## 4 Numerical Test
### Rothyp Function Test  
The second test function is the Rothyp function, whose expression is shown in Equation (29):  
\[y(\mathbf{x})=\sum_{i=1}^{d}\sum_{j=1}^{i}\mathbf{x}_{j}^{2}\qquad-65.536\leq\mathbf{x}_{ i}\leq 65.536\qquad i=1,\cdots,d \tag{29}\]  
For the 50-dimensional and 100-dimensional problems of this function, 110 and 220 sampling points were used for testing. Its testing process is consistent with that  
Figure 5: (**a**) Modeling time and its standard deviation for the 30D Griewank function. (**b**) RMSE and its standard deviation for the 30D Griewank function. Figure 6: (**a**) Modeling time and its standard deviation for the 70D Griewank function. (**b**) RMSE and its standard deviation for the 70D Griewank function. From Table 4 and Figures 7-9, the following characteristics can be observed. (1) The advantage of the modeling time for the KMDS method is particularly obvious. For example, in the case of 100 dimensions, the modeling time of KPLS-3 is approximately 33.28 times that of KMDS. Even the KPLS-1 method that retains only one principal component takes 5.19 times as much time as the proposed method, which fully illustrates the advantages of the KMDS method in terms of modeling efficiency. (2) Although the accuracy of the KMDS method is not as good as that of the KPLS methods, it is already very close to them. After all, any method has two sides. Increased modeling speed usually leads to lower accuracy. Significantly, this work has greatly improved the modeling efficiency under the condition of close accuracy, which is what we expect. (3) From the perspective of the error bar graph for modeling time and modeling accuracy and the trend chart for the standard deviation, the KMDS method also has good stability and effectiveness compared to the other four methods. This shows that the proposed dimensionality reduction method is effective in actual tests. Figure 8: (a) Modeling time and its standard deviation for the 50D Rothyp function. (b) RMSE and its standard deviation for the 50D Rothyp function. Figure 9: (a) Modeling time and its standard deviation for the 100D Rothyp function. (b) RMSE and its standard deviation for the 100D Rothyp function. From Table 5, it is easy to see that the test results for the Michalewicz function are similar to those of the Griewank function and the Rothyp function. However, in the case of 40D and 110 sampling points, the test results of the Michalewicz function are slightly different: the modeling time of KMDS is still the shortest, while the modeling error of the ordinary Kriging method is slightly smaller than that of the KMDS method. The complexity of the function itself and the uniformity of the proportions of each dimension may cause this problem. Figures 10-12 also show the advantages of the KMDS method. Comparing KMDS and KPLS, it can be seen that KMDS is better than KPLS in terms of modeling time and its standard deviation. In some cases, the model accuracy or the average accuracy of the proposed method is better than the KPLS method, which further illustrates the effectiveness and stability of the KMDS method. In summary, it can be seen from the test results of all benchmark functions that the modeling efficiency of the KMDS method is the highest. In terms of modeling accuracy, test results for the KMDS method are superior to those of Kriging method in most cases. Compared with the KPLS methods, the accuracy of KMDS method is slightly worse, but it is also relatively close, and even better than them in some cases. Moreover, with the increase in the dimensionality and sample size, the KMDS method can show its absolute  
Figure 11: (**a**) Modeling time and its standard deviation for the 40D Michalewicz function. (**b**) RMSE and its standard deviation for the 40D Michalewicz function. Figure 12: (**a**) Modeling time and its standard deviation for the 80D Michalewicz function. (**b**) RMSE and its standard deviation for the 80D Michalewicz function. advantage in terms of time spent with a small loss of precision.