# Multivariate Multiscale Dispersion Entropy of Biomedical Times Series

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## Abstract

**:**

## 1. Introduction

**X**is more complex than the multivariate time series

**Y**, if for the most temporal scales, the mvSE measures for

**X**are larger than those for

**Y**; (2) a monotonic fall in the multivariate entropy values along the temporal scale factors shows that the signal only includes useful information at the smallest scale factors; and (3) a multivariate signal illustrating long-range correlations and complex creating dynamics is characterized by either a constant mvSE or this demonstrates a monotonic rise in mvSE with the temporal scale factor [8].

## 2. Multivariate Multiscale Dispersion Entropy (mvMDE)

#### 2.1. Coarse-Graining Process for Multivariate Signals

#### 2.2. Background Information for the mvDE

_{I}to III and mvDE) until we arrive at a preferred (or optimal) one, i.e., mvDE. However, here, we present all the simpler alternatives (mvDE

_{I}to mvDE

_{III}), since they can still be useful in some settings and allow for clearer comparisons with other current approaches.

#### 2.2.1. mvDE_{I}

_{I}of the multi-channel coarse-grained time series $\mathbf{X}={\left\{{x}_{k,i}\right\}}_{k=1,2,\cdots ,p}^{i=1,2,\cdots ,N}$, which is based on the mvMPE algorithm [19], is calculated as follows:

_{I}is calculated as follows:

_{I}is obtained, which has a value of $ln\left({c}^{m}\right)$. In contrast, if there is only one $p\left({\pi}_{{v}_{0}\cdots {v}_{m-1}}\right)$ different from zero, which demonstrates a completely regular/certain signal, the smallest value of mvDE

_{I}is obtained. In the algorithm of mvDE

_{I}, we compare $Np$ dispersion patterns of a p-channel signal with ${c}^{m}$ potential patterns. Thus, at least ${c}^{m}+Np$ elements are stored.

_{I}counts the dispersion patterns for every channel of a multivariate time series, it is suggested ${c}^{m}<\u230a\frac{pL}{{\tau}_{max}}\u230b$. mvDE

_{I}extracts the dispersion patterns from each of channels regardless of their cross-channel information. Thus, mvDE

_{I}works appropriately when the components of a multivariate signal are statistically independent. However, the mvDE

_{I}algorithm, like mvPE [19], does not consider the spatial domain of time series. To overcome this problem, we propose mvDE

_{II}based on the Taken’s theorem [17,35].

#### 2.2.2. mvDE_{II}

_{II}is as follows:

_{I}, $\mathbf{X}={\left\{{x}_{k,i}\right\}}_{k=1,2,\cdots ,p}^{i=1,2,\cdots ,N}$ are mapped to $\mathbf{Z}={\left\{{z}_{k,i}\right\}}_{k=1,2,\cdots ,p}^{i=1,2,\cdots ,N}$ based on the NCDF.

**Z**is defined as:

_{II}is calculated as follows:

_{II}, at least ${c}^{mp}+Np$ elements are stored. Thus, when p is large, the algorithm needs huge space of memory to store elements. To work with reliable statistics to calculate mvMDE

_{II}, it is recommended ${c}^{mp}<\u230a\frac{L}{{\tau}_{max}}\u230b$. Thus, although mvDE

_{II}deals with both the spatial and time domains, the length of a signal and its number of channels should be very large and small, respectively, to reliably calculate mvDE

_{II}values. To alleviate the problem, we propose mvDE

_{III}.

#### 2.2.3. mvDE_{III}

_{III}is as follows:

_{I}and mvDE

_{II}approaches, $\mathbf{X}={\left\{{x}_{k,i}\right\}}_{k=1,2,\cdots ,p}^{i=1,2,\cdots ,N}$ are mapped to $\mathbf{Z}={\left\{{z}_{k,i}\right\}}_{k=1,2,\cdots ,p}^{i=1,2,\cdots ,N}$.

**m**. For simplicity, we assume ${m}_{k}=m$ and ${d}_{k}=d$.

_{II}, we have more reliable results for a signal with a small number of samplthan those fore points, as shown later.

_{III}is calculated as follows:

_{III}assumes embedding dimension 1 for all signals except one, which might limit the potential to explore the dynamics. Moreover, in the algorithm of mvDE

_{III}, at least ${c}^{m+p-1}+Np$ elements are stored. Although this number is noticeably smaller than that for mvDE

_{II}, the algorithm still needs to have large memory space for a signal with a large number of channels. To work with reliable statistics to calculate mvMDE

_{III}, it is recommended ${c}^{m+p-1}<\u230a\frac{pL}{{\tau}_{max}}\u230b$. Therefore, albeit mvDE

_{III}takes into account both the spatial and time domains and needs to smaller number of sample points in comparison with mvDE

_{II}, there is a need to have a large enough number of samples and small number of channels. To alleviate these deficiencies, we propose mvDE.

#### 2.3. Multivariate Dispersion Entropy (mvDE)

_{I}to III, the multivariate signal $\mathbf{X}={\left\{{x}_{k,i}\right\}}_{k=1,2,\cdots ,p}^{i=1,2,\cdots ,N}$ is mapped to $\mathit{c}$ classes with integer indices from 1 to $\mathit{c}$.

_{II}, to consider both the spatial and time domains, multivariate embedded vectors ${Z}_{\mathbf{m}}\left(j\right),1\le j\le N-(m-1)d$ are created based on the Taken’s embedding theorem [35]. For simplicity, we assume ${d}_{k}=d$ and ${m}_{k}=m$.

_{II}to III, leading to more stable results for signals with a short length and a large number of samples. As the number of patterns obtained by the mvMDE method is $(N-(m-1)d)\left(\genfrac{}{}{0pt}{}{mp}{m}\right)$, it is suggested ${c}^{m}<\u230a\frac{L\left(\genfrac{}{}{0pt}{}{mp}{m}\right)}{{\tau}_{max}}\u230b$ to work with reliable statistics. It is worth mentioning that if the order of channels in a multi-channel time series changes, although the assignment to each dispersion pattern obtained by the mvMDE-based methods may change, the entropy value will stay the same.

#### 2.4. Parameters of the mvMDE, mvMSE, and mvMFE Methods

**m**, number of classes c, and time delay vector $\mathbf{d}$. Although some information with regard to the frequency of signals may be ignored for ${d}_{k}>1$, it is better to set ${d}_{k}>1$ for oversampled time series. However, like previous studies about multivariate entropy methods [2,8], we set ${d}_{k}=1$ for simplicity. Nevertheless, when the sampling frequency is considerably larger than the highest frequency component of a time series, the first minimum or zero crossing of the autocorrelation function or mutual information can be utilized for the selection of an appropriate time delay [36]. We need $1<c$ to keep away the trivial case of having only one dispersion pattern. For simplicity, we use $c=5$ and ${m}_{k}=2$ for all signals used in this study, although the range $2<c<9$ leads to similar findings. For more information about c, ${m}_{k}$, and ${d}_{k}$, please refer to [13,30].

## 3. Evaluation Signals

#### 3.1. Synthetic Signals

**H**. Afterwards,

**H**was multiplied with the standard deviation (hereafter, sigma) and then, the value of the mean (hereafter, mu) was added. Next,

**H**was multiplied by the upper triangular matrix

**L**obtained from the Cholesky decomposition of a defined correlation matrix

**R**(which is positive and symmetric) to set the correlation. Here, we set $\mathbf{R}=\left[\begin{array}{cc}1& 0.95\\ 0.95& 1\end{array}\right]$ according to [8,17]. An in-depth study on the effect of correlated and uncorrelated $1/f$ noise and WGN on multiscale entropy approaches can be found in [8,10].

^{th}sample of a bidimensional time series, ${\mathbf{A}}_{\gamma}$ denotes the $2\times 2$ matrix of parameters corresponding to lag order $\gamma $, and ${\mathbf{e}}_{n}$ is the $2\times 1$ vector of error terms assumed to be WGN [38].

#### 3.2. Real Biomedical Datasets

## 4. Results and Discussions

#### 4.1. Synthetic Signals

#### 4.1.1. Uncorrelated White Gaussian and $1/f$ Noises

_{I}, mvMDE

_{II}, mvMDE

_{III}, mvMDE, mvMSE, and mvMFE are depicted in Figure 1a–f, respectively. Using all the existing and proposed methods, the entropy values of trivariate WGN signals are higher than those of the other trivariate time series at low scale factors. However, the entropy values for the coarse-grained trivariate $1/f$ noise signals stay almost constant or decrease slowly along the temporal scale factor, while the entropy values for the coarse-grained WGN signal monotonically decreases with the increase of scale factors. When the length of WGN signals, obtained by the coarse-graining process, decreases (i.e., the scale factor increases), the mean value of inside each signal converges to a constant value and the SD becomes smaller. Therefore, no new structures are revealed at higher temporal scales. This demonstrates a multivariate WGN time series has information only in small temporal scale factors. In contrast, for trivariate $1/f$ noise signals, the mean value of the fluctuations inside each signal does not converge to a constant value.

_{II}, mvMDE

_{II}, and mvMDE

_{I}, respectively. Overall, the smallest CV values for trivariate $1/f$ noise and WGN profiles are reached by the mvMDE methods, showing the superiority of the mvMDE methods over mvMSE and mvMFE in terms of stability of results.

_{I}is able to distinguish these four different kinds of noise signals at scale factor 1. For the higher temporal scale factors, mvMDE

_{I}and mvMDE distinguish these time series, showing a limitation of mvMDE for the discrimination of white from $1/f$ noise at lower scale factors and also the importance of considering higher temporal scales for the mvMDE technique. As can be seen in Figure 2a,d, the mvMDE

_{I}and mvMDE methods better discriminate different dynamics of the noise signals. However, the mvMSE values are undefined at higher scale factors. It is worth mentioning that we compared mvMDE with the original algorithms of mvMSE and mvMFE. However, more recent studies on entropy estimation of short physiological signals provided methods to deal with this issue [17,42].

_{II}-based values are defined at all scale factors, they cannot distinguish the dynamics of different noise signals. The profiles obtained by mvMDE

_{III}are more distinguishable than mvMDE

_{II}, as mentioned that mvMDE

_{III}needs a smaller number of sample points. Nevertheless, the profiles obtained by mvMDE

_{III}have overlaps at several scale factors. Overall, the results show the superiority of mvMDE

_{I}and mvMDE over mvMDE

_{II}, mvMDE

_{III}, mvMSE, and mvMFE for short uncorrelated signals.

#### 4.1.2. Computational Time

_{I}to III, and mvMDE, we use uncorrelated multivariate WGN time series with different lengths, changing from 100 to 10,000 sample points, and different number of channels, changing from 2 to 8. The results are depicted in Table 3. The simulations have been carried out using a PC with Intel (R) Core (TM) i7-7820X CPU, 3.6 GHz and 16-GB RAM by MATLAB R2018b. The results show that the computation times for mvMSE and mvMFE are close. The slowest algorithm is mvMDE

_{II}, while the fastest ones are mvMDE

_{I}and mvMDE, in that order. For an 8-channel signal with 10,000 samples, using mvMDE

_{II}, the array exceeded the memory available. Overall, in terms of computation time and memory space, mvMDE outperforms the other methods that take into account both the time and spatial domains. We used the mvMSE code provoided in [8] and the mvMDE, mvMSE, and mvMFE Matlab codes have not been optimized.

#### 4.1.3. Correlated white Gaussian and $1/f$ Noises

_{I}, mvMDE

_{II}, mvMDE

_{III}, and mvMDE to model both the within- and cross-channel properties in multivariate signals.

_{I}cannot discriminate the correlated from uncorrelated WGN or $1/f$ noise. This fact is revealed in Figure 3a. Therefore, mvMDE

_{I}should only be used when the components of a multi-channel time series are statistically independent. Multivariate multiscale entropy-based methods at scale factor 1 show the irregularity of multi-channel signals [8]. The mvMDE

_{II}, mvMDE

_{III}, and mvMDE values at scale 1 show that the uncorrelated WGN is the most irregular and unpredictable time series in agreement with [10], while the most irregular signals using mvMFE and mvMSE are the correlated WGN [8,17], in contrast with the fact that correlated multi-channel WGN signals are more predictable and regular than uncorrelated WGN ones [10,27]. Although mvMDE was able to distinguish all four different kinds of noises at the small scale factors, there are some overlaps between the results for the correlated and uncorrelated bivariate WGN time series at the high scale factors showing the importance both low and high temporal scale factors in mvMDE.

_{II}, mvMDE

_{III}, and mvMDE. The second most complex signal is the uncorrelated bivariate $1/f$ noise, as can be seen in Figure 3. The decreases of the uncorrelated bivariate WGN profiles using mvMDE

_{II}, mvMDE

_{III}, and mvMDE are the largest, evidencing the fact that the uncorrelated WGN is the least complex time series. These facts are also in agreement with the previous studies [8,14,17]. Therefore, as desired, the mvMDE

_{II}, mvMDE

_{III}, and mvMDE deal with both the cross- and within-channel correlations.

#### 4.1.4. Bivariate AR Processes

_{I}, mvMDE

_{II}, mvMDE

_{III}, and mvMDE methods are shown in Figure 4. As expected, when the lag order increases, the complexity of the corresponding time series using the mvMDE approaches increases, in agreement with the fact that a larger lag order denotes a more complex time series [8]. As the elements of ${\mathbf{A}}_{{\gamma}_{1}}$ are smaller than those of ${\mathbf{A}}_{{\gamma}_{2}}$ and ${\mathbf{A}}_{{\gamma}_{3}}$, the behaviour of the profiles obtained by the mvMDE methods are more similar to the results for WGN (see Figure 1). In fact, the smaller the elements of ${\mathbf{A}}_{\gamma}$, the less complex the BAR, leading to lower entropy values at higher scale factors.

**A**are constant and those of anti-diagonal linearly increase from 0 to 0.17, leading to more complex series. We moved a bivariate window—termed temporal window—with length 2000 samples and $20\%$ overlap along this BAR(3) signal. The entropy of each bivariate temproal window is caculated. The results, depicted in Figure 5 show that when the time window is occupied at the beginning of the BAR(3) ($\mathbf{A}=\left[\begin{array}{cc}0.17& 0\\ 0& 0.17\end{array}\right]$), the mvMDE

_{I}, mvMDE

_{II}, mvMDE

_{III}, and mvMDE values at higher scale factors are the smallest, showing the least complexity of BAR(3) in lower temporal windows, while their corresponding entropy values in the end of BAR(3) process ($\mathbf{A}=\left[\begin{array}{cc}0.17& 0.17\\ 0.17& 0.17\end{array}\right]$) are the largest. It is worth noting that as described before, mvMDE

_{II}needs a larger number of sample points to appropriately characterize the dynamics of signals. This fact can be observed in Figure 5, showing mvMDE

_{II}is the least able to distinguish such changes.

#### 4.2. Real Biomedical Datasets

_{I}for biomedical signals, because it does not take into account both the spatial and time domains at the same time.

_{III}, mvMDE, and mvMFE, respectively depicted in Figure 6a–c, show that the self-paced unconstrained walk’s fluctuations have more complexity and greater long-range correlations than the metronomically-paced walk’s series, in agreement with those reportred in [2]. We did not use mvMDE

_{II}, as the signals do not follow the typical number of samples required for mvMDE

_{II}. To compare the results, the CV values for both the metronomically- and self-paced walk (MPW and SPW) at scale factor 4, as a trade-off between the long and short scales, are shown in Table 4. The CV values for the mvMDE

_{III}- and mvMDE-based profiles are smaller than those for mvMFE, showing the superiority of the proposed methods over mvMFE in terms of the stability of results. The smallest CV values are achieved by the mvMDE.

_{II}, mvMDE

_{III}, mvMDE, and mvMFE, respectively depicted in Figure 7a–d, show that the focal time series are less complex than the non-focal ones, in agreement with previous studies [40,43]. The CV values for the focal- and non-focal-based results at scale 6 are shown in Table 5. All the mvMDE-based CV values are smaller than those using mvMFE, showing more stability of the results obtained by the proposed methods. Moreover, the CV values for mvMDE are smaller than those for mvMDE

_{III}, and the latter ones are smaller than those for mvMDE

_{II}, suggesting that the mvMDE leads to more stable profiles.

_{II}and mvMDE

_{III}.

_{II}, mvMDE

_{III}, and mvMDE over mvMFE to discriminate different types of dynamics of multi-channel signals as well as the superiority of mvMDE over mvMFE in terms of ability to discriminate various dynamics of time series, computational time, and memory cost. As mentioned before, mvMPE does not consider the spatial domain. We have also refined the mvMPE [19] on the basis of mvMDE

_{II}, mvMDE

_{III}, and mvMDE. These approaches have the following advantages over the first version of mvMPE [19]: (1) they take into account both the spatial and time domains; (2) their results were more stable than the mvMPE-based ones; and (3) better distinguished different dynamics of multivariate signals. However, since the mvMDE methods are considerably faster, result in more stable profiles, and lead to larger differences between physiological conditions of recordings, for simplicity, we did not report the mvMPE-based results.

## 5. Conclusions

_{I}to mvDE

_{III}) may still be useful in some settings.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Mean value and SD of the results using (

**a**) mvMDE

_{I}, (

**b**) mvMDE

_{II}, (

**c**) mvMDE

_{III}, (

**d**) mvMDE, (

**e**) mvMSE, and (

**f**) mvMFE computed from 40 different uncorrelated trivariate WGN and $1/f$ noise time series with length 15,000 sample points.

**Figure 2.**Mean value and SD of the results obtained by (

**a**) mvMDE

_{I}, (

**b**) mvMDE

_{II}, (

**c**) mvMDE

_{III}, (

**d**) mvMDE, (

**e**) mvMSE, and (

**f**) mvMFE computed from 40 different uncorrelated trivariate WGN and $1/f$ noise time series with length 300 sample points.

**Figure 3.**Mean value and SD of the results obtained by (

**a**) mvMDE

_{I}, (

**b**) mvMDE

_{II}, (

**c**) mvMDE

_{III}, and (

**d**) mvMDE computed from 40 different correlated and uncorrelated bivariate WGN and $1/f$ noise time series with length 20,000 sample points.

**Figure 4.**Mean and SD values of the results using mvMDE

_{I}, mvMDE

_{II}, mvMDE

_{III}, and mvMDE computed from 40 different BAR(1), BAR(3), and BAR(5) time series with ${\mathbf{A}}_{{\gamma}_{1}}$ (first row), ${\mathbf{A}}_{{\gamma}_{2}}$ (second row), and ${\mathbf{A}}_{{\gamma}_{3}}$ (third row).

**Figure 5.**Results obtained by the mvMDE methods using a bivariate temporal window with length 2000 sample points moving along the BAR(3) signal, which the elements of anti-diagonal of the matrix

**A**linearly increase from 0 to 0.17, leading to more complex series.

**Figure 6.**Mean value and SD of the results using (

**a**) mvMDE

_{III}, (

**b**) mvMDE, and (

**c**) mvMFE for self-paced vs. metronomically-paced stride interval fluctuations.

**Figure 7.**Mean value and SD of the results using (

**a**) mvMDE

_{II}, (

**b**) mvMDE

_{III}, (

**c**) mvMDE, and (

**d**) mvMFE for focal vs. non-focal time series.

**Figure 8.**Mean value and SD of the results obtained by mvMDE computed from 36 AD patients versus 26 elderly controls for all the 148 channels. Red and blue respectively indicate AD patients and controls. The scales with p-values smaller than 0.001 are shown with *.

**Figure 9.**Mean value and SD of the results obtained by (

**a**) mvMDE and (

**b**) mvMFE computed from 36 AD patients versus 26 elderly age-matched controls over five scalp regions. Red and blue indicate AD patients and controls, respectively. The scale factors with p-values smaller than 0.001 are shown with *.

**Table 1.**Ability to deal with spatial domain and characterization of short signals (300 sample points), typical number of elements to be stored, and typical number of samples needed for each of the mvSE, mvFE, and mvDE algorithms for a p-channel signal with length N sample points.

Methods | Spatial Domain | Short Signals | Typical Number of Elements Stored | Typical Number of Samples |
---|---|---|---|---|

mvSE [3] | yes | undefined | $\left(\genfrac{}{}{0pt}{}{Np}{2}\right)+Np(pm+1)$ | ${10}^{m}<N$ |

mvFE [17] | yes | unreliable | $\left(\genfrac{}{}{0pt}{}{Np}{2}\right)+Np(pm+1)$ | ${10}^{m}<N$ |

mvPE [19] and mvWPE [20] | no | reliable | $m!+Np$ | $m!<N$ |

mvDE_{I} | no | reliable | ${c}^{m}+Np$ | $\frac{{c}^{m}}{p}<N$ |

mvDE_{II} | yes | unreliable | ${c}^{mp}+Np$ | ${c}^{mp}<N$ |

mvDE_{III} | yes | unreliable | ${c}^{m+p-1}+Np$ | $\frac{{c}^{m+p-1}}{p}<N$ |

mvDE | yes | reliable | ${c}^{m}+Np$ | $\frac{{c}^{m}}{\left(\genfrac{}{}{0pt}{}{mp}{m}\right)}<N$ |

**Table 2.**CV values of the proposed and existing multivariate multiscale entropy-based analyses at scale factor 10 for the uncorrelated trivariate $1/f$ noise and WGN.

Time Series | mvMDE_{I} | mvMDE_{II} | mvMDE_{III} | mvMDE | mvMSE | mvMFE |
---|---|---|---|---|---|---|

All three channels contain $1/f$ noise | 0.0028 | 0.0025 | 0.0037 | 0.0022 | 0.0405 | 0.0355 |

Two channels contain $1/f$ noise and one contains WGN | 0.0042 | 0.0032 | 0.0036 | 0.0044 | 0.0283 | 0.0274 |

One channel contains $1/f$ noise and two contain WGN | 0.0066 | 0.0052 | 0.0058 | 0.0061 | 0.0305 | 0.0292 |

All three channels contain WGN | 0.0072 | 0.0080 | 0.0092 | 0.0101 | 0.0232 | 0.0211 |

Number of Channels and Samples | mvMSE | mvMFE | mvMDE_{I} | mvMDE_{II} | mvMDE_{III} | mvMDE |
---|---|---|---|---|---|---|

2 channels and 1000 samples | 0.051 s | 0.066 s | 0.014 s | 0.023 s | 0.026 s | 0.020 s |

2 channels and 3000 samples | 0.237 s | 0.296 s | 0.035 s | 0.057 s | 0.068 s | 0.052 s |

2 channels and 10,000 samples | 1.821 s | 2.016 s | 0.111 s | 0.190 s | 0.223 s | 0.181 s |

5 channels and 1000 samples | 0.209 s | 0.223 s | 0.028 s | 43.096 s | 0.490 s | 0.050 s |

5 channels and 3000 samples | 1.129 s | 1.204 s | 0.080 s | 82.246 s | 1.137 s | 0.137 s |

5 channels and 10,000 samples | 9.432 s | 9.801 s | 0.260 s | 218.553 s | 3.343 s | 0.491 s |

8 channels and 1000 samples | 0.489 s | 0.501 s | 0.042 s | out of memory error | 65.560 s | 0.086 s |

8 channels and 3000 samples | 2.973 s | 2.906 s | 0.124 s | out of memory error | 150.122 s | 0.243 s |

8 channels and 10,000 samples | 27.993 s | 25.951 s | 0.398 s | out of memory error | 363.752 s | 0.824 s |

**Table 4.**CV values of the entropy results at scale factor 4 using mvMDE

_{III}, mvMDE, and mvMFE for self-paced walk (SPW) vs. metronomically-paced walk (MPW).

Stride Interval Fluctuations | mvMFE | mvMDE_{III} | mvMDE |
---|---|---|---|

Self-paced walk | 0.040 | 0.005 | 0.002 |

Metronomically-paced walk | 0.116 | 0.025 | 0.019 |

**Table 5.**CV values of the entropy results at scale factor 6 using mvMDE

_{II}, mvMDE

_{III}, mvMDE, mvMSE, and mvMFE for focal vs. non-focal EEG recordings.

Signals | mvMSE | mvMFE | mvMDE_{II} | mvMDE_{III} | mvMDE |
---|---|---|---|---|---|

focal EEGs | 0.019 | 0.019 | 0.006 | 0.003 | 0.002 |

Non-focal EEGs | 0.021 | 0.015 | 0.008 | 0.003 | 0.002 |

**Table 6.**Differences between results for AD patients’ vs. healthy controls’ MEGs obtained by mvMFE and mvMDE for five main brain regions based on the Hedges’ g effect size.

Region-Method | Scale 1 | Scale 2 | Scale 3 | Scale 4 | Scale 5 | Scale 6 | Scale 7 | Scale 8 | Scale 9 | Scale 10 |
---|---|---|---|---|---|---|---|---|---|---|

Anterior-mvMFE | 0.36 | 0.73 | 0.57 | 0.04 | 0.33 | 0.53 | 0.63 | 0.70 | 0.72 | 0.73 |

Central-mvMFE | 0.68 | 0.67 | 0.49 | 0.10 | 0.23 | 0.48 | 0.65 | 0.76 | 0.79 | 0.83 |

Left lateral-mvMFE | 0.53 | 0.64 | 0.34 | 0.18 | 0.60 | 0.83 | 0.92 | 0.98 | 0.97 | 0.98 |

Posterior-mvMFE | 0.46 | 0.72 | 0.58 | 0.16 | 0.30 | 0.57 | 0.73 | 0.78 | 0.82 | 0.85 |

Right lateral-mvMFE | 0.30 | 0.50 | 0.22 | 0.18 | 0.53 | 0.71 | 0.84 | 0.92 | 0.97 | 0.95 |

Anterior-mvMDE | 0.18 | 0.37 | 0.36 | 0.03 | 0.49 | 0.80 | 0.95 | 1.02 | 1.06 | 1.04 |

Central-mvMDE | 0.29 | 0.45 | 0.29 | 0.48 | 0.78 | 0.88 | 0.97 | 1.01 | 1.03 | 1.04 |

Left lateral-mvMDE | 0.37 | 0.40 | 0.24 | 0.24 | 0.77 | 1.07 | 1.17 | 1.20 | 1.19 | 1.19 |

Posterior-mvMDE | 0.05 | 0.19 | 0.18 | 0.24 | 0.67 | 0.90 | 1.015 | 1.05 | 1.06 | 1.06 |

Right lateral-mvMDE | 0.15 | 0.19 | 0.00 | 0.51 | 0.90 | 1.05 | 1.14 | 1.18 | 1.20 | 1.16 |

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**MDPI and ACS Style**

Azami, H.; Fernández, A.; Escudero, J. Multivariate Multiscale Dispersion Entropy of Biomedical Times Series. *Entropy* **2019**, *21*, 913.
https://doi.org/10.3390/e21090913

**AMA Style**

Azami H, Fernández A, Escudero J. Multivariate Multiscale Dispersion Entropy of Biomedical Times Series. *Entropy*. 2019; 21(9):913.
https://doi.org/10.3390/e21090913

**Chicago/Turabian Style**

Azami, Hamed, Alberto Fernández, and Javier Escudero. 2019. "Multivariate Multiscale Dispersion Entropy of Biomedical Times Series" *Entropy* 21, no. 9: 913.
https://doi.org/10.3390/e21090913