## 4 Error Analysis Method  
Through parameterized numerical model analysis, we obtained a large number of stress data of incompletely similar nodes. Based on these data, this paper used regression analysis to obtain similar errors. The flow chart of modeling and regression analysis of incompletely similar nodes is as follows (Figure 23):  
Figure 22: Stress distribution of the end-plate.  
The method of regression analysis is as follows:  
Regression analysis is a predictive modeling technique, which studies the relationship between a dependent variable (target) and an independent variable (feature). This technique is often used for predictive analysis and to find causal relationships between variables. Usually a curve is used to fit the data points, and the goal is to minimize the difference in distance between the curve and the data points. Linear regression is a regression problem which assumes that the target value and the characteristics are linearly related, that is, to satisfy a multivariate linear equation. By constructing the loss function, the parameters w and b when the loss function is minimum are solved. Usually, we can express it as the following formula:  
\[y=wx+b \tag{12}\]  
where \(y\) is the predicted value, and the independent variable \(x\) and the dependent variable \(y\) are known. What we want to achieve is the prediction of the value of \(y\) when adding a new \(x\). Therefore, in order to construct this functional relationship, we need to solve the two parameters \(w\) and \(b\) in the linear model through known data points.  
Solving the best parameters requires a standard to measure the results. For this, we need to quantify an objective function formula so that the computer can continuously optimize during the solution process.  
For the error-solving problem of incompletely similar nodes, the loss function can be defined as follows:  
\[L=\frac{1}{n}\sum_{i=1}^{n}\left(y_{pi}-y_{i}\right)^{2} \tag{13}\]  
where \(y_{pi}\) is the predicted value of the node, \(y_{i}\) is the known value of the node, and \(n\) is the number of node models.  
Figure 23: Analysis flow chart.  
Figure 24: Similarity errors of different end-plate thicknesses at specified points.  
Figure 25: Data points and the regression curve at B\({}_{\rm f}\).  
Figure 26: Data points and the regression curve at C\({}_{\rm f}\).  
Figure 28: Data points and the regression curve at C\({}_{\rm W}\).  
Figure 27: Data points and the regression curve at C\({}_{\rm P}\).  
Figure 29: Data points and the regression curve at N\({}_{\rm c}\).  
Therefore, the linear regression can realize the prediction and correction of most similar errors, and the residual similar errors after correction are shown in Table 7. Except for the error of the stress data in the middle and upper parts of the end-plate, most of the similar errors have been greatly improved. In this way, the similarity error problem of incompletely similar models has been solved.  
Figure 31: Linear regression-corrected value.  
Figure 30: Data points and the regression curve at \(\mathrm{E_{P}}\).  
It is the main purpose of this paper to reduce the similarity error by looking for the law of data change. The authors did not pursue accurate linear stress change laws, but attempted to reduce similar errors through linear regression. It can be seen that linear regression is at least an effective method applied in this field.