# Multivariate Generalized Multiscale Entropy Analysis

## Abstract

**:**

**G**MSE and M

**G**rcMSE, respectively) are first analyzed through the processing of synthetic signals. We reveal that M

**G**rcMSE shows better performance than M

**G**MSE for short multivariate data. We then study the performance of M

**G**rcMSE on two sets of short multivariate electroencephalograms (EEG) available in the public domain. We report that M

**G**rcMSE may show better performance than MrcMSE in distinguishing different types of multivariate EEG data. M

**G**rcMSE could therefore supplement MMSE or MrcMSE in the processing of multivariate datasets.

## 1. Introduction

**G**MSE) and the multivariate generalized rcMSE (M

**G**rcMSE), and evaluate their performance on both synthetic and real-world multivariate processes.

## 2. Materials and Methods

#### 2.1. Multiscale Entropy

- A coarse-graining procedure. For a monovariate discrete signal of length N $\{{x}_{1},\dots ,{x}_{i},\dots ,{x}_{N}\}$, the coarse-grained time series $\{{y}^{\left(\tau \right)}\}$ is computed as$${y}_{j}^{\left(\tau \right)}=\frac{1}{\tau}\sum _{i=(j-1)\tau +1}^{j\tau}{x}_{i},\phantom{\rule{2.em}{0ex}}1\le j\le \lfloor N/\tau \rfloor .$$
- Computation of the sample entropy for each coarse-grained time series. The sample entropy quantifies the regularity of finite length time series [3]. A low value for the sample entropy reflects a high degree of regularity, while a random signal has a relatively higher value of sample entropy. Sample entropy is a conditional probability measure that quantifies the likelihood that a sequence of m consecutive data points—that matches another sequence of the same length (match within a tolerance of r)—will still match the other sequence when their length is increased by one sample (sequences of length $m+1$); m therefore defines the length of the patterns that are compared to each other [3]. For a time series $\{{x}_{1},\dots ,{x}_{i},\dots ,{x}_{N}\}$, the sample entropy is computed as$$SampEn(x,m,r)=-ln\frac{{n}^{m+1}}{{n}^{m}},$$$${n}^{m}=\frac{1}{N-m}\sum _{i=1}^{N-m}{B}_{i}^{m}\left(r\right),$$$${B}_{i}^{m}\left(r\right)=\frac{{n}_{i}^{m}\left(r\right)}{N-m-1},$$

#### 2.2. Refined Composite Multiscale Entropy

- For a scale factor τ, τ coarse-grained time series are generated. The k-th coarse-grained time series for a scale factor τ is defined as ${y}_{k}^{\left(\tau \right)}={\{{y}_{k,j}\}}_{j=1}^{N/\tau}$, where [12]$${y}_{k,j}^{\left(\tau \right)}=\frac{1}{\tau}\sum _{i=(j-1)\tau +k}^{j\tau +k-1}{x}_{i},\phantom{\rule{2.em}{0ex}}1\le j\le \frac{N}{\tau},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1\le k\le \tau .$$
- For each scale factor τ, and for all τ coarse-grained time series, the number of matched vector pairs ${n}_{k,\tau}^{m+1}$ and ${n}_{k,\tau}^{m}$ is computed, where ${n}_{k,\tau}^{m}$ represents the total number of m-dimensional matched vector pairs and is computed from the k-th coarse-grained time series at a scale factor τ
- rcMSE is then defined as [5]$$rcMSE(x,\tau ,m,r)=-ln\left(\frac{{\sum}_{k=1}^{\tau}{n}_{k,\tau}^{m+1}}{{\sum}_{k=1}^{\tau}{n}_{k,\tau}^{m}}\right).$$

#### 2.3. Multivariate (Refined Composite) Multiscale Entropy

**G**MSE${}_{\mu}$ in what follows—relies on the same steps as MSE [7,8]: a coarse-graining procedure and a sample entropy computation for each coarse-grained time series. Each of these two steps is adapted to multivariate data. Thus, for the coarse-graining procedure, temporal scales are defined by averaging a p-variate time series ${\{{x}_{l,i}\}}_{i=1}^{N}$ ($l=1,\dots ,p$ is the channel index and N is the number of samples in every channel) over time segments of increasing length. Thus, for a scale factor τ, a multi-channel coarse-grained time series is computed as (here the channel index l goes from 1 to p)

**G**rcMSE${}_{\mu}$) has also recently been proposed [9]. It is based on the same steps as rcMSE, but each of the steps is adapted to multivariate signals. It has been reported that M

**G**rcMSE${}_{\mu}$ outperforms M

**G**MSE${}_{\mu}$ because it leads to lower standard deviation values than M

**G**MSE${}_{\mu}$ [9].

#### 2.4. Multivariate Generalized (Refined Composite) Multiscale Entropy

**G**MSE) and generalized rcMSE (M

**G**rcMSE) that we propose, the same steps as the ones of MMSE (resp. MrcMSE) are found, except in the coarse-graining procedure, where the averaging is replaced by the computation of the second moment (M

**G**MSE${}_{{\sigma}^{2}}$ and M

**G**rcMSE${}_{{\sigma}^{2}}$) or by the computation of the third moment (M

**G**MSE${}_{\mathrm{skewness}}$ and M

**G**rcMSE${}_{\mathrm{skewness}}$), and so on. M

**G**MSE${}_{{\sigma}^{2}}$ and M

**G**rcMSE${}_{{\sigma}^{2}}$ quantify the dynamics of the volatility over multiple time scales [6]. Thus, for M

**G**MSE${}_{{\sigma}^{2}}$, the coarse-grained multivariate time series at scale factor τ is computed as

**G**MSE${}_{\mathrm{skewness}}$ is determined as in Equation (9) where the computation of the variance is replaced by the computation of the skewness (similar steps are used to compute M

**G**rcMSE${}_{{\sigma}^{2}}$ and M

**G**rcMSE${}_{\mathrm{skewness}}$, see Figure 1). From Equation (9), we observe that M

**G**MSE${}_{{\sigma}^{2}}$ and M

**G**rcMSE${}_{{\sigma}^{2}}$ are 0 for scale factor $\tau =1$. In what follows, we will therefore concentrate on scale factors larger than 1 for M

**G**MSE${}_{{\sigma}^{2}}$ and M

**G**rcMSE${}_{{\sigma}^{2}}$. For the same reason, we will concentrate on scale factors larger than 2 for M

**G**MSE${}_{\mathrm{skewness}}$ and M

**G**rcMSE${}_{\mathrm{skewness}}$. Hereafter we will study M

**G**MSE${}_{{\sigma}^{2}}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, M

**G**MSE${}_{\mathrm{skewness}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$ on synthetic and real datasets.

#### 2.5. Datasets Acquisition

## 3. Experimental Results and Discussion

#### 3.1. Results for Synthetic Signals

**G**MSE${}_{{\sigma}^{2}}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, M

**G**MSE${}_{\mathrm{skewness}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$ of a trivariate time series, where all the data channels were originally realizations of mutually independent white noise [7]. We then gradually decreased the number of variates representing white noise (from 3 to 0), and simultaneously increased the number of data channels representing independent $1/f$ noise (from 0 to 3), as already proposed in [7,8,9]. The total number of variates was always three. The embedding dimension ${m}_{k}$ was chosen to be equal to 2, and the threshold r was fixed to 0.15×(standard deviation of the normalized time series) for each data channel [9]. For each kind of trivariate data, 10 independent realizations were simulated. For each realization, 15,000 samples were generated in each variate. Scales were lower than 40. Therefore, all the coarse-grained time series were longer than 300 samples, as recommended by others [8,18].

**G**MSE${}_{\mu}$ and M

**G**rcMSE${}_{\mu}$ for similar kinds of data have been previously reported [7,9]: the multivariate sample entropy decreases with increasing scale factors, whatever the composition of the multivariate data (see Figure 2a,b). Moreover, for scale factors larger than 2, the higher the number of variates representing $1/f$ noise, the higher the multivariate entropy value. We observe that the results obtained with M

**G**MSE${}_{\mu}$ and M

**G**rcMSE${}_{\mu}$ are similar. However, for the largest scales, we observe some wandering on M

**G**MSE${}_{\mu}$ that is not visible on M

**G**rcMSE${}_{\mu}$.

**G**MSE${}_{{\sigma}^{2}}$ and M

**G**rcMSE${}_{{\sigma}^{2}}$ are shown in Figure 2c,d. We note a slight increase and then a decrease of the multivariate sample entropy with increasing scale factors, except for the trivariate data containing only $1/f$ noise, where a nearly constant multivariate sample entropy is observed for increasing scale factors. For scales lower than 40, the multivariate data with channels containing only white noise show higher multivariate sample entropy than multivariate data with channels containing only $1/f$ noise. Others have reported similar findings for univariate data [19]. Therefore, the scale for which the multivariate data containing only $1/f$ noise shows larger entropy values than those of multivariate data containing only white noise is larger than what was observed with M

**G**MSE${}_{\mu}$: for scales lower than 40, trivariate data containing channels with only $1/f$ noise have a lower multivariate sample entropy than trivariate data containing channels with only white noise; for M

**G**MSE${}_{\mu}$, for scales larger than 2, trivariate data containing channels with only $1/f$ noise have a larger multivariate sample entropy than trivariate data containing channels with only white noise (see Figure 2a,b). If we focus on the decrease rate observed for M

**G**MSE${}_{{\sigma}^{2}}$ and M

**G**rcMSE${}_{{\sigma}^{2}}$, we can say that the results are consistent with what was expected: the more the multivariate data contain $1/f$ noise, the slower the decrease of the multivariate sample entropy values with increasing scale factors ($1/f$ data are theoretically more complex than white noise, because $1/f$ data contain long-range correlations [2]). Moreover, we observe that the results obtained with M

**G**MSE${}_{{\sigma}^{2}}$ and M

**G**rcMSE${}_{{\sigma}^{2}}$ are rather similar.

**G**MSE${}_{\mathrm{skewness}}$ and M

**G**rcMSE${}_{\mathrm{skewness}}$ are shown in Figure 2e,f. We observe that—whatever the composition of the multivariate dataset—the multivariate sample entropy first increases until scale factor $\tau =5$, and then decreases. For M

**G**rcMSE${}_{\mathrm{skewness}}$, the most rapid decrease is observed for data with channels containing only white noise. The slowest decrease is observed for data with channels containing only $1/f$ noise. For scale factors larger than 20, the highest multivariate sample entropy is observed for data with channels containing only $1/f$ noise, and the lowest for data with channels containing only white noise. The global trend for M

**G**MSE${}_{\mathrm{skewness}}$ is similar to the one of M

**G**rcMSE${}_{\mathrm{skewness}}$, but the different rates of decrease for each variate are not as clear as for M

**G**rcMSE${}_{\mathrm{skewness}}$.

**G**MSE and M

**G**rcMSE for two different synthesized bivariate data: one containing 15,000 samples and another containing 1000 samples. Our goal is to analyze if the results obtained for the two lengths (15,000 and 1000 samples) are similar. For this purpose, we considered only scales lower than 10, as data with only 1000 samples do not reasonably allow us to go beyond this limit. Thus, for data with 15,000 samples, the shortest coarse-grained time series has a length of 1500 samples. For data with 1000 samples, the shortest coarse-grained time series has a length of 100 samples. For this last case, the length of the coarse-grained time series will therefore be lower than 300, as recommended by others [8,18]. With a length of 100 samples, the results obtained with M

**G**rcMSE should still be acceptable, as shown in [9]. For each length (15,000 samples and 1000 samples), 10 independent realizations were simulated.

**G**MSE${}_{\mu}$ and M

**G**rcMSE${}_{\mu}$ show consistent results with those observed for trivariate datasets (see Figure 3a,b). However, for data with 1000 samples, we observe that M

**G**MSE${}_{\mu}$ leads to undefined entropy values (see Figure 3c). M

**G**rcMSE${}_{\mu}$ shows similar values to those obtained with 15,000 samples—except for the bivariate data containing only $1/f$ noise, where wandering are observed (see Figure 3d). However, for scales lower than 10, the results obtained with M

**G**rcMSE${}_{\mu}$ are still acceptable.

**G**MSE${}_{{\sigma}^{2}}$ and M

**G**rcMSE${}_{{\sigma}^{2}}$ (see Figure 4), the bivariate data containing only $1/f$ time series present almost constant multivariate sample entropy values with increasing scale factors. However, for the bivariate data containing only white noise and for the bivariate data containing both $1/f$ noise and white noise, we observe an increase and then a decrease of the multivariate sample entropy for increasing scale factors. We also observe that M

**G**MSE${}_{{\sigma}^{2}}$ shows different results for data of 15,000 samples and for data of 1000 samples. However, M

**G**rcMSE${}_{{\sigma}^{2}}$ shows close findings for the two lengths.

**G**MSE${}_{\mathrm{skewness}}$ and M

**G**rcMSE${}_{\mathrm{skewness}}$, the results show that—whatever the composition of the bivariate dataset—the behavior is the same as scales increase (see Figure 5). We observe an increase of the multivariate sample entropy with increasing scale factors and then a progressive decrease. However, as above, M

**G**rcMSE${}_{\mathrm{skewness}}$ shows more stable results for short time series than M

**G**MSE${}_{\mathrm{skewness}}$ where wandering are observed.

**G**MSE. When M

**G**rcMSE is used, the differences are much less pronounced. For scales lower than 10, M

**G**rcMSE computed with 1000 samples leads to consistent results with those obtained with 15,000 samples. This is true for M

**G**rcMSE${}_{\mu}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$. In what follows, we will therefore concentrate on the refined versions of the multivariate generalized entropy measures: M

**G**rcMSE${}_{\mu}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$.

#### 3.2. Results for Biomedical Datasets

**G**rcMSE${}_{\mu}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$ are presented in Figure 7. We observe that M

**G**rcMSE${}_{\mu}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$ show different patterns for each of the five cases. However, the highest differences between the five cases are given by M

**G**rcMSE${}_{{\sigma}^{2}}$. In order to more easily quantify these differences, we propose the computation of a complexity index, defined here as the sum of the (multivariate) sample entropy for all scales studied. The computation of a similar index has been performed in other studies (e.g., [20,21,22,23]). For the univariate case, the sum was computed for each channel independently, and then the mean of the two sums was considered. The results for each case are presented in Figure 8. We observe that M

**G**rcMSE${}_{{\sigma}^{2}}$ leads to a better distinction of the different cases than M

**G**rcMSE${}_{\mu}$. Data B, C, D, and E all contain spike-wave discharges. Therefore, we could have expected to obtain similar signatures for M

**G**rcMSE${}_{{\sigma}^{2}}$ and/or for M

**G**rcMSE${}_{\mathrm{skewness}}$. However, we obtain different results on these datasets. This is consistent with other synchronization measures which were also able to distinguish them [15]. Contents of data B, C, D, and E are therefore different. Our findings suggest that the multivariate multiscale complexity of the volatility may be more interesting than M

**G**rcMSE${}_{\mu}$ when studying multivariate datasets.

**G**rcMSE${}_{\mu}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$ are presented in Figure 10. We observe that M

**G**rcMSE${}_{\mu}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$ facilitate the differentiation between the five cases. However, the highest differences between the five cases are given by M

**G**rcMSE${}_{{\sigma}^{2}}$. As above (and with the same definition as above), we computed the complexity index. The results for each case are presented in Figure 11. As previously, we observe that M

**G**rcMSE${}_{{\sigma}^{2}}$ leads to a better distinction of the five cases than M

**G**rcMSE${}_{\mu}$.

**G**MSE and M

**G**rcMSE. However, the first findings presented herein are encouraging. Nevertheless, our study presents limitations. A first limitation is the non-existence of an inverse transform for this multiscale transform. Another limitation is the computational time of M

**G**MSE (and even more, the one of M

**G**rcMSE): this computational time prevents real-time analyses. This is particularly annoying, as we need to record long time series if large scales are to be analyzed.

**G**MSE and M

**G**rcMSE based on these faster codes. Furthermore, the original MSE relies on a sample entropy-based approach, but can be used with different types of entropic measures: permutation entropy, cross-approximate entropy, compression entropy, etc. The multivariate generalization proposed herein could be extended to MSE based on other entropic measures. In addition to rcMSE, several other variants to MSE have been proposed: refined MSE [11], composite MSE [12], modified MSE for short-term time series [10], and short time MSE [32], to cite only a few. The multivariate approach of these variants could now be proposed, as well as their generalization to higher moments. For our simulated time series, M

**G**rcMSE${}_{\mathrm{skewness}}$ leads to close values, whatever the composition of the multivariate dataset. These results have to be studied more thoroughly in the future. However, we observe that different findings are obtained for EEG data. For the univariate case, MSE${}_{\mathrm{skewness}}$ has already been shown to give interesting results for the biomedical field [33].

## 4. Conclusions

**G**MSE and M

**G**rcMSE on synthetic signals, and processed two different publicly available bivariate EEG data sets. Our results show that M

**G**rcMSE may present better performance than MrcMSE in differentiating different types of EEG signals. M

**G**rcMSE could therefore become an interesting signal processing tool for multivariate datasets. Moreover, we have shown that M

**G**MSE and M

**G**rcMSE lead to similar results when long data are analyzed. However, M

**G**rcMSE shows better performance than M

**G**MSE when short data are processed. The time complexity of M

**G**rcMSE being worse than the one of M

**G**MSE, we suggest the use of M

**G**MSE for long data and M

**G**rcMSE for short data.

**G**(rc)MSE could help in the understanding and diagnosis of some cardiovascular diseases. M

**G**(rc)MSE could also be used for many other kinds of data: financial time series, chemical data, etc.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Flow chart describing the computation of multivariate extension of generalized refined composite multiscale entropy (M

**G**rcMSE${}_{{\sigma}^{2}}$) for a p-variate time series ${\{{x}_{l,i}\}}_{i=1}^{N}$ ($l=1,\dots ,p$ is the channel index, and N is the number of samples in every channel).

**Figure 2.**(

**a**) multivariate extension of the generalized MSE (M

**G**MSE${}_{\mu}$); (

**b**) M

**G**rcMSE${}_{\mu}$; (

**c**) M

**G**MSE${}_{{\sigma}^{2}}$; (

**d**) M

**G**rcMSE${}_{{\sigma}^{2}}$; (

**e**) M

**G**MSE${}_{\mathrm{skewness}}$; and (

**f**) M

**G**rcMSE${}_{\mathrm{skewness}}$ average values for ten trivariate data containing white Gaussian noise (WGN) and $1/f$ noise, each with 15,000 samples.

**Figure 3.**M

**G**MSE${}_{\mu}$ (left column) and M

**G**rcMSE${}_{\mu}$ (right column) average values for ten bivariate data containing white Gaussian noise (WGN) and $1/f$ noise, each with 15,000 samples (upper panels) and 1000 samples (lower panels).

**Figure 4.**M

**G**MSE${}_{{\sigma}^{2}}$ (left column) and M

**G**rcMSE${}_{{\sigma}^{2}}$ (right column) average values for ten bivariate data containing white Gaussian noise (WGN) and $1/f$ noise, each with 15,000 samples (upper panels) and 1000 samples (lower panels).

**Figure 5.**M

**G**MSE${}_{\mathrm{skewness}}$ (left column) and M

**G**rcMSE${}_{\mathrm{skewness}}$ (right column) average values for ten bivariate data containing white Gaussian noise (WGN) and $1/f$ noise, each with 15,000 samples (upper panels) and 1000 samples (lower panels).

**Figure 7.**From top to bottom: M

**G**rcMSE${}_{\mu}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$ for five pairs of rat EEG signals—see Figure 6.

**Figure 8.**Complexity index for five EEG cases—see Figure 6.

**Figure 9.**MSE values for five EEG cases (data A to E, from top to bottom). For each subplot, each curve corresponds to a data channel. See text for details on each data set, and [17] for the data acquisition procedure.

**Figure 10.**From top to bottom: M

**G**rcMSE${}_{\mu}$, M

**G**rcMSE${}_{{\sigma}^{2}}$, and M

**G**rcMSE${}_{\mathrm{skewness}}$ for five pairs of human EEG signals—see Figure 9.

**Figure 11.**Complexity index for five EEG cases—see Figure 9.

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Multivariate Generalized Multiscale Entropy Analysis. *Entropy* **2016**, *18*, 411.
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Multivariate Generalized Multiscale Entropy Analysis. *Entropy*. 2016; 18(11):411.
https://doi.org/10.3390/e18110411

**Chicago/Turabian Style**

Humeau-Heurtier, Anne.
2016. "Multivariate Generalized Multiscale Entropy Analysis" *Entropy* 18, no. 11: 411.
https://doi.org/10.3390/e18110411