## 3 Results and Discussion
### Hardness Distribution
#### 3.2.2 Regression Models and 3D Plots of Hardness Distribution  
Equation (6) was used to express the hardness response's linear regression model, which was dependent on the amount of pressure (P), the H/D ratio, and the wt.% of Al\({}_{2}\)O\({}_{3}\), as well as the interactions among these variables. In the present research, the R\({}^{2}\) and adjusted R\({}^{2}\) had values that were quite near the value of 1.  
\[\begin{array}{l}\mathrm{H_{v}}=+12.37184\ +\ 0.079765\ \mathrm{P}\ +\ 11.94883\ \mathrm{H/D}\ +\ 4.21279\ \mathrm{Al_{2}O_{3}}\ -\ 0.012500\ \mathrm{P}\ x\ \mathrm{H/D}\ +\ 0.001250\ \mathrm{P}\ x\ \mathrm{A}\mathrm{L_{2}O_{3}}\\ -2.10000\ \mathrm{H/D}\ *\ \mathrm{Al_{2}O_{3}}-0.000022\ \mathrm{P}^{2}\ -1.19594\ \mathrm{(H/D)}^{2}\ +\ 0.224554\ \mathrm{(Al_{2}O_{3})}^{2} \end{array} \tag{6}\]  
Figure 8: (**a**) Evaluation of predicted versus experimental hardness for the training and testing groups. (**b**) Predicted train, predicted test, and experimental datasets for the anticipated hardness as a function of sample index.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**ML**} & \multicolumn{2}{c}{**Training Set**} & \multicolumn{2}{c}{**Testing Set**} \\ \cline{2-5}  & **RMSE** & **R\({}^{2}\)** & **RMSE** & **R\({}^{2}\)** \\ \hline Linear regression & 2.81 & 0.95 & 2.21 & 0.93 \\ Regression trees & 1.12 & 0.99 & 2 & 0.94 \\ Random forest regression & 2.04 & 0.97 & 2.99 & 0.87 \\ Gaussian process regression & 1.13 & 0.99 & 2.15 & 0.95 \\ Artificial neural networks & 0.818 & 0.996 & 2.19 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Evaluation metrics for the training and testing groups of hardness for ML models.  
Figure 9 illustrates the relationship between the hardness experimental results for the parameters of the Al-4 Cu\(/\)(Al\({}_{2}\)O\({}_{3}\))\({}_{\rm x}\) composite and their corresponding anticipated values. It is obvious from these data that the suggested regression models were effective since the experimental and forecast values match up well. The following conclusion (Table 6) was drawn from the fact that the majority of the intersections among the experimental and predicted values are located near the median line.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Input** & \multicolumn{4}{c}{**Hardness (HV)**} \\ \hline
**Pressure** & **H/D** &
\begin{tabular}{c} **Al\({}_{2}\)O\({}_{3}\)** \\ **(wt. \%)** \\ \end{tabular} & **Exp.** & **Predict** & **Error** \\ \hline
800 & 1.5 & 2.5 & 69.5 & 69.5 & \(5.97\times 10^{-13}\) \\
600 & 1.5 & 2.5 & 64 & 64 & \(-1.07\times 10^{-12}\) \\
600 & 1 & 2.5  
Figure 10 shows how the AMC parameters had an effect on the hardness as measured by the regression models (Equation (6)). It was observed that the parameters of the AMC, specifically the pressure and Al\({}_{2}\)O\({}_{3}\)%, had a proportional relationship with the hardness values, while a drop in the specimen's hardness was noticed with a modest increase in the H/D. The highest Vickers hardness was achieved at 900 MPa and a 0.75 H/D for all Al\({}_{2}\)O\({}_{3}\)% percentages tested. Figure 10e displays the highest feasible Vickers hardness that was achieved at 10 Al\({}_{2}\)O\({}_{3}\)%, 900 MPa, and a 0.75 H/D with a value of 126.27 Hv. This hardness was superior to the hardness that was produced experimentally with a value of 108 Hv at 700 MPa, a 1.25 H/D, and 10 Al\({}_{2}\)O\({}_{3}\)%.  
Figure 10: Three-dimensional response plot of hardness at (**a**) 0% (**b**) 2.5%, (**c**) 5%, (**d**) 7.5%, and (**e**) 10% of Al\({}_{2}\)O\({}_{3}\).