Refined Multiscale Fuzzy Entropy based on Standard Deviation for Biomedical Signal Analysis

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1 Refined Multiscale Fuzzy Entropy based on Standard Deviation for Biomedical Signal Analysis Hamed Azami Institute for Digital Communications, School of Engineering, University of Edinburgh Edinburgh, King s Buildings, EH9 3JL, United Kingdom hamed.azami@ed.ac.uk Alberto Fernández Departamento de Psiquiatría Psicología Médica Universidad Complutense de Madrid, Madrid, Spain aferlucas@med.ucm.es Javier Escudero * Institute for Digital Communications, School of Engineering, University of Edinburgh Edinburgh, King s Buildings, EH9 3JL, United Kingdom javier.escudero@ed.ac.uk Multiscale entropy (MSE) has been a prevalent algorithm to quantify the complexity of fluctuations in the local mean value of biomedical time series. Recent developments in the field have tried to improve the MSE by reducing its variability in large scale factors. On the other hand, there has been recent interest in using other statistical moments than the mean, i.e. variance, in the coarse-graining step of the MSE. Building on these trends, here we introduce the so-called refined composite multiscale fuzzy entropy based on the standard deviation (RCMFEσ) to quantify the dynamical properties of spread over multiple time scales. We demonstrate the dependency of the RCMFEσ, in comparison with other multiscale approaches, on several straightforward signal processing concepts using a set of synthetic signals. We also investigate the complementarity of using the standard deviation instead of the mean in the coarse-graining process using magnetoencephalograms in Alzheimer s disease and publicly available electroencephalograms recorded from focal and non-focal areas in epilepsy. Our results indicate that RCMFEσ offers complementary information to that revealed by classical coarse-graining approaches and that it has superior performance to distinguish different types of physiological activity. Keywords: Complexity; multiscale entropy; sample entropy; fuzzy entropy; biomedical signal; statistical moments.. Introduction One of the most important challenges in biomedical signal processing is to quantify the dynamical complexity of time series []. There are a number of approaches, such as entropies and fractal dimensions, to understand the dynamics of signals. Entropy is an appealing and powerful tool that has been widely used in physiological signal analysis [, 3]. One of the most popular entropybased approaches is sample entropy (SaEn), which is relatively robust to noise []. Another widely used complexity quantifier is fuzzy entropy (FuzEn) [4]. These two entropy-based approaches have attracted a great deal of attention during the recent years []. Although SaEn is slightly faster than FuzEn, the latter has stronger relative consistency and less dependency on the data length [4, 5]. The traditional methods to quantifying the complexity of biomedical recordings may fail to account for the multiple time scales inherent in such time series and may yield to contradictory and misleading findings. For instance, even though the SaEn of white Gaussian noise (WGN) time series is higher than of that of /f noise, /f noise has more complex structures than WGN due to the presence of long-range correlations [6, 7]. To address this problem, Costa et al. introduced the multiscale (sample) entropy (MSE), which is based on assessing the entropy of signals at multiple time scales [6]. In the MSE method, the original signal is first divided into non-overlapping segments of length τ, termed the scale factor. Next, the mean of each segment is estimated to derive the coarsegrained signals. Finally, the entropy measure, using SaEn, is calculated for each coarse-grained sequence [6]. * Corresponding author.

2 The complexity evaluation of time series with MSE is rooted in the concept that complexity is associated with meaningful structural richness, which is in contrast with regularity measures defined from classical entropy algorithms. This is because the output of entropy-based metrics grows monotonically with the degree of randomness of the analyzed time series. Therefore, these measures assign the highest entropy values to uncorrelated random signals like white noise, which are highly unpredictable but not structurally complex, and, at a global level, permit a very simple description. Thus, when applied to biomedical signals, traditional entropybased methods may lead to misleading outputs. For instance, they assign high entropy values to certain pathologic cardiac rhythms that generate erratic outputs whereas healthy cardiac rhythms that are exquisitely regulated by multiple interacting control mechanisms are given low values of entropy. In this context, the complexity of biomedical signals reflects their ability to adapt and function in an ever-changing environment because physiological signals require to operate across multiple temporal and spatial scales. Thus, substantial attention has been concentrated on defining a quantitative measurement of complexity, i.e. MSE, that vanishes for both deterministic/predictable and uncorrelated random/unpredictable time series [6, 7]. Extensive analyses have shown that abnormal and disease states, which decrease the adaptive capacity of the subject, appear to degrade the multiscale entropy metrics [6, 7]. Costa and Goldberger have very recently introduced a new MSE approach using the variance, instead of the mean, in the coarse-graining process of MSE. This was named MSE σ [8]. Note that, in order to discriminate MSE σ and basic MSE, we will denote the latter as MSE µ. MSE σ revealed that the dynamics of the volatility (variance) of heartbeat signals obtained from healthy young subjects are highly complex[8]. A recent review about multiscale entropy-based methods can be seen in [9]. Nonetheless, since the standard deviation (σ) has the same dimension as the signal and its mean values (as used in the original definition of the MSE: MSE µ), we propose to use σ in the coarse-graining process, as an alternative to MSE µ and MSE σ. Furthermore, one of the most important problems of MSE µ is that, when applied to short biological signals, the results may be undefined and inaccurate [0, ]. To alleviate this problem, the refined composite MSE µ (RCMSE µ) has been recently introduced [0] using the average of the sample entropy values of several coarse-grained signals in each scale factor. Simulation results showed that the RCMSE µ had better stability for all temporal scales than MSE µ [0]. We build on these recent developments to combine their advantages, and propose the refined composite multiscale fuzzy entropy (RCMFE) based on μ and σ: RCMFE μ and RCMFE σ, respectively. We hypothesize these measures will be more accurate, robust and stable than previous entropy metrics. Furthermore, we exemplify the behavior of these measures for different kinds of classical signal concepts (e.g., frequency, nonlinearity) to demonstrate the dependency of RCMFE σ and RCMFE µ on these concepts. Finally, we illustrate their application to two real biomedical datasets: focal and non-focal electroencephalograms (EEGs) and restingstate magnetoencephalogram (MEG) activity in Alzheimer s disease (AD).. Methods.. Entropy approaches... Sample Entropy (SaEn) Assume we have a real-valued discrete time series of y { y, y,..., y } length N: N. At each time t of y, a vector including the m-th subsequent values is m constructed as: Y t { yt, yt,..., yt m, y t m} for t=,,,n-(m-), where m, termed embedding dimension, determines how many samples are contained in each vector. Define the distance between such vectors as the maximum difference of their corresponding scalar components, m m m m d Y,Y t max Y Y : 0 - and t t k t k k m t t m m A match happens when the distance d Y,Y t t is smaller than a predefined tolerance r. The probability m B () r shows the total number of m-dimensional matched vectors []. Similarly, B m () r is defined for embedded dimension of m. The SaEn is defined as follows []: m m SaEn( y, m, r) ln B ( r) B ( r ) ()... Fuzzy entropy (FuzEn) In this case, for the time series y { y, y,..., yn}, embedding dimension m, and tolerance r,

3 U { y, y,..., y } y 0 is formed where m t t t t m t y0 m t j t j0 y m. The distance between each of m U t and m m m U t is defined as d, t t d U t U t max U U : 0 k m - and t t. Given FuzEn m m tk tk power n and tolerance r, the similarity degree calculated through a fuzzy function d,, tt n exp ( tt ) /. d tt is n r as d r The function m is then defined as: N m N m m n ( y, n, r) exp ( dtt ) / r () N m N m t t, t t Finally, the FuzEn of the signal is defined as the negative natural logarithm of the ratio of m and m (computed following the same procedure for embedding dimension m+) [4]: (,,, ) ln m FuzEn y m n r m.. Coarse-graining for multiscale entropy A coarse-graining process is applied to a time series { x, x,..., x,..., x } where C is the length of the signal. b C Each element of the coarse-grained time series for MSE µ/mfe µ is defined as: ( ) i C yi xb i ( ) N (4) b i where is the time scale factor [7]. This means that these coarse grained sequences are computed as the average of consecutive samples. Costa et al. [8] also has recently proposed to use the variance, instead of mean value, as follows: ( ) ( ) i C yi ( xb i ), i ( ) N (5) b i The dimension of variance is not the same as the samples of the original signal, and the quadratic behavior of variance causes the differences between the data points and their corresponding average become larger and smaller, respectively, for those differences which are larger and smaller than. To alleviate these shortcomings, we propose to use σ in the coarse graining process as a measure of spread via: C y ( x ), i N (6) ( ) ( ) i i b i b( i).3. Refined composite multiscale fuzzy entropy The traditional application of the coarse graining procedure in MSE µ leads to two main shortcomings. First, the MSE µ is not symmetric in its dependency on the samples of the original time series. According to Fig. (a), for example in scale 3, we could rationally expect the measure to behave the same for x 3 and x 4, in comparison with x and x 3. However, at scale 3, x, x and x 3 are separated from x 4, x 5 and x 6. The second shortcoming is the variability of the MSE µ results for high scale factors. When the MSE µ is computed, in the coarse-graining process, the number of samples of the resulting coarse-grained sequence is C/ N. When the scale factor is high, the number of time points in the coarse-grained sequence decreases. This may yield an unstable measure of entropy. To alleviate these drawbacks, the improved multiscale permutation entropy and RCMSE µ algorithm were proposed [0, ]. Here, considering the advantages of FuzEn over SaEn, and RCMSE µ over MSE µ, we introduce RCMFE σ and RCMFE µ. The RCMFE σ is calculated in two main steps: ( ) ( ) ( ) ) First, z { y, y,...} are generated, where i i, i, y ( x ), i N, where i j ( ) i, j b i bi ( j) i j x b bi ( j). As can be seen in Fig. (b), in the i RCMFE σ algorithm, for each scale factor τ, we have τ ( different time series z ) ( i,..., ), while in the MSE/MFE methods, only i z ( ) is considered. ) For a defined scale factor τ and embedding dimension m m m,, k ( k,..., ) and, k ( k,..., ) for each of ( z ) ( k,..., ) are separately calculated. Next, k the average of values of m, k and m, k on k τ are computed, respectively. Finally, the RCMFE σ is computed as follows: m RCMFE (,,,, ) ln x m n r m (7) It should be mentioned the difference between RCMFE σ and RCMFE µ is that the latter one uses Eq. 4 whereas the first one uses Eq. 6 in their first step of algorithm.

4 3. Evaluation Signals 3.. Synthetic signals In this subsection, the signals used to study the mentioned multiscale approaches, and their interpretability in terms of classical signal processing concepts are described. We consider the performance of the multiscale entropy metrics on WGN and /f noise. The number of sample points of each of the WGN and /f noise was The other synthetic signals have a sampling frequency (f s) of 50 Hz and a length of 00 s (5000 sample points). The time plots of these synthetic signals, and their corresponding spectrograms, and two zooms (for each kind of signal) on their start and end, to show the changes in their characteristics, are illustrated in Fig.. All of them have been employed to inspect the Lempel- Scale Ziv complexity measure and/or auto-mutual information function rate of decrease and have been explained in [3] and [4], respectively, where additional details can be found. ) RCMFE σ and RCMFE µ Versus Noise: The dependency between the above-mentioned multiscale entropy-based methods and /f noise and WGN is considered in this paper. WGN has a constant power spectral density as WGN is a signal whose samples are randomly drawn from a Gaussian distribution and uncorrelated [5]. The power spectral density of a stochastic process appropriates to model evolutionary or developmental systems is characterized by equal energy per octave as /f [6]. x x x3 x4 x5 x6 xi xi+ xi+ y () y () y3 () y(i+)/ () Scale 3 x x x3 x4 x5 x6 xi xi+ xi+ y y y(i+)/3 (a) Scale x x x 3 x 4 x 5 x 6 x i x i+ x i+ x 7 z () y, () y, () y,3 () y,(i+)/ () x x x 3 x 4 x 5 x 6 x i x i+ x i+ x 7 x i z () y, () y, () y,3 () y,(i+)/ ()

5 Scale 3 x x x 3 x 4 x 5 x 6 x 7 x i x i+ x i+ x i+3 x i+4 z y, y, y,(i+)/3 x x x 3 x 4 x 5 x 6 x 7 x i x i+ x i+ x i+3 x i+4 z y, y, y,(i+)/3 x x x 3 x 4 x 5 x 6 x 7 x i x i+ x i+ x i+3 x i+4 z 3 y 3, y 3, y 3,(i+)/3 (b) Fig.. The scheme demonstrating (a) the existing and (b) proposed coarse-graining of a sequence for scale factor τ= and τ=3. ) RCMFE σ and RCMFE µ Versus Frequency: In order to clarify how the RCMFE σ/rcmfe µ changes when the frequency of sinusoidal signals is changed, a constant amplitude chirp signal whose frequency is swept logarithmically from 0. Hz to 30 Hz in 00 s is considered [, 3]. RCMFE σ and the other multiscale entropy methods were applied to this signals using a moving window of 000 samples (3.33 s) with 90% overlap. Fig. (a) demonstrates the constant chirp signal. 3) RCMFE σ and RCMFE µ Versus Spectral Content of Colored Noise: In order to find the relationship between the RCMFE σ or RCMFE µ and the spectral content of colored noise, an autoregressive (AR) process of order, AR(), was generated varying the model parameter, ρ, linearly from +0.9 to 0.9. Its energy hence moved from low to high frequencies. In case of ρ=0, the sequence corresponded to WGN, in the center of the synthetic time series. Fig. (b) shows the corresponding spectrogram, time plot and zoom views. 4) RCMFE σ and RCMFE µ Versus Changes from Randomness to Orderliness: In order to consider how the RCMFE σ and RCMFE µ change when a stochastic sequence progressively turns into a periodic deterministic time series, we created a MIX process employed by [4, 7, 8]. It is defined as follows: MIX ( z) x zy (8) where z denotes a random variable which is equal to with probability p and is equal to 0 with probability p, x depicts a periodic synthetic signal generated by x sin k / and y is a uniformly distributed k variable on 3, 3 [4, 7]. The signal was based on a MIX process whose parameter linearly changed between 0.0 and Thus, this signal, depicted in Fig. (c), evolved from randomness to orderliness. 5) RCMFE σ and RCMFE µ Versus Changes from periodicity to non-periodic non-linearity: In order to clarify the dependence of the multiscale entropies on these changes, the logistic map is employed. This analysis is dependent on the model parameter [4, 7] as follows: x x ( x ) (9) k k k The synthetic signal x was created varying the parameter α linearly from 3.5 to With α=3.5, the signal oscillated among four values. For α between 3.5 and 3.57, the signal is periodic and the number of values doubles progressively. For 3.57 α 3.99, the time series is chaotic, although it has windows of periodic behavior (e.g., α 3.8, as seen in Fig. (d)) [9]. 6) RCMFE σ and RCMFE µ Versus different Nonlinear regimes: In order to investigate the changes in the behavior of a nonlinear system, the Lorenz attractor is used here as: x ( y x) y x( z) y z xy z (0) where, and denote the system parameters [9, 0]. The first segment of this time series has a length of 7500 sample points and it was created with 0, 8/ 3 and 8. Therefore, it has a chaotic

6 H. Azami et al. behavior. The second segment, which has 7500 sample points, was generated with 0, 8 / 3 and It exhibits a torus knot [3, 9]. Both segments were created by the use of a fixed step-size first-order integration technique without pre-integration and with the step-size set to /fs. It should be noted that these two segments were normalized with SD of, after these segments had been generated. The coordinate x, which is the signal analyzed in this article, appears in Fig. (e). (a) (b) (c) (d)

7 (e) Fig.. Spectrograms, time plots and zoom views on the first and last time intervals of the synthetic signals used in this study. (a) Chirp signal with constant amplitude (f =0. Hz, f =30 Hz). (b) AR() process with variable parameter. (c) MIX process evolving from randomness to periodic oscillations. (d) Logistic map signal. (e) Lorenz system with two different nonlinear dynamics. Red corresponds to high power, and blue corresponds to low power. 3.. Real biomedical signals The ability of the newly proposed RCMFE μ and RCMFE σ to distinguish different types of physiological activity was tested on the following real datasets MEG data Resting state MEG activity was recorded from 36 patients with probable AD [] (4 women; age = ± 6.95 years, mean ± standard deviation; MMSE score = 8.06 ± 3.36) and 6 age-matched controls (7 women; age = 7.77 ± 6.38 years; MMSE score = 8.88 ±.8). The subjects laid on a hospital bed in a relaxed state with eyes closed. For each participant, five minutes of MEG resting state activity were recorded at a sampling frequency (f s) of 69.54Hz. The signals were divided into segments of 0s (695 samples per channel) and visually inspected using an automated thresholding procedure to discard segments significantly contaminated with artefacts []. The effect of cardiac artifact was reduced from the recordings using a constraint blind source separation procedure. Finally, a band-pass FIR filter with cut-offs at.5hz and 40Hz was applied to the data. For more information about the dataset, please refer to [3]. For each subject and each channel, we analyzed each epoch of 0s individually and the average of results is reported EEG data The intracranial EEG signals were recorded from 5 patients suffering from pharmacoresistant focal-onset epilepsy leading to two main separate sets of signals. The first one was recorded from brain regions where the primarily ictal EEG recordings changes were detected as judged by expert visual inspection ( focal signals ). The second set of signals was recorded from brain regions not involved at seizure onset ( non-focal signals ). Each set includes five patients. Each patient consists of 750 pair signals, which the length of each of them was 040 sample points or 0 seconds. The sampling frequency was 5 Hz. Each pair includes two EEG time series which are recorded from adjacent channels which here we consider the first time series. They also provided a subset of the recordings containing the first 50 signals for each set. We use this subset to evaluate the proposed methods. For more information about the dataset, please refer to [4]. Before computing the multiscale entropy approaches, all signals were digitally filtered employing an FIR band-pass filter with cut-off frequencies at 0.5 Hz and 40 Hz. 4. Results and Discussions 4.. Synthetic signals First, we consider WGN and /f noise as two widely used signals tested in multiscale entropy methods [6, 0]. The results for MSE µ, MFE µ, RCMSE µ, RCMFE µ, MSE σ, MSE σ, MFE σ, MFE σ, RCMSE σ, RCMSE σ, RCMFE σ, and RCMFE σ are depicted in Fig. 3 (a) to (l), respectively. Note that the embedding dimension m, FuzEn power n and tolerance r for all of the approaches were respectively chosen as,, and 0.5 multiplied by the standard deviation of the original time series [, 4, 5]. It should be added the results obtained for parameter

8 r, used in [8], are similar to those with r=0.5 multiplied by the SD of that time series, employed in [5]. It is worth mentioning that the scale factor is very important parameter in multiscale entropy-based methods, because the actual length of the coarse-grained signal is equal to the length of the original signal divided by scale factor τ. As it can be observed in Fig. 3, for WGN, the entropy values of all multiscale approaches, except MSE σ and RCMSE σ, decreases monotonically with scale factor. However, for /f noise, the entropy values become approximately constant over larger scale factors. These facts are in agreement with the MSE-based results in [6, 0]. Note that each error bar of each scale factor depicts the standard deviation, SD, of the results of 40 signals for each WGN or /f noise. Comparing results obtained by MSE µ (Fig. 3 (a)) and MFE µ (Fig. 3 (b)) shows, as expected theoretically, the MFE µ leads to a smaller variability in the results. Statistical tests confirmed the smaller variability of the MFE µ results (pvalue 0.05) as assessed with a Levene's test at τ=60. In addition, the RCMSE µ/rcmfe µ profiles have smaller SDs than MSE µ/mfe µ. Although the MSE σ values for WGN are larger than /f noise for scale factors to 60, according to Fig. 3 (f), it might be predicted that this measure for WGN will become smaller than those of /f noise for large enough scale factors. For MSE σ and for scale factors to 37, the larger entropy values are assigned to WGN signal in comparison with /f noise, while for scale factors larger than 37, the SaEns for /f noise are larger than those of WGN. /f noise is considered more structurally complex across multiple scales [7, 6]. Comparing the results shows that crossing between WGN and /f noise does not happen at short levels of scale factor for coarse-graining process based on variance and standard deviation, unlike mean. The patterns for MFE σ and MFE σ are relatively the same as MSE σ and MSE σ, respectively. However, as expected theoretically, the SD of MFE σ/mfe σ values for each scale is comparatively smaller than that of MSE σ/mse σ measures. The FuzEn for /f noise is larger than those of WGN when 33<τ. It shows another relative advantage of MFE σ over MSE σ. Although the curves for RCMSE σ/rcmse σ have smaller SDs than MSE σ/mse σ for each scale factor, these have larger SDs in comparison with RCMFE σ/rcmfe σ. This fact confirms our theoretical expectation about RCMFE σ and RCMFE µ producing the most stable results among these multiscale entropy methods. In brief, FuzEn-based multiscale methods are more stable than SaEn-based algorithms. Furthermore, the refined composite coarsegraining techniques can improve the stability of MSE or MFE. In addition, for /f and WGN time series, the multiscale methods based on standard deviation may have better performance in shorter temporal scales than those based on variance. (a) (b) (c)

9 (d) (e) (f) (g) (h) (i) (j) (k) (l) Fig. 3. Mean value and SD of results of the (a) MSE µ, (b) MFE µ, (c) RCMSE µ, (d) RCMFE µ, (e) MSE σ, (f) MSE σ, (g) MFE σ, (h) MFE σ, (i) RCMSE σ, (j) RCMSE σ, (k) RCMFE σ, and (l) RCMFE σ computed from 40 different /f noise test signals. Red and blue indicate /f and WGN results, respectively. To understand the effect of frequency on multiscale entropy-based methods, we employed a sliding window moving along a chirp signal with constant amplitude. Then, for each scale factor, the multiscale entropy-based method of that part of the signal was computed. Because the length of the window is 000 sample points, we consider the scale factor from to 5, to ensure the length of the coarse-grained signals is enough for m= [7]. The RCMFE σ, RCMFE µ, MSE σ and MSE µ results are respectively shown in Fig. 4(a), 4(b), 4(c), and 4(d). It is worth noting that since the SD/variance, unlike mean value, of one single number is 0, the entropy measure in the first scale factor variance is undefined. This fact can be seen in Fig. 4(a) and 4(c) in comparison with Fig. 4(b) and 4(d). When the time window is occupied the beginning of the signal, which has smaller frequency, the FuzEn/SaEn is low across all τ. All the RCMFE σ, RCMFE µ, MSE σ and MSE µ values go up in case of increasing frequency, which happens in later temporal windows (TWs). Another important point is that the relative increase in entropy values is more abrupt for RCMFE σ as suggested by the steeper change from blue to red. In addition, refined composite multiscale entropybased approaches, i.e. RCMFE σ and RCMFE µ are more stable than their basic counterparts (MSE σ and MSE µ). It is worth noting that the results are in agreement with [0] about the fact that the results depend on the main frequency of a signal.

10 In Fig. 4 (e), 4(f), 4(g), and 4(h), it can be observed generally, using an AR() process with variable parameter, the entropy measures of RCMFE σ, MSEσ and MSEσ, unlike RCMFE µ, increase in higher TWs in every scale factor. This fact demonstrates the importance of using both of the SD/variance and mean to complement each other. For RCMFE σ, unlike RCMFE µ, when the frequency of signal is very low, σ of each segment is very low, so in this case, the entropy of the coarse-grained sequences is very low. In addition, results obtained by MSE σ and MSE σ have similar patterns, although MSE σ is relatively more variable than MSE σ. As expected theoretically, refined composite technique could reduce the variability of the results. Fig. 4(i), 4(j), 4(k), and 4(l) respectively show the results obtained by RCMFE σ, RCMFE µ MSE σ, and MSE µ using the abovementioned MIX process. When the TW moves from a stochastic signal to periodic deterministic sequence, the entropy measures for all these methods decrease. In addition, moving from τ= to τ=5, the entropy measures go down. Although all these approaches generally demonstrate the same behavior, the RCMFE σ and RCMFE µ results are more stable than their more basic counterparts. Fig. 4(m), 4(n), 4(o) and 4(p) illustrate the results obtained by RCMFE σ, RCMFE µ, MFE σ, and MFE µ, respectively, using the logistic map which the parameter α changes linearly from 3.5 to As expected, the entropy measures, obtained by the all the entropy-based methods, generally increase along the signal, for each scale factor, except for the downward spikes in the windows of periodic behavior. This fact is in agreement with Fig. 4.0 (page 87 in [9]). It is also supported by Fig.. (d) which shows that the frequency of the signal for t=70-75 s is lower than for its adjacent time samples. In case of increasing scale factor, the RCMFE σ and MFE σ results decrease, whereas the RCMFE µ and MFE µ results first increase respectively until τ= and τ=4 then decrease. It is worth noting that the changes obtained by RCMFE σ and RCMFE µ in two subsequent scale factors or TWs are smaller than MFE σ and MFE µ. The results again demonstrate the importance of RCMFE σ and RCMFE µ to complement each other. Using the Lorentz system, we assess whether RCMFE σ, RCMFE µ, MSEµ, and RCMSE σ respectively shown in Fig. 4(q), 4(r), 4(s), and 4(t), can distinguish two different nonlinear dynamics. The results show although in smaller scale factors, the entropy measures for these three algorithms are very low, two different segments are distinguishable in larger scale factors. This fact depicts the importance of multiscale entropy methods in signal processing. As can be seen in Fig. 4(q), 4(r), 4(s), and 4(t), the results obtained by the RCMFEµ are more stable than RCMSE µ, and RCMSE µ results are more stable than MSE µ ones. It demonstrates the importance of fuzzy entropy and refined composite algorithm to improve the stability of the results. 4.. Real biomedical signals 4... MEG data We also assess the suitability of the RCMFE µ and RCMFE σ methods to characterize AD in MEG signals. The profiles are shown in Fig. 5. The average of RCMFE σ values for AD patients is smaller than for controls for all scale factors. This is in agreement with [8, 9]. In contrast, the average of RCMFE µ values for AD patients is smaller than for controls for only τ 3. The results in Fig. 5 show increases in entropy for wide ranges of τ. This emphasizes the suitability of multiscale evaluations for the assessment of biomedical data as these approaches managed to reveal different dynamics associated with pathology (AD in this case), despite previous claims by some authors that the coarse graining procedure in MSE had the shortcoming that it tended to artificially decrease the entropy values as a function of the time scale [30]. It should be added that the entropy parameters used for biomedical signals are exactly similar to those mentioned for synthetic time series. We tested different r values from 0.05 to 0. for this kind of signals and for all of them the results had similar patterns and the conclusions do not change when the parameters are varied. False discovery rate (FDR)-adjusted [3] p-values of a student s t-test assuming unequal variances for each MEG channel and temporal scale factor to evaluate the differences between the values of entropy for AD patients and controls are shown in Fig. 5 in a logarithmically scale. The FDR-adjusted p-values obtained by RCMFE µ, unlike those of RCMFE σ, initially increase and then decrease along the temporal scale factor for almost all channels. Moreover, the most significant differences can be seen around temporal scales 7 9 and 5-7 using RCMFE µ and RCMFE σ, respectively. This fact demonstrates RCMFE µ and RCMFE σ may complement each other in the characterization of physiological activity in disease. (The significance level of p-value tests was 0.05.)

11 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) (s) (t) Fig. 4. Results of the tests performed to understand better diverse multiscale entropy approaches and their interpretation. Relationships between chirp signal with constant amplitude and (a) RCMFEσ, (b) RCMFEµ, (c) MSEσ, and (d) MSEµ. Relationships between AR() process with variable parameter and (e) RCMFEσ, (f) RCMFEµ, (g) MSEσ, and (h) MSEσ. Relationships between the abovementioned MIX process and (i) RCMFEσ, (j) RCMFEµ, (k) MSEσ, and (l) MSEµ. Relationships between the logistic map and (m) RCMFEσ, (n) RCMFEµ, (o) MFEσ, and (p) MFEµ. Relationships between Lorenz system with two different nonlinear dynamics and (q) RCMFEσ, (r) RCMFEµ, (s) MSEµ and (t) RCMSEµ.

12 We also classify the AD subjects and controls using a naive Bayes classifier [3]. For each individual, 5 and 4 features (temporal scale factors) are extracted by averaging the RCMFE µ and RCMFE σ, results across all channels, respectively. We ran 00 repetitions of a 0- fold cross-validation (CV). The average classification accuracies were 7.8% and 78.%, respectively for RCMFE µ and RCMFE σ. This shows that RCMFE σ features lead to higher classification accuracy than RCMFE µ ones. The classification was done with the WEKA data mining software [33] EEG data We also show the behavior of RCMFE µ and RCMFE σ in focal and non-focal EEG time series. The error bars illustrating the distributions of the RCMFE µ and RCMFE σ values computed from focal and non-focal EEG signals are shown in Fig. 6 until scale factor 30. For each scale factor, the average of entropy values of focal EEG signals is smaller than that of non-focal ones. It illustrates the non-focal EEG recordings are generally more complex than the focal ones and it is in agreement with [4] and [34]. Since all EEG signals were band-pass filtered between 0.5 and 40 Hz, there is no relevant information left for analysis of frequencies higher than 40 Hz. However, the frequency that corresponds to the analysis of scale is 5/ Hz. This may be the reason why SaEn is so low for short time scales. In addition, a t- test was performed for focal vs. non-focal EEG recordings. We adjusted the FDR independently for each of RCMFE µ and RCMFE σ. The results show the RCMFE σ method achieves significant differences at all scales factors whereas the RCMFE µ algorithm yields significant differences at scale factors between and 0. This suggests a better performance of the RCMFE σ algorithm over the RCMFE µ method for the distinction of physiological activity related to focal and non-focal EEG signals. It should be added that the significance level of p-value tests was 0.05 for the EEG time series. We also applied the same classification scheme to distinguish the focal and non-focal signals. The average classification accuracies were 7.58% and 79.6%, respectively for RCMFE µ and RCMFE σ. As mentioned before, when the length of the signal is short, the results obtained by the MSE-based methods, especially in high scale factors, may be unreliable and unstable [, 35]. To alleviate this shortcoming, the coarse-graining process is used here. Note that we kept the value of r fixed across temporal scales. Other authors suggested recalculating the tolerance r at each scale factor separately []. Using several physiological datasets, they found that recalculating r produced similar results to those obtained by not recomputing r, as in the original description of MSE proposed by Costa et al.[7]. Considering that there was not evidence of the fact that recomputing r for each scale improved the results, we decided to keep r fixed so that we retained the advantages of the original formulation of multiscale entropy by which the entropy of WGN decreases with τ. (Note that renormalizing r for each scale will lead to flatter MSE curves for WGN, contrary to theoretical expectations.) Nevertheless, in future work, we will investigate the effect of calculating tolerance r, separately, at each scale factor on the results when standard deviation is used in the coarse-graining step of MSE.

13 Fig. 5. Error bars illustrating the mean ± SD of the (A) RCMFEµ and (B) RCMFEσ values for AD subjects and controls. Base-0 logarithm of the FDR-adjusted p-values for the differences in (C) RCMFEµ and (D) RCMFEσ at each channel and temporal scale between AD patients and controls.. (a) (b) Fig. 6. Error bars illustrating the mean ± SD of the (a) RCMFEµ and (b) RCMFEσ values computed from focal and non-focal EEG signals. 5. Conclusions As an alternative to MSE µ, a widely used method to characterize biomedical signals, MSE σ has been very recently introduced. Building on recent trends in the field, this article introduces the use of standard deviation in the coarse-graining process and proposes RCMFE σ and RCMFE µ as complexity metrics of fluctuations in the standard deviation and mean, respectively, of a time series that overcome several limitations of previous approaches and provide complementary information. We illustrated the behavior of these multiscale entropy-based approaches versus WGN, /f noise, and several straightforward concepts in signal processing. We also demonstrated their use in the separation of different kinds of physiological activity as recorded in MEG and EEG signals. The results have shown that MSE σ and MFE σ had better performance than, respectively, MSE σ and MFE σ for /f noise and WGN time series. The FuzEn-based multiscale methods were more stable than SaEn-based algorithms and, furthermore, the refined composite technique noticeably improved the stability of MSE σ and MFE σ. Overall, the results obtained by the RCMFE σ and RCMFE µ algorithms have had the smallest SD values among the existing and proposed multiscale entropy-based approaches. The classification results, obtained using simple classification methods, showed that RCMFE σ based features lead to higher classification accuracies in comparison with the RCMFE µ-based ones. Since the aim of this paper is mainly methodological, future work will address the physiological interpretation of the results. To sum up, the RCMFE σ and RCMFE µ led to more stable results in comparison with other multiscale entropy approaches and both RCMFE σ and RCMFE µ

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