## 3 Ore Bonding Parameter Determination Method
So the function is as follows:  
\[L=\sum_{i=1}^{n}\varepsilon_{i}^{2}=\varepsilon\epsilon^{2}=(y-X\beta)^{T}(y- X\beta) \tag{11}\]  
and its objective function can be expressed as follows:  
Taking the partial derivatives of the objective function and setting it to 0, we get the following:  
\[\frac{\partial L}{\partial\beta}\bigg{|}_{\hat{\beta}}=-2X^{T}y+2X^{T}X\hat{ \beta}=0 \tag{12}\]  
The solution of the least square method is as follows:  
\[\hat{\beta}=\left(X^{T}X\right)^{-1}X^{T}y \tag{13}\]  
Therefore, the multiple regression model of the response surface is as follows:  
\[\hat{y}=X\hat{\beta} \tag{14}\]  
In order to verify the validity and accuracy of the resulting quadratic response surface multiple regression model, it is necessary to employ ANOVA [28] and significance tests in order to assess the significance of the overall model as well as the significance of the individual model coefficients. The test of significance is employed to ascertain the existence of a linear relationship between the independent variable and the dependent variable. It is typically conducted using the \(F\)-test, whereby the \(F\)-statistic is calculated according to the following formula:  
\[F=\frac{\frac{SS_{E}}{m}}{\frac{SS_{R}}{n-1-m}} \tag{15}\]  
where \(SS_{E}\) is the residual sum of squares; \(SS_{R}\) is the model sum of squares. Add up to the sum of squares \(S_{yy}\) The formula is as follows:  
\[SS_{E}=\sum\limits_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}=y^{T}y-\beta^{T}X^{T}y \tag{16}\]  
\[SS_{R}=\hat{\beta}^{T}X^{T}y-\frac{\left(\sum\limits_{i=1}^{n}y_{i}\right)^{ 2}}{n} \tag{17}\]  
\[S_{yy}=SS_{E}+SS_{R}=y^{T}y-\frac{\left(\sum\limits_{i=1}^{n}y_{i}\right)^{2}} {n} \tag{18}\]  
The critical value \(F_{\alpha}\) corresponding to the given significance level \(\alpha\) can be obtained by consulting the distribution table with degrees of freedom. Should the calculated F-value exceed \(F_{\alpha}\), it can be inferred that the regression equation is statistically significant. Furthermore, the accuracy of the model can be evaluated through the use of a modified coefficient of determination. The formula is as follows:  
\[R_{a\beta i}^{2}=1-\frac{\frac{SS_{E}}{n-p}}{\frac{S_{yy}}{n-1}}=1-\left( \frac{n-1}{n-p}\right)\left[1-\left(1-\frac{SS_{E}}{S_{yy}}\right)^{2}\right] \tag{19}\]  
One method of assessing the extent to which independent variables affect a model is through significance testing of the regression coefficients. This can also be carried out through F-tests. The statistic of the F-test is calculated as follows:  
\[F_{\beta i}=\frac{\frac{\hat{\beta}_{i}}{C_{ij}}}{\frac{SS_{E}}{n-1-m}} \tag{20}\]  
where \(C_{ij}\) denotes the diagonal element of the matrix \(\left(X^{T}X\right)^{-1}\). \(x_{i}\) can be shown to have a significant effect on the model by the given test level \(\alpha\) when \(F_{\beta_{i}}>F_{\alpha,1,n-m-1}\). The Box-Behnken design (BBD) is a common experimental design methodology employed to assess nonlinear relationships between factors and indicators. The BBD design is a central synthetic design characterized by the selection of a central point and appropriate level values within the range of values of each factor, with the objective of achieving balance and efficiency in the design. The BBD is typically employed in cases involving three or more factors, with a maximum of three levels for each factor. This approach ensures that no single factor is represented by a single level, allowing the construction of a quadratic response surface model to predict the relationship between the response variable and each factor. The objective of this paper is to determine the optimal bonding parameters through a combination of response surface methodology and BBD. Using compressive strength as the dependent variable, we consider unit area normal stiffness, unit area shear stiffness, critical normal stress, and critical shear stress as independent variables [14].