#### 2.1. Multivariate Empirical Mode Decomposition

Since EMD was proposed by Huang

et al. [

16], it has been widely used for non-linear, non-stationary data analysis based on the intrinsic characteristics of time series. The intrinsic mode functions (IMFs) are decomposed by the mean m(t) of upper and down envelops of a time series X(t), expressed as follows:

where

c_{i} and

r_{n} are the

ith IMF and the last residue of time series. In order to expand the applications of EMD, MEMD, a multivariate extension of EMD, was presented to decompose multivariate nonlinear and nonstationary signals [

20,

21]. MEMD not only overcomes the single input limitation of EMD, but also solves the problem of mode mixing through addition of white noise to different channels. Furthermore, it is similar to EMD that acts as a dyadic filter bank on each channel of the multivariate input, and has the advantage of aligning the corresponding IMFs from different channels across the same frequency range [

20]. Therefore, the more frequencies that exist in different channels, the more IMFs that are decomposed in each channel.

In computation of MEMD, the mean m(t) is calculated by means of the multivariate envelope curves of a set of K direction vectors, shown as equation (2):

where

${\left\{{e}^{{\theta}_{k}}\left(t\right)\right\}}_{k=1}^{K}$ are the multivariate envelope curves of all vectors that are the projections of multichannel input s(t) along K directions. Then, the multivariate IMF is calculated by s(t)-m(t) and the stoppage criterion. This process is repeated until the stoppage criteria in standard EMD is fulfilled by all the projected signals.

#### 2.2. Multivariate Multiscale Entropy

In the entropy family, MSE is an effective method to evaluate the complexity of signals over different time scales, and has been applied effectively in the analysis of physiology, biology, and geosciences data [

22,

23,

24]. There are two procedures in the MSE, the first one is coarse-grained procedure that computes the

${y}_{j}^{\left(\tau \right)}$ of original time series

$\mathrm{}\{{\mathrm{x}}_{1},{\mathrm{x}}_{2},\cdots ,{\mathrm{x}}_{\mathrm{N}}\}$ based on the scale factor

$\mathsf{\tau}$, which is according to equation (3):

Then the sample entropy (SampEn) [

25] is employed to measure the degree of randomness of the finite length time series

${\mathrm{y}}_{j}^{\left(\mathsf{\tau}\right)}$, in which

$1\le j\le N/\tau $. SampEn is considered as a conditional probability that is computed according to the following steps: firstly, two sequences of

m consecutive data points such as

${Y}_{m}\left(i\right)=\left\{{y}_{i},\dots ,{y}_{i+m-1}\right\}$ and

${Y}_{m}\left(j\right)=\left\{{y}_{j},\dots ,{y}_{j+m-1}\right\}$ are selected to compute the number of

${Y}_{m}\left(j\right)$ that meets the condition

$d\left[{Y}_{m}\left(i\right),{Y}_{m}\left(j\right)\right]\le \mathsf{\gamma}*S.D.$ (

$j\in \left[1,N\u2019-m\right],\mathrm{}i\ne jand\gamma \in \left(0,1\right)$), in which the

S.D. is the standard deviation of the time series

$\mathrm{}\{{\mathrm{y}}_{1},{\mathrm{y}}_{2},\cdots ,{y}_{\mathrm{N\u2019}}\}$ and

$d\left[{Y}_{m}\left(i\right),{Y}_{m}\left(j\right)\right]$ is the maximum distance between

${Y}_{m}\left(i\right)$ and

${Y}_{m}\left(j\right)$:

Then

${B}_{i}^{m}\left(\gamma \right)$ is the amount of all

${Y}_{m}\left(i\right)$ similar to

${Y}_{m}\left(j\right)$, and its average for

$i\in \left[1,\text{N\u2019}-m\right]$ is:

Thus, the

${B}^{m}\left(\gamma \right)$ is the average similarity of

m consecutive data points over whole

$N\prime -m$ data, and

${B}^{m+1}\left(\gamma \right)$ is that of m+1 consecutive data points. Finally, the SampEn is obtained using equation (6) :

Like MSE, the MMSE is used to calculate the relative complexity of normalized multichannel signals by plotting multivariate SampEn as a set of the scale factor. Its first step is to define temporal scales of the increased length by coarse-graining the p-variate time series

$\mathrm{}{\left\{{\mathrm{x}}_{\mathrm{k},\mathrm{i}}\right\}}_{\mathrm{i}=1}^{\mathrm{N}},\mathrm{k}=1,2,\dots ,\mathrm{p}$. For a scale factor

$\mathsf{\tau}$, the multivariate coarse-grained time series

${y}_{k,j}^{\left(\tau \right)}$ are calculated as equation (7), where

$\mathrm{}1\le j\le N/\tau $:

But the second step is different from that in MSE, the multivariate SampEn (MSampEn) is calculated for each coarse-grained multivariate

$\mathrm{}{y}_{k,j}^{\left(\tau \right)}$, and it is plotted as a function of the scale factor

$\mathsf{\tau}$. In order to obtain the MSampEn, the multivariate emdedded vectors

${Y}_{m}\left(i\right)\in {\mathbb{R}}^{p}$ must be constructed firstly, which is shown as:

where

$1\le i\le {\mathrm{N}}^{\prime}-n\mathrm{and}{\mathrm{N}}^{\prime}=N/\tau ,\mathrm{}n=max\left\{M\right\}\times max\left\{\epsilon \right\}.$ $M=\left[{m}_{1},{m}_{2},\dots ,{m}_{p}\right]\in {\mathbb{R}}^{p}$ is the embedding vector, while

$\epsilon =\left[{\epsilon}_{1},{\epsilon}_{1},\dots ,{\epsilon}_{p}\right]$ is the time lag vector and

$m={\displaystyle \sum}_{k=1}^{p}{m}_{k}$. Then the maximum norm is defined by Chebysev distance between any two composite delay vectors

${Y}_{m}\left(i\right)$ and

${Y}_{m}\left(j\right)$, that is expressed as:

where

$j\in \left[1,{\mathrm{N}}^{\prime}-n\mathrm{}\right],j\ne i$. For a given

${Y}_{m}\left(i\right)$,

P_{i} is the number of vector pairs that meets

$d\left[{Y}_{m}\left(i\right),{Y}_{m}\left(j\right)\right]\le \gamma \mathrm{*}S.D.$ (

$j\in \left[1,N\u2019-n\right],\mathrm{}i\ne jand\gamma \in \left(0,1\right)$) where

S.D. is the standard deviation of the multivariate emdedded vectors

${\mathrm{Y}}_{\mathrm{m}}\left(\mathrm{i}\right)$, so that

${A}_{i}^{m}\left(\gamma \right)={P}_{i}/\left({\mathrm{N}}^{\prime}-n-1\right)$, where

$n=max\left\{M\right\}\times max\left\{\tau \right\}$. For all

i,

${A}^{m}\left(\gamma \right)={\left({\mathrm{N}}^{\prime}-n\right)}^{-1}{\displaystyle \sum}_{i=1}^{{\mathrm{N}}^{\prime}-n}{A}_{i}^{m}\left(\gamma \right)$.

Finally, the average similarity

${A}^{m}\left(\gamma \right)$ over all

$i\in \left[1,{\mathrm{N}}^{\prime}-n\mathrm{}\right]$ and the

${A}^{m+1}\left(\gamma \right)$ over all

$i\in \left[1,\text{p*}\left({\mathrm{N}}^{\prime}-n\right)\mathrm{}\right]$ are used to gain the MSampEn, as shown in equation (10):

where

$\gamma $ is the tolerance level and

${\mathrm{N}}^{\prime}$ is the length of the time series

${y}_{k,j}^{\left(\tau \right)}$.

In MMSE, the multivariate time series are considered more complex than another if the MSampEn values are higher than other for majority of the scales, which is the same as the original MSE. The embedding vector $M=\left[{m}_{1},{m}_{2},\dots ,{m}_{p}\right]\in {\mathbb{R}}^{p}$ and the tolerance level $\gamma $ in MMSE have the equivalent function with parameters m and $\gamma $ in MSE. In the following anaysis, m is 2 and $\gamma $ is 0.2 in calculation of SampEn, and $M=\left[{m}_{1},\dots ,{m}_{i},\dots ,{m}_{p}\right],\epsilon =\left[{\epsilon}_{1},\dots ,{\epsilon}_{i},\dots ,{\epsilon}_{p}\right]({m}_{i}=2,{\epsilon}_{i}=1,i\in \left[1,p\right])$, and $\gamma $=0.2 are set in MMSE so as to compare the performace between MSE and MMSE with the same parameters.