## 2 Methodology  
### Multiscale Dispersion Entropy Based on RMS (MDE_rms_)  
MDE_rms_ is an interval method to measure the complexity and regularity of time series. Based on the original definition of dispersion entropy, the paper further develops the MDE_rms_ algorithm, which consists of five steps as below:  
(1) Coarse-graining  
A coarse-graining procedure is applied to the time series \(U=\{u_{1},u_{2},\cdots u_{L}\}\) with the length of \(L\), in order to reduce high frequency parameters [15]. By introducing the scale factor \(\tau\) ranged from to \(\tau_{\max}\) in the coarse-graining processing, we obtain the representation of time series at different time scales and generate the corresponding coarse-grained sequence \(X=\left\{{{x_{1}}^{\left({\tau}\right)}},{{x_{2}}^{\left({\tau}\right)}},\cdots {{x_{N}}^{\left({\tau}\right)}}\right\}\) where \(N\) is the length. For each time scale \(\tau\), the element of the coarse-grained time sequence is calculated as follows:  
\[{{x_{j}}^{\left({\tau}\right)}}=\frac{1}{\tau}\sum\limits_{b=\left(j-1\right) \tau+1}^{j\tau}{{u_{b}}}\ \ \ \ \ \left(1\leq j\leq\left\lfloor\frac{L}{\tau}\right\rfloor=N\right) \tag{1}\]  
(2) Mapping  
\({{x_{j}}^{\left({\tau}\right)}}\) is mapped to \(c\) classes and the corresponding labels range from 1 to \(c\). To realize the purpose, the paper applies the normal cumulative distribution function (NCDF) to the coarse-grained series and maps \({{x_{j}}^{\left({\tau}\right)}}\) into \({{y_{j}}}\) from 0 to 1 as follows:  
\[{{y_{j}}}=\frac{1}{{\sigma\sqrt{2\pi}}}\int\limits^{{{x_{j}}}}{{e^{-\left({x -rms}\right)^{2}}}dt} \tag{2}\]  
For all the scale factors, choose the \(rms\) and \(\sigma\) of the original series as the coefficient of the NCDF, where the \(rms\) is the root mean square (RMS) and \(\sigma\) is the standard deviation (SD). Compared with the average of NCDF, the NCDF based \(rms\) is better suitable for the distribution character of vibration signal and can overcome the extremes of mapping the \({{x_{j}}^{\left({\tau}\right)}}\) to only a few classes. Then a linear algorithm is employed to portion \({{y_{j}}}\) to an integer which ranges from 1 to \(c\):  
\[{{z_{j}}^{c}}=round(c\cdot{{y_{j}}}+0.5) \tag{3}\]  
For each element of the mapped signal, \({{y_{j}}}\) is linearly assigned to the \({{z_{j}}^{c}}\) class, where rounding involves either increasing or decreasing a number to the next integer [16]. (3) Introduce the embedding dimension \(m\) and time delay \(d\), and construct the time series \({{z_{i}}^{m,c}}\) as follows [17; 18]:  
\[{{z_{i}}^{m,c}}=\left\{{{z_{i}}^{c}},{{z_{i+d}}^{c}}\cdots{{z_{i+\left({m-1} \right)}}^{c}}\right\} \tag{4}\]  
(4) Assign each embedding vector \({{z_{i}}^{m,c}}\) to a dispersion pattern \({{\pi_{{{\tau_{0}}}}{{v_{1}}}\cdots{{v_{m-1}}}}}\) where \({{z_{i}^{c}}}={{v_{0}}},{{z_{i+d}^{c}}}={{v_{1}}}\cdots{{z_{i+\left({m-1} \right)d}^{c}}}={{v_{m-1}}}\)[19]. As the signal with \(m\) member is divided into \(c\) classes, the number of dispersion patterns corresponds to \(c^{m}\). To improve the calculation reliability of MDE\(rms\), the number of dispersion patterns is suggested as less than the original signal length (\(c^{m}<L\)). Furthermore, the introduction of the coarse-graining procedure will reduce the signal length, hence \(c^{m}\) should be less than \(\left\lfloor\frac{L}{{{\tau_{\max}}}}\right\rfloor\) (\(c^{m}<\left\lfloor\frac{L}{{{\tau_{\max}}}}\right\rfloor\)).