Algorithm A1 Multivariate Multiscale Sample Entropy

The steps of standard Multivariate Multiscale Sample Entropy are given below, for a multi-variate data set ${\{x_{c,i}\}}_{i=1}^N,1\le c\le P$ of length N and number of channel P. The manually selected parameters are the embedding dimension $M=[m_1,m_2,\dots ,m_P]$, tolerance r, time delay $L=[l_1,l_2,\dots ,l_P]$, and scale factor $\tau$: Normalize the original multi-variate data sets by subtracting the mean and dividing by the standard deviation. Perform Coarse Graining Process to obtain the scaled multi-channel time series ${\{y_{c,i}^{(\tau )}\}}_{j=1}^{N/\tau }$, according to $y_{c,i}^{(\tau )}(j)=\frac{1}{\tau }{\sum }_{i=j-\tau /2-1}^{j+\tau /2-1}{x}_c(i),1\le j\le \frac{N}{\tau },c=1,2,\dots ,P$. Form the Composite Delay Vectors (CDV) ${\mathbf{Y}}_M(i)$ according to M and L in the form $$\begin{array}{cc}\hfill {\mathbf{Y}}_M(i)=& [y_{1,i},y_{1,i+l_1},\dots ,y_{1,i+(m_1-1)l_1)},\hfill \\ & y_{2,i},y_{2,i+l_2},\dots ,y_{2,i+(m_2-1)l_2)},\hfill \\ & ⋮\hfill \\ & y_{c,i},y_{c,i+l_c},\dots ,y_{c,i+(m_c-1)l_c)},]\hfill \end{array}$$ Compute the similarity for all pairwise CDVs, $Y_M(i)\&Y_M(j)$, based on the Chebyshev distance as $D(i,j)=max\{Y_M(i+k)-Y_M(j+k)||0\le k\le (({\sum }_{c=1}^P{m}_c)-1),i\ne j\}$. Calculate the number of matching patterns, defined as similar pairs $B(i)$ that satisfy the criterion $D(i,j)\le r$. Compute the local probability, $C(i)$, and global probability, $\mathsf{\Phi }(i)$, of $B(i)$ as $C(i)=\frac{B(i)}{N-n-1},\mathsf{\Phi }=\frac{{\sum }_{i=1}^{N-n}C(i)}{N-n},n=max(M)\ast max(L)$. Repeat Steps 3–6 with an increased embedding dimension, $m_c$+1, and obtain the updated global probability as ${\mathsf{\Phi }}^{\ast }=\frac{{\sum }_{i=1}^{N-n}{C}^{\ast }(i)}{N-n},n=max(M^{\ast })\ast max(L)$. Recall that there are P ways to increase the embedding dimension and the modified global probability, ${\mathsf{\Phi }}^{\ast }$, is the averaged result. Multivariate Multiscale Sample Entropy is defined as $MMSE=-ln[\frac{{\mathsf{\Phi }}^{\ast }}{\mathsf{\Phi }}]$.