# Univariate and Multivariate Generalized Multiscale Entropy to Characterise EEG Signals in Alzheimer’s Disease

^{1}

^{2}

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## Abstract

**:**

_{σ2}) to multichannel signals, termed multivariate MSE

_{σ2}(mvMSE

_{σ2}), to take into account both the spatial and time domains of time series. Then, we investigate the mvMSE

_{σ2}of EEGs at different frequency bands, including the broadband signals filtered between 1 and 40 Hz, θ, α, and β bands, and compare it with the previously-proposed multiscale entropy based on mean (MSE

_{µ}), multivariate MSE

_{µ}(mvMSE

_{µ}), and MSE

_{σ2}, to distinguish different kinds of dynamical properties of the spread and the mean in the signals. Results from 11 AD patients and 11 age-matched controls suggest that the presence of broadband activity of EEGs is required for a proper evaluation of complexity. MSE

_{σ2}and mvMSE

_{σ2}results, showing a loss of complexity in AD signals, led to smaller p-values in comparison with MSE

_{µ}and mvMSE

_{µ}ones, suggesting that the variance-based MSE and mvMSE can characterise changes in EEGs as a result of AD in a more detailed way. The p-values for the slope values of the mvMSE curves were smaller than for MSE at large scale factors, also showing the possible usefulness of multivariate techniques.

## 1. Introduction

_{µ}, has been recently introduced [22]. The mvMSE

_{µ}algorithm was validated on both illustrative benchmark signals and on real-world multivariate physiological and non-physiological datasets [22,24].

_{µ}and MSE

_{σ2}, respectively. MSE

_{σ2}was used to analyse heartbeat signals from healthy young and older subjects and patients with congestive heart failure syndrome. It was demonstrated that the dynamics of the volatility of heartbeat signals obtained from healthy young subjects are highly complex. The results also showed that MSE

_{σ2}values decrease with ageing and pathology [25].

_{µ}for magnetoencephalograms (MEGs) in AD [32]. They consider five subsets of channels and not all the channels as a whole. Multiscale approaches using the variance in the coarse-graining process have yet to be applied to EEG analysis. Therefore, there is a need to investigate the usefulness of MSE

_{σ2}and mvMSE where the coarse graining process uses variance (mvMSE

_{σ2}) in comparison with the more broadly used methods based on mean (MSE

_{µ}and mvMSE

_{µ}) to characterise EEGs in AD.

_{µ}, MSE

_{σ2}, mvMSE

_{µ}, and mvMSE

_{σ2}algorithms. Results are presented in Section 3. The discussions and conclusions are explained in Section 4 and Section 5, respectively.

## 2. Materials and Methods

#### 2.1. Subjects

#### 2.2. EEG Recordings

#### 2.3. Methods

#### 2.3.1. Multiscale Entropy Based on Mean and Variance

- (I)
- Assume we have a signal $\{{x}_{1},{x}_{2},\dots ,{x}_{b},\dots ,{x}_{C}\}$ with length C. Each element of the coarse-grained time series for MSE
_{µ}and recently proposed MSE_{σ2}are respectively calculated as:$${}^{\mu}{y}_{i}{}^{(\lambda )}=\frac{1}{\lambda}{\displaystyle \sum _{b=(i-1)\lambda +1}^{i\lambda}{x}_{b}}\text{\hspace{1em}\hspace{1em}}1\le i\le \lfloor \frac{C}{\lambda}\rfloor =N$$$${}^{{\sigma}^{2}}{y}_{i}{}^{(\lambda )}=\frac{1}{\lambda}{\displaystyle \sum _{b=(i-1)\lambda +1}^{i\lambda}{({x}_{b}-{}^{\mu}{y}_{i}{}^{(\lambda )})}^{2}}\text{\hspace{1em}\hspace{1em}}1\le i\le \lfloor \frac{C}{\lambda}\rfloor =N$$_{µ}and MSE_{σ2}are respectively the mean and variance values of consecutive sample points [17,25]. Note that the coarse-graining process based on the mean and variance start from scale factor 1 and 2, respectively [17,25]. - (II)
- At each scale factor, the SampEn of the coarse-grained signal $y=\{{y}_{1},{y}_{2},\dots ,{y}_{N}\}$ is calculated in the next step. For the sake of conciseness, here, we use y
_{i}for both the coarse-grained signals ${}^{{\sigma}^{2}}{y}_{i}{}^{(\lambda )}$ and ${}^{\mu}{y}_{i}{}^{(\lambda )}$. At each time t of**y**, a vector ${\mathrm{Y}}_{t}^{m}=\{{y}_{t},{y}_{t+1},\dots ,{y}_{t+m-2},{y}_{t+m-1}\}$ for t = 1, 2, …, N−(m−1), including the m-th subsequent values is constructed, where m, named embedding dimension, stands for how many samples are contained in each vector. Next, the distance between such vectors as the maximum difference of their corresponding scalar components, $d\text{}\left[{\mathrm{Y}}_{{t}_{1}}^{m},{\mathrm{Y}}_{{t}_{2}}^{m}\right]=\mathrm{max}\left\{\left|{\mathrm{Y}}_{{t}_{1}+k}^{m}-{\mathrm{Y}}_{{t}_{2}+k}^{m}\right|:0\le k\le m-1\text{}\mathrm{and}\text{}{t}_{1}\ne {t}_{2}\right\}$ are calculated. A match happens when the distance $d\left[{\mathrm{Y}}_{{t}_{1}}^{m},{\mathrm{Y}}_{{t}_{2}}^{m}\right]$ is smaller than a predefined tolerance r. The probability B^{m}(r) shows the total number of m-dimensional matched vectors [15]. Similarly, B^{m}^{+1}(r) is defined for embedded dimension of m + 1. Finally, the SampEn is defined as follows [15]:$$SampEn(y,m,r)=-\mathrm{ln}\left({B}^{m+1}(r)/{B}^{m}(r)\right)$$

#### 2.3.2. Multivariate Multiscale Entropy Based on Mean and Variance

- (I)
- Assume we have a p-channel (multivariate) time series $Z={\left\{{z}_{q,b}\right\}}_{b=1}^{C}$, q = 1, …, p, where C is the length of each channel’s signal. Each element of the coarse-grained time series is calculated as follows:$${}^{\mu}{u}_{q,i}{}^{(\lambda )}=\frac{1}{\lambda}{\displaystyle \sum _{b=(i-1)\lambda +1}^{i\lambda}{z}_{q,b}}\text{\hspace{1em}\hspace{1em}}1\le i\le \lfloor \frac{C}{\lambda}\rfloor =N,\text{}1\le q\le p$$
_{σ2}[25] to multi-channel signals, we propose to use variance, instead of mean value, in the coarse-graining process as follows:$${}^{{\sigma}^{2}}{u}_{q,i}{}^{(\lambda )}=\frac{1}{\lambda}{\displaystyle \sum _{b=(i-1)\lambda +1}^{i\lambda}{({z}_{q,b}-{}^{\mu}{u}_{q,i}{}^{(\lambda )})}^{2}}\text{\hspace{1em}\hspace{1em}}1\le i\le \lfloor \frac{C}{\lambda}\rfloor =N,\text{}1\le q\le p$$ - (II)
- Second, for the defined scale factor λ, the mvSE of the coarse-grained signal is calculated [24,37,38]. To calculate the mvSE, multivariate embedded vectors are initially generated [24]. In [39], the Takens embedding theorem for multivariate concept is described. Using the p-channel signal $U={\left\{{u}_{q,i}\right\}}_{q=1,\text{}i=1}^{q=p,\text{}i=N\text{}}$ where N is the length of each coarse-grained time series ${\left\{{u}_{q}\right\}}_{q=1}^{p}$, the multivariate embedded reconstruction is defined as:$${U}_{m}(i)=[{u}_{1,i},{u}_{1,i+{\tau}_{1}},\dots ,{u}_{1,i+({m}_{1}-1){\tau}_{1}},{u}_{2,i},{u}_{2,i+{\tau}_{2}},\dots ,{u}_{2,i+({m}_{2}-1){\tau}_{2}},\dots ,{u}_{p,i},{u}_{p,i+{\tau}_{p}},\dots ,{u}_{p,i+({m}_{p}-1){\tau}_{p}}]$$

- Form multivariate embedded vectors ${U}_{m}(i)\in {R}^{m}$ where $i=1,2,\dots ,N-n$ and $n=max\left\{m\right\}\times max\left\{\mathsf{\tau}\right\}$.
- Calculate the distance between any two composite delay vectors ${U}_{m}(i)$ and ${U}_{m}(j)$ as the maximum norm.
- For a given ${U}_{m}(i)$ and a threshold r, count the number of instances P
_{i}where $d[{U}_{m}(i),{U}_{m}(j)]\le r,\text{}i\ne j$. Next, calculate the frequency of occurrence as ${\varphi}_{i}^{m}(r)=\frac{1}{N-n}{P}_{i}$ and define a global quantity ${\varphi}^{m}(r)=\frac{1}{N-n}{\displaystyle \sum _{i=1}^{N-n}{\varphi}_{i}^{m}(r)}$. - Extend the dimensionality of the multivariate delay vector in (6) from m to (m + 1) (keep the dimension of the other variables unchanged).
- Repeat steps 1–4 and find ${\varphi}_{i}^{({m}_{q}+1)}(r)$. Next, calculate ${\varphi}_{i}^{(m+1)}(r)$ which denotes the average over all n of ${\varphi}_{i}^{({m}_{q}+1)}(r)$. Finally, find ${\varphi}_{i}^{(m+1)}(r)$ which stands for the average over all i of ${\varphi}_{i}(r)$ in an (m + 1)-dimensional space.
- Finally, mvSE is defined as:$$mvSE(Z,m,\mathsf{\tau},r)=-\mathrm{ln}\left(\frac{{\varphi}^{(m+1)}(r)}{{\varphi}^{m}(r)}\right)$$
_{k}, τ_{k}, and r for all of the approaches were, respectively, chosen as 2, 1, and 0.15 multiplied by the SD of the original time series according to [15,24]. Note that the number of sample points is at least 10^{m}, or preferably at least 30^{m}, to robustly estimate SampEn and mvSE, according to [24,40,41].

#### 2.4. Experimental Procedures

_{µ}, mvMSE

_{µ}, MSE

_{σ2}, and mvMSE

_{σ2}on different EEG frequency bands, including θ, α, and β. Note that δ and γ, respectively, have too low and high frequency to be considered here based on the fact that the MSE

_{µ}and mvMSE

_{µ}methods at scale factor λ can be considered as a low-pass filter with cut-off frequency $\frac{{f}_{s}}{2\lambda}$ [42].

## 3. Results

#### 3.1. Global Evaluation of Multivariate and Univariate Multiscale Entropies

_{µ}, mvMSE

_{µ}, MSE

_{σ2}, and mvMSE

_{σ2}methods are, respectively, shown in Figure 1, Figure 2, Figure 3 and Figure 4. For each of Figure 1, Figure 2, Figure 3 and Figure 4, (a)–(d) show the results at frequency bands 1–40 Hz, θ, α, and β, respectively. As can be seen in (b)–(d) of Figure 1, Figure 2, Figure 3 and Figure 4, the results obtained at frequency bands θ, α, and β do not show that controls’ signals are more complex than AD patients’ ones. This fact suggests that complexity changes are best highlighted considering broadband activity.

_{µ}and mvMSE

_{µ}are different for the control individuals and AD patients at short- and long-time scale factors. In comparison with the AD group, controls’ signals have more irregularity at short-time scales, whereas the AD patients’ time series are more irregular at long-time scales.

_{µ}highlights differences between groups at individual scales better than the averaging of univariate MSE

_{µ}profiles. However, the opposite seems to happen for MSE

_{σ2}(Figure 3) when compared with mvMSE

_{σ2}(Figure 4). This might be because the variance coarse-grained sequences have too little variability and the multivariate implementation leads to values that are too low (notice that the output values are in 1/100s of the unit).

_{σ2}- and mvMSE

_{σ2}-based profiles show that MSE

_{σ2}leads to significant differences at all scale factors, while the significant differences based on mvMSE

_{σ2}are seen at scale factors 8 and 10. In comparison with MSE

_{µ}, MSE

_{σ2}discriminates better AD group and controls, while compared with mvMSE

_{σ2}, mvMSE

_{µ}discriminates better these two groups. It shows that the mean- and variance-based complexity measures can complement each other to characterise EEGs in AD. It is worth noting that the results obtained by different values of r (0.2, 0.25, and 0.3) and m (1 and 2) employed in other complexity studies are similar to our results [11,32,44,45].

_{µ}results of brain signals [46], we evaluated all MSE and mvMSE methods on 40 different univariate and uncorrelated multivariate WGN time series band-pass filtered at 1–40 Hz, 4–8 Hz, 8–13 Hz, and 13–30 Hz, to investigate whether the entropy profiles of brain signals are linked to their power content. The length of the time series and the number of channels of the filtered multivariate WGN were respectively 1280 sample points (equal to the length of the EEG time series) and 16 (equal to the number of channels of EEG time series), and the parameter values for the multiscale methods were equal to those used for the EEG dataset. The results, shown in Figure 1, Figure 2, Figure 3 and Figure 4, show that the shape of MSE

_{µ}and MSE

_{σ2}curves are linked to the power spectral density of the corresponding filtered signals. In fact, to some extent, the MSE curves are determined by the (low and high cut-offs of the) filtering process, especially for frequency bands 4–8 Hz, 8–13 Hz, and for 13–30 Hz to a lesser extent. However, it is important to note that the entropy profiles for EEG signals of AD patients and controls do not overlap with the curves of the filtered WGN at most scale factors for the frequency band of 1–40 Hz. It also evidences the need to have broadband EEGs, instead of narrow band activity, for the evaluation of multiscale complexity. In contrast with the results of univariate entropy techniques, the mvMSE

_{µ}- and mvMSE

_{σ2}-based curves for AD patients and controls have clearly dissimilar shapes with those for filtered uncorrelated multichannel WGN, suggesting that the multivariate entropy-based values of AD patients’ and controls’ time series are completely different to those of filtered uncorrelated sixteen-channel WGN.

_{µ}, mvMSE

_{µ}, MSE

_{σ2}, and mvMSE

_{σ2}methods for one of the 16-channel AD signals with the length of 1280 sample points are shown in Table 1. Note that the running time for the MSE-methods is the sum of computation time values for each of the 16 channels. In this study, the simulations have been carried out using a PC with Intel (R) Xeon (R) CPU, E5420, 2.5 GHz and 8-GB RAM by MATLAB R2010a. Since the MSE

_{σ2}and mvMSE

_{σ2}start from scale factor 2 and SampEn has a computational cost of O(N

^{2}), the computation time of this kind of algorithms is noticeably smaller than that of the MSE

_{µ}or mvMSE

_{µ}algorithms. The mvMSE methods deal with both the spatial and time domains, albeit the MSE algorithms consider only the time domain. Thus, as can be seen in Table 1, the MSE techniques are significantly faster than their corresponding mvMSE methods.

#### 3.2. Regional Evaluation with Univariate Metrics

_{µ}and MSE

_{σ2}values and their p-values for each channel of EEGs, band-pass-filtered between 1 and 40 Hz, are presented in Figure 5 and Figure 6, respectively. The p-values show the superiority of MSE

_{σ2}over MSE

_{µ}for characterising AD. Moreover, the lowest p-values for MSE

_{σ2}that obtained by the channels O1, O2, and P3 were equal to 0.0058, 0.0086, and 0.0087, respectively, in agreement with [11].

#### 3.3. Features (Slopes) from Univariate and Multivariate Multiscale Profiles

_{µ}and mvMSE

_{µ}methods, the curves increase until a scale factor of 4. Then, the slope decreases and the SampEn and mvSE values are nearly constant or decrease slightly. Therefore, we can divide each of the MSE and mvMSE curves into two segments: (I) the first part corresponds to the steep increasing slope (small scale factors, i.e., $1\le \lambda \le 4$), and (II) the second one contains the scale factors in which the slope of the SampEn and mvSE values is smoother (large scale factors, i.e., $5\le \lambda \le 10$). For MSE

_{σ2}and mvMSE

_{σ2}profiles, because the curves are always ascending and their slope values do not change noticeably, we consider one slope from the scale factor 2–10 (the entropy values for MSE

_{σ2}and mvMSE

_{σ2}methods are undefined at a scale factor of 1). Note that the slope values of both parts were calculated based on the least-square approach.

_{µ}and mvMSE

_{µ}, whereas the differences between these groups are significant (MSE

_{µ}) and very significant (mvMSE

_{µ}) when we consider the large scale factors. This demonstrates the importance of mvMSE method to characterise EEG signals in AD. Moreover, both the MSE

_{σ2}and mvMSE

_{σ2}methods lead to the significant differences for AD patients and controls.

_{µ}profiles with scale factors $5\le \lambda \le 10$. Table 3 also shows that the p-values for all channels at large scale factors leads to (very) significant differences for several channels. The average ± SD of slope values for MSE

_{σ2}curves are shown in Table 4. The p-values derived by electrodes O1, O2, F4, P3, and T5 for both the MSE

_{µ}with scale factors $5\le \lambda \le 10$ and MSE

_{σ2}profiles are (very) significant. The p-values for F4 and O1 are smaller than 0.05 for MSE

_{µ}with scale factors $5\le \lambda \le 10$, while electrode O2 leads to the significant difference using the MSE

_{σ2}method. This suggests that variance- and mean-based MSE offer complementary approaches to characterise AD.

## 4. Discussion and Conclusions

#### 4.1. Global Evaluation of Multivariate and Univariate Multiscale Entropies

_{µ}, mvMSE

_{µ}, MSE

_{σ2}, and mvMSE

_{σ2}to characterise the complexity of EEG signals in AD. This was done for conventional frequency bands θ, α, and β, and also for the broadband EEG signals after band-pass filtering between 1 and 40 Hz. The results obtained for frequency bands θ, α, and β were in contradiction with the widely reported higher complexity in control subjects than in AD patients, which could nevertheless be observed when estimating the complexity of the broadband EEGs. This suggests that the presence of broadband activity of EEGs may be needed for a comprehensive evaluation of complexity with multiscale entropy-based methods. Furthermore, we have related these findings with a very recent article providing guidelines on the interpretation of MSE results of brain signals [46] and showed that the profile of multivariate multiscale entropy of EEG signals at different frequency bands is not determined by the band-pass filtering process in comparison with the univariate multiscale entropy.

_{µ}and mvMSE

_{µ}curves, the slope of the curve increasing or decreasing at different bands can be predicted based on the sampling frequency and the effect of coarse-graining process on the frequency of signals. Since MSE

_{µ}and mvMSE

_{µ}at scale factor λ can be considered as a low-pass filter with cut-off frequency $\frac{{f}_{s}}{2\lambda}$ [42], scales 9 and 10, and 4–10 of the broadband analysis corresponds with α and β, respectively, with θ falling off the range.

_{µ}-based profiles (Figure 2a) were similar to MSE

_{µ}-based ones (Figure 1a), although the crossing point for mvMSE

_{µ}results was located at a smaller scale factor compared with that obtained by MSE

_{µ}. These results are in agreement with [11,32,35,38,43,44]. Unlike MSE

_{µ}and mvMSE

_{µ}, MSE

_{σ2}, and mvMSE

_{σ2}of the controls’ EEGs had more complexity values at all scale factors and smaller p-values. This suggests that both the multivariate and univariate multiscale methods based on the variance may characterise changes in EEGs in AD patients in a more detailed way than methods based on the mean.

#### 4.2. Regional Evaluation with Univariate Metrics

_{σ2}and MSE

_{µ}to characterise EEGs in each channel. The lowest p-values for MSE

_{σ2}and MSE

_{µ}were obtained for the channels O1, O2, P3, and P4 and O1, P3, F3, and F4, respectively. This shows that when MSE

_{μ}(or mvMSE

_{μ}) cannot distinguish different types of dynamics of a particular time series (channel), MSE

_{σ2}(or mvMSE

_{σ2}) may do so, and vice versa. It is worth noting that the electrodes with the lowest p-values are similar to most of our previous research using this database, such as [36,51].

#### 4.3. Features (Slopes) from Univariate and Multivariate Multiscale Profiles

_{µ}) and very significant (mvMSE

_{µ}) when the large scale factors were considered. This also illustrates the prominence of the mvMSE

_{µ}approach over MSE

_{µ}. In addition, significant differences between AD patients and controls were found with both the MSE

_{σ2}and mvMSE

_{σ2}. The p-values at electrodes O1, O2, F4, P3, and T5 for both the MSE

_{µ}and MSE

_{σ2}were significant or very significant.

#### 4.4. Limitations

## 5. Conclusions

_{µ}, mvMSE

_{µ}, MSE

_{σ2}, and mvMSE

_{σ2}to characterise the complexity of different frequency bands of EEG signals in AD was investigated. MSE

_{μ}and mvMSE

_{μ}, MSE

_{σ2}, and mvMSE

_{σ2}quantify the dynamical properties of average and spread, respectively, over multiple time scales. They extract different kinds of information from signals. The results indicated that when MSE

_{μ}or mvMSE

_{μ}cannot distinguish different types of dynamics of a particular time series, MSE

_{σ2}or mvMSE

_{σ2}may do so, and vice versa. The multivariate entropy methods may lead to more significant differences between groups by taking into account both the spatial and time domains. However, they cannot characterise the multivariate time series for single channels. Our results also evidenced that the presence of broadband activity in EEGs is required for a comprehensive evaluation of complexity with univariate and multivariate multiscale entropy approaches. From a clinical perspective, MSE

_{σ2}and mvMSE

_{σ2}results were associated with a loss of complexity in AD time series and showed that the variance-based MSE and mvMSE better discriminate the AD patients’ signals from the controls’ ones in comparison with mean-based multiscale methods. The p-values for the slope values of mvMSE curves were smaller than for MSE, showing the possible usefulness of multivariate approaches. Overall, our results support the relevance of multivariate and univariate multiscale complexity analyses for the characterisation of EEG signals in AD.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

_{1}+ m

_{2}+ ... + m

_{p}, where p denotes the number of channels of a time series. For example, for a trivariate time series with the embedding dimension

**m**= [2, 2, 2], the length of embedded vectors is 6. Then, the conditional probability that sequences with the embedding dimension

**m**= [2, 2, 2] close to each other for six data points will also be close to each other for seven data points, associated with the embedding dimensions [2, 2, 3], [2, 3, 2], or [3, 2, 2], is calculated. Note that the length of the newly embedded vectors is 7. Therefore, the proportion of unseen samples over the number of total samples in previous patterns for the embedding dimension

**m**= [2, 2, 2] is 16.66%. Likewise, for a four-channel time series with the embedding dimension

**m**= [2, 2, 2, 2], the proportion of unseen samples over the number of samples of previous patterns is 12.5%. Consequently, the proportion of new samples decreases proportionally to the number of channels, thus decreasing the likelihood of the longer new pattern not being a match with the shorter ones.

**Figure A1.**Multivariate Multiscale entropy values for the uncorrelated 1- to 16-channel WGN noise time series.

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**Figure 1.**Plots illustrating the mean ± SD (as error bars) of the MSE

_{µ}values computed from 11 AD, 11 control subjects’ signals, and 40 filtered univariate WGN time series at frequency bands (

**a**) 1–40 Hz; (

**b**) θ (4–8 Hz); (

**c**) α (8–13 Hz); and (

**d**) β (13–30 Hz). The scales with the p-values between 0.01 and 0.05, and less than 0.01, are shown with + and *, respectively.

**Figure 2.**Plots illustrating the mean ± SD (as error bars) of the mvMSE

_{µ}values computed from 11 AD, 11 control subjects’ signals, and 40 filtered uncorrelated sixteen-channel WGN time series at frequency bands (

**a**) 1–40 Hz; (

**b**) θ (4–8 Hz); (

**c**) α (8–13 Hz); and (

**d**) β (13–30 Hz). The scales with the p-values between 0.01 and 0.05, and less than 0.01, are shown with + and *, respectively.

**Figure 3.**Plots illustrating the mean ± SD (as error bars) of the MSE

_{σ2}values computed from 11 AD, 11 control subjects’ signals, and 40 filtered univariate WGN time series at frequency bands (

**a**) 1–40 Hz; (

**b**) θ (4–8 Hz); (

**c**) α (8–13 Hz); and (

**d**) β (13–30 Hz). The scales with the p-values between 0.01 and 0.05, and less than 0.01, are shown with + and *, respectively.

**Figure 4.**Plots illustrating the mean ± SD (as error bars) of the mvMSE

_{σ2}values computed from 11 AD, 11 control subjects’ signals, and 40 filtered uncorrelated sixteen-channel WGN time series at frequency bands (

**a**) 1–40 Hz; (

**b**) θ (4–8 Hz); (

**c**) α (8–13 Hz); and (

**d**) β (13–30 Hz). The scales with the p-values between 0.01 and 0.05, and less than 0.01, are shown with + and *, respectively.

**Figure 5.**Plots illustrating the mean ± SD (as error bars) of the MSE

_{µ}values computed from 11 AD and 11 control subjects for each channel of 1–40 Hz band-pass-filtered EEG signals. Sixteen electrodes of the international 10–20 system were analysed. (

**a**) C3; (

**b**) C4; (

**c**) F3; (

**d**) F4; (

**e**) F7; (

**f**) F8; (

**g**) Fp1; (

**h**) Fp2; (

**i**) O1; (

**j**) O2; (

**k**) P3; (

**l**) P4; (

**m**) T3; (

**n**) T4; (

**o**) T5; and (

**p**) T6. The scales with the p-values between 0.01 and 0.05, and less than 0.01 are shown with + and *, respectively.

**Figure 6.**Plots illustrating the mean ± SD (as error bars) of the MSE

_{σ2}values computed from 11 AD and 11 control subjects for each channel of 1–40 Hz bandpass filtered EEG signals. Sixteen electrodes of the international 10–20 system were analysed. (

**a**) C3; (

**b**) C4; (

**c**) F3; (

**d**) F4; (

**e**) F7; (

**f**) F8; (

**g**) Fp1; (

**h**) Fp2; (

**i**) O1; (

**j**) O2; (

**k**) P3; (

**l**) P4; (

**m**) T3; (

**n**) T4; (

**o**) T5; and (

**p**) T6. The scales with the p-values between 0.01 and 0.05, and less than 0.01 are shown with + and *, respectively.

**Table 1.**The computation time of the univariate and multivariate multiscale entropy based on the mean and variance.

MSE_{µ} | mvMSE_{µ} | MSE_{σ2} | mvMSE_{σ2} |
---|---|---|---|

4.77 s | 21.78 s | 2.4 s | 8.05 s |

**Table 2.**Average ± SD of slope values of the MSE and mvMSE profiles, and p-values and classification accuracies for AD patients versus controls over all channels and subjects. The scales with the p-values between 0.01 and 0.05, and less than 0.01 are shown with

^{+}and *, respectively.

Method | AD Patients | Controls | p-Value | Classification Ratio |
---|---|---|---|---|

MSE_{µ} $(1\le \lambda \le 4)$ | 0.4107 ± 0.0226 | 0.4185 ± 0.0238 | 0.3933 | 63.64% |

MSE_{µ} $(5\le \lambda \le 10)$ _{+} | 0.0022 ± 0.0195 | −0.0216 ± 0.0240 | 0.0215 | 72.73% |

MSE_{σ2} $(2\le \lambda \le 10)$ ^{+} | 0.1130 ± 0.0154 | 0.1301 ± 0.0137 | 0.0151 | 72.73% |

mvMSE_{µ} $(1\le \lambda \le 4)$ | 0.0074 ± 0.0088 | 0.0077 ± 0.0092 | 0.5114 | 31.82% |

mvMSE_{µ} $(5\le \lambda \le 10)$ * | −0.0048 ± 0.0037 | −0.0099 ± 0.0033 | 0.0071 | 72.73% |

mvMSE_{σ2} $(2\le \lambda \le 10)$ ^{+} | 0.0030 ± 0.0009 | 0.0041 ± 0.0012 | 0.0302 | 63.64% |

**Table 3.**Average ± SD of slope values of the MSE

_{µ}profiles and p-values for controls versus AD patients at scale factors $(5\le \lambda \le 10)$ for each channel. The scales with the p-values between 0.01 and 0.05, and less than 0.01 are shown with

^{+}and *, respectively.

Electrode | AD Patients | Controls | p-Value |
---|---|---|---|

C3 | −0.006 ± 0.0302 | −0.0360 ± 0.0356 | 0.0762 |

C4 | −0.018 ± 0.0264 | −0.0160 ± 0.0472 | 0.8955 |

F3 | −0.001 ± 0.0171 | −0.0209 ± 0.0238 | 0.0878 |

F4 * | 0.0076 ± 0.0244 | −0.0318 ± 0.0220 | 0.0031 |

F7 | 0.0018 ± 0.0219 | −0.0206 ± 0.0317 | 0.1150 |

F8 ^{+} | −0.007 ± 0.0285 | −0.0279 ± 0.0149 | 0.0418 |

Fp1 | −0.001 ± 0.0174 | −0.0136 ± 0.0443 | 0.1007 |

Fp2 | 0.0029 ± 0.0115 | −0.0099 ± 0.0378 | 0.0660 |

O1 * | 0.0162 ± 0.0285 | −0.0306 ± 0.0256 | 0.0031 |

O2 ^{+} | 0.0194 ± 0.0277 | −0.0136 ± 0.0415 | 0.0418 |

P3 ^{+} | 0.0276 ± 0.0238 | −0.0040 ± 0.0453 | 0.0488 |

P4 | 0.0177 ± 0.0303 | −0.0151 ± 0.0399 | 0.0660 |

T3 | −0.019 ± 0.0379 | −0.0267 ± 0.0390 | 0.8438 |

T4 | −0.029 ± 0.0496 | −0.0324 ± 0.0297 | 0.7427 |

T5 ^{+} | 0.0139 ± 0.0278 | −0.0246 ± 0.0312 | 0.0126 |

T6 | 0.0120 ± 0.0361 | −0.0213 ± 0.0494 | 0.0660 |

**Table 4.**Average ± SD of slope values of the MSE

_{σ2}profiles and p-values for controls versus AD patients at scale factors $(\lambda \le 10)$ for each channel. The scales with the p-values between 0.01 and 0.05, and less than 0.01 are shown with

^{+}and *, respectively.

Electrode | AD Patients | Controls | p-Value |
---|---|---|---|

C3 ^{+} | 0.1163 ± 0.0178 | 0.1296 ± 0.0119 | 0.0488 |

C4 | 0.1213 ± 0.0186 | 0.1289 ± 0.0135 | 0.2643 |

F3 | 0.1127 ± 0.0127 | 0.1257 ± 0.0173 | 0.0660 |

F4 ^{+} | 0.1139 ± 0.0164 | 0.1278 ± 0.0123 | 0.0418 |

F7 | 0.1161 ± 0.0133 | 0.1273 ± 0.0204 | 0.1891 |

F8 | 0.1165 ± 0.0183 | 0.1326 ± 0.0154 | 0.0569 |

Fp1 | 0.1079 ± 0.0212 | 0.1253 ± 0.0182 | 0.1486 |

Fp2 | 0.1085 ± 0.0148 | 0.1226 ± 0.0214 | 0.1310 |

O1 ^{+} | 0.1080 ± 0.0186 | 0.1342 ± 0.0208 | 0.0126 |

O2 * | 0.1078 ± 0.0195 | 0.1358 ± 0.0219 | 0.0071 |

P3 ^{+} | 0.1023 ± 0.0193 | 0.1231 ± 0.0183 | 0.0356 |

P4 ^{+} | 0.1038 ± 0.0180 | 0.1255 ± 0.0201 | 0.0215 |

T3 | 0.1267 ± 0.0218 | 0.1406 ± 0.0201 | 0.1486 |

T4 | 0.1301 ± 0.0314 | 0.1389 ± 0.0210 | 0.4307 |

T5 ^{+} | 0.1069 ± 0.0203 | 0.1307 ± 0.0215 | 0.0418 |

T6 ^{+} | 0.1091 ± 0.0216 | 0.1327 ± 0.0187 | 0.0256 |

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**MDPI and ACS Style**

Azami, H.; Abásolo, D.; Simons, S.; Escudero, J.
Univariate and Multivariate Generalized Multiscale Entropy to Characterise EEG Signals in Alzheimer’s Disease. *Entropy* **2017**, *19*, 31.
https://doi.org/10.3390/e19010031

**AMA Style**

Azami H, Abásolo D, Simons S, Escudero J.
Univariate and Multivariate Generalized Multiscale Entropy to Characterise EEG Signals in Alzheimer’s Disease. *Entropy*. 2017; 19(1):31.
https://doi.org/10.3390/e19010031

**Chicago/Turabian Style**

Azami, Hamed, Daniel Abásolo, Samantha Simons, and Javier Escudero.
2017. "Univariate and Multivariate Generalized Multiscale Entropy to Characterise EEG Signals in Alzheimer’s Disease" *Entropy* 19, no. 1: 31.
https://doi.org/10.3390/e19010031