Generalized Approach to Analysis of Multifractal Properties from Short Time Series

—The paper considers a generalized approach to the time series multifractal analysis. The focus of research is on the correct estimation of multifractal characteristics from short time series. Based on numerical modeling and estimating, the main disadvantages and advantages of the sample fractal characteristics obtained by three methods: the multifractal fluctuation detrended analysis, wavelet transform modulus maxima and multifractal analysis using discrete wavelet transform are studied. The generalized Hurst exponent was chosen as the basic characteristic for comparing the accuracy of the methods. A test statistic for determining the monofractal properties of a time series using the multifractal fluctuation detrended analysis is proposed. A generalized approach to estimating the multifractal characteristics of short time series is developed and practical recommendations for its implementation are proposed. A significant part of the study is devoted to practical applications of fractal analysis. The proposed approach is illustrated by the examples of multifractal analysis of various real fractal time series.


I. INTRODUCTION
In the last years, there has been a growing interest in complex systems that have a fractal structure: informational, biological, physical, technological, financial and other. The dynamics of such systems generate time series with fractal (self-similar) properties. Time series fractal analysis is used to simulate, analyze and control complex systems in various fields [1]- [5]. Processes with fractal properties can be divided into two groups: monofractal and multifractal. Monofractal processes are homogeneous in the sense of fractal properties and have single scaling exponent. Multifractal processes have heterogeneous scale properties and are characterized by a set of scaling exponents.
There are a large number of methods for estimating the parameters of self-similar and multifractal processes from time series [6]- [11]. Methods based on wavelet transforms are of particular importance among the research methods of fractal nonstationary processes [2], [12], [13]. Currently, the two most popular tools of time series multifractal analysis are the method of multifractal detrended fluctuation analysis (MFDFA) and method of wavelet transform modulus maxima (WTMM). The MFDFA method is focused on time series with trend components and has been widely used due to [6], [14]. The WTMM method is based on continuous wavelet transform and was originally proposed in [15], [16]. Both methods are a powerful tool for the statistical processing of nonstationary processes. A significantly smaller number of studies are carried out using the method of multifractal analysis based on discrete wavelet transform [2], [17], [18].
In recent years, many studies were focused on the statistical properties of fractal characteristics estimates obtained by the above methods. However, most of the studies done have certain limitations. The main attention was paid to the statistics of estimates of self-similarity degree obtained by different methods [7], [8], [12], [18]. When studying the properties of multifractal characteristics, time series of great length were usually considered. So, for example, in [14], [19], when conducting a comparative analysis between MFDFA and WTMM, the length of the realizations was over 60 thousand values. In this case, the sample fractal characteristics are quite close to the theoretical ones. At the same time, the time series obtained in practice have a much smaller range of values. In [9], [20], aimed at studying the accuracy of estimating short time series, the minimum series length was over 1000 values, which is also quite large.
Many studies also compare the estimates of multifractal characteristics calculated using the same method [9], [14], [19]. The most common in practice is the MFDFA method. Quite a few researchers use several methods to evaluate the multifractal properties of time series. Despite numerous publications related to the practical application of fractal analysis, still, there is no universal approach to the estimation of fractal characteristics by time series. Besides, some practical questions of estimation from short non-stationary time series remain unsolved.
This work aims to study the application of multifractal analysis methods to short time series and to propose a generalized approach to the analysis of the multifractal properties of time series. *Corresponding Author Submission Date: April 20, 2020 Acceptance Date: May 12, 2020 The parameter H is called Hurst exponent. It is a measure of self-similarity and together with that a measure of longrange dependence of process. In this case, the correlation function () K  of the process () Xt decreases hyperbolically: A process with discrete time X is called self-similar with the parameter H if the expression is true.
that is, the process does not change the distribution laws after averaging over blocks of length m .
The Hurst parameter H is a measure of the long-term dependence duration. The case 0. 5 1 H  means that the process has persistent behavior or long memory. In other words, if positive (or negative) process increments have been observed for some time in the past, then with the probability close to the value H , this trend will continue.
The case 0 0.5 H  means an antipersistent process. Here, high process values follow low ones, and vice versa. In other words, the probability that the process will change its trend in the opposite direction is as great as the parameter H closer to 0. When 0.5 H  the deviations of the process from the mean are indeed random and do not depend on the previous values, that corresponds to the case of the ordinary Brownian motion.
The q-moments of the self-similar random process () Xt can be expressed as: In particular, for the value 2 q  , we have a scaling relation for the variance.
Most methods for estimating the Hurst exponent are based on scaling change of some series characteristics close to variance. For example, in the method of R/S analysis change of the normalized range of the cumulative series (

B. Methods for Estimating the Hurst Exponent
To estimate the Hurst exponent of a time series, many methods have been proposed in [2], [6]- [8]. Consider the most popular.
Rescaled range method. This method was proposed by H. Hurst and is currently the most well-known and popular method of fractal analysis [7], [8]. It is widely used in telecommunication technologies, in the study of the selfsimilar properties of information traffic, the study of the dynamics of financial markets, the study of biomedical signals, etc.
In this method in the study of a time series () xt of length  , the following relation is determined.  Fig. 1.
The method of detrended fluctuation analysis (DFA). This method was originally proposed for the analysis of long-term correlations in the structure of the heart rhythm. Currently, it is one of the most widely used methods for studying various non-stationary time series [6], [14], [19]- [23]. 185 | P a g e www.ijacsa.thesai.org In the DFA method, for the initial time series () xt , the cumulative series is constructed into N segments with a length  , and for each segment () yt , the fluctuation function is calculated: Yt is a local m -polynomial trend that limited by this segment.
The function () F  averaged throughout the entire series () yt has a scaling dependence on the length of a series segment: The plot of dependence log ( ) F  on log in a certain range of values is straight line approximated by the least squares method. The value Н is calculated as the tangent of straight line angle of dependence log ( ) F  on log , as it is shown in Fig. 2.
Discrete Wavelet Transform Method. The main development of the method was in the papers [2], [12]. Currently, it is used in the analysis of both stationary and nonstationary time series in various fields of research.
The method of wavelet estimation of self-similarity degree H is based on the properties of the detail wavelet coefficients of time series decomposition by Discrete Wavelet Transform. The basis of this method is a statement that the wavelet energy value j E at the wavelet decomposition level j satisfies the scaling relation:

A. Definitions and Properties of Multifractal Stochastic Processes
Multifractal stochastic processes also are invariant in distribution, but in this case, the change of quantitative characteristics of the process depends on the magnitude of the stretching in time.
where () a M is a random function which independent of   Xt. In the case of a self-similar (monofractal) process For multifractal processes, the following relationship of qmoments holds:  Thus, before investigating the multifractal properties of the time series, it is necessary to determine the existence of selfsimilar properties. A more detailed analysis of the selfsimilarity of time series is presented in [21]. Consider some popular methods of time series estimation of multifractal properties.

B. The Method of Multifractal Detrended Fluctuation
Analysis When performing multifractal fluctuation detrended analysis (MFDFA), the dependence of the fluctuation function () q Fs on the parameter q is investigated: Fs is obtained by raising the expression (7) to the power q and then averaging over all segments. Since at 0 q  (12) contains uncertainty, the following expression is usually used instead:  We can offer the following step-by-step algorithm for the estimation of the multifractal characteristics of time series The value  increases and steps 3-5 are repeated.
by the least squares method. The value k is equal to the value of () hq for a given value q .
7. Performing steps 2-6 for all given values of the parameter q , the function () hq is obtained.

C. Wavelet Transform Modulus Maxima Method
The method of wavelet transform modulus maxima (WTMM) is based on the mathematical apparatus of wavelet analysis. The continuous wavelet transform of the function () Xt is described by Xt . On the other hand, an increase in the number of zero moments leads to an increase in the number of local extrema lines and the appearance of a large number of additional lines terminating on small scales [24]. Such lines are too short for estimating power dependences and become hindrances in the numerical analysis of singularities.
Usually, wavelets of the Gauss family are used as wavelet functions [15], [19], [24]. If high order derivatives of the Gauss function are taken, then the number of additional short lines of local maxima increases, which is due to oscillating "tails" of wavelet functions. Therefore, when conducting a multifractal analysis, it is advisable to limit the value 2 m  .
The basic information about possible local features of () Xt at a point 0 t lies in the asymptotic behavior of the coefficients 0 ( , ) W a t at small scales a . If the coefficients on a small scale diverge, () Xt has a feature in 0 t . If the coefficients 0 ( , ) W a t are close to zero in a neighborhood 0 t on a small scale, then () Xt is regular at this point. The first stage of the WTMM algorithm is completed by construction of the "skeleton" of the coefficients ( , ) The second stage of the WTMM algorithm is to calculate the partition function: where () La is the set of all lines maxima l of the wavelet coefficients modules of the scale a ; () l xa is the location of the maximum at this scale. To calculate ( , ) Z q a the maximum value of the wavelet coefficients modulus along each line on scales smaller than the specified scale a is chosen. In this case, the following relation holds: where () q  is scaling exponent that is related to the generalized Hurst exponent () hq by the ratio [14]: The multifractal analysis based on the wavelet transform allows studying singularities with negative values of q [15,16]. Partial functions ( , ) Z q a at 0 q  characterizing scaling features for weak singularities (small fluctuations), and at 0 q  for strong singularities (large fluctuations). Z q a of a realization of the multifractal process for the values { 5, 2,0, 2,5} q    are shown in Fig. 6(a). The scaling exponents () q  of multifractal and monofractal realizations are shown in Fig. 6

The partial function ( , )
Z q a is calculated by (15) for given q and a . 6. The linear regression log ( , ) log( ) Z q a k a b  is constructed by the least squares method. The value k is equal the value of () q  for a given value q . 7. By performing steps 5-6 for all given values of the parameter q , we obtain the function () q  . 8. The values of the generalized Hurst exponent () hq are determined by (17).

D. Multifractal Analysis based on Discrete Wavelet
Transform In subsection 2.2, a method of determining the degree of self-similarity using DWT is considered. It is based on the properties of the detail coefficients obtained by the time series decomposition. Similarly, in [17,18], a method of estimating multifractal characteristics based on DWT was proposed (MFDWT).

If for a multifractal process ()
Xt there are q -th moments, then for the detail wavelet coefficients obtained by DWT, the following relation holds: is a nonlinear function of a parameter q associated with the generalized Hurst exponent () hq considering (17) by the expression: Based on the expressions (18)(19) and taking into account the properties of the detail wavelet coefficients, the following step-by-step algorithm can be offered to estimate the multifractal characteristics of time series ( ), 1, 2,..., X t t n  by MFDWT.
1. The original series has to be converted to cumulative series kn  . If the original time series is cumulative, this step is skipped. 2. The range q is specified, in which it is required to determine the multifractal characteristics () q  and () hq . 3. The range of decomposition levels of DWT is selected. The following value is calculated for each decomposition level j :  Fig. 7(a). The functions () q  of different multifractal realizations are shown in Fig. 7

A. Experiment Description
The section presents the results of a numerical experiment, during which the realizations of different fractal stochastic processes were modeled: fractal Brownian motion (monofractal process),   stable process (bi-fractal process) and binomial multiplicative stochastic cascade (multifractal process). The length of the realizations were chosen to be 250, 500, 1000 and 2000 values. For each generated time series, using the methods of MFDFA, WTMM and MFDWT the functions of the generalized Hurst exponent () hq were calculated. Then functions () hq were averaged over a set of realizations. The parameter q had values in the range 55 q    .
The following characteristic was chosen as a measure of proximity to theoretical values: Stable process with independent increments. A random process is a   stable process if its finite-dimensional distributions are stable random variables [26]. The parameter  is called the stability index and determines the heavy tails of the distribution. For a   stable process with independent increments, equality holds.
It is shown that such processes are bi-fractal. The corresponding generalized Hurst exponent is described by: Stochastic binomial cascade. The simplest model of a multifractal process is the deterministic binomial cascade [25]. When constructing it, the initial unit segment is divided into two equal intervals, weight coefficients 1 p and 21 1 pp  are assigned to each of them. As a result, in the second step, there are four intervals with weights 2 1 p , 12 pp , 21 pp and 2 2 p . With an increase in the number of iterations, we obtain an ordered set of weights, which has multifractal properties. When constructing stochastic cascades, the weight coefficients are values of some random variable [27]. If a random variable with a beta distribution ( , ) Beta a b is used as a random variable, then in the case of ab  , the scaling exponent () q  can analytically determine over the interval of values The theoretical value of the function () hq is determined in accordance with (17).

B. The Results of the Generalized Hurst Estimation Exponent by Time Series
Estimation of multifractal characteristics for monofractal realizations.
A typical FBM realization with H=0.8 is shown in Fig. 8(a) and the corresponding realization of increments (fractal Gaussian noise) is shown in Fig. 8(b). The length of realizations is 1000 values.
The sample values of the generalized Hurst exponent calculated by the methods MFDFA (a), WTMM (b) and MFDWT (c) are presented in Fig. 9. A straight line is the theoretical values () hq . It is shown that with an increase in the length of the realization, sample characteristics tend to theoretical values. However, at small lengths, the obtained estimates demonstrate false multifractal properties. Therefore, in the case of the realization of a small length, it is necessary to conduct additional studies to confirm the monofractal properties. Table I represents the values of the deviations h  , calculated in accordance with (20). The estimates of the generalized Hurst exponent obtained by the MFDFA method are much closer to their theoretical values than the estimates calculated using wavelet transforms. Also, when using the WTMM method, determining the wavelet spectrum, local maxima lines and choosing the appropriate scale range is a www.ijacsa.thesai.org more complicated task than determining the polynomial degree and the segment lengths in MFDFA. The MFDWT method demonstrates incorrect estimation of () hq with negative q , however, for positive q the estimates by MFDWT method are close to the estimates by WTMM.
Estimation of multifractal characteristics for bi-fractal realizations.   Let us consider the estimation of () hq for the realizations of a   stable process. A typical realization of a   stable process with 1.5   is shown in Fig. 10(a) and the corresponding realization of increments is shown in Fig. 10(b). The length of realizations is 1000 values.
The sample functions of the generalized Hurst exponent obtained using the MPDFA, WTMM and MFDWT methods are shown in Fig. 11(a), (b) and (c), respectively.
The obtained estimates of () hq show false multifractal properties. The estimates obtained by the MFDFA method are closer to their theoretical values than the estimates calculated using wavelet transforms, as evidenced by Table II. The MFDWT method, as well as in the monofractal case, demonstrates incorrect estimation of multifractal properties. However, for positive q the estimates of () hq by the MFDWT method have the smallest difference with the theoretical values.
Estimation of multifractal characteristics for realizations of a stochastic binomial cascade.
Let consider the estimation of multifractal properties for realizations of stochastic multiplicative binomial cascades whose weights have a beta distribution. For comparison with analytical characteristics (21), calculations were performed in the range -15 q .
The typical realization of a binomial stochastic cascade with a length of 1000 values is shown in Fig. 12(a). In this case the weights have a uniform distribution (i.e. ). The corresponding sample values of the generalized Hurst exponents are shown in Fig. 12 The numerical values of the deviations are given in Table III. The WTMM method shows the worst estimation of the stochastic multifractal cascades. The MFDWT method shows good results and it is not inferior to the MFDFA when evaluating multifractal realizations.

C. Testing the Hypothesis about Monofractality of Time Series
The results of the multifractal analysis of model time series demonstrate the need to develop a tool to distinguish between mono-and multifractal time series. Since the MFDFA method showed the best results in evaluating monofractal realizations, a method based on estimates obtained by MFDFA is proposed. It allows to accept or reject the hypothesis of the presence of time series monofractal properties.
For self-similar processes, the question of the estimates Hurst exponent Н distribution was considered in several publications [2], [8], [19], where it was shown, that estimates are normal random variables. The analysis of the sample distribution of the generalized Hurst exponent () hq showed that the estimates at 0 q  have a normal distribution, the parameters of which depend on the value q . For 0 q  the sample values () hq in the general case are not normal random variables.

D. A Generalized Approach to Estimating the Multifractal Properties of Small Length Time Series
Summing up the research results, the following scheme for multifractal analysis of some random process represented by a time series () Xt of length N can be proposed. It should be noted that before proceeding to fractal analysis, it is necessary to find out from a priori known information whether the series is cumulative (for example, the exchange rate) or increments (for example, information traffic).
The main stages of fractal analysis can involve various methods of estimation. Since the application of the wavelet transform apparatus requires appropriate software and experience, the description of the scheme is structured in such way that the use of wavelet estimation methods is a desirable but not necessary element. However, the application of the MFDFA method is necessary for two reasons: this method has sufficient accuracy and is designed for non-stationary time series. Consider the step by step implementation of the generalized approach to the estimation of the multifractal properties of self-similar time series.

1)
A preliminary study of the time series structure is a necessary stage of multifractal analysis, its full description is given in [21] This stage includes determining intervals of various scaling, detecting and removing short-term autoregressive dependence, estimation of the Hurst exponent Н .
2) All three considered methods MFDFA, WTMM, and MFDWT are intended for analysis non-stationary series. But before proceeding to an estimation of multifractal properties, it is necessary to investigate time series structure by correlation function, Fourier spectrum or spectrum of wavelet energy, which allow identifying the trend and cyclical www.ijacsa.thesai.org components of the series. The results of the preliminary study allow selecting the input parameters correctly for the multifractal analysis method.
3) When estimating the generalized Hurst exponent using the MFDFA method, it is necessary to first carry out a rough estimation of the Hurst exponent Н = (2) h using local polynomials of increasing degree and determine the degree to estimation. After that, evaluate the function () hq , removing the local polynomial trend of this degree. 4) When evaluating the scaling exponent () q  by the WTMM method, it needs to select the appropriate scale range 12 a a a  on which the local extremums of the function ( , ) W a b are determined. The right choice of the scale range allows ignoring the influence of the trend components of the time series. Then it is necessary to carry out a rough estimation of the partial function ( , ) Z q a to make sure that there are intervals of linear dependence log ( , ) Z q a on log a . 5) When estimating the scaling exponent () q  using the MFDWT method, it needs to first select the appropriate range of values of levels 12 j j j  at which there is a linear dependence 2 log j E on the level number j . 6) With weakly expressed multifractal properties, it is necessary to test the hypothesis of the monofractal properties of time series. 7) If estimates of multifractal characteristics were obtained by different methods, then to increase accuracy, it is desirable to conduct comparative analysis taking into account a priori information and the results of numerical studies.

A. Fractal Analysis of Temperature Series
Well-known time series with the property of self-similarity are temperature series. The specialized site [28] presents daily temperature series (maximum, minimum and average values) for different cities over 50 years from 1942 to 1992. Fig. 13 shows a fragment of the temperature series for 1952-1956 (a) and the corresponding series of daily increments (b). Before the multifractal analysis, the fluctuation function () F  was investigated. The presence of a section with a linear dependence of () F  corresponds to the self-similar behavior of the time series. If the function () F  has several linear sections, this implies the several time scaling for different time intervals. Fig. 14(a) shows a graph () F  plotted over a time interval from 5 2 (month) to 9.5 2 (two years) days.
It is obvious that the graph has two sections close to linear with different tilt angles. It is worth noting the high expulsion in the second section when the value of the argument is equal to 8.5 2 which is almost equal to one year and corresponds to the annual seasonal component of the time series. Multifractal analysis by the MFDFA method was carried out separately for each section. For the first section, the Hurst exponent 0.21 H  , for the second 0.92 H  . Fig. 14(b) shows the multifractal spectrum functions for both sections.
Thus, it can be concluded that at intervals from two months to six months (Section 1), the series of temperature dependence is antipersistent, and at time values from six months to one and a half years (Section 2), the series has a strong long-term dependence.
The research of the monofractal property, carried out by the method presented in subsection 4.3, has shown that in both cases the series has almost monofractal properties.

B. Fractal Analysis of Atmospheric Layer Return Echo
One of the methods for studying processes occurring in the atmospheric boundary layer is acoustic sounding and subsequent analysis of both echo signals and ensembles of sequentially obtained echo signal profilesechograms. Let's conduct a fractal analysis of some of the experimental data obtained at the existing acoustic sounding station of the Kharkiv National University of Radio Electronics.
In this case, the maximum depth of the sounding atmospheric layer 372 Zm  ; the value of the resolution in depth 5 Zm  ; the carrier frequency of the sounding pulse 1800 f Hz  , sounding pulse repetition period 2 sec T  . As seen from the figure, at low depth, where the proximity of buildings, structures, tall trees and other objects of the metropolis are affecting, the values of the Hurst exponent do not exceed 0.6. Then, due to the influx of heat in the lower layers of the atmosphere, the values of the Hurst exponent increase to 0.75, which indicates the presence of long-term dependence in atmospheric processes at low depth. As the depth and distance from the megapolis increase, the value of Hurst exponent decreases and tends to 0.5, which corresponds to weakly correlated random processes.
A multifractal analysis carried out by MFDWT confirms that with increasing depth of the reflecting atmosphere layer, the correlation structure of the processes changes, which is reflected in a decrease in the values of the generalized Hurst exponent () hq . Fig. 18 shows the values of the generalized Hurst exponent for signals reflected at 50, 150, and 200 m depth. In this case, the values of the parameter q changed in the range 1 10 q  .   Monofractal property testing presented in Subsection 4.3 has shown that signals reflected at different depth have weak multifractal properties. Table VII has the results of the sample generalized Hurst exponent analysis for signals of various depths. The critical value h  was obtained using calculated data for significance level 0.05   and corresponding series length.

C. Electroencephalogram Signals Research
Fractal geometry has been used in biology for over a quarter of a century. The use of fractal methods opens up new opportunities in studying the functional organization of living systems. Numerous experimental and clinical data lead to the conclusion that research on the fractal properties of different biological systems will lay the foundation for fractal diagnostics [29], [30].
Multifractal characteristics of electroencephalogram records of different activity.
In [31], the study of how the multifractal characteristics of the electroencephalogram records (EEG) change when a person performs a physical action, and when he imagines that he is doing it, was carried out. Experimental data were taken from the [32], where are the EEG of persons, who, with one signal clasped their hand into a fist, and with another signal, they only imagined that they were doing it. Fig. 19 shows typical EEG realizations in cases where the respondent squeezed his fist (left) and when the respondent imagines it (right).
Multifractal characteristics of EEG records for the abovementioned cases were studied using the method of wavelet transform modulus maxima (WTMM). Fig. 20 shows the multifractal spectrum functions  () f corresponding to considered EEG records. Spectrum  () f for real fist squeezed states are represented by line 1, and line 2 corresponds to an imaginary squeezed case.
Therefore, the functions of the multifractal spectrum obtained from the EEG realization make it possible to differentiate the real and imaginary actions of the respondent.
Multifractal characteristics of EEG for different phases of wakefulness and sleep.
The EEG realizations of laboratory animals, in different phases of wakefulness and sleep, have been researched. Fig. 21 shows typical EEG for various phases: wakefulness (awake), slow-wave sleep (sws) and rapid eye movement sleep (rem).   Before the multifractal analysis, the research of the selfsimilar properties of the EEG realization was carried out, which identified the long-term dependence for the awake, ant persistence for the slow sleep phase and weak autocorrelation dependence for the fast sleep phase. Fig. 22 shows a fluctuation function () F  typical for the EEG realizations in the wakefulness.
Multifractal analysis conducted by the MFDFA method detected significant differences in fractal properties of EEG for wakefulness and sleep. Fig. 23 shows the functions of the generalized Hurst exponent () hq for the EEG realizations presented above.   The analysis also showed that there is an undoubted longterm dependence for the EEG realizations in the awake phase: in this case, Hurst exponent H is significantly exceeded 0.5. The slow sleep phase is characterized by ant persistence, Hurst exponent takes values in the range of less than 0.5. For REM sleep, the Hurst exponent H is close to 0.5, it takes values greater or less than 0.5. In this case, the EEG realizations are characterized by very weak autocorrelation dependence.

D. Multifractal Analysis of Seismic Wave
One of the known examples of time series with clearly expressed fractal properties is seismic waves. Nowadays realtime seismic monitoring is not only important but also is the most complicated task of seismological practice. There are various seismic signal detection methods based on amplitude ratio analysis, spectral characteristics, wavelet analysis, etc. Some methods based on the estimation of fractal characteristics of the seismic process such as Hurst exponent and fractal dimension [33], [34].
Numerical studies show that seismic waves have multifractal properties and different stages of their development have different functions of generalized Hurst exponent () hq . Fig. 24 (top) shows a seismic time series where two windows are highlighted: before the activity burst and after. Obviously, in these windows, the function of the generalized Hurst exponent () hq , shown in Fig. 24 (bottom) has significant differences. The results of the analysis allow asserting that seismic wave realizations have strong multifractal properties that allow to distinguish them from white noise which is monofractal. These properties can be used when detecting seismic signals.

E. Analysis Fractal Properties of Community Activities Time Series in Social Networks
In recent years, there have been studies of community dynamics in social groups, which show that the corresponding time series have self-similarity properties [35], [36]. The comparative fractal analysis for two groups on the social network Facebook found on keywords related to cyber threats was carried out. Two groups ThreatPost and ThreatSignal were selected to research each with 14000 and 84000 users respectively. For each of these social groups, data on the number of likes, comments, and involvement levels over the past five years has been collected.   Fig. 24. Seismic Series and () hq before Burst (Line 1) and after (Line 2).   Research has shown that a series of likes are quite strongly correlated with the series of involvement levels, so their multifractal characteristics are very close. The series of comments have a rather close to each other multifractal structure. Thus, the research confirmed that many of the time series of social network activity indicators have fractal properties and the application of fractal analysis allows to detect differences and to reveal characteristic features of different social groups' dynamics.

VI. CONCLUSION
The work has considered the features of the numerical implementation of multifractal analysis methods: multifractal detrended fluctuation analysis, wavelet transform modulus maxima, multifractal analysis using discrete wavelet transform. The properties of generalized Hurst exponent estimates obtained by these methods from short time series have been investigated. For this testing, each method on realizations of fractal processes of different types has been carried out. The advantages and disadvantages of each method were considered.
It is shown that the estimates of the generalized Hurst exponent, obtained from short time series, have a bias that decreases with increasing series length. It is shown that the method of multifractal detrended fluctuation analysis has significant advantages in accuracy when evaluating the characteristics of monofractal processes. A method has been developed that allows to accept or reject the hypothesis of the presence of monofractal properties in a time series. It is based on the study of sample values of the generalized Hurst exponent, obtained by the method of multifractal detrended fluctuation analysis.
Summing up the results of research, a generalized approach to the estimation of multifractal properties of time series and practical recommendations for the implementation of this approach are proposed. Examples of the practical use of the considered approach and methods for multifractal analysis of biomedical signals, natural phenomena and social networks are presented.
It should be noted certain limitations of the proposed approach. Non-stationary time series are considered in the work, but multifractal properties are assumed to be practically constant during the time realization. This requirement is consistent with the short length of the studied time series. In multifractal analysis of sufficiently long time series, it makes sense to conduct additional studies of fractal characteristics using the sliding window method.