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Scaling features of intermittent dynamics: Differences of characterizing correlated and anti-correlated data sets
Physica A 536 (2019) 122586
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Scaling features of intermittent dynamics: Differences of characterizing correlated and anti-correlated data sets O.N. Pavlova a , A.N. Pavlov a,b , a b
∗
Saratov State University, Astrakhanskaya Str. 83, Saratov 410012, Russia Yuri Gagarin State Technical University of Saratov, Politechnicheskaya Str. 77, Saratov 410054, Russia
article
info
Article history: Received 6 June 2019 Received in revised form 25 July 2019 Available online 4 September 2019 Keywords: Detrended fluctuation analysis Long-range correlations Intermittent dynamics Chaotic oscillations
a b s t r a c t Using the detrended fluctuation analysis (DFA), we consider the effect of repetitive switching between different states in complex systems, including random processes, intermittent behavior of coupled chaotic oscillators, and the dynamics of blood pressure in rats. We address the problem of diagnosing the state of a system based on time series, when the latter includes data segments with distinct correlation properties, and show significant distinctions in the diagnostics of correlated and anti-correlated data sets. We demonstrate that anti-correlated dynamics are highly sensitive to switching between different states of the system, and the presence of several ‘‘alien’’ segments can provide a much stronger displacement of the scaling exponent unlike the case of correlated dynamics. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Characterization of complex systems based on experimental data is often complicated by the presence of artifacts, noise and time-varying dynamics, which may include changes in control parameters and transitions between distinct types of oscillations [1]. Additive fluctuations in the time series can be completely or partly eliminated during the preprocessing stage, such as noise reduction, provided for both localized artifacts and non-localized fluctuations related, e.g., to measuring noise [2–4]. Despite the existence of many filtering techniques, extracting artifacts from experimental recordings is still a challenging problem, especially if the frequency bands of the noise-free signal and artifacts overlap in the frequency domain, and simple excluding corrupted data segments is usually carried out. This case occurs, for example, in neurophysiological studies, where the percentage of fragments without artifacts can contain about 10% of the total data set [5–7], and only these fragments are selected for further processing. When the remaining parts of the data set are short and do not reliably characterize the state of the system, they are brought together to obtain the required amount of data. However, this procedure destroys the long-range correlation in the original data set, which leads to significant changes in quantitative measures. According to recent studies [8–10], such changes are much stronger for anti-correlated processes, where the elimination of a relatively small percentage of the data set is crucial for characterizing scaling features. Correlated signals, on the other hand, are less sensitive to missing parts of records, and reliable characterization of the dynamics is ensured even in the case of extreme data loss [8]. Unlike corrupted segments of time series related to artifacts or failures of the recording equipment which can be easily recognized even by visual monitoring of data, transitions between distinct oscillatory modes are less distinguishable ∗ Corresponding author at: Saratov State University, Astrakhanskaya Str. 83, Saratov 410012, Russia. E-mail address: [email protected] (A.N. Pavlov). https://doi.org/10.1016/j.physa.2019.122586 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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during the pre-processing stage. These modes are often characterized by nearly similar statistical measures, such as mean value and standard deviation, and the acquired time series may be wrongly associated with a stationary process. Thus, bifurcations of chaotic attractors are often accompanied by changes in the power spectrum, complexity and predictability measures, although a visual inspection of the related changes may not allow detecting the moments of time when one dynamical mode is replaced by another type of oscillations. In the case of rare transitions, recognition can be performed by using a windowed processing of experimental data, i.e., by characterizing the features of the signal within a ‘‘floating’’ window translated along a time series. When dealing with relatively short records containing intermittent oscillatory dynamics, the latter approach becomes ineffective, since the errors in computing numerical measures become comparable with changes of such measures for distinct modes of complex oscillations. In this paper, we consider the effect of repetitive switching between various types of oscillatory behavior in benchmark models of nonlinear dynamics on the correlation properties of signals reflecting their functioning. In addition, we discuss transitions between stochastic processes characterized by distinct statistical properties and between separate states of the cardio-vascular system in order to establish general phenomena in quantifying the scaling features of intermittent dynamics for correlated and anti-correlated data sets. Taking into account that such transitions are typically associated with non-stationary signals, we use a fairly universal method of data processing, namely the detrended fluctuation analysis (DFA) [11,12], which quantifies the power-law correlations in the dynamics of systems with both, stable and time-varying parameters. Unlike many other approaches, the DFA operates with a random walk created from the original signal and analyzes the fluctuations around the local trend, representing slow variations in the mean value. The paper is organized as follows: in Section 2, we briefly discuss the DFA approach applied for signal processing, the benchmark models used to produce signals with intermittent behavior, and experimental data reflecting the stress-induced transient in cardiovascular dynamics. A comparative analysis of the scaling features of intermittent dynamics for correlated and anti-correlated data sets is performed in Section 3. Section 4 summarizes main results of the study. 2. Methods and models 2.1. DFA-approach Detrended fluctuation analysis proposed by Peng et al. [11,12] and widely applied for signal processing in various studies [13–22], involves the construction of a signal profile y(k) =
k ∑
[x(i) − ⟨x⟩] ,
i=1
⟨x ⟩ =
N 1 ∑
N
x(i),
(1)
i=1
its segmentation into non-overlapping parts of size n and fitting the local trend yn (k) within each segment. Often, the piece-wise linear function is used to describe the nonstationarity of the profile, although the DFA-approach has no restrictions with respect to the type of fitting, and the analysis can be performed using polynomial functions of the order 1, 2, 3, etc. The root mean-square fluctuation of the profile around the local trend
N 1 ∑ [y(k) − yn (k)]2 F (n) = √ N
(2)
k=1
typically increases with n, and there is a power-law dependence F (n) ∼ nα ,
(3)
where α is the scaling exponent closely related to the quantities describing the decay of the correlation function or spectral density. Based on α , we can distinguish between anti-correlated dynamics (α < 1/2, large and small values of time series alternate), uncorrelated dynamics (α = 1/2, large and small values have equal probability), power-law correlations (1/2 < α < 1, large values mainly follow large values and vice versa) and correlations which may differ from power-law statistics (α > 1). 2.2. Intermittent noise Aiming to characterize how missing data affects the scaling characteristics of random signals quantified by the DFAapproach, Ma et al. [8] selected noise with different statistical properties and compared correlated and anti-correlated time series. Unlike their study, here we analyze intermittent behavior in stochastic processes. A simple example is a simulated data set that includes repeating segments of noises with distinct statistics (Fig. 1). The data set shown in Fig. 1 involves segments of correlated 1/f -noise described by α = 1.0, and anti-correlated noise with α ∼ 0.12. The duration of the segments and the times of switching are randomly selected. The relative amount of data belonging to each process (N2 /N1 ) varies in order to assess how intermittent behavior with various durations of the main stochastic process (N1 ) with respect to the additional data set (N2 ) changes the scaling exponent α compared to the case when a single process is
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Fig. 1. An example of a stochastic process consisting of segments with different correlation properties (correlated data sets with α = 1.0, and anti-correlated data sets with α ∼ 0.12).
considered. Equal amounts of N1 = N2 mean that correlated and anti-correlated noises are taken under similar conditions when there is no dominant random process. The case when the total duration of one process exceeds the duration of another process enables to decide how stable the estimates of α -exponent are if switching appear. To avoid the effects of short samples for simulated data sets, we used sequences about 100,000 return times, and the average duration of the included segments ranged from 100 to 1000. We tried not to consider longer segments that can be detected using the floating window approach for analyzing data sets. 2.3. Coupled chaotic oscillators Two models of coupled chaotic oscillators are chosen because they demonstrate a wider range of complex dynamical phenomena as compared to single chaotic systems, and therefore, diverse intermittent behavior can be considered for a more generalized conclusion. The first model is a system of two diffusively coupled Rössler oscillators, described by six ordinary differential equations dx1,2 dt dy1,2 dt dz1,2 dt
= −ω1,2 y1,2 − z1,2 + γ (x2,1 − x1,2 ), = ω1,2 x1,2 + ay1,2 , = b + z1,2 (x1,2 − c),
(4)
ω1,2 = ω0 ± ∆
with four fixed parameters ω0 = 1, a = 0.15, b = 0.2, γ = 0.02, and two variable parameters, c and ∆, the choice of which provides transitions among different types of regular, chaotic and hyper-chaotic oscillations. The complete bifurcation diagram of the model (4) is given in Ref. [23], where transitions to and between various types of in-phase and out-of-phase oscillations are considered. We have chosen the following types of complex dynamics:
• • • •
synchronous chaotic attractor (SA); asynchronous chaotic attractor (AA); quasiperiodic attractor (QA); hyperchaotic attractor (HA).
The values of the parameters related to these modes were selected near the bifurcation lines, and further studies were carried out using sequences of return times to the Poincaré section x1 = 0. Such sequences reflect information about a multiscale structure of chaotic attractors. The second model is a system of two coupled Lorenz oscillators dx1,2 dt dy1,2 dt dz1,2 dt
= σ (y1,2 − x1,2 ) + γ (x2,1 − x1,2 ), = r1,2 x1,2 − x1,2 z1,2 − y1,2 ,
(5)
= x1,2 y1,2 − z1,2 b
with fixed parameters σ = 10, r1 = 28.8, r2 = 28, b = 8/3 and variable coupling strength γ . This model produces chaotic oscillations with two distinct time scales: a slow time scale related to switching between oscillations
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Fig. 2. The difference (∆) of the oscillation frequencies in the model of two coupled Lorenz oscillators (5) versus the coupling strength.
around unstable focus points, which can be interpreted as transitions between two states ‘‘+1’’ (x1,2 > 0) and ‘‘1’’ (x1,2 < 0), if each oscillator is considered as a bistable system [24], and a fast time scale related to the frequency of the oscillations themselves. Both time scales vary depending on the parameters of each oscillator and the interaction between subsystems. The model (5) provides a quite atypical behavior depending on the coupling strength [24]. Unlike the expected entrainment of the oscillation frequencies of the subsystems, there is the phenomenon of desynchronization with an increase in γ to a value of γ ∼ 2, which is further replaced by the adjustment of frequencies in the region of γ > 2 (Fig. 2). The dynamics of model (5) will be studied based on the sequences of return times to the Poincaré section z1 = 30. 2.4. Blood pressure dynamics In addition to the simulated data sets, transients associated with the stress-induced response of the cardiovascular system in healthy rats are considered. Experimental studies were conducted on adult male rats weighing from 250 to 300 g in accordance with the Guide for the Care and Use of Laboratory Animals (NIH Publication No. 85–23, revised 1996) and protocols approved by the local ethics committee of Saratov State University (Saratov, Russia). Animals were kept at 25 ± 2 ◦ C, humidity 55% and light/dark cycle of 12:12 h. Blood pressure (BP) was measured directly using an intra-arterial polyethylene catheter. The data were recorded using the PowerLab/400ML401 measuring complex and Chart 4 software (ADInstruments Ltd., Australia). The sampling step was chosen to be 200 Hz. The model of immobilization stress was analyzed when a rat was placed in a special case that did not allow it to move. Each experiment included a control measurement (10 min), immobilization (10 min) and the recovery process after stress-off (10 min). The analysis was performed using sequences of time intervals between local maxima of the BP signal, representing an analogue of the return times in simulated studies with chaotic oscillators. In order to clearly identify these maxima and, therefore, time intervals, spline interpolation of the measured time series was provided. Additionally, we used a band-pass filter with cutoff frequencies of 0.05 Hz and 30 Hz during the pre-processing stage in order to exclude high-frequency fluctuations and slow floating of the mean value. According to the earlier study [25], typical values of α are between 0.9 and 1.3 for control measurements. Stress usually reduces the scaling exponent, but the dynamics remain correlated in all states. In few animals, however, stress provided stronger changes in cardiovascular dynamics, when BP becomes uncorrelated or anti-correlated with α , taking values of about 0.4–0.5 during stress. We selected two such animals, and the corresponding records were chosen to study the transitions between correlated and anti-correlated dynamics. 3. Results 3.1. Intermittent noise The study of intermittent dynamics was started with a relatively simple example of a stochastic signal made up of segments with distinct statistical properties shown in Fig. 1. The results of the DFA-analysis for such a data set clearly depend on the relative amount of data belonging to each process. In the case when 1/f noise is considered as the main process, and the total duration of the anti-correlated noise is less or comparable, there is no significant shift in the scaling exponent (Fig. 3). The value of α fluctuates nearly α = 1.0, and such variations are close to computing errors for time series of finite length. This situation changes dramatically for an anti-correlated process that contains short fragments of noise with power-law correlations. Even the appearance of segments with a total duration of about 5% of the main process provides an increase in α by 0.5–0.6, i.e. almost 5-fold compared with the scaling exponent of the main stochastic process (α = 0.12). Therefore, the processing of anti-correlated data sets is very sensitive to the presence of parts with other statistical properties. When the N2 /N1 ratio exceeds 20%, the anti-correlated dynamics can be treated as correlated one (Fig. 3).
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Fig. 3. Dependence of the scaling exponent α on the N2 /N1 ratio for the stochastic signal shown in Fig. 1. Hereinafter, N1 is the number of samples of the main process, and N2 is the number of samples of the additional data set. Results are shown for cases of anti-correlated and correlated signals, considered as the main process, respectively.
Fig. 4. Dependencies of α on the N2 /N1 ratio for transitions between (a) synchronous and asynchronous chaotic oscillations, (b) hyperchaotic and asynchronous chaotic oscillations, (c) periodic and quasiperiodic oscillations in the model of two coupled Rössler systems (4). All designations are given in accordance with Fig. 3. Regardless of the type of transition, different sensitivity of correlated and anti-correlated dynamics to the appearance of fragments with distinct correlation properties is shown.
3.2. Coupled chaotic oscillators Transitions between different types of regular, chaotic and hyperchaotic dynamics in coupled oscillators confirm the distinctions for correlated and anti-correlated data sets (Fig. 4). Consider switching between the SA (c = 6.8, ∆ = 0.0097) and AA (c = 6.8, ∆ = 0.0098) attractors in the model of two diffusively coupled Rössler oscillators (4) – Fig. 4a. The series of return times to the Poincaré section for these attractors are described by scaling exponents α ≃ 0.02 (SA) and α ≃ 0.80 (AA). The presence of anti-correlations in the first series is a consequence of the period-doubling route to chaos, which produces the alternation of large and small return times [26]. Asynchronous chaotic oscillations arise
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Fig. 5. Dependencies of the scaling exponent α on the N2 /N1 ratio for transitions between (a) synchronous chaotic and hyperchaotic oscillations, (b) quasiperiodic and asynchronous chaotic oscillations in the model of two coupled Rössler systems. In both cases, the transition between data sets with the same correlation properties is considered (anti-correlations in a, and correlations in b). In this regard, there are no strong changes in the value of α .
due to the destruction of quasiperiodic oscillations, which are characterized by relatively slow modulation of amplitude and frequency and do not show such an alternation. If the series of return times includes a random switching between SA and AA, and the total duration of the data sets related to each mode is almost the same, then the scaling exponent takes a value α ≃ 0.63. Therefore, the shift in the scaling exponent is asymmetric; such shift is about 3.6-fold greater for the anti-correlated dynamics of SA (∆α = 0.61) than for the correlated dynamics of AA (∆α = 0.17). In this paper, we show results for segments of the mean duration of 500 return times. The results do not strongly depend on the duration of such segments if the N2 /N1 ratio is fixed. For example, the difference between the values of α for segments with the duration of 200 and 1000 takes about 3%–4%. Note also that the phase multistability does not essentially affect the results. Thus, in-phase and out-of-phase periodic or chaotic attractors are characterized by different values of α . However, such distinctions are quite small compared with the distinctions between the values of α for correlated and anti-correlated data sets. The abovementioned is also confirmed for other types of intermittent behavior, which included the alternation of correlated and anti-correlated data sets. Fig. 4b illustrates very similar results for random switching between hyperchaotic oscillations (HA, c = 7.2, ∆ = 0.0094, which appears due to the merging of synchronous chaotic modes and inherits the anti-correlated structure of return times with α ≃ 0.03) and asynchronous chaos considered for parameters c = 7.2, ∆ = 0.0096, which produces a correlated dynamics of return times with α ≃ 0.79). Again, there is an asymmetric shift in the scaling exponent if the time series under study is a mixture of segments formed by both types of complex oscillations. The dependences of α (N2 /N1 ) nearly coincide in both switching variants (Fig. 4a and b). Consideration of the transition between simpler types of oscillations — periodic oscillations related to the cycle of period 4 (c = 6.2, ∆ = 0.0090, α ≃ 0.0) and quasiperiodic oscillations (QA, c = 6.2, ∆ = 0.0100, α ≃ 0.92) leads to a similar conclusion. Thus, regardless of the complexity of the oscillations, there is an essentially different sensitivity of correlated and anti-correlated data sets to additive segments with another type of correlation. Transitions between attractors with correlation properties of the same type can be detected, however, they do not provide strong changes in the value of α , and similar results are obtained for both, correlated and anti-correlated sequences of return times (Fig. 5). Asymmetric changes in the scaling exponent, caused by the presence of segments with different correlation properties, are observed for other examples of complex oscillations, e.g., for the coupled Lorenz oscillators, where the distinction between the two types of behavior is due to the variation of coupling strength (Fig. 6). Unlike interacting Rössler systems, demonstrating a 3.6-fold greater shift for anti-correlated dynamics, an analogous shift in α is more than 10-fold greater in the example, shown in Fig. 6, compared with a correlated sequence of return times that is distorted by segments with distinct correlation properties. Results remain unchanged when using other Poincaré sections or polynomial functions to fit the local trend. 3.3. Blood pressure dynamics When considering relatively long signals recorded in different conditions, the analysis of experimental data using a ‘‘floating’’ window approach clearly identifies distinctions in the data set. A more complicated problem is when the short segments related to different states are mixed. In the case of a physiological system, the latter means that transitions between physiological states occur frequently, and the time when the system attends one of the states is not enough to reliably diagnose the condition of the system. The application of DFA to BP signals in rats in the control condition and under stress enables a separation of these states with the scaling exponent (α = 0.93 and α = 0.45, respectively, in the example shown in Fig. 7). In an effort to address the more complicated separation problem, we generated a simulated
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Fig. 6. Dependencies of the scaling exponent α on the N2 /N1 ratio for transitions between synchronous and asynchronous chaotic oscillations in the model of two coupled Lorenz systems (5). They are in accordance with the results for the model of the coupled Rössler oscillators (4).
Fig. 7. Dependencies of the scaling exponent α on the N2 /N1 ratio estimated for BP signals in rats, related to transitions between control and stressed conditions. Again, a different sensitivity of correlated and anti-correlated data sets to the presence of ‘‘alien’’ segments is shown.
signal, which consists of mixed segments of data related to both physiological states. Depending on the selection of the main state (control or stress), this signal included a different number of segments of another state. Estimates verify that the displacement of the scaling exponent due to signal mixing highly depends on the main type of statistical properties, and transitions between physiological states have a much stronger influence on the diagnosis of anti-correlated dynamics. All the examples discussed in our paper confirm this conclusion. In addition to DFA, other approaches can be applied, e.g., the wavelet-based multifractal formalism [27,28]. Measures based on singularity spectra confirm the main results and conclusions of this study. 4. Discussion and conclusion Distinctions in the diagnosis of positively correlated and anti-correlated time series, based on experimental data, have been discussed in several recent studies. Thus, Ma et al. [8] showed that the correlation measures of such data sets exhibit different sensitivity to data loss due to artifacts, failures of recording equipment, etc. For this purpose, they analyzed random processes characterized by various values of the scaling exponent. These results were also confirmed for the complex dynamics of chaotic oscillators [9,10]. In this paper, we discussed the different sensitivity of correlated and anti-correlated processes to intermittent behavior, when switching between states produces a significant variation of the scaling exponent. Our analysis was based on several examples of data sets, including noise, sequences of return times for chaotic oscillators, and physiological time series. All these examples provided a similar effect – a significantly higher sensitivity of anti-correlated time series to the appearance of segments with distinct correlation properties. This effect is clearly observed for different amounts of data sets, and an asymmetric shift in the scaling exponent was shown for similar numbers of segments with correlated and anti-correlated statistics. We also compared variations of scaling properties, when there are only few ‘‘alien’’ segments and the relative amount of the data sets related to each state is clearly different. Although changes in the numerical values of the scaling exponent vary between the systems with a relative amount of data belonging to each process, they are significantly (up to 10-fold) stronger for anti-correlated time series.
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It should be noted that the different sensitivity of intermittent dynamics depends not only on the type of correlations, but also on the value of α . Thus, processes with α ∼ 0 show a stronger variation of the scaling exponent in the presence of segments with correlated data sets in comparison with anti-correlated processes with α close to 0.5. In this regard, the case of a smaller α is more complicated when performing diagnostics of complex dynamical modes using experimental data. When dealing with such data sets, a preliminary analysis of the ability to use the floating window approach should be considered. In the case of several segments of sufficiently long length that have other correlation properties, they can be easily detected using this analysis, and it is possible to ensure the correct diagnosis of the underlying dynamics. Therefore, a comparative analysis of the global quantity estimated for the entire data set and the local quantities computed for its different parts is a natural way to verify the results of signal processing. Short and frequently occurring segments with distinct correlation properties create more problems when analyzing data, as their detection becomes more complicated and not always successful. In this case, we may not separate these segments and should know the possible impact of switching between different data sets on the quality of diagnostics of the system behavior. In this study we discussed the deterministic dynamics of simulated models of coupled oscillators. The presence of measurement noise creates additional difficulties, since it also provides a different effect depending on the type of correlations. In summary, correlated and anti-correlated data sets show different sensitivity to several circumstances, including data loss, additive fluctuations, and the presence of segments with other types of correlations. All these circumstances complicate the diagnosis of anti-correlated data sets in comparison with correlated signals. Acknowledgment This work was supported by the Russian Science Foundation (Agreement 19-12-00037). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
J.S. Bendat, A.G. Piersol, RandOm Data Analysis and Measurement Procedures, fourth ed., Wiley, New Jersey, 2010. N.G. Kingsbury, J. Appl. Comput. Harmonic Anal. 10 (2001) 234. I.W. Selesnick, R.G. Baraniuk, N.G. Kingsbury, IEEE Signal Process. Mag. 22 (6) (2005) 123. A. Barri, A. Dooms, P.J. Schelkens, Math. Anal. Appl. 389 (2012) 1303. Y. Tran, A. Craig, P. Boord, D. Craig, Med. Biol. Eng. Comput. 42 (2004) 627. N.P. Castellanos, V.A. Makarov, J. Neurosci. Methods 158 (2006) 300. J.A. Urigüen, B. Garcia-Zapirain, J. Neural Eng. 12 (3) (2015) 031001. Q.D.Y. Ma, R.P. Bartsch, P. Bernaola-Galván, M. Yoneyama, P. Ch. Ivanov, Phys. Rev. E 81 (2010) 031101. A.N. Pavlov, O.N. Pavlova, A.S. Abdurashitov, O.A. Sindeeva, O.V. Semyachkina-Glushkovskaya, J. Kurths Chaos 28 (1) (2018) 013124. O.N. Pavlova, A.S. Abdurashitov, M.V. Ulanova, N.A. Shushunova, A.N. Pavlov, Commun. Nonlinear Sci. Numer. Simul. 66 (2019) 31. C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49 (1994) 1685. C.-K. Peng, S. Havlin, H.E. Stanley, A.L. Goldberger, Chaos 5 (1995) 82. H.E. Stanley, L.A.N. Amaral, A.L. Goldberger, S. Havlin, P.Ch. Ivanov, C.-K. Peng, Physica A 270 (1999) 309. P. Talkner, R.O. Weber, Phys. Rev. E 62 (2000) 150. P.Ch. Ivanov, L.A.N. Amaral, A.L. Goldberger, S. Havlin, M.G. Rosenblum, H.E. Stanley, Z.R. Struzik, Chaos 11 (2001) 641. K. Hu, P.Ch. Ivanov, Z. Chen, P. Carpena, H.E. Stanley, Phys. Rev. E 64 (2001) 011114. Z. Chen, P.Ch. Ivanov, K. Hu, H.E. Stanley, Phys. Rev. E 65 (2002) 041107. A.L. Goldberger, L.A.N. Amaral, J.M. Hausdorff, P.Ch. Ivanov, C.-K. Peng, H.E. Stanley, Proc. Natl. Acad. Sci. USA 99 (2002) 2466. R.M. Bryce, K.B. Sprague, Sci. Rep. 2 (2012) 315. L. Kristoufek, Commun. Nonlinear Sci. Numer. Simul. 50 (2017) 193. A.N. Pavlov, A.E. Runnova, V.A. Maksimenko, O.N. Pavlova, D.S. Grishina, A.E. Hramov, Physica A 509 (2018) 777. N.S. Frolov, V.V. Grubov, V.A. Maksimenko, A. Lüttjohann, V.V. Makarov, A.N. Pavlov, E. Sitnikova, A.N. Pisarchik, J. Kurths, A.E. Hramov, Sci. Rep. 9 (2019) 7243. D.E. Postnov, T.E. Vadivasova, O.V. Sosnovtseva, A.G. Balanov, V.S. Anishchenko, E. Mosekilde, Chaos 9 (1999) 227. V.S. Anishchenko, A.N. Silchenko, I.A. Khovanov, Phys. Rev. E 57 (1998) 316. A.N. Pavlov, A.R. Ziganshin, O.A. Klimova, Chaos Solitons Fractals 24 (2005) 57. A.N. Pavlov, O.V. Sosnovtseva, A.R. Ziganshin, N.-H. Holstein-Rathlou, E. Mosekilde, Physica A 316 (2002) 233. J.-F. Muzy, E. Bacry, A. Arneodo, Phys. Rev. Lett. 67 (1991) 3515. J.-F. Muzy, E. Bacry, A. Arneodo, Int. J. Bifurcation Chaos 4 (1994) 245.
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Physica A journal homepage: www.elsevier.com/locate/physa
Scaling features of intermittent dynamics: Differences of characterizing correlated and anti-correlated data sets O.N. Pavlova a , A.N. Pavlov a,b , a b
∗
Saratov State University, Astrakhanskaya Str. 83, Saratov 410012, Russia Yuri Gagarin State Technical University of Saratov, Politechnicheskaya Str. 77, Saratov 410054, Russia
article
info
Article history: Received 6 June 2019 Received in revised form 25 July 2019 Available online 4 September 2019 Keywords: Detrended fluctuation analysis Long-range correlations Intermittent dynamics Chaotic oscillations
a b s t r a c t Using the detrended fluctuation analysis (DFA), we consider the effect of repetitive switching between different states in complex systems, including random processes, intermittent behavior of coupled chaotic oscillators, and the dynamics of blood pressure in rats. We address the problem of diagnosing the state of a system based on time series, when the latter includes data segments with distinct correlation properties, and show significant distinctions in the diagnostics of correlated and anti-correlated data sets. We demonstrate that anti-correlated dynamics are highly sensitive to switching between different states of the system, and the presence of several ‘‘alien’’ segments can provide a much stronger displacement of the scaling exponent unlike the case of correlated dynamics. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Characterization of complex systems based on experimental data is often complicated by the presence of artifacts, noise and time-varying dynamics, which may include changes in control parameters and transitions between distinct types of oscillations [1]. Additive fluctuations in the time series can be completely or partly eliminated during the preprocessing stage, such as noise reduction, provided for both localized artifacts and non-localized fluctuations related, e.g., to measuring noise [2–4]. Despite the existence of many filtering techniques, extracting artifacts from experimental recordings is still a challenging problem, especially if the frequency bands of the noise-free signal and artifacts overlap in the frequency domain, and simple excluding corrupted data segments is usually carried out. This case occurs, for example, in neurophysiological studies, where the percentage of fragments without artifacts can contain about 10% of the total data set [5–7], and only these fragments are selected for further processing. When the remaining parts of the data set are short and do not reliably characterize the state of the system, they are brought together to obtain the required amount of data. However, this procedure destroys the long-range correlation in the original data set, which leads to significant changes in quantitative measures. According to recent studies [8–10], such changes are much stronger for anti-correlated processes, where the elimination of a relatively small percentage of the data set is crucial for characterizing scaling features. Correlated signals, on the other hand, are less sensitive to missing parts of records, and reliable characterization of the dynamics is ensured even in the case of extreme data loss [8]. Unlike corrupted segments of time series related to artifacts or failures of the recording equipment which can be easily recognized even by visual monitoring of data, transitions between distinct oscillatory modes are less distinguishable ∗ Corresponding author at: Saratov State University, Astrakhanskaya Str. 83, Saratov 410012, Russia. E-mail address: [email protected] (A.N. Pavlov). https://doi.org/10.1016/j.physa.2019.122586 0378-4371/© 2019 Elsevier B.V. All rights reserved.
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during the pre-processing stage. These modes are often characterized by nearly similar statistical measures, such as mean value and standard deviation, and the acquired time series may be wrongly associated with a stationary process. Thus, bifurcations of chaotic attractors are often accompanied by changes in the power spectrum, complexity and predictability measures, although a visual inspection of the related changes may not allow detecting the moments of time when one dynamical mode is replaced by another type of oscillations. In the case of rare transitions, recognition can be performed by using a windowed processing of experimental data, i.e., by characterizing the features of the signal within a ‘‘floating’’ window translated along a time series. When dealing with relatively short records containing intermittent oscillatory dynamics, the latter approach becomes ineffective, since the errors in computing numerical measures become comparable with changes of such measures for distinct modes of complex oscillations. In this paper, we consider the effect of repetitive switching between various types of oscillatory behavior in benchmark models of nonlinear dynamics on the correlation properties of signals reflecting their functioning. In addition, we discuss transitions between stochastic processes characterized by distinct statistical properties and between separate states of the cardio-vascular system in order to establish general phenomena in quantifying the scaling features of intermittent dynamics for correlated and anti-correlated data sets. Taking into account that such transitions are typically associated with non-stationary signals, we use a fairly universal method of data processing, namely the detrended fluctuation analysis (DFA) [11,12], which quantifies the power-law correlations in the dynamics of systems with both, stable and time-varying parameters. Unlike many other approaches, the DFA operates with a random walk created from the original signal and analyzes the fluctuations around the local trend, representing slow variations in the mean value. The paper is organized as follows: in Section 2, we briefly discuss the DFA approach applied for signal processing, the benchmark models used to produce signals with intermittent behavior, and experimental data reflecting the stress-induced transient in cardiovascular dynamics. A comparative analysis of the scaling features of intermittent dynamics for correlated and anti-correlated data sets is performed in Section 3. Section 4 summarizes main results of the study. 2. Methods and models 2.1. DFA-approach Detrended fluctuation analysis proposed by Peng et al. [11,12] and widely applied for signal processing in various studies [13–22], involves the construction of a signal profile y(k) =
k ∑
[x(i) − ⟨x⟩] ,
i=1
⟨x ⟩ =
N 1 ∑
N
x(i),
(1)
i=1
its segmentation into non-overlapping parts of size n and fitting the local trend yn (k) within each segment. Often, the piece-wise linear function is used to describe the nonstationarity of the profile, although the DFA-approach has no restrictions with respect to the type of fitting, and the analysis can be performed using polynomial functions of the order 1, 2, 3, etc. The root mean-square fluctuation of the profile around the local trend
N 1 ∑ [y(k) − yn (k)]2 F (n) = √ N
(2)
k=1
typically increases with n, and there is a power-law dependence F (n) ∼ nα ,
(3)
where α is the scaling exponent closely related to the quantities describing the decay of the correlation function or spectral density. Based on α , we can distinguish between anti-correlated dynamics (α < 1/2, large and small values of time series alternate), uncorrelated dynamics (α = 1/2, large and small values have equal probability), power-law correlations (1/2 < α < 1, large values mainly follow large values and vice versa) and correlations which may differ from power-law statistics (α > 1). 2.2. Intermittent noise Aiming to characterize how missing data affects the scaling characteristics of random signals quantified by the DFAapproach, Ma et al. [8] selected noise with different statistical properties and compared correlated and anti-correlated time series. Unlike their study, here we analyze intermittent behavior in stochastic processes. A simple example is a simulated data set that includes repeating segments of noises with distinct statistics (Fig. 1). The data set shown in Fig. 1 involves segments of correlated 1/f -noise described by α = 1.0, and anti-correlated noise with α ∼ 0.12. The duration of the segments and the times of switching are randomly selected. The relative amount of data belonging to each process (N2 /N1 ) varies in order to assess how intermittent behavior with various durations of the main stochastic process (N1 ) with respect to the additional data set (N2 ) changes the scaling exponent α compared to the case when a single process is
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Fig. 1. An example of a stochastic process consisting of segments with different correlation properties (correlated data sets with α = 1.0, and anti-correlated data sets with α ∼ 0.12).
considered. Equal amounts of N1 = N2 mean that correlated and anti-correlated noises are taken under similar conditions when there is no dominant random process. The case when the total duration of one process exceeds the duration of another process enables to decide how stable the estimates of α -exponent are if switching appear. To avoid the effects of short samples for simulated data sets, we used sequences about 100,000 return times, and the average duration of the included segments ranged from 100 to 1000. We tried not to consider longer segments that can be detected using the floating window approach for analyzing data sets. 2.3. Coupled chaotic oscillators Two models of coupled chaotic oscillators are chosen because they demonstrate a wider range of complex dynamical phenomena as compared to single chaotic systems, and therefore, diverse intermittent behavior can be considered for a more generalized conclusion. The first model is a system of two diffusively coupled Rössler oscillators, described by six ordinary differential equations dx1,2 dt dy1,2 dt dz1,2 dt
= −ω1,2 y1,2 − z1,2 + γ (x2,1 − x1,2 ), = ω1,2 x1,2 + ay1,2 , = b + z1,2 (x1,2 − c),
(4)
ω1,2 = ω0 ± ∆
with four fixed parameters ω0 = 1, a = 0.15, b = 0.2, γ = 0.02, and two variable parameters, c and ∆, the choice of which provides transitions among different types of regular, chaotic and hyper-chaotic oscillations. The complete bifurcation diagram of the model (4) is given in Ref. [23], where transitions to and between various types of in-phase and out-of-phase oscillations are considered. We have chosen the following types of complex dynamics:
• • • •
synchronous chaotic attractor (SA); asynchronous chaotic attractor (AA); quasiperiodic attractor (QA); hyperchaotic attractor (HA).
The values of the parameters related to these modes were selected near the bifurcation lines, and further studies were carried out using sequences of return times to the Poincaré section x1 = 0. Such sequences reflect information about a multiscale structure of chaotic attractors. The second model is a system of two coupled Lorenz oscillators dx1,2 dt dy1,2 dt dz1,2 dt
= σ (y1,2 − x1,2 ) + γ (x2,1 − x1,2 ), = r1,2 x1,2 − x1,2 z1,2 − y1,2 ,
(5)
= x1,2 y1,2 − z1,2 b
with fixed parameters σ = 10, r1 = 28.8, r2 = 28, b = 8/3 and variable coupling strength γ . This model produces chaotic oscillations with two distinct time scales: a slow time scale related to switching between oscillations
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Fig. 2. The difference (∆) of the oscillation frequencies in the model of two coupled Lorenz oscillators (5) versus the coupling strength.
around unstable focus points, which can be interpreted as transitions between two states ‘‘+1’’ (x1,2 > 0) and ‘‘1’’ (x1,2 < 0), if each oscillator is considered as a bistable system [24], and a fast time scale related to the frequency of the oscillations themselves. Both time scales vary depending on the parameters of each oscillator and the interaction between subsystems. The model (5) provides a quite atypical behavior depending on the coupling strength [24]. Unlike the expected entrainment of the oscillation frequencies of the subsystems, there is the phenomenon of desynchronization with an increase in γ to a value of γ ∼ 2, which is further replaced by the adjustment of frequencies in the region of γ > 2 (Fig. 2). The dynamics of model (5) will be studied based on the sequences of return times to the Poincaré section z1 = 30. 2.4. Blood pressure dynamics In addition to the simulated data sets, transients associated with the stress-induced response of the cardiovascular system in healthy rats are considered. Experimental studies were conducted on adult male rats weighing from 250 to 300 g in accordance with the Guide for the Care and Use of Laboratory Animals (NIH Publication No. 85–23, revised 1996) and protocols approved by the local ethics committee of Saratov State University (Saratov, Russia). Animals were kept at 25 ± 2 ◦ C, humidity 55% and light/dark cycle of 12:12 h. Blood pressure (BP) was measured directly using an intra-arterial polyethylene catheter. The data were recorded using the PowerLab/400ML401 measuring complex and Chart 4 software (ADInstruments Ltd., Australia). The sampling step was chosen to be 200 Hz. The model of immobilization stress was analyzed when a rat was placed in a special case that did not allow it to move. Each experiment included a control measurement (10 min), immobilization (10 min) and the recovery process after stress-off (10 min). The analysis was performed using sequences of time intervals between local maxima of the BP signal, representing an analogue of the return times in simulated studies with chaotic oscillators. In order to clearly identify these maxima and, therefore, time intervals, spline interpolation of the measured time series was provided. Additionally, we used a band-pass filter with cutoff frequencies of 0.05 Hz and 30 Hz during the pre-processing stage in order to exclude high-frequency fluctuations and slow floating of the mean value. According to the earlier study [25], typical values of α are between 0.9 and 1.3 for control measurements. Stress usually reduces the scaling exponent, but the dynamics remain correlated in all states. In few animals, however, stress provided stronger changes in cardiovascular dynamics, when BP becomes uncorrelated or anti-correlated with α , taking values of about 0.4–0.5 during stress. We selected two such animals, and the corresponding records were chosen to study the transitions between correlated and anti-correlated dynamics. 3. Results 3.1. Intermittent noise The study of intermittent dynamics was started with a relatively simple example of a stochastic signal made up of segments with distinct statistical properties shown in Fig. 1. The results of the DFA-analysis for such a data set clearly depend on the relative amount of data belonging to each process. In the case when 1/f noise is considered as the main process, and the total duration of the anti-correlated noise is less or comparable, there is no significant shift in the scaling exponent (Fig. 3). The value of α fluctuates nearly α = 1.0, and such variations are close to computing errors for time series of finite length. This situation changes dramatically for an anti-correlated process that contains short fragments of noise with power-law correlations. Even the appearance of segments with a total duration of about 5% of the main process provides an increase in α by 0.5–0.6, i.e. almost 5-fold compared with the scaling exponent of the main stochastic process (α = 0.12). Therefore, the processing of anti-correlated data sets is very sensitive to the presence of parts with other statistical properties. When the N2 /N1 ratio exceeds 20%, the anti-correlated dynamics can be treated as correlated one (Fig. 3).
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Fig. 3. Dependence of the scaling exponent α on the N2 /N1 ratio for the stochastic signal shown in Fig. 1. Hereinafter, N1 is the number of samples of the main process, and N2 is the number of samples of the additional data set. Results are shown for cases of anti-correlated and correlated signals, considered as the main process, respectively.
Fig. 4. Dependencies of α on the N2 /N1 ratio for transitions between (a) synchronous and asynchronous chaotic oscillations, (b) hyperchaotic and asynchronous chaotic oscillations, (c) periodic and quasiperiodic oscillations in the model of two coupled Rössler systems (4). All designations are given in accordance with Fig. 3. Regardless of the type of transition, different sensitivity of correlated and anti-correlated dynamics to the appearance of fragments with distinct correlation properties is shown.
3.2. Coupled chaotic oscillators Transitions between different types of regular, chaotic and hyperchaotic dynamics in coupled oscillators confirm the distinctions for correlated and anti-correlated data sets (Fig. 4). Consider switching between the SA (c = 6.8, ∆ = 0.0097) and AA (c = 6.8, ∆ = 0.0098) attractors in the model of two diffusively coupled Rössler oscillators (4) – Fig. 4a. The series of return times to the Poincaré section for these attractors are described by scaling exponents α ≃ 0.02 (SA) and α ≃ 0.80 (AA). The presence of anti-correlations in the first series is a consequence of the period-doubling route to chaos, which produces the alternation of large and small return times [26]. Asynchronous chaotic oscillations arise
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Fig. 5. Dependencies of the scaling exponent α on the N2 /N1 ratio for transitions between (a) synchronous chaotic and hyperchaotic oscillations, (b) quasiperiodic and asynchronous chaotic oscillations in the model of two coupled Rössler systems. In both cases, the transition between data sets with the same correlation properties is considered (anti-correlations in a, and correlations in b). In this regard, there are no strong changes in the value of α .
due to the destruction of quasiperiodic oscillations, which are characterized by relatively slow modulation of amplitude and frequency and do not show such an alternation. If the series of return times includes a random switching between SA and AA, and the total duration of the data sets related to each mode is almost the same, then the scaling exponent takes a value α ≃ 0.63. Therefore, the shift in the scaling exponent is asymmetric; such shift is about 3.6-fold greater for the anti-correlated dynamics of SA (∆α = 0.61) than for the correlated dynamics of AA (∆α = 0.17). In this paper, we show results for segments of the mean duration of 500 return times. The results do not strongly depend on the duration of such segments if the N2 /N1 ratio is fixed. For example, the difference between the values of α for segments with the duration of 200 and 1000 takes about 3%–4%. Note also that the phase multistability does not essentially affect the results. Thus, in-phase and out-of-phase periodic or chaotic attractors are characterized by different values of α . However, such distinctions are quite small compared with the distinctions between the values of α for correlated and anti-correlated data sets. The abovementioned is also confirmed for other types of intermittent behavior, which included the alternation of correlated and anti-correlated data sets. Fig. 4b illustrates very similar results for random switching between hyperchaotic oscillations (HA, c = 7.2, ∆ = 0.0094, which appears due to the merging of synchronous chaotic modes and inherits the anti-correlated structure of return times with α ≃ 0.03) and asynchronous chaos considered for parameters c = 7.2, ∆ = 0.0096, which produces a correlated dynamics of return times with α ≃ 0.79). Again, there is an asymmetric shift in the scaling exponent if the time series under study is a mixture of segments formed by both types of complex oscillations. The dependences of α (N2 /N1 ) nearly coincide in both switching variants (Fig. 4a and b). Consideration of the transition between simpler types of oscillations — periodic oscillations related to the cycle of period 4 (c = 6.2, ∆ = 0.0090, α ≃ 0.0) and quasiperiodic oscillations (QA, c = 6.2, ∆ = 0.0100, α ≃ 0.92) leads to a similar conclusion. Thus, regardless of the complexity of the oscillations, there is an essentially different sensitivity of correlated and anti-correlated data sets to additive segments with another type of correlation. Transitions between attractors with correlation properties of the same type can be detected, however, they do not provide strong changes in the value of α , and similar results are obtained for both, correlated and anti-correlated sequences of return times (Fig. 5). Asymmetric changes in the scaling exponent, caused by the presence of segments with different correlation properties, are observed for other examples of complex oscillations, e.g., for the coupled Lorenz oscillators, where the distinction between the two types of behavior is due to the variation of coupling strength (Fig. 6). Unlike interacting Rössler systems, demonstrating a 3.6-fold greater shift for anti-correlated dynamics, an analogous shift in α is more than 10-fold greater in the example, shown in Fig. 6, compared with a correlated sequence of return times that is distorted by segments with distinct correlation properties. Results remain unchanged when using other Poincaré sections or polynomial functions to fit the local trend. 3.3. Blood pressure dynamics When considering relatively long signals recorded in different conditions, the analysis of experimental data using a ‘‘floating’’ window approach clearly identifies distinctions in the data set. A more complicated problem is when the short segments related to different states are mixed. In the case of a physiological system, the latter means that transitions between physiological states occur frequently, and the time when the system attends one of the states is not enough to reliably diagnose the condition of the system. The application of DFA to BP signals in rats in the control condition and under stress enables a separation of these states with the scaling exponent (α = 0.93 and α = 0.45, respectively, in the example shown in Fig. 7). In an effort to address the more complicated separation problem, we generated a simulated
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Fig. 6. Dependencies of the scaling exponent α on the N2 /N1 ratio for transitions between synchronous and asynchronous chaotic oscillations in the model of two coupled Lorenz systems (5). They are in accordance with the results for the model of the coupled Rössler oscillators (4).
Fig. 7. Dependencies of the scaling exponent α on the N2 /N1 ratio estimated for BP signals in rats, related to transitions between control and stressed conditions. Again, a different sensitivity of correlated and anti-correlated data sets to the presence of ‘‘alien’’ segments is shown.
signal, which consists of mixed segments of data related to both physiological states. Depending on the selection of the main state (control or stress), this signal included a different number of segments of another state. Estimates verify that the displacement of the scaling exponent due to signal mixing highly depends on the main type of statistical properties, and transitions between physiological states have a much stronger influence on the diagnosis of anti-correlated dynamics. All the examples discussed in our paper confirm this conclusion. In addition to DFA, other approaches can be applied, e.g., the wavelet-based multifractal formalism [27,28]. Measures based on singularity spectra confirm the main results and conclusions of this study. 4. Discussion and conclusion Distinctions in the diagnosis of positively correlated and anti-correlated time series, based on experimental data, have been discussed in several recent studies. Thus, Ma et al. [8] showed that the correlation measures of such data sets exhibit different sensitivity to data loss due to artifacts, failures of recording equipment, etc. For this purpose, they analyzed random processes characterized by various values of the scaling exponent. These results were also confirmed for the complex dynamics of chaotic oscillators [9,10]. In this paper, we discussed the different sensitivity of correlated and anti-correlated processes to intermittent behavior, when switching between states produces a significant variation of the scaling exponent. Our analysis was based on several examples of data sets, including noise, sequences of return times for chaotic oscillators, and physiological time series. All these examples provided a similar effect – a significantly higher sensitivity of anti-correlated time series to the appearance of segments with distinct correlation properties. This effect is clearly observed for different amounts of data sets, and an asymmetric shift in the scaling exponent was shown for similar numbers of segments with correlated and anti-correlated statistics. We also compared variations of scaling properties, when there are only few ‘‘alien’’ segments and the relative amount of the data sets related to each state is clearly different. Although changes in the numerical values of the scaling exponent vary between the systems with a relative amount of data belonging to each process, they are significantly (up to 10-fold) stronger for anti-correlated time series.
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It should be noted that the different sensitivity of intermittent dynamics depends not only on the type of correlations, but also on the value of α . Thus, processes with α ∼ 0 show a stronger variation of the scaling exponent in the presence of segments with correlated data sets in comparison with anti-correlated processes with α close to 0.5. In this regard, the case of a smaller α is more complicated when performing diagnostics of complex dynamical modes using experimental data. When dealing with such data sets, a preliminary analysis of the ability to use the floating window approach should be considered. In the case of several segments of sufficiently long length that have other correlation properties, they can be easily detected using this analysis, and it is possible to ensure the correct diagnosis of the underlying dynamics. Therefore, a comparative analysis of the global quantity estimated for the entire data set and the local quantities computed for its different parts is a natural way to verify the results of signal processing. Short and frequently occurring segments with distinct correlation properties create more problems when analyzing data, as their detection becomes more complicated and not always successful. In this case, we may not separate these segments and should know the possible impact of switching between different data sets on the quality of diagnostics of the system behavior. In this study we discussed the deterministic dynamics of simulated models of coupled oscillators. The presence of measurement noise creates additional difficulties, since it also provides a different effect depending on the type of correlations. In summary, correlated and anti-correlated data sets show different sensitivity to several circumstances, including data loss, additive fluctuations, and the presence of segments with other types of correlations. All these circumstances complicate the diagnosis of anti-correlated data sets in comparison with correlated signals. Acknowledgment This work was supported by the Russian Science Foundation (Agreement 19-12-00037). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
J.S. Bendat, A.G. Piersol, RandOm Data Analysis and Measurement Procedures, fourth ed., Wiley, New Jersey, 2010. N.G. Kingsbury, J. Appl. Comput. Harmonic Anal. 10 (2001) 234. I.W. Selesnick, R.G. Baraniuk, N.G. Kingsbury, IEEE Signal Process. Mag. 22 (6) (2005) 123. A. Barri, A. Dooms, P.J. Schelkens, Math. Anal. Appl. 389 (2012) 1303. Y. Tran, A. Craig, P. Boord, D. Craig, Med. Biol. Eng. Comput. 42 (2004) 627. N.P. Castellanos, V.A. Makarov, J. Neurosci. Methods 158 (2006) 300. J.A. Urigüen, B. Garcia-Zapirain, J. Neural Eng. 12 (3) (2015) 031001. Q.D.Y. Ma, R.P. Bartsch, P. Bernaola-Galván, M. Yoneyama, P. Ch. Ivanov, Phys. Rev. E 81 (2010) 031101. A.N. Pavlov, O.N. Pavlova, A.S. Abdurashitov, O.A. Sindeeva, O.V. Semyachkina-Glushkovskaya, J. Kurths Chaos 28 (1) (2018) 013124. O.N. Pavlova, A.S. Abdurashitov, M.V. Ulanova, N.A. Shushunova, A.N. Pavlov, Commun. Nonlinear Sci. Numer. Simul. 66 (2019) 31. C.-K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49 (1994) 1685. C.-K. Peng, S. Havlin, H.E. Stanley, A.L. Goldberger, Chaos 5 (1995) 82. H.E. Stanley, L.A.N. Amaral, A.L. Goldberger, S. Havlin, P.Ch. Ivanov, C.-K. Peng, Physica A 270 (1999) 309. P. Talkner, R.O. Weber, Phys. Rev. E 62 (2000) 150. P.Ch. Ivanov, L.A.N. Amaral, A.L. Goldberger, S. Havlin, M.G. Rosenblum, H.E. Stanley, Z.R. Struzik, Chaos 11 (2001) 641. K. Hu, P.Ch. Ivanov, Z. Chen, P. Carpena, H.E. Stanley, Phys. Rev. E 64 (2001) 011114. Z. Chen, P.Ch. Ivanov, K. Hu, H.E. Stanley, Phys. Rev. E 65 (2002) 041107. A.L. Goldberger, L.A.N. Amaral, J.M. Hausdorff, P.Ch. Ivanov, C.-K. Peng, H.E. Stanley, Proc. Natl. Acad. Sci. USA 99 (2002) 2466. R.M. Bryce, K.B. Sprague, Sci. Rep. 2 (2012) 315. L. Kristoufek, Commun. Nonlinear Sci. Numer. Simul. 50 (2017) 193. A.N. Pavlov, A.E. Runnova, V.A. Maksimenko, O.N. Pavlova, D.S. Grishina, A.E. Hramov, Physica A 509 (2018) 777. N.S. Frolov, V.V. Grubov, V.A. Maksimenko, A. Lüttjohann, V.V. Makarov, A.N. Pavlov, E. Sitnikova, A.N. Pisarchik, J. Kurths, A.E. Hramov, Sci. Rep. 9 (2019) 7243. D.E. Postnov, T.E. Vadivasova, O.V. Sosnovtseva, A.G. Balanov, V.S. Anishchenko, E. Mosekilde, Chaos 9 (1999) 227. V.S. Anishchenko, A.N. Silchenko, I.A. Khovanov, Phys. Rev. E 57 (1998) 316. A.N. Pavlov, A.R. Ziganshin, O.A. Klimova, Chaos Solitons Fractals 24 (2005) 57. A.N. Pavlov, O.V. Sosnovtseva, A.R. Ziganshin, N.-H. Holstein-Rathlou, E. Mosekilde, Physica A 316 (2002) 233. J.-F. Muzy, E. Bacry, A. Arneodo, Phys. Rev. Lett. 67 (1991) 3515. J.-F. Muzy, E. Bacry, A. Arneodo, Int. J. Bifurcation Chaos 4 (1994) 245.