# Parameter Analysis and Optimization of Annular Jet Pump Based on Kriging Model
## 4 Hybrid Algorithm and Optimization Process
### Approximate Model  
The Kriging model is widely used in predicting spatially and temporally related data. It was originally used in geostatistics which is an interdisciplinary field of statistics, mathematics, geology, engineering and mining. Due to its stochastic assumption, the Kriging approximation is superior to traditional RSM in high dimensional nonlinear problems and high precision prediction [29]. The Kriging model is described by:  
\[y(x)=f(x)+Z(x) \tag{14}\]  
where \(y(x)\) is the unknown response function, \(f(x)\) is a known polynomial function of \(x\) called the trend, and \(Z(x)\) is a stochastic process with nonzero covariance, variance \(\sigma^{2}\), and mean zero. The term of \(f(x)\) in the equation is similar to the polynomial model in a response surface, serving as a \"global\" model of the design space. In many cases, \(f(x)\) is taken to be a constant term \(\beta_{0}\). When \(f(x)\) \"globally\" approximates the design space, \(Z(x)\) localized deviation, so the Kriging approximate model interpolates the sampling data points. The covariance matrix of \(Z(x)\) that represents the local deviations is:  
\[Cov\big{[}Z(x_{i}),Z\big{(}x_{j}\big{)}\big{]}=\sigma^{2}\text{R}\Big{(}\big{[} R\big{(}x_{i},x_{j}\big{)}\big{]}\Big{)} \tag{15}\]  
where R is the correlation matrix, and \(R(x_{i},x_{j})\) is the correlation function between any two of the \(n_{\text{s}}\) sampled data points \(x_{i}\) and \(x_{j}\). \(R\) is a \([n_{\text{s}}\times n_{\text{s}}]\) symmetric, positive definite matrix with ones along the diagonal. Many different correlation functions are available for the Kriging model, such as the linear correlation function, exponential correlation function and Matern Cubic correlation function. However, the Gauss correlation function is more popular in Kriging metamodels. The Gaussian correlation model is used in this paper, defined as:  
\[R(\theta,\omega,x)=\exp\left(-\sum_{j=1}^{n}\theta_{j}\big{(}\omega_{j}-x_{j} \big{)}^{2}\right) \tag{16}\]  
where \(\omega_{j}\) and \(x_{j}\) represent the \(jth\) component of sample points \(\omega\) and \(x\), respectively, \(\theta_{j}\) is the unknown correlation parameters used to fit the model and \(n\) is the dimension of the design variable.  
Figure 5: Design of experiments (DOE) method. (**a**) Latin hypercube sampling algorithm (LHS); (**b**) OSF.  
Once the correlation function has been selected and the best \(\theta_{j}\) estimated, the Kriging model can be used to predict the response \(y\) (\(x\)) at an untried location \(x\) using:  
\[\hat{y}(x)=\hat{\beta}+r^{T}(x)R^{-1}\Big{(}Y-f\hat{\beta}\Big{)} \tag{17}\]  
where, \(\hat{y}(x)\) is the vector of estimated response values at each sample point, \(\hat{\beta}\) is a constant, \(r^{T}(x)\) is the vector of correlation values between the untried location \(x\) and the sample data points, and \(f\) is the vector with values of the trend function evaluated at each sample point.  
\(\hat{\beta}\) can be calculated as  
\[\hat{\beta}=\Big{(}f^{T}R^{-1}f\Big{)}^{-1}f^{T}R^{-1}Y \tag{18}\]  
The estimate of the variance is  
\[\hat{\sigma}^{2}=\frac{1}{n_{s}}\Big{(}Y-f\hat{\beta}\Big{)}^{T}R^{-1}\Big{(} Y-f\hat{\beta}\Big{)} \tag{19}\]  
After selecting the form of the regression model and the correlation function, the correlation matrix \(R\), the unknown parameter \(\beta_{0}\), and \(\sigma^{2}\) depend on the correlation parameter \(\theta\). Therefore, the Kriging metamodel can only be fully established after determining the value of \(\theta\). Maximum likelihood estimation (MLE) is a commonly used method for calculating related parameter values. This optimization can be translated into an unconstrained global optimization, as follows:  
\[\left\{\begin{array}{l}\text{Minimize}\left\{\psi(\theta)=\sigma\Big{(} \theta^{2}\Big{)}|R_{\theta}|^{\frac{1}{n_{s}}}\right\}\\ \text{Subjectto}\ \ \theta>0\end{array}\right. \tag{20}\]