## 2 Methodology  
### Refined Symbolic Diversity Entropy  
In this section, a new entropy method called refined symbolic diversity entropy (RSDE) is developed to evaluate the dynamic complexity of bearing vibration signals. Unlike existing entropy methods, RSDE combines a nonlinear mapping to symbolize the time series. Additionally, RSDE takes the state probability distribution of the patterns between the phase space trajectories of the resulting symbolic series as an evaluation indicator, which can well reflect the health status of the rolling bearings. The specific steps of the defined RSDE are as follows:  
1. The time series \(X=\{x_{1},\cdots,x_{i},\cdots,x_{N}\}\) mapped to \(Y=\{y_{1},\cdots,y_{i},\cdots,y_{N}\}\) by using a normal cumulative distribution function (NCDF). This step deals with the shortcoming of the amplitude values of the series may be dominated by the components of vectors coming from the largest amplitudes [22]. The NCDF is defined as follows: \[y_{i}=\frac{1}{\sigma\sqrt{2\pi}}\int\limits_{-\infty}^{x_{i}}e^{-\frac{t-\mu} {2\mu^{2}}}dt\] (1) where \(\mu\) is the expected value and \(\sigma\) is the standard deviation. 2. \(Y\) is mapped to \(Z=\{z_{1},\cdots,z_{i},\cdots,z_{N}\}\) (from 1 to c) by a linear transformation: \[z_{i}=\text{round}(c\cdot y_{i}+0.5)\] (2) where \(c\) is an integer which refers to the number of symbols, and round (\(\cdot\)) represents the rounding function. This procedure completes the symbolization of the time series. To clearly illustrate the improvement in noise immunity due to symbolization, a schematic diagram is shown in Figure 1. In Figure 1, \(Y_{1}=\{y_{11},\,y_{12},\,y_{13}\}\) denotes the initial sequence. The values of the elements in this sequence represent the true amplitude. \(Y_{2}=\{y_{21},\,y_{22},\,y_{23}\}\) and \(Y_{3}=\{y_{31},\,y_{32},\,y_{33}\}\) indicate the sequence after adding noise interference. Due to the interference of noise, the amplitude of the sequence fluctuates around the true value. By symbolization, if \(c\) is taken as 3, both sequences become \(\{z_{2},\,z_{2},\,z_{2}\}\). Therefore, this step can effectively reduce the influence of noise interference in the process of fault feature extraction. At the same time, the symbolization of time series has merits in information reservation [23]. In addition, this step can transfer the signal into symbol series by using a few number symbols so that the subsequent calculation can be simplified and the commutating efficiency can be greatly enhanced. Based on the phase space embedding theory, the embedded dimension \(m\) needs to be pre-given. According to the symbol sequence \(Z\), a vector space with \(m\) dimensions is constructed as follows: \[z(i)=[z_{i},z_{i+1},\cdots,z_{i+m-1}]\] (3) where \(i=1,2,3,\cdots,N-m\). 4. Calculate the cosine similarity between adjacent symbolic orbits, and the set of cosine similarities \(\{d_{1},d_{2},\cdots,d_{N-m}\}\) is obtained. The cosine similarity between two orbits is defined as follows: \[ \begin{array}{l}D(m)=\{d_{1},d_{2},\cdots,d_{N-m}\}\\ =\{d[z_{1}(m),z_{2}(m)],d[z_{2}(m),z_{3}(m)],\cdots,d[z_{N-m}(m),z_{N-m+1}(m)]\} \end{array} \] (4) \[ \begin{array}{l}d\big{[}z_{i}(m),z_{j}(m)\big{]}=\frac{\sum\limits_{k=1}^{ m}z_{i}(k)\times z_{j}(k)}{\sqrt{\sum\limits_{k=1}^{m}z_{i}(k)^{2}\times \sqrt{\sum\limits_{k=1}^{m}z_{j}(k)^{2}}}}\end{array} \] (5) where the value domain of the cosine similarity \(d\) is \([-1,1]\). The geometric meaning of cosine similarity is the cosine of the angle between the orbits in the phase space. When \(d\) tends to \(1\), it indicates a similar, predictable, or periodic dynamical behavior between two orbits. On the contrary, when \(d\) tends to \(-1\), it indicates that the dynamical behavior of the two orbits is random and divergent. In addition, calculating only the cosine similarity of adjacent orbits can obtain high computational efficiency. 5. The value domain \([-1,1]\) of the cosine similarity \(d\) is divided into \(\varepsilon\) intervals, denoted as \([I_{1},I_{2},\cdots,I_{\varepsilon}]\).