## 4 Numerical Test
### Griewank Function Test  
The first test function used is the Griewank with greater complexity, and its specific expression is shown in Equation (28):  
\[y(\mathbf{x})=\sum_{i=1}^{d}\frac{\mathbf{x}_{i}^{2}}{4000}-\prod_{i=1}^{d}\cos(\frac {\mathbf{x}_{i}}{\sqrt{i}})+1\qquad-600\leq\mathbf{x}_{i}\leq 600\qquad i=1,\cdots,d \tag{28}\]  
For the case where the dimensions of the function are 30 and 70, 110 and 220 sampling points are used for testing in order to effectively verify the effect and stability of the KMDS method in different dimensions and with different sample sizes. The Kriging method, KMDS, KPLS-1, KPLS-2 and KPLS-3 were used to test the modeling time and convergence accuracy of the Griewank function. Table 3 shows the average modeling time and its standard deviation (STD). Furthermore, the average modeling error and its standard deviation were obtained by testing the Griewank function in five different modeling methods, each of which represents the average of 10 test results. The test results show that the KMDS method always takes the shortest time to build the Kriging model under the conditions of different dimensions and different samples. With the increase in the number of dimensions and the number of sample points, the efficiency advantage of the KMDS method becomes more obvious. The main reason is because the KMDS method transforms high-dimensional modeling into three-dimensional modeling, so the amount of time spent calculating the correlation matrix and hyper-parameters in the modeling process is greatly reduced, thereby improving the modeling efficiency of Kriging. In terms of accuracy, the accuracy of the KMDS method is very close to that of ordinary Kriging. Compared with the KPLS method, when the condition of the test function is 30D and 110 samples, the test time of the KMDS method is only 0.1188 s faster \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Griewank**} & \multicolumn{1}{c}{**Statistic**} & \multicolumn{1}{c}{**Kriging**} & \multicolumn{1}{c}{**KMDS**} & \multicolumn{1}{c}{**KPLS-1**} & \multicolumn{1}{c}{**KPLS-2**} & \multicolumn{1}{c}{**KPLS-3**} \\ \hline \multirow{8}{*}{30D} & \multirow{4}{*}{110} & t(s) & 21.3954 & **1.1498** & 1.2686 & 2.5323 & 5.7447 \\  & & STD & 0.7999 & **0.0578** & 0.2175 & 0.1594 & 0.1042 \\  & & RMSE & 14.0585 & 13.3769 & 12.6699 & 12.7925 & **12.6464** \\  & & STD & 0.8021 & 0.9089 & **0.4453** & 1.0586 & 0.4736 \\ \cline{2-8}  & \multirow{4}{*}{220} & t(s) & 169.0297 & **6.8024** & 10.8794 & 25.7107 & 58.5037 \\  & & STD & 1.8902 & **0.2296** & 0.5263 & 0.3441 & 1.2995 \\  & & RMSE & 10.0340 & 9.6798 & 8.4632 & **7.9802** & 8.3273 \\  & & STD & 0.5234 & 0.2891 & 0.4772 & **0.2215** & 0.5698 \\ \hline \multirow{8}{*}{70D} & \multirow{4}{*}{110} & t(s) & 55.8389 & **1.1410** & 2.6329 & 6.3931 & 15.2219 \\  & & STD & 1.1604 & **0.0359** & 0.1836 & 0.4135 & 0.5666 \\ \cline{1-1}  & & RMSE & 21.2036 & 20.1981 & 20.0653 & 20.0093 & **20.0010** \\ \cline{1-1}  & & STD & 1.3604 & **0.7127** & 0.7714 & 1.0447 & 1.2915 \\ \cline{1-1} \cline{2-8}  & \multirow{4}{*}{220} & t(s) & 504.3929 & **5.7970** & 21.3492 & 59.6396 & 138.2662 \\ \cline{1-1}  & & STD & 1.4190 & **0.1934** & 1.0168 & 2.1632 & 2.9013 \\ \cline{1-1}  & & RMSE & 14.9673 & 14.5079 & 13.5973 & 13.5935 & **13.5839** \\ \cline{1-1}  & & STD & 0.5226 & 0.3136 & **0.3038** & 0.5506 & 0.6101 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results for time and RMSE for Griewank function. The symbols 30D and 70D indicate that the dimensions of benchmark functions are 30 and 70, respectively. than that of the KPLS-1 method, but the error score of KPLS-1 is 0.707 lower than that of KMDS. The KPLS-1 method is better at this time. The reason may be that the KPLS method considers the relationship between design variables and response variables to make the accuracy higher. In addition, retaining a principal component greatly reduces the amount of calculation of hyper-parameters, thereby improving the efficiency of Kriging modeling.