## 4 Multi-Objective Geometric Uncertainty Pneumatic Robustness Optimization
### Pearson Correlation Coefficient  
Correlation analysis is the analysis of two or more elements of a variable that are correlated to measure the closeness of the correlation between the two variables and the linearity of the relationship between the elements. The small diagonal flow fan studied in this paper involve many study parameters. If all of them are considered design variables for optimal design, the number of sample spaces is too large. It is necessary to reduce the sample size by removing the less influential parameters according to their degree of influence on the performance of the fans [14]. The Pearson correlation coefficient is the most common method used when studying the correlation between two variables and requires the calculation of the following two coefficients, the sample correlation coefficient (r) and the overall correlation coefficient (\(\rho\)). The sample correlation coefficient, replacing the overall covariance and standard deviation with the sample's covariance and standard deviation calculated. Pearson correlation coefficients range from [\(-1\), \(+1\)], with less than 0 representing a negative correlation, greater than 0 representing a positive correlation, and equal to 0 representing no correlation. The closer the correlation coefficient is to 0, the weaker the correlation is; the closer it is to \(-1\) or \(+1\), the stronger the correlation is. In this paper, Pearson correlation analysis was performed with Spss software to detect the degree of linear correlation between several parameter variables of the rear guide vane of the diagonal flow fan in order to avoid the influence of the design sample size on the judgement of the degree of correlation between the optimization variables and the optimization objective. Figure 8 shows the correlation between the optimization variables of the rear guide vane and the optimization objective, ranking the optimization variables in order of the magnitude of the correlation coefficient as speed, chord length, mounting angle, bending angle and guide vane thickness [15]. Based on the 60 samples in Table 1, the optimal Latin hypercube experimental design was used. The selected sampling points were more complete in spatial distribution, and there was no problem of missing sample points locally. The final sampling ratio was determined through numerous trainings to ensure the accuracy of the proxy model, as shown in Figure 9.  
Figure 7: A schematic diagram of guide vane geometric parameters. The response value sound pressure level and full pressure were obtained with numerical calculation and Pearson correlation analysis. The kriging proxy model is established with the guide vane installation angle as the angle, the guide vane bending angle as the angle, C as the guide vane chord length and the speed \(n\) as the optimization variables, and the sound pressure level SPL and full pressure P of the diagonal flow fan as the optimization targets. The R\({}_{2}\) value (\(0\leq\text{R}_{2}\leq 1\)) is above 0.9, and the predicted value matches the calculated value. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Design Parameters** & **Upper Limit** & **Lower Limit** \\ \hline Bend angle & \(\theta\) & \(85^{\circ}\) & \(76^{\circ}\) \\ Length of string & \(C\) & \(84\) mm & \(62\) mm \\ Mounting angle & \(\beta\) & \(19^{\circ}\) & \(10^{\circ}\) \\ Design speed & \(n\) & \(2300\) rpm & \(2000\) rpm \\ Guide leaf thickness & \(b\) & \(1\) mm & \(3\) mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The value range of design parameters. Figure 8: The correlation between optimization variables and optimization objectives. Figure 9: A sampling of 60 original samples. Where the design speed is 2250 rpm, only increasing or decreasing the speed is not in line with the original intention of optimizing the structural design of the fan, and the guide vane thickness has the most negligible effect on the optimization objective.