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Limits of permutation-based entropies in assessing complexity of short heart period variability

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2015 Physiol. Meas. 36 755

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Physiological Measurement

Limits of permutation-based entropies in assessing complexity of short heart period variability

Alberto Porta1,2, Vlasta Bari3, Andrea Marchi4, Beatrice De Maria5, Paolo Castiglioni6, Marco di Rienzo6, Stefano Guzzetti7, Andrei Cividjian8 and Luc Quintin8

1 Department of Biomedical Sciences for Health, University of Milan, Milan, Italy2 IRCCS Galeazzi Orthopedic Institute, Milan, Italy3 Department of Cardiothoracic, Vascular Anesthesia and Intensive Care, IRCCS Policlinico San Donato, Milan, Italy4 Department of Anesthesia and Intensive Care Unit, Humanitas Clinical and Research Center, Rozzano, Milan, Italy5 IRCCS Maugeri Foundation, Milan, Italy6 IRCCS Fondazione Don C. Gnocchi, Milan, Italy7 Department of Emergency, L. Sacco Hospital, Milan, Italy8 Physiology (EA 4612: Neurocardiology), University of Lyon, Lyon, France

E-mail: alberto.porta@unimi.it

Received 31 July 2014, revised 10 September 2014Accepted for publication 13 October 2014Published 23 March 2015

AbstractThe study compares permutation-based and coarse-grained entropy approaches for the assessment of complexity of short heart period (HP) variability recordings. Shannon permutation entropy (SPE) and conditional permutation entropy (CPE) are computed as examples of permutation-based entropies, while the k-nearest neighbor conditional entropy (KNNCE) is calculated as an example of coarse-grained conditional entropy. SPE, CPE and KNNCE were applied to ad-hoc simulated autoregressive processes corrupted by increasing amounts of broad band noise and to real HP variability series recorded after complete vagal blockade obtained via administration of a high dose of atropine (AT) in nine healthy volunteers and during orthostatic challenge induced by 90° head-up tilt (T90) in 15 healthy individuals. Over the simulated series the performances of SPE and CPE degraded more rapidly with the amplitude of the superimposed broad band noise than those of KNNCE. Over real data KNNCE identified the expected decrease of the HP variability complexity both after AT and during T90. Conversely SPE and CPE detected the decrease of HP variability complexity solely during T90 as a likely result of the more favorable signal-to-noise ratio during T90 than after AT. Results derived from

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both simulations and real data indicated that permutation-based entropies had a larger susceptibility to broad band noise than KNNCE. We recommend caution in applying permutation-based entropies in presence of short HP variability series characterized by a low signal-to-noise ratio.

Keywords: permutation entropy, conditional entropy, atropine, head-up tilt, heart rate variability, autonomic nervous system, cardiovascular control

1. Introduction

Entropy-based complexity analysis of heart period (HP) variability provides markers helpful in typifying cardiac control in physiological conditions. Indeed, entropy-based complexity indexes of cardiac regulation progressively decrease with age (Kaplan et al 1991, Goldberger et al 2002, Porta et al 2014) and tilt table  inclination (Porta et al 2007, Turianikova et al 2011) as a result of the gradual increase of the sympathetic drive and progressive vagal with-drawal with ageing and relevance of the orthostatic challenge respectively. Entropy-based markers of complexity of the cardiac control were assessed in pathological conditions as well (Goldberger et al 2002) and found helpful in distinguishing pathological individuals from healthy age-matched subjects (Javorka et al 2008, Schulz et al 2010).

Two main classes of methods are currently exploited for the assessment of entropy-based indexes of HP variability complexity. The first technique requires coarse graining of the origi-nal dynamics (Pincus 1995, Porta et al 1998 2013, Richman and Moorman 2000), while the second one is based on permutations (Bandt and Pompe 2002, Parlitz et al 2012, Riedl et al 2013). Techniques based on permutations (Amigò et al 2013) are gaining attention because they do not require any coarse graining of the original dynamics, thus avoiding the setting of parameters defining the degree of approximation of the dynamics. In spite of their large utili-zation in assessing complexity of short HP variability series, permutation-based and coarse-grained entropy approaches were never compared directly. We hypothesize that, despite the undeniable ability of ordinal pattern statistics to extract features helpful in classification and symbolic representation (Parlitz et al 2012, Amigò et al 2013), their capability of providing global indexes of dynamical complexity might be biased by broad band noise superimposed to data, thus being inadequate compared to coarse-grained approaches that intrinsically filter out the series according to the adopted coarse graining strategy.

The aim of this study is to compare permutation-based entropies (Bandt and Pompe 2002), i.e. Shannon permutation entropy (SPE) and conditional permutation entropy (CPE), with a coarse-grained entropy, i.e. the k-nearest-neighbor conditional entropy (KNNCE) (Porta et al 2013), over ad-hoc simulated autoregressive (AR) processes and short HP variability series recorded during two paradigmatic experimental protocols characterized by different signal-to-noise ratios: (i) high dose administration of atropine (AT) pharmaco-logically blocking the contribution of the vagal modulation to sinus node and dramatically reducing the HP variance (Parlow et al 1995); (ii) 90° head-up tilt (T90) inducing a vagal withdrawal with preservation of the HP variance (Porta et al 2000). Permutation-based and coarse grained techniques are expected to indicate a decrease of the complexity of the HP variability after AT and T90 as a consequence of the loss of fast temporal scales in cardiac regulation associated to the vagal blockade or withdrawal (Pomeranz et al 1985, Montano et al 1994).

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2. Methods

2.1. Permutation-based entropies

Given the series HP = {HP(i), i = 1, … , N}, where i is the progressive beat counter and N is the series length, and defined as HPL(i) = [HP(i),HP(i − 1), … ,HP(i − L + 1)] the ordered sequence formed by L consecutive delayed HP values, we term as permutation of HPL(i) the pattern formed by substituting the components of HPL(i) with their rank in HPL(i). Ranking was performed according to an ascending order. Equal ranks were avoided. In case of equal HP values, leading to identical ranks, they were differentiated according to the different posi-tions of the equal HP samples in HPL(i) with larger ranks attributed to more recent HP val-ues. The number of possible permutations was L!. Two types of permutation-based entropies were computed: the SPE computed as the Shannon entropy of the permutations (Bandt and Pompe 2002) and the CPE computed as the variation of SPE at L with respect to SPE at L − 1 (Unakafov and Keller 2014). Since SPE and CPE were functions of L, SPE and CPE were calculated at L = 3 and these values were indicated as SPE(3) and CPE(3) in the following. The setting L = 3 is one of the most common choice in entropy calculation over short HP variability recordings using metrics necessitating the assignment of the pattern length (Pincus 1995, Richman and Moorman 2000).

2.2. Coarse-grained entropy

As an example of coarse-grained conditional entropy we computed the KNNCE (Porta et al 2013). KNNCE was computed directly over the samples of HP as the Shannon entropy of the current sample, HP(i), given its L − 1 previous samples, HPL − 1(i − 1) = [HP(i − 1),…,HP(i − L + 1)] averaged over all HPL − 1(i − 1). The distribution of HP(i) given HPL − 1(i − 1) was built by considering the next value of the k patterns nearest to HPL − 1(i − 1) regardless of their actual distance from it. The value of k set the level of coarse-graining: the larger the value of k, the more coarse the graining because it was more likely that distant patterns were considered to be indistinguishable. The distance was computed according to the Euclidean norm. The course of KNNCE was the balance between two opposite tendencies (Porta et al 2013): (i) if past values were helpful in reducing the uncertainty associated to future samples, KNNCE decreased with L; (ii) in presence of noise patterns progressively took apart with L in the multidimensional embedding space and, consequently, uncertainty in predicting future values raised with L. The two opposite tendencies resulted in a minimum in presence of repetitive patterns embedded in noise. The value of KNNCE at the minimum was taken as a complexity index (KNNCEmin). Normalized KNNCEmin (NKNNCEmin) was obtained by dividing KNNCEmin by Shannon entropy of HP (Porta et al 2013). NKNNCEmin ranged between 0 and 1, where 0 and 1 indi-cated null and maximal complexity of the HP series. The procedure of minimization allowed the calculation of NKNNCEmin without fixing a priori L but NKNNCEmin needed the assign-ment of k. We set k = 30 according to applications of KNNCE over short HP variability record-ings (Porta et al 2013 2014).

2.3. Simulated AR processes with additive broad band noise

We built 20 realizations (256 samples) of three types of second order AR processes with two complex and conjugated poles with phases equal to ±0.05, ±0.1 and ±0.25 cycles/beat respectively. These AR processes were all characterized by a dominant spectral peak but its central frequency was different, thus simulating very low, low and high frequency (VLF, LF

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and HF) oscillations usually present in short-term HP variability (Task Force 1996). A pole modulus equal to 0.9 guaranteed random fluctuations and the degree of predictability typical of short HP variability (Porta et al 1998). These realizations were normalized to have zero mean and unit variance. The AR partial predictability was gradually masked by adding a white Gaussian noise independent of the AR process with variance progressively increasing from 0 to 0.5 with steps of 0.05, thus obtaining noisy series with levels of corruption rang-ing from 0 to 50%. Each AR series was shuffled to obtain a surrogate sequence preserving the distribution of the original AR process but being completely irregular (i.e. the original autocorrelation was destroyed). The shuffled version allowed the evaluation of the degree of complexity in absence of deterministic and repetitive patterns. The performance of complex-ity indexes (i.e. NKNNCEmin, SPE(3) and CPE(3)) in distinguishing the complexity of the original series compared to a completely random version was assessed by calculating the difference, Δ, between the index computed over each surrogate and original AR series (i.e. ΔNKNNCEmin, ΔSPE(3) and ΔCPE(3)). The mean value of Δ and its 95% confidence interval (i.e. mean of Δ ± two times the standard deviation of Δ) was computed. If the 95% confidence interval crossed 0, the considered index was not able to differentiate the original AR process from its shuffled version (i.e. the original series was indistinguishable from broad band noise).

3. Experimental protocols and data analysis

3.1. Pharmacological protocol

We exploited data recorded during an experimental protocol planned to study the effects of pharmacological blockades of the autonomic nervous system on the cardiac baroreflex sen-sitivity (Parlow et al 1995). We recorded electrocardiogram (ECG) from lead II in 9 healthy male physicians aged from 25 to 46 years familiar with the study setting. None had any abnor-mal finding in history, physical examination or electrocardiography or was receiving any med-ication. They were instructed to avoid tobacco, alcohol and caffeine for 12 h and strenuous exercise for 24 h before each experiment. ECG was sampled at 500 Hz. All subjects gave their written informed consent. Subjects remained at rest in supine position in a quiet darkened room during the protocol. Each experiment consisted of 15–20 min of baseline recording at rest in supine position (B) followed by 15–20 min of recording after parasympathetic block-ade with AT (40 μg kg−1 intravenous atropine sulfate) to block muscarinic receptors. During the protocol the subjects breathed spontaneously but were not allowed to talk. The proto-col adhered to the principles of the Declaration of Helsinki. The human research and ethical review board of the Hospices Civils de Lyon, Lyon, France, approved the protocol.

3.2. Head-up tilt protocol

We exploited data recorded during an experimental protocol planned to study the complex-ity of the cardiac autonomic regulation (Porta et al 2000). We recorded ECG (lead II) in 15 healthy nonsmoking humans (age from 24 to 54, 9 males and 6 females). A detailed medical history and examination excluded the evidence of any disease. The subjects did not take any medication and consume any caffeine or alcohol-containing beverages in the 24 h before the recording. ECG was sampled at 300 Hz. Informed written consent was collected during the enrollment. Subjects were on the tilt table supported by two belts at the level of thigh and waist respectively and with both the feet touching the footrest of the tilt table. During the protocol the subjects breathed spontaneously but were not allowed to talk. After 10 min of familiar-ization with the setting of the laboratory, ECG was recorded at B for 7 min and during T90

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maintained for 10 min. All subjects were able to complete the T90 session without experienc-ing any sign of pre-syncope. The study adheres to the principles of the Declaration of Helsinki for medical research involving human subjects. The human research and ethical review boards of the L. Sacco Hospital, Milan, Italy, approved the protocol.

3.3. Extraction of the beat-to-beat HP variability

The R-wave apex was located over the ECG with minimum jitters using parabolic interpola-tion. HP was approximated as the time distance between two consecutive R-wave peaks. The occurrences of R-wave peaks were carefully checked to avoid erroneous detections or missed beats. If isolated ectopic beats affected HP values, these measures were linearly interpolated using the closest values unaffected by ectopic beats. According to the standards of short-term analysis of cardiovascular variability (Task Force 1996) sequences of 256 consecutive HP values were randomly selected from each experimental sessions. The stationarity of the selected sequence was tested according to (Magagnin et al 2011) over the original series after linear detrending. If the test for the steadiness of mean and variance was not fulfilled, a new random selection was carried out again until the fulfillment of the restricted weak stationarity test (Magagnin et al 2011). The mean and variance of HP were indicated as µHP and σHP

2 and expressed in ms and ms2 respectively.

3.4. Statistical analysis

One way analysis of variance (Tukey test for multiple comparisons), was applied to check the significance of the differences between complexity of simulated VLF, LF and HF processes in absence of superimposed noise. If the normality test (Kolmogorov–Smirnov test) was not fulfilled, Kruskal–Wallis one way analysis of variance on ranks (Tukey test for multiple com-parisons) was applied. We performed paired t-test to evaluate the effect of the pharmaco-logical challenge (i.e. AT) or experimental maneuver (i.e. T90) over NKNNCEmin, SPE(3) and CPE(3). If the normality test was not fulfilled, Wilcoxon signed rank test was utilized. Statistical analysis was carried out using a commercial statistical program (Sigmastat, SPSS, ver.3.0.1). A p < 0.05 was always considered as significant.

4. Results

4.1. Simulation results

Figure 1 depicts the bargraphs relevant to the NKNNCEmin (figure 1(a)), SPE(3) (figure 1(b)), and CPE(3) (figure 1(c)) as a function of the type of simulated AR process (i.e. VLF, LF and HF processes). Results are given as mean plus standard deviation. All considered indexes indicated that the complexity of the AR process increases significantly as a function of the fre-quency of the dominant rhythm. This conclusion was reached by all considered entropy-based methods (i.e. KNNCE, SPE and CPE).

Figure 2 shows the course of ΔNKNNCEmin (figures 2(a)–(c)), ΔSPE(3) (figures 2(d)–(f)) and ΔCPE(3) (figures 2(g)–(i)) as a function of the percentage of the noise variance. Multiple straight line plots report the mean (solid line), 95% confidence limits (dashed lines) com-puted as the mean  ±  two times the standard deviation and the zero line (dotted line). If the zero line crossed the region delimited by the confidence limits, the relevant index (i.e. NKNNCEmin in figures 2(a)–(c), SPE(3) in figures 2(d)–(f) and CPE(3) in figures 2(g)–(i)) would not be able to distinguish the original from the surrogate series. The leftmost column

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of panels are relevant to the results of VLF simulations (figures 2(a), (d), (g)), while the indexes computed over LF and HF processes are shown in the central (figures 2(b), (e), (h)) and rightmost ( figures  2(c), (f), (i)) column of panels respectively. All three indexes (i.e. ΔNKNNCEmin, ΔSPE(3) and ΔCPE(3)) exhibited a tendency to decrease towards the zero line with the variance of the additive noise, thus suggesting that the performance of all indexes in

Figure 1. Bargraphs show NKNNCEmin (a), SPE(3) (b) and CPE(3) (c) as a function of the type of the simulated AR process (i.e. VLF, LF and HF series). Entropy values were obtained from a set of 20 realizations. The symbol * indicates a significant difference with p < 0.05. Values are shown as mean plus standard deviation. VLF, LF and HF series exhibit different complexity values regardless of the method exploited for calculation of entropy (a–c).

Figure 2. Multiple straight line plots show the courses of ΔNKNNCEmin (a–c), ΔSPE(3) (d–f) and ΔCPE(3) (g–i) as a function of the percentage of the noise variance superposed to the original realizations. The lines indicate the mean (solid line), confidence limits (dashed lines) computed as the mean plus/minus two times the standard deviation and the zero line (dotted line). The leftmost, central and rightmost panels are relevant to the results of VLF (a, d, g), LF (b, e, h) and HF (c.f.i) processes respectively. If the inferior confidence limit crosses the zero line, the correspondent entropy method cannot differentiate the series from a broad band noise.

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distinguishing the original process from its shuffled version decreased more and more with the noise variance. This conclusion was independent of the shape of the power spectrum (i.e. on the type of the simulated series). The trend of ΔSPE(3) and ΔCPE(3) towards the zero line was much faster than that of ΔNKNNCEmin. Indeed, while in the case of ΔNKNNCEmin the zero line never crossed the region delimited by the 95% confidence limits (figures 2(a)–(c)), this situation always occurred in the case of ΔSPE(3) and ΔCPE(3) (figures 2(d)–(f) and figures 2(g)–(i) respectively). The crossing occurred at the smallest percentage of noise vari-ance (i.e. 15%) in the case of the VLF process (figures 2(d), (g)). It is remarkable that SPE(3) and CPE(3) performed similarly.

4.2. Experimental results

Figure 3 depicts the bargraphs relevant to HP mean, µHP, (figures 3(a), (c)) and variance, σHP2

(figures 3(b), (d)) in both experimental protocols. In the upper panels (figures 3(a), (b)) the bargraphs show the effects of AT on µHP (figure 3(a)) and σHP

2 (figure 3(b)), while in the lower panels (figures 3(c), (d)) the ones of T90. Results are given as mean plus standard deviation. Both AT and T90 induced an important tachycardic response (figures 3(a), (c)) but, while AT led to a dramatic reduction of the magnitude of the HP fluctuations (figure 3(b)), σHP

2 was not significantly modified by T90 (p = 0.064, figure 3(d)).

Figure 4 depicts the bargraphs relevant to NKNNCEmin (figures 4(a), (d)), SPE(3) (figures 4(b), (e)), and CPE(3) (figures 4(c), (f)) in both experimental protocols. In the upper panels (figures 4(a)–(c)) the bargraphs show the effects of AT on NKNNCEmin (figure 4(a)), SPE(3)

Figure 3. Bargraphs show µHP and σHP2 before (i.e. B) and after parasympathetic

blockade (i.e. AT) in (a) and (b) respectively and before (i.e. B) and after orthostatic challenge (i.e. T90) in (c) and (d) respectively. The symbol * indicates a significant difference with p < 0.05. Values are shown as mean plus standard deviation. Both AT and T90 induce a tachycardic response (a, c) but only AT abolishes HP fluctuations (b).

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(figure 4(b)) and CPE(3) (figure 4(c)), while in the lower panels (figures 4(d)–(f)) the changes induced by T90 are depicted. Results are given as mean plus standard deviation. NKNNCEmin significantly decreased both after AT and during T90 (figures 4(a), (d)), thus suggesting that both AT and T90 caused a loss of complexity of the cardiac control. Conversely, SPE(3) and CPE(3) detected the decrease of the complexity of the cardiac control only during T90 ( figures 4(e), (f)). It is remarkable that SPE(3) and CPE(3) performed similarly.

5. Discussion

The main finding of this study is that permutation-based entropies are more susceptible to broad band noise compared to coarse-grained conditional entropy, thus suggesting caution when they are applied for the assessment of dynamical complexity of time series characterized by a low signal-to-noise ratio.

5.1. Permutation-based entropies are more susceptible to broad band noise than coarse-grained conditional entropy over simulated processes

Results from simulated AR processes with a dominant spectral peak at different frequencies indicated that entropy-based estimates of dynamical complexity increased with the rate of the dominant rhythm. This finding, previously observed with a coarse grained conditional entropy (Porta et al 1998), is here confirmed by permutation-based entropies. However, permutation-based indexes are more sensitive to broad band noise than the coarse-grained one. Indeed, if entropy is estimated by SPE and CPE, a small amount of noise (i.e. as small as 15% of the variance of the uncorrupted process in the case of VLF series) makes the complexity of the time series indistinguishable from the complexity of its shuffled version. Conversely, KNNCE

Figure 4. Bargraphs show NKNNCEmin, SPE(3) and CPE(3) before (i.e. B) and after parasympathetic blockade (i.e. AT) in (a–c) respectively and before (i.e. B) and after orthostatic challenge (i.e. T90) in (d–f) respectively. The symbol * indicates a significant difference with p < 0.05. Values are shown as mean plus standard deviation. SPE(3) and CPE(3) remain unmodified after AT (b, c), while they significantly decrease during T90 (e, f). NKNNCEmin decreases both after AT (a) and during T90 (d).

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allows the separation of the original time series from its shuffled version even in presence of a level of noise equal to 50% of the variance of the uncorrupted process. It is worth stressing that SPE and CPE performed similarly, thus suggesting that none of the two permutation-based metrics had superior performances in rejecting broad band noise. Interestingly, the per-formances of SPE and CPE in rejecting broad band noise depend on the shape of the power spectrum of the uncorrupted series, with processes characterized by VLF oscillations becom-ing indistinguishable from completely unpredictable series at a smaller amount of superim-posed noise than processes with faster oscillations. These findings suggest that performance of permutation-based indexes over real HP variability recordings may depend on the experi-mental condition, state of sympatho-vagal balance, pathology and autonomic nervous system impairment. This finding contributes to better define the limits of permutation-based entropies in estimating complexity of short times series.

5.2. Permutation-based entropies are more susceptible to broad band noise than coarse-grained conditional entropy over real HP variability recordings

It is well known that parasympathetic blockade induced by AT and vagal withdrawal associ-ated to T90 decreased complexity of the cardiac control as measured from short HP variability series. This finding was quite robust being confirmed by approaches estimating entropy, pre-dictability and fractal properties (Yamamoto et al 1995, Porta et al 2000 2007 2012, Tulppo et al 2001, Tan et al 2009, Castiglioni et al 2011, Turianikova et al 2011) and was explained as the result of the loss of high frequency rhythms of HP variability due to the vagal blockade after AT and vagal withdrawal during T90 (Porta et al 2000). The decrease of the complexity of the cardiac control observed as a consequence of the parasympathetic blockade and pas-sive orthostatic challenge is associated to completely different levels of HP variance. Indeed, while the HP variance is negligible after AT, during T90 it was similar to B (Montano et al 1994, Pomeranz et al 1985). It is reasonable to hypothesize that the amount of noise affecting HP series due to random jitters in the delineation of the R-wave peak was not significantly different after AT and during T90 compared to B. Conversely, the capability of the autonomic nervous system in modulating sinus node activity is remarkably dissimilar, being greatly lim-ited after AT and largely preserved during T90, thus leading to completely different values of HP variance. As a consequence the signal-to-noise ratio is very low after AT and high during T90. Given this consideration, and in presence of simulations showing the limited ability of permutation-based entropies to reject broad band noise, it is not surprising to find out that SPE and CPE were able to detect the decrease of complexity of the cardiac control during T90 but they failed to identify the same behavior after AT. Conversely, KNNCE was able to detect the decrease of HP variability complexity both after AT and during T90 due to the intrinsic low-pass filtering ability of the coarse graining procedure. Given the higher susceptibility of the permutation-based entropies to broad band noise, the complexity of HP variability estimated via SPE and CPE might be biased towards the complexity of the superposed noise and, thus, largely above that of the cardiac control.

6. Conclusions

When quantifying the complexity of the cardiac control from short HP variability record-ings with low signal-to-noise ratio, coarse-grained entropies should be preferred with respect to entropies based on ranks of the original values. Conversely, if permutation-based entro-pies were applied to short series with low signal-to-noise ratio, the complexity of the cardiac

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control would be largely overestimated due to their relevant susceptibility to broad band noise. Conclusions of this study might be dependent on the adopted level of coarse graining (here the most widely utilized level in HP variability analysis). Reducing the level of coarse grain-ing, leading to a better approximation of the original dynamics, might worsen the perfor-mance of coarse grained methods and render it more similar to that of permutation entropies. Conversely, increasing the level of coarse graining, leading to a worse approximation of the original dynamics, might deteriorate the performance of coarse grained entropies as well. Future studies should compare more systematically complexity indexes derived from the two classes of entropy estimators as a function of the level of coarse graining imposed to the coarse grained entropies.

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