## 2 Development of a Servo Valve Prediction Model
### Servo Valve Reliability Model Based on Copula Function
#### 2.2.5 Joint Probability Density Distribution  
This paper uses the two-dimensional joint probability density distribution as the study sample, and its calculation method is divided according to the different correlation coefficients used. The construction method based on the Pearson correlation coefficient is called the approximate method \(P\); the other one is based on the Spearman correlation coefficient, called the approximate method \(S\).  
Approximate Method \(P\)  
Suppose the Pearson's correlation coefficient between variables \(Y_{1}\) and \(Y_{2}\) is \(r_{PY}\), then the joint probability density functions of variables \(Y_{1}\) and \(Y_{2}\) are \(f_{1}(y_{1})\) and \(f_{2}(y_{2})\), respectively. The joint probability density function of the variables \(Y_{1}\) and \(Y_{2}\) can be found using the Nataf model, which is expressed by the following equation:  
\[f_{1,2}^{p}(y_{1},y_{2};r_{PY})=f_{1}(y_{1})f_{2}(y_{2})\frac{\phi_{2}\Big{(} x_{1},x_{2};r_{PY}^{p}\Big{)}}{\phi(x_{1})\phi(x_{2})} \tag{19}\]  
where \(\phi(x)\)is denoted as the probability density function of a one-dimensional standard normal variable, and \(\phi_{2}(x_{1}\),\(x_{2}\),\(r^{p}\),\(x)\) is the probability density function of a two-dimensional standard normal distribution with a _Pearson_ correlation coefficient of \(r^{p}\)\({}_{px}\).  
Approximate Method \(S\)  
Suppose the _Spearman_ correlation coefficient between variables \(Y_{1}\) and \(Y_{2}\) is \(r_{S,Y}\), and the _Spearman_ correlation coefficient between variables \(X_{1}\) and \(X_{2}\) is \(r_{S,X}\); according to the definition, we can get  
\[r_{S,X}=r_{S,Y} \tag{20}\]  
The joint probability density function of the variables \(Y_{1}\) and \(Y_{2}\) constructed by the approximation method \(S\) can be expressed as  
\[f_{1,2}^{S}(y_{1},y_{2};r_{S,Y})=f_{1}(y_{1})f_{2}(y_{2})\frac{\phi_{2}\Big{(}x _{1},x_{2};r_{P,X}^{S}\Big{)}}{\phi(x_{1})\phi(x_{2})} \tag{21}\]  
where \(\phi_{2}(x_{1},x_{2};r^{P},x)\) is the probability density function of a two-dimensional standard normal distribution with a _Pearson_ correlation coefficient of \(r^{P}{}_{p,x}\). The _Pearson_ correlation coefficient between variables \(X_{1}\) and \(X_{2}\) is \(r^{P}{}_{p,x}\). The equation for the relationship between \(r^{P}{}_{p,x}\) and _Spearman_ correlation coefficient \(r_{S,Y}\) is expressed as  
\[r_{P,X}^{S}=2\sin\Big{(}\frac{\pi}{6}r_{S,X}\Big{)} \tag{22}\]  
Using both approximation methods, the reliability of the life prediction system can be effectively calculated to a reasonable level of accuracy. If the probabilistic information associated with a lifetime prediction system needs to be more comprehensive, it is more convenient to use an approximation to evaluate its accuracy. On the other hand, if the relevant probabilistic information is more complete, it is preferable to construct joint probability distribution functions for the variables using simpler approximations. In theory, these methods can ignore the dependence of probabilistic data and allow accurate evaluation of the lifetime prediction system, albeit with a slightly different computational procedure.