Algorithm 1 Variational Embedding Multiscale Sample Entropy

Assume that there are P channels measured from a system, where a signal, recorded from the cth channel, is denoted by $x_c(i)$ and is of length N, where $1\le c\le P,1\le i\le N$. Parameters involved in the veMSE algorithm are the tolerance quotient $(r)$, embedding dimension $(m)$, scale factor $(\tau )$, and time lag $(L)$. The detailed steps of veMSE are shown below. Coarse graining is firstly applied to the original datasets for all the channels, whereby the scaled time series are calculated as $y^{(\tau )}(j)=\frac{1}{\tau }{\sum }_{i=(j-1)\tau +1}^{j\tau }x(i)$, $1\le j\le N_t,N_t=\frac{N}{\tau }$. For each channel, the embedding dimension, m is set as a variable. The dimension for the cth channel is calculated as $m(c)=m+c-1$, as listed in Table 1. Therefore, combined with the time delay, L, the embedding delay vector of data $y^{(\tau )}(i)(1\le i\le N_t)$ for channel c is designated as a template, $Y_c^{(\tau )}(i)$, and calculated as $Y_c^{(\tau )}(i)=[y^{(\tau )}(i),y^{(\tau )}(i+L),\dots ,y^{(\tau )}(i+$$n_c$$)],n_c=(m(c)-1)L$. Compute the Chebyshev distance between templates, $Y_c^{(\tau )}$, where the distance is denoted according to the amplitude of the embedding vector, as $d=max\{Y_c^{(\tau )}(i+k))-Y_c^{(\tau )}(j+k)\}$, where $0\le k\le m(c)-1,1\le i,j\le N_t-n_c,i\ne j$. For each channel, the number of segments, $Y_c^{(\tau )}(j)$, within the tolerance level, r, of $Y_c^{(\tau )}(i)$, is recorded as $B_c(i)$. In other words, $B_c(i)$ is the number of templates matched or the number of similar patterns in the dataset, based on the boundary set, where the boundary is defined by the tolerance, r. Therefore, the local probability of the occurrence of template match for channel c is $R_c(i)=\frac{B_c(i)}{(N_t-n_c-1)}$. The global probability of the occurrence of template match for a channel c is calculated as $\mathsf{\Phi }(c)=\frac{1}{(N_t-n_c)}{\sum }_{i=1}^{N_t-n_c}{R}_c(i)$. For all data channels, compute the sum of the global probability as $\mathsf{\Phi }={\sum }_{c=1}^P\mathsf{\Phi }(c)$. Recall that, unlike the MMSE algorithm (see Appendix A), the embedding dimension, $m(c)$, is not a fixed parameter and is varied, along with the index of channel, c. Modify the embedding dimension to $(m(c)+1)$. Hence, $n_c$ is adjusted to $m(c)\times L$, and Steps 2–6 are repeated to obtain the global probability for the so-increased embedding dimension $\mathsf{\Phi }(m+1)$. Variational embedding multiscale sample entropy is finally obtained, as $veMSE=-ln\frac{\mathsf{\Phi }(m+1)}{\mathsf{\Phi }(m)}$.