# Application Research of a New Adaptive Noise Reduction Method in Fault Diagnosis
## 2 Basic Theory
### Multi-Scale Sample Entropy (MSE)  
Fault signals detected in rotating machinery are usually embedded in multi-scale structures. However, the sample entropy (SE) [33; 34] only analyzes the single scale signal and ignores much useful information. This limits its performance in extracting embedded fault features [35]. In order to solve the shortcomings of sample entropy, [PERSON] [36] proposed multi-scale sample entropy as an effective complexity measure of time series on different time scales. For a time series {\(x_{1}\), \(x_{2}\),..., \(x_{N-1}\), \(x_{N}\)}, MSE constructs multiple coarse-graining time series. The process of coarse-graining scale factor \(Q\) is obtained by averaging the internal structure of the time series, but without overlapping the length of the scale factor. Figure 2 shows the coarse-graining examples of processes. Therefore, the coarse-graining time series can be obtained from Equation (4):  
\[y_{j}^{Q}=\frac{1}{Q}\sum_{i=(j-1)Q+1}^{jQ}x_{i},\qquad 1\leq j\leq\frac{N}{Q} \tag{4}\]  
where \(Q\) is the scale factor and is a positive integer. When \(Q=1\), the coarse-graining time series \(y_{j}^{Q}\) is the original time series. \(j\) is the length of the coarse-graining time series.  
The next step is to calculate the multi-scale sample entropy and scale factor \(Q\) for each newly reconstructed coarse-graining time-series signal \(y^{Q}\). The multi-scale sample entropy is calculated as follows:  
1. From \(N/Q\)-\(m+1\) to vector \(y_{m}^{(Q)}(i)\) \[y_{m}^{(Q)}(i)=\Big{[}y^{Q}(i),y^{Q}(i+1),\cdots,y^{Q}(i+m-1)\Big{]},\quad i  
Calculate the maximum norm between two vectors \(d_{m}[y_{m}^{(Q)}(i),y_{m}^{(Q)}(j)]\).  
\[d_{m}\Big{[}y_{m}^{Q}(i),y_{m}^{Q}(j)\Big{]}=\max\biggl{[}\big{|}y_{m}^{Q}(i+k)- y_{m}^{Q}(j+k)\Big{|}\biggr{]},\quad 0\leq k\leq m-1 \tag{6}\]  
Define function B  
\[B_{i}^{m}(r)=\frac{1}{N/Q-m+1}u^{m}(i),\quad i=1,\cdots,N/Q-m+1 \tag{7}\]  
where \(r\) is the tolerance of the acceptance matrix, \(u^{m}(i)\) is equal to \(d_{m}[y_{m}^{(Q)}(i),y_{m}^{(Q)}(j)]\), and \(d_{m}[y_{m}^{(Q)}(i),y_{m}^{(Q)}(j)]\leq r,i\
eq j\).  
Define function A  
\[A_{i}^{m}(r)=\frac{1}{N/Q-m+1}v^{m+1}(i),\quad i=1,\cdots,N/Q-m+1 \tag{8}\]  
where \(v^{m+1}\) is equal to \(d_{m+1}[y_{m+1}^{(Q)}(i),y_{m+1}^{(Q)}(j)]\), and \(d_{m+1}[y_{m+1}^{(Q)}(i),y_{m+1}^{(Q)}(j)]\leq r,i\
eq j\).  
Then calculate the probability of matching points  
\[B^{m}(r)=\frac{1}{N/Q-m+1}\sum_{i=1}^{N/Q-m+1}B_{i}^{m}(r) \tag{9}\]  
\[A^{m}(r)=\frac{1}{N/Q-m}\sum_{i=1}^{N/Q-m}A_{i}^{m}(r) \tag{10}\]  
where \(B^{m}(r)\) and \(A^{m}(r)\) represent the probability of two sequences matching \(m\) and \(m+1\) points, respectively.  
The theoretical value of multiscale sample entropy is defined as  
\[SampEnt(m,r)=\lim_{N/Q\rightarrow\infty}\biggl{\{}-\ln\frac{A^{m}(r)}{B^{m}( r)}\biggr{\}}. \tag{11}\]  
When the data point N/Q is a finite number, the estimate of sample entropy is given by the following formula:  
\[SampEnt(m,r,N)=\biggl{[}-\ln\frac{A^{m}(r)}{B^{m}(r)}\biggr{]}. \tag{12}\]  
It can be seen from Equation (12) that the value of \(SampEnt\) is related to the value of \(m\),\(r\). At present, there is no definite value, and the empirical value is \(m=2\), \(r\in(0.1 SD\)\(\sim\)\(0.5 SD)\). \(SD\) is the standard deviation of the original data. In this paper, when studying the complexity of time series, \(m=1\), \(r=0.25 SD\) is taken.