## 2 Background of MP-SAX  
For the whole life cycle of equipment, the performance degradation of equipment often leads to the change of trend component of monitoring signal, while the change of operating condition or environment often causes the change of detail component of monitoring signal. The similarity measurement method based on the morphological pattern and symbolic aggregate approximation (MP-SAX), which measured the similarity of time series by measuring the similarity of trend component and detail component respectively, can effectively measure the similarity of the time series with the changes both in trend and detail [26]. Therefore, the MP-SAX can be used to measure the similarity of monitoring signal during equipment degradation under multiple operating conditions. For two time series \(X=(x_{1},x_{2},\cdots,x_{n})\) and \(Y=(y_{1},y_{2},\cdots,y_{m})\), the MP-SAX of the two-time series can be obtained as follows:  
Step 1: According to empirical mode decomposition (EMD) [27; 28], the time series \(X=(x_{1},x_{2},\cdots,x_{n})\) and \(Y=(y_{1},y_{2},\cdots,y_{m})\) are decomposed to IMFs \(\left\{IMF_{i}^{X}\right\}_{i=1}^{N_{1}}\) and \(\left\{IMF_{j}^{Y}\right\}_{j=1}^{N_{2}}\). Step 2: According to multi-scale permutation entropy (MPE) [29], the MPE\(\left\{MPE_{i}^{X}\right\}_{i=1}^{N_{1}}\) and \(\left\{MPE_{j}^{Y}\right\}_{j=1}^{N_{2}}\) of each IMFs can be calculated. Then, the IMFs \(\left\{IMF_{i}^{X}\right\}_{i=1}^{N_{1}}\) and \(\left\{IMF_{j}^{Y}\right\}_{j=1}^{N_{2}}\) are reconstructed into trend component (\(TR^{X}\) and \(TR^{Y}\)) and detail component (\(DE^{X}\) and \(DE^{Y}\)) according to the MPEs as follows:  
\[TR^{X}=\sum_{i\in\left\{i|MPE_{i}^{X}<0.4\right\}}IMF_{i}^{X} \tag{1}\]  
\[TR^{Y}=\sum_{j\in\left\{j|MPE_{j}^{Y}<0.4\right\}}IMF_{j}^{Y} \tag{2}\]  
\[DE^{X}=\sum_{i\in\left\{i|MPE_{i}^{X}\geq 0.4\right\}}IMF_{i}^{X} \tag{3}\]  
\[DE^{Y}=\sum_{j\in\left\{j|MPE_{j}^{Y}\geq 0.4\right\}}IMF_{j}^{Y} \tag{4}\]  
where \(TR^{X}=(tx_{1},tx_{2},\cdots,tx_{n})\), \(TR^{Y}=(ty_{1},ty_{2},\cdots,ty_{m})\), \(DE^{X}=(dx_{1},dx_{2},\cdots,dx_{n})\), \(DE^{Y}=(dy_{1},dy_{2},\cdots,dy_{m})\). Step 3: The trend component (\(TR^{X}\) and \(TR^{Y}\)) are converted into morphological pattern (MP) symbol sequences \(MC^{X}\) and \(MC^{Y}\) as follows (take \(MC^{X}\) as an example):  
\[MC^{X}_{i}=\left\{ \begin{array}{rl}&3,\ (tx_{i}-tx_{i-1})/t>1\\ &2,\ (tx_{i}-tx_{i-1})/t=1\\ &1,\ (tx_{i}-tx_{i-1})/t<1\\ &0,\ \ tx_{i}=tx_{i-1}\\ -1,\ (tx_{i}-tx_{i-1})/t>-1\\ -2,\ (tx_{i}-tx_{i-1})/t=-1\\ -3,\ (tx_{i}-tx_{i-1})/t<-1\end{array} \right. \tag{5}\]  
where \(t\) is the time interval between two consecutive sample points. Step 4: The detail component (\(DE^{X}\) and \(DE^{Y}\)) are converted into symbolic aggregate approximation (SAX) symbol sequences \(SC^{X}\) and \(SC^{Y}\) as follows (take \(SC^{X}\) as an example):  
Step 4.1: The detail component \(DE^{X}=(dx_{1},dx_{2},\cdots,dx_{n})\) is first normalized as follows:  
\[NDE^{X}=\frac{DE^{X}-\mu}{\sigma} \tag{6}\]where \(NDE^{X}\) is the normalized series of \(DE^{X}\), \(\mu\) is the mean value of \(DE^{X}\), and \(\sigma\) is its standard deviation. Step 4.2: The normalized series \(NDE^{X}\) is divided into \(w\) equal-sized segments, then the normalized series can be represented by the average of each segment as follows:  
\[\overline{Ndx_{k}}=\frac{w}{n}\sum_{i=\frac{n}{b}(k-1)+1}^{\frac{n}{n}k}Ndx_{i} \tag{7}\]  
where \(\overline{Ndx_{k}}\) is the average of the \(k\)th segment of normalized series \(NDE^{X}\). Step 4.3: The distribution space of \(\overline{Ndx_{k}}\) on the amplitude is divided into \(\alpha\) equiprobable regions, the breakpoints \(\beta\) refers to the lookup table in [30; 31]. Step 4.4: The SAX symbol sequences \(SC^{X}\) can be obtained by assigning symbols to each region which is determined by breakpoints. Step 5: The symbol sequences similarity of trend component \(STR\) and detail component \(SDE\) are measured by the longest common subsequence (LCS) [32] respectively.