# Selection Strategy of Vibration Feature Target under Centrifugal Pumps Cavitation
## 2 Grey Relation Entropy Analysis Method
### Calculation Process  
The concrete steps of grey slope correlation with weight entropy methods are listed as follows. **Step 1:** Define reference sequence (RS) and comparative sequence (CS)  
Suppose \(N=\{N(P_{1}),N(P_{2}),N(P_{3}),\cdots,N(P_{n})\}\) as the reference sequence, which indicates the sequence of tabulated data of NPSH, where \(P_{n}\) represents the real-time pressure on \(n\)th times. Vibration features are taken as the comparative sequence, that is \(V_{i}=\{V_{i}(P_{1})\,,\,V_{i}(P_{2})\,,\,V_{i}(P_{3})\,,\,\cdots\,,\,V_{i}(P_ {n})\}\,,\,\,i=1\,,\,2\,,\,\cdots\,,\,k\), which denotes the comparative sequence. **Step 2:** Make sequence be dimensionless  
Due to the values' physical scale difference, the maximum value treatment can be used to normalize the data. This mathematical process can enable us to obtain more accurate results in the grey correlation analysis. The preprocessing can express as:  
\[\overset{\text{max}}{P}_{n}=\frac{P_{n}}{\max\{P_{n}\}}\,\,n=1\,,\,2\,,\,3\,, \,\cdots\,,\,m \tag{1}\]  
\[X(\overset{\text{max}}{P}_{n})=\frac{N(P_{n})}{\max\{N(P_{n})\}}\,\,n=1\,,\,2\,, \,3\,,\,\cdots\,,\,m \tag{2}\]  
\[Y_{i}(\overset{\text{max}}{P}_{n})=\frac{V_{i}(P_{n})}{\max\{V_{i}(P_{n})\}}\, \,n=1\,,\,2\,,\,3\,,\,\cdots\,,\,m \tag{3}\]  
**Step 3:** Calculate the coefficient of grey slope correlation  
For the unequal interval sequence, define the grey slope correlation coefficient as  
\[\xi_{i}(\overset{\text{max}}{P}_{n})=\text{sgn}(\Delta X(\overset{\text{max} }{P}_{n}),\Delta Y_{i}(\overset{\text{max}}{P}_{n}))\cdot\Theta \tag{4}\]  
where,  
\[\text{sgn}(\Delta X(\overset{\text{max}}{P}_{n}),\Delta Y_{i}(\overset{\text{ max}}{P}_{n}))=\left\{ \begin{array}{l}1\,,\,\,\Delta X(\overset{\text{max}}{P}_{n}) \Delta Y_{i}(\overset{\text{max}}{P}_{n})\geq 0\\ -1\,,\,\Delta X(\overset{\text{max}}{P}_{n})\Delta Y_{i}(\overset{\text{max}}{ P}_{n})<0\end{array} \right. \tag{5}\]  
\[\Theta=\frac{1+\left|\frac{1}{X}\frac{\Delta X(\overset{\text{max}}{P}_{n})}{ \Delta\overset{\text{max}}{P}_{n}}\right|}{1+\left|\frac{1}{X}\frac{\Delta X( \overset{\text{max}}{P}_{n})}{\Delta\overset{\text{max}}{P}_{n}}\right|}+ \left|\frac{1}{X}\frac{\Delta X(\overset{\text{max}}{P}_{n})}{\Delta\overset{ \text{max}}{P}_{n}}-\frac{1}{\overset{\text{max}}{Y}_{i}}\frac{\Delta Y \[\left\{ \begin{array}{l}\Delta X(\overset{\max}{P_{n}})=X(\overset{\max}{P_{n}})-X (\overset{\max}{P_{n-1}})\\ \Delta Y_{i}(\overset{\max}{P_{n}})=\Delta Y_{i}(\overset{\max}{P_{n}})-\Delta Y _{i}(\overset{\max}{P_{n-1}})\,\ \ n\geq 2\\ \Delta\overset{\max}{P_{n}}=\overset{\max}{P_{n}}-\overset{\max}{P_{n-1}}\\ \end{array} \right. \tag{7}\]  
\[\overline{Y}_{i}=\frac{1}{m}\overset{m}{\underset{n-1}{\sum}}Y_{i}(\overset{ \max}{P_{n}}) \tag{8}\]  
\[\overline{X}=\frac{1}{m}\overset{m}{\underset{n-1}{\sum}}X(\overset{\max}{P_ {n}}) \tag{9}\]  
**Step 4:** Standardize the target matrix \(\xi_{i}(\overset{\max}{P_{n}})\)  
As the uncertainty of positive and negative value exist in \(\xi_{i}(\overset{\max}{P_{n}})\), therefore, the transmitting to the same sign is necessary for this paper.