## 3 KMDS Method
### Determination of the Space Dimension \(d^{\prime}\) after Dimensionality Reduction  
Multidimensional scaling can freely determine the low-dimensional space dimension \(d^{\prime}\) when performing dimensionality reduction on high-dimensional data. In this work, the dimension \(d^{\prime}=3\) is selected. For specific reasons, we use the 50-dimensional Griewank function as an example to explain. The expression of the Griewank function is shown in Equation (25):  
\[y(\mathbf{x})=\sum_{i=1}^{50}\frac{\mathbf{x}_{i}^{2}}{4000}-\prod_{i=1}^{50}\cos( \frac{\mathbf{x}_{i}}{\sqrt{i}})+1\qquad-600\leq\mathbf{x}_{i}\leq 600\qquad i=1, \ldots,50 \tag{25}\]  
Table 1 is the modeling result of the 50-D Griewank function through Kriging and KMDS. Among them, KMDS-1, KMDS-2, KMDS-3, KMDS-4 and KMDS-5 refer to reducing the high-dimensional data to 1-D (one dimension), 2-D, 3-D, 4-D and 5-D data, respectively. During the modeling process, the Latin hypercube sampling (LHS) method is used to generate \(N\) (\(N=10\) or \(N=20\)) initial sample points. The modeling time and accuracy of Kriging and KMDS methods when adding 10\(N\) sample points are compared. Each method was run 10 times separately, and finally the average value was taken as the result so as to ensure the accuracy of the results. The RMSE (root mean square error) values in Table 1 are obtained by using leave-one-out cross-validation [44]. The concrete expression for finding the RMSE is as follows:  
\[\text{RMSE}=\frac{1}{k}\sqrt{\sum_{i=1}^{k}\hat{s}_{i}^{2}} \tag{26}\]  
The parameter \(k\) here represents the number of samples in the current data. If the Kriging model is used to estimate the variance of point \(\mathbf{x}_{i}\), we first need to reconstruct the Kriging model with the remaining \(k-1\) sampling points, except point \(\mathbf{x}_{i}\). Then, we \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Griewank (50D)** & **Kriging** & **KMDS-1** & **KMDS-2** & **KMDS-3** & **KMDS-4** & **KMDS-5** \\ \hline \multirow{2}{*}{110} & t (s) & 29.2118 & 0.5247 & 0.5648 & 1.1065 & 2.0409 & 3.0003 \\  & RMSE & 17.9327 & 187.3951 & 85.5546 & 18.0510 & 17.9595 & 17.8875 \\ \hline \multirow{2}{*}{220} & t (s) & 279.4006 & 2.9883 & 3.1055 & 5.7695 & 11.6390 & 15.5642 \\  & RMSE & 12.8756 & 120.5884 & 73.9394 & 12.9491 & 12.7546 & 12.6127 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modeling time and accuracy of Kriging and KMDS methods. calculate the estimated variance \(\hat{s}_{i}^{2}\) of point \(\mathbf{x}_{i}\) by using the newly built Kriging model and Formula (8). After repeating \(k\) times to complete the variance estimation of these \(k\) sampling points, the average value can be calculated to obtain the RMSE following Equation (26). To visually prove the stability and effectiveness of KMDS, the Griewank function was tested 10 times with different sample points, and the results are shown in the box plots in Figures 1 and 2. The middle line (solid red line) of each box plot shows the median value of the 10 data. The blue solid line above the box is the upper quartile line, and the blue solid line below the box is the lower quartile line. The top horizontal line is drawn on the basis of the upper quartile line at a distance of 1.5 times the interquartile range (the upper quartile minus the lower quartile) or the limit of the data (if the limit of the data is within 1.5 times the interquartile range). The bottom horizontal line is drawn in the same way. Abnormal data points outside the horizontal line are represented by \"+\". Judging from the median time (red solid line) and the size of the box (the area formed by the upper quartile and the lower quartile), the median value of the KMDS method is very low, and the box area is also very small. It can be seen from the RMSE box plot that the RMSE of the other four methods are ideal except for the larger median value and frame area of the KMDS-1 and KMDS-2 methods. This proves that the KMDS method can guarantee accuracy and has a greater advantage with regard to speed.