## 2 Materials and Methods  
### Refined Composite Multiscale Cross-Sample Entropy (RCMCSE) Approach  
Cross-sample entropy (Cross-SampEn) is typically applied to evaluate the degree of dissimilarity between two time series. However, Cross-SampEn is not applicable to all problems for a more accurate value. Hence, RCMCSE was proposed to solve some problems such as the statistical reliability of Cross-SampEn was reduced as time scale increased [12]. RCMCSE, which is based on Cross-SampEn and MSE, is applicable to more problems compared with Cross-SampEn because of modifications to its calculation procedures. The RCMCSE and Cross-SampEn procedures are summarized as follows.  
First, regarding Cross-SampEn, consider two time series with N points: \(\{X_{i}\}=\{x_{1},\,\ldots,\,x_{i},\,\ldots,\,x_{N}\}\) and \(\{Y_{j}\}=\{y_{1},\,\ldots,\,y_{j},\,\ldots,\,y_{N}\}\). These time series can be divided into templates of length m.  
\[u_{m}(i)=\{x_{i},\,x_{i+1},\,\ldots,\,x_{i+m-1}\},\,1\,\leq\,i\,\leq\,N-m\,+\,1 \tag{1}\]  
\[\
u_{m}(i)=\{y_{j^{\prime}},\,y_{j+1},\,\ldots,\,y_{j+m-1}\},\,1\,\leq\,j\, \leq\,N-m\,+\,1 \tag{2}\]  
The number of similarities is \(n_{i}^{m}(r)\), and it can be defined as the maximum distance between two templates \(i\) and \(j\). The formula of \(n_{i}^{m}(r)\) is expressed as follows:  
\[n_{i}^{m}(r)=\sum_{j\,=\,1}^{N-m}d[u_{m}(i),\,v_{m}(j)] \tag{3}\]  
\[d[u_{m}(i),\,v_{m}(j)]\,=\,\max\{|\!|x(i+k)-y(j+k)|\!|;\,0\,\leq\,k\,\leq\,m\, -\,1\} \tag{4}\]  
\[d[u_{m}(i),\,v_{m}(j)]\,\leq\,r,\,1\,\leq\,j\,\leq\,N-m \tag{5}\]  
The probability of similarity between the templates is calculated through the following equations  
\[U_{i}^{m}(r)(v\|u)=\frac{n^{m}(r)}{(N\,-\,m)} \tag{6}\]  
The average probability of similarity of the templates with the length m is calculated as follows:  
\[U^{m}(r)(v\|u)=\frac{1}{(N\,-\,m)}\sum_{i\,=\,1}^{N-m}U_{i}^{m}(r)(v\|u) \tag{7}\]  
Next, new templates are created with different template lengths \(m\)+1. The average probability of similarity \(U^{m+1}(r)(v\|u)\) is the procedure of averaging the probability of similarity. Cross-SampEn is defined as follows:  
\[\text{CS}_{E}(m,\,r,\,N)=\,-\ln\!\left\{\frac{U^{m+1}(r)(v\|u)}{U^{m}(r)(v\|u )}\right\} \tag{8}\]  
Second, RCMCSE comprises more procedures than Cross-SampEn. It includes a coarse-graining procedure that is applied to analyze two time series at a given scale factor. Each point in the coarse-grained time series is defined as follows:  
\[x_{j}^{(\tau)}=\frac{1}{\tau}\sum_{i\,=\,(j\,-1)+1}^{j\tau}x_{i},\,1\,\leq\, j\,\leq\,\frac{N}{\tau} \tag{9}\]  
\[y_{j}^{(\tau)}=\frac{1}{\tau}\sum_{i\,=\,(j\,-1)+1}^{j\tau}y_{i},\,1\,\leq\, j\,\leq\,\frac{N}{\tau} \tag{10}\]For MSCE, the length of the time series decreases when the coarse-graining procedure is applied to the original time series. Therefore, using MSCE to analyze a short time series may lead to inaccurate Cross-SampEn results at large time scales. To resolve this problem, CMSE was previously proposed [5]. On the basis of CMSE, CMCSE can be defined as follows:  
\[CMCSE(x,y,\tau,m,r)=\frac{1}{\tau}\sum_{k=1}^{\tau}SampEn\bigg{(}x_{k}^{(\tau)},y_{k}^{(\tau)},m,r\bigg{)} \tag{11}\]  
The CMCSE value is the mean of \(\tau\) Cross-SampEn values. However, the CMCSE value is undefined if one of the values of \(n_{k,\tau}^{m}\) or \(n_{k,\tau}^{m+1}\) is zero. Therefore, the RCMCSE method was proposed [6] to address the problem of undefined entropy values. In contrast to CMCSE, RCMCSE involves the assumption that \(n_{k,\tau}^{m}\) and \(n_{k,\tau}^{m+1}\) represent the means of \(n_{k,\tau}^{m}\) and \(n_{k,\tau}^{m+1}\), respectively. Instead of averaging entropy values, RCMCSE averages the sum of the number of similarities.  
\[\overline{n}_{k,\tau}^{m}=\frac{1}{\tau}\sum_{k=1}^{\tau}n_{k,\tau}^{m} \tag{12}\]