intersected by the graph of the time series. The solution of neither the second problem nor the first one is evident. There is no unique procedure to reproduce a graph of a function having only values on a fixed grid. In our algorithms we simply used the piecewise linear interpolation (another possible approach, for example, is to use self-similar constructions). Moreover, to mark atomic squares intersected by the graph of the time series, it is reasonable to perform a preliminary scaling. This means that the extended

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counting method should be applied not to the time series x(ih) directly but to a time series y(i) = Cx(ih). Therefore, we have to introduce additional parameter C depending on N and the range of values x(ih), i = 1, . . . , N . To choose C we used the following rule: C[maxi x(ih) − minj x(jh)] = Cy . N −1 In that case the graph of the piecewise linear function with nodes (i, y(i)) belongs to a rectangle of size (N − 1) × Cy (N − 1). The constant Cy will be called scaling factor. In numerical experiments we used two variants of the extended counting method (1) applied to the scaled time series y(i), i = 1, . . . , N : the first variant (we shall denote it by method X1) with parameters d1 = 1, x0 = 0.5, y0 = y(1) − 0.5, and the second variant (method X2) with parameters d1 = 2, x0 = 0.5, y0 = y(1) − 0.5. For the second variant the atomic squares are twice larger (the corresponding advise can be found at the beginning of Section 5 in [13]; see, in addition, Section 5.3 in [15]).

3

Local variance-dimension of a random process and the variance counting method

In addition to the extended counting method we consider another algorithm designed to estimate the Hausdorff dimension of a random process. The algorithm is based on the simple idea to exploit the formula for variance of increments, see for example [3], Section 9.4. The variance counting method is based on the fact that (see [1] and [6, 11]) the Hausdorff dimension of a Gaussian random process x0 (t) with zero mean, stationary increments, and a continuous covariance function is equal to 2 − α/2 if E|x0 (t + h) − x0 (t)|2 ∼ |h|α , α ∈ (0, 2] (2) for h → 0 (to be more precise, we should say that the realizations of the process x0 (t) have the Hausdorff dimension 2 − α/2 with probability one). This fact inspires us to introduce the concept of local variance-dimension of a random process x(t) at point t: vdim x(t) = 2 −

V[x(t + γh) − x(t)] 1 lim logγ . 2 h→0 V[x(t + h) − x(t)]

4

(3)

Here V denotes the variance and it is assumed that the limit in formula (3) exists and does not depend on γ > 0. It is obvious, that at least for random processes with stationary increments and property (2) the limit in (3) exists, it does not depend on t and γ, and the value of vdim x(t) is equal to the Hausdorff dimension of the Gaussian process x(t). Actually, the expectation of a process x(t) with stationary increments is a linear function, Ex(t) = a + bt, the Hausdorff dimensions of the processes x(t) and x0 (t) = x(t) − Ex(t) are equal, and lim logγ

h→0

V[x(t + γh) − x(t)] E[x0 (t + γh) − x0 (t)]2 = logγ lim = h→0 E[x0 (t + h) − x0 (t)]2 V[x(t + h) − x(t)] |γh|α = logγ = logγ γ α = α. α |h|

To get a numerical algorithm to compute the (local) variance-dimension of a time series x(i) one can simply substitute estimates for the variances in formula (3). For the processes with stationary increments the local variancedimension does not depend on point t and the standard estimates for the variances can be exploited. In our numerical experiments we used the following formula corresponding to (3) with γ = 2: vdim ≈ 2 −

1 V2 log2 , 2 V1

(4)

where V1 , V2 are estimates of variances. To calculate the variances we applied the unbiased estimates for the increments with unknown average:   N 1 1 X 2 2 V1 = [x(i) − x(i − 1)] − E1 , (5) 1 − N 1−1 N − 1 i=2 1 V2 = 1 − N 1−2



 X 1 2 2 [x(i) − x(i − 2)] − E2 , N − 2 i=3,4,...,N

(6)

N

1 X E1 = [x(i) − x(i − 1)], N − 1 i=2 X 1 E2 = [x(i) − x(i − 2)], N − 2 i=3,4,...,N and for the increments with zero mean: N

1 X V1 = [x(i) − x(i − 1)]2 , N − 1 i=2 5

(7)

V2 =

X 1 [x(i) − x(i − 2)]2 . N − 2 i=3,4,...,N

(8)

Thus, we studied two versions of the variance counting method developed for random processes with stationary increments with unknown and zero average according to formulas (4, 5, 6) and (4, 7, 8).

4

Fractional Brownian motion: properties and simulation

The generalized Wiener process (also called the fractional Brownian motion) of order α is a Gaussian process wα (t), t > 0, with mean zero and correlation function of the following type (see, for example, [4]) Kα (t, s) = Ewα (t)wα (s) = σ 2 (|t|α + |s|α − |t − s|α )/2,

α ∈ (0, 2].

Here σ is an additional parameter that determines the variance of the generalized Wiener process: Vwα (t) = Kα (t, t) = σ 2 |t|α . If α = 1, then it is an “ordinary” Wiener process with correlation function K1 (t, s) = σ 2 min(t, s). For α = 2 we have K2 (t, s) = σ 2 ts and the realizations of the generalized Wiener process are the straight lines: wα (t) = σξt, where ξ is a standard normal variable. The generalized Wiener process has stationary increments (see, for example, [12]): the random process δ(t) = wα (t + h) − wα (t) is Gaussian, stationary, with mean zero, correlation function Eδ(t + τ )δ(t) =

|τ + h|α + |τ − h|α − |τ |α , 2

and spectral density A2 1 (1 − cos λh), π |λ|α+1 6

λ ∈ (−∞, +∞).

For α = 1 the constant A is equal to 1, and in general case it is defined from the equalities )−1 ( )−1 ( Z∞ 1 2 2 απ A2 = (1 − cos λ)dλ = − Γ(−α) cos . π λα+1 π 2 0

The fractional Brownian motion is the process with independent increments only for α = 1 (i.e., only in the case of the “ordinary” Brownian motion). The spectral representations for the correlation function and the process of the fractional Brownian motion have the forms [12]
A2 Kα (t, s) = |t|α + |s|α − |t − s|α /2 = 2π

Z+∞

1 (eiλt − 1) (e−iλs − 1) dλ |λ|α+1

−∞

A2 = π

Z+∞

1  λα+1


 cos λ(t − s) − cos(λt) − cos(λs) + 1 dλ,

0

A wα (t) = √ 2π A =√ π

( Z+∞ 0

1 λ

α+1 2

Z+∞ −∞

1 |λ|

α+1 2

(eiλt − 1) dz(λ) Z+∞

(cos λt − 1) dξ(λ) − 0

λ

1 α+1 2

) sin λt dη(λ) .

Here z(λ) is the complex standard Wiener process, and ξ(λ), η(λ) are independent real-valued standard Wiener processes: Ez(λ) = Eξ(λ) = Eη(λ) = 0, E|dz(λ)|2 = E[dξ(λ)]2 = E[dη(λ)]2 = dλ,

z(−λ) = z¯(λ),

1 1 z(λ) = √ ξ(λ) + i √ η(λ). 2 2 Formally the generalized Wiener process of order α can be considered as a derivative of fractional order (1 − α)/2 of the “ordinary” Brownian motion or as the white noise integrated with order (1 + α)/2 (see, for example, [12] and references there). In particular, for α −→ −1 the fractional Brownian motion formally “tends to” the white noise. The Hausdorff dimension of the generalized Wiener process is equal to 2 − α/2, α ∈ (0, 2] (see, for instance, [4, 5]). 7

To simulate a random vector x of values x(i) = wα (ih), i = 0, . . . , N , for the generalized Wiener process wα we used a standard method (see, for example, [10, 12]) based on a factorization of covariance matrix R: R(i, j) = Kα (ih, jh),

i, j = 0, . . . , N,

R = AAT ,

x = Aε, where A is a triangular or symmetric square matrix, and ε is a vector of independent standard normal random variables. The corresponding software can be found in Internet, see http://sergeim.prigarin.googlepages.com/ or http://osmf.sscc.ru/˜smp.

5

Results of Monte Carlo simulation

Numerical experiments were performed for a set of time series x(i), i = 0, . . . , N , simulated as realizations of the fractional Brownian motion, x(i) = wα (i/N ),

i = 0, . . . , N,

(9)

for different values of parameter α and the Hausdorff dimension d = 2 − α/2. To estimate the Hausdorff dimension of the time series we applied two versions of the variance counting method (according to formulas (4, 5, 6), (4, 7, 8)) and two versions of the extended counting method described in Section 2: method X1 and method X2 with parameters M and Cy which are called exploratory and scaling factors, respectively. The estimates of the Hausdorff dimension obtained by the extended counting method we shall call x-dimension and the estimates obtained by the variance counting method we shall call v-dimension.

5.1

Dependence of x-dimension on the exploratory factor

On Figures 1-3 we present realizations of the fractional Brownian motion for α = 1.8, 1, 0.2 with Hausdorff dimension equal to 1.1, 1.5, 1.9, respectively, and the dependence of x-dimension of the realizations on the exploratory factor M . Parameters N = 1000, Cy = 1 were used for this numerical experiment. Results for similar numerical experiments but on a more rare (N = 200) and a more fine (N = 5000) grid are presented on Figures 4-6 and Figures 7-9, respectively. 8

5.2

Dependence of x-dimension on the scaling factor

Figures 10-12 illustrate the dependence of x-dimension on the scaling factor Cy for methods X1 and X2 for the three realizations of the fractional Brownian motion presented on Figs. 1-3 (N = 1000). Method X1 was applied with exploratory factor M = 200 and method X2 was applied with exploratory factor M = 100. Figures 13-15 illustrate the dependence of x-dimension on the scaling factor Cy for methods X1 and X2 for the three realizations of the fractional Brownian motion presented on Figs. 4-6 (N = 200). Method X1 was applied with exploratory factor M = 40 and method X2 was applied with exploratory factor M = 20. Figures 16-18 illustrate the dependence of x-dimension on the scaling factor Cy for methods X1 and X2 for the three realizations of the fractional Brownian motion presented on Figs. 7-9 (N = 5000). Method X1 was applied with exploratory factor M = 1000 and method X2 was applied with exploratory factor M = 500.

5.3

Statistical properties of the estimates of the dimension

To study statistical properties of x-dimension and v-dimension we used Monte Carlo method simulating many independent identically distributed realizations of the fractional Brownian motion on the interval [0, 1] according to the description presented at the end of Section 4. The computations were performed for different Hausdorff dimension of the fractional Brownian motion with the amplitude parameter σ = 1 and the grid parameter N equal to 100, 200 and 1000. Tables 1 - 3 show the results of Monte Carlo simulation for v-dimension: the bias (v-dimension minus real dimension), mean square deviation (root of the variance), and mean square deviation of the error (root of the second moment of the difference between v-dimension and real dimension). Similarly, Table 4 shows the results of Monte Carlo simulation for x-dimension.

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Table 1. Statistical properties of v-dimension of the generalized Wiener process, N = 100 (Monte Carlo sample size is equal to one million). estimate dimension bias MSD error MSD (4, 5, 6) 1.1 0.087 0.051 0.1 (4, 7, 8) 1.1 0.028 0.058 0.064 (4, 5, 6) 1.5 0.0111 0.0739 0.0747 (4, 7, 8) 1.5 0.0037 0.0731 0.0732 (4, 5, 6) 1.9 -0.008 0.082 0.0826 (4, 7, 8) 1.9 -0.008 0.082 0.0827

Table 2. Statistical properties of v-dimension of the generalized Wiener process, N = 200 (Monte Carlo sample size is equal to one million). estimate dimension bias MSD error MSD (4, 5, 6) 1.1 0.067 0.037 0.076 (4, 7, 8) 1.1 0.021 0.046 0.051 (4, 5, 6) 1.5 0.006 0.052 0.053 (4, 7, 8) 1.5 0.002 0.052 0.052 (4, 5, 6) 1.9 -0.0023 0.0633 0.064 (4, 7, 8) 1.9 -0.0024 0.0633 0.064

Table 3. Statistical properties of v-dimension of the generalized Wiener process, N = 1000 (Monte Carlo sample size is equal to 100000). estimate dimension bias MSD error MSD (4, 5, 6) 1.1 0.0106 0.0307 0.0325 (4, 7, 8) 1.1 0.0106 0.0306 0.0324 (4, 5, 6) 1.5 0.0011 0.023 0.023 (4, 7, 8) 1.5 0.0004 0.023 0.023 (4, 5, 6) 1.9 0.0000 0.03 0.03 (4, 7, 8) 1.9 0.0000 0.03 0.03

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Table 4. Statistical properties of x-dimension of the generalized Wiener process, N = 200 (Monte Carlo sample size is equal to 100000). estimate dimension bias MSD error MSD X1, Cy = 1, M = 40 1.1 0.1755 0.0690 0.189 X2, Cy = 1, M = 20 1.1 0.233 0.0823 0.2471 X1, Cy = 1, M = 40 1.3 0.104 0.0652 0.123 X2, Cy = 1, M = 20 1.3 0.182 0.0774 0.198 X1, Cy = 1, M = 40 1.5 0.0304 0.0484 0.0571 X2, Cy = 1, M = 20 1.5 0.127 0.0577 0.140 X1, Cy = 1, M = 40 1.7 -0.0547 0.0337 0.0642 X2, Cy = 1, M = 20 1.7 0.0583 0.0403 0.0709 X1, Cy = 1, M = 40 1.9 -0.152 0.0217 0.1531 X2, Cy = 1, M = 20 1.9 -0.026 0.0255 0.0367 Figures 19 and 20 display box-whisker plots with 0.05 and 0.95 boxquantiles for distributions of v-dimension and x-dimension obtained for time series (9) with N = 200, 500 and different values of the Hausdorff dimension.

5.4

Conclusions and remarks

1. In the computational experiments the variance counting method turned out to be significantly more effective than the extended counting method. It is considerably faster, more precise, and has no uncertain parameters that influence the results (like in case of the extended counting method). But the reader should keep in mind that such direct comparison of the methods is biased. The extended counting method originally is destined to estimate the Hausdorff dimension of an arbitrary inhomogeneous fractal (i.e. consisting of parts with different dimensions) in a multidimensional space. The considered versions of the variance counting method (4-8) are appropriate for a restricted class of objects and a generalization for inhomogeneous fractals in spaces of larger dimensions will result in more complicated algorithms. 2. Formulas (4-8) theoretically can give values less than 1 and larger than 2 and it is reasonable to substitute values 1 and 2 for such improper “dimension”. 3. For curves with a low Hausdorff dimension the extended counting method tends to overestimate the value of the dimension. One can easily understand that x-dimension can be equal to 1 only in a very specific case (for horizontal or vertical straight lines, for example). V-dimension of a straight line is always equal to 1. 4. According to Figures 1-18 the values of x-dimension are larger for larger scaling factors and decrease for increasing exploratory factor. Table 4 11

and Figure 19 show that the bias of x-dimension can be considerably large and it essentially depends on the real dimension of the time series and the parameters of the algorithm. Therefore, it is difficult to propose any exact rule for tuning the algorithm’s parameters. The quality of the estimates becomes better for a longer time series (see the results for N=5000), but even for comparatively long time series a good estimation of the Hausdorff dimension by the studied versions of the extended counting method may be questionable, particularly for high values. As a very rough approximation we would recommend to choose exploratory factor M according to the rule N/(d1 M ) ≈ 5 and to use scaling factor Cy in range [1,3]. 5. Method X2 has an advantage compared to X1: for higher dimensions method X2 has considerably smaller bias and mean squared error (see Table 4 and Figure 20). On the other hand, method X2 usually has the larger variance. 6. According to the results presented in Tables 1-3 the bias of v-dimension is comparatively small and it slightly increases when the Hausdorff dimension becomes smaller. With respect to the Hausdorff dimension the variance of vdimension is stable for the longer time series (N = 1000), while it appreciably increases for a shorter time series (N = 100, N = 200). One can notice the inverse property of x-dimension, which variance decreases when the Hausdorff dimension grows (see Table 4 and Figures 19, 20). The variance counting method provides reasonable results even for short time series of the fractional Brownian motion. 7. The variance counting method (4, 5, 6) that “does not know” the real value of mathematical expectation for the increments has practically the same good characteristics as method (4, 7, 8) if the time series is not too short. 8. Finally, we would like to point out the following important problem concerning numerical estimation of the Hausdorff dimension or other characteristics of fractal or multifractal structures. Any numerical method includes quantization (or discrete approximation) of the object to be studied. This quantization immediately kills all fractal properties which are revealed only in limit. That antinomy can be solved by developing a specific theory for numerical fractal analysis based on methods of fractal interpolation and approximation (see, for example, [2, 4]).

Acknowledgments We would like to thank Prof. Konrad Sandau, Dr. Peter Massopust, Dr. Angela Kempe, and Martin Kazmierczak for helpful discussions. 12

References [1] R. J. Adler, The Geometry of Random Fields, Wiley, New York, 1981 [2] M. Barnsley, Fractals Everywhere, Academic Press, London, 1988 [3] R.M. Crownover, Introduction to Fractals and Chaos. Johns and Bartlett Publishers, Inc., Boston, London, 1995 [4] K. Falconer, Fractal Geometry. Wiley, New York, 2003 [5] J. Feder, Fractals. Plenum Press, New York, 1988 [6] T. Gneting and M. Schlather, Stochastic models that separate fractal dimension and the Hurst effect. SIAM Review, V.46, No.2, 2004, P.269– 282 [7] B.B. Mandelbrot, The Fractal Geometry of Nature. W. H. Freeman and Co., New York, 1982. [8] B.B. Mandelbrot, Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Springer, New York, 2005 [9] P. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets. Academic Press, 1995 [10] V.A. Ogorodnikov and S.M. Prigarin, Numerical Modelling of Random Processes and Fields: Algorithms and Applications. VSP, Utrecht, 1996 [11] S. Orey, Gaussian sample functions and the Hausdorff dimension of the level crossings. Z. Warhscheinlichkeitstheorie und Verw. Gebeite, 15 (1970), P.249–256 [12] S. M. Prigarin, Spectral Models of Random Fields in Monte Carlo Methods. VSP, Utrecht, 2001 [13] K. Sandau, A note on fractal sets and the measurement of fractal dimension. Physica A, V.233, 1996, P.1–18 [14] K. Sandau and H.Kurz, Measuring fractal dimension and complexity - an alternative approach with an application. Journal of Microscopy, V.186, Pt.2, 1997, P.164–176 [15] D. Stoyan and H. Stoyan, Fractals, Random Shapes and Point Fields. Wiley, Chichester, 1994

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Figure 1: Dependence of x-dimension of a realization of the Brownian motion (presented on top), N = 1000, with Hausdorff dimension 1.5 (dash lines) on the exploratory factor for methods X1 (in the middle) and X2 (at the bottom). V-dimension of the time series is equal to 1.48. 14

Figure 2: Dependence of x-dimension of a realization of the fractional Brownian motion (presented on top), N = 1000, with Hausdorff dimension 1.1 (dash lines) on the exploratory factor for methods X1 (in the middle) and X2 (at the bottom). The values of v-dimension for this time series is equal to 1.14 for (4, 5, 6) and 1.11 for (4, 7, 8). 15

Figure 3: Dependence of x-dimension of a realization of the fractional Brownian motion (presented on top), N = 1000, with Hausdorff dimension 1.9 (dash lines) on the exploratory factor for methods X1 (in the middle) and X2 (at the bottom). V-dimension of the time series is equal to 1.88. 16

Figure 4: Dependence of x-dimension of a realization of the Brownian motion (presented on top), N = 200, with Hausdorff dimension 1.5 (dash lines) on the exploratory factor for methods X1 (in the middle) and X2 (at the bottom). V-dimension of the time series is equal to 1.453. 17

Figure 5: Dependence of x-dimension of a realization of the fractional Brownian motion (presented on top), N = 200, with Hausdorff dimension 1.1 (dash lines) on the exploratory factor for methods X1 (in the middle) and X2 (at the bottom). V-dimension of the time series is equal to 1.205. 18

Figure 6: Dependence of x-dimension of a realization of the fractional Brownian motion (presented on top), N = 200, with Hausdorff dimension 1.9 (dash lines) on the exploratory factor for methods X1 (in the middle) and X2 (at the bottom). V-dimension of the time series is equal to 1.936. 19

Figure 7: Dependence of x-dimension of a realization of the Brownian motion (presented on top), N = 5000, with Hausdorff dimension 1.5 (dash lines) on the exploratory factor for methods X1 (in the middle) and X2 (at the bottom). V-dimension of the time series is equal to 1.50. 20

Figure 8: Dependence of x-dimension of a realization of the fractional Brownian motion (presented on top), N = 5000, with Hausdorff dimension 1.1 (dash lines) on the exploratory factor for methods X1 (in the middle) and X2 (at the bottom). V-dimension of the time series is equal to 1.11. 21

Figure 9: Dependence of x-dimension of a realization of the fractional Brownian motion (presented on top), N = 5000, with Hausdorff dimension 1.9 (dash lines) on the exploratory factor for methods X1 (in the middle) and X2 (at the bottom). V-dimension of the time series is equal to 1.895. 22

Figure 10: Dependence of x-dimension on the scaling factor for method X1 with exploratory factor M = 200 (solid line) and for method X2 with exploratory factor M = 100 (dot line) for the realization of the fractional Brownian motion presented on Fig.1 with the Hausdorff dimension 1.5 (dash line), N = 1000.

Figure 11: Dependence of x-dimension on the scaling factor for method X1 with exploratory factor M = 200 (solid line) and for method X2 with exploratory factor M = 100 (dot line) for the realization of the fractional Brownian motion presented on Fig.2 with the Hausdorff dimension 1.1 (dash line), N = 1000.

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Figure 12: Dependence of x-dimension on the scaling factor for method X1 with exploratory factor M = 200 (solid line) and for method X2 with exploratory factor M = 100 (dot line) for the realization of the fractional Brownian motion presented on Fig.3 with the Hausdorff dimension 1.9 (dash line), N = 1000.

Figure 13: Dependence of x-dimension on the scaling factor for method X1 with exploratory factor M = 40 (solid line) and for method X2 with exploratory factor M = 20 (dot line) for the realization of the fractional Brownian motion presented on Fig.4 with the Hausdorff dimension 1.5 (dash line), N = 200.

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Figure 14: Dependence of x-dimension on the scaling factor for method X1 with exploratory factor M = 200 (solid line) and for method X2 with exploratory factor M = 100 (dot line) for the realization of the fractional Brownian motion presented on Fig.5 with the Hausdorff dimension 1.1 (dash line), N = 200.

Figure 15: Dependence of x-dimension on the scaling factor for method X1 with exploratory factor M = 40 (solid line) and for method X2 with exploratory factor M = 20 (dot line) for the realization of the fractional Brownian motion presented on Fig.6 with the Hausdorff dimension 1.9 (dash line), N = 200.

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Figure 16: Dependence of x-dimension on the scaling factor for method X1 with exploratory factor M = 1000 (solid line) and for method X2 with exploratory factor M = 500 (dot line) for the realization of the fractional Brownian motion presented on Fig.7 with the Hausdorff dimension 1.5 (dash line), N = 5000.

Figure 17: Dependence of x-dimension on the scaling factor for method X1 with exploratory factor M = 1000 (solid line) and for method X2 with exploratory factor M = 500 (dot line) for the realization of the fractional Brownian motion presented on Fig.8 with the Hausdorff dimension 1.1 (dash line), N = 5000.

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Figure 18: Dependence of x-dimension on the scaling factor for method X1 with exploratory factor M = 1000 (solid line) and for method X2 with exploratory factor M = 500 (dot line) for the realization of the fractional Brownian motion presented on Fig.9 with the Hausdorff dimension 1.9 (dash line), N = 5000.

Figure 19: Box-whisker plots with 0.05 and 0.95 box-quantiles for distributions of v-dimension (4, 5, 6) and x-dimension (method X1, Cy = 1, M = 40) obtained for time series (9) with N = 200 and the values of the Hausdorff dimension (shown by black points) 1.1, 1.3, 1.5, 1.7, 1.9. Sample size for one box-whisker plot is 104 .

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Figure 20: Box-whisker plots with 0.05 and 0.95 box-quantiles for distributions of v-dimension (4, 5, 6) and x-dimension (methods X1 and X2) obtained for time series (9) with N = 500 and the values of the Hausdorff dimension (shown by black points) 1.1, 1.3, 1.5, 1.7, 1.9. Sample size for one box-whisker plot is 103 . The following parameters were used for x-dimension: Cy = 1, M = 100 for method X1, and Cy = 1, M = 50 for method X2.

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