## 3 KMDS Method
### Determination of the Space Dimension \(d^{\prime}\) after Dimensionality Reduction
In addition, this method has very little fluctuation in terms of the time spent on these ten modeling scenarios, and the error convergence of KMDS-3, KMDS-4 and KMDS-5 is relatively stable. The following characteristics can be further explained on the basis of the data shown in Algorithm 1 and Figures 1 and 2. (1) The running time of the KMDS method is greatly shortened compared to Kriging modeling. When adding 100 sample points, its modeling time is already 10-60 times faster than the Kriging method. When 200 sample points are added, the time spent during the KMDS method is about 18-94 times shorter than that of Kriging method. (2) In the low-dimensional space after dimensionality reduction, the higher the dimensionality, the longer the modeling time required by the KMDS method. For example, KMDS-5 took 15.5642 s to model the 220 sampling points, while KMDS-1 only took 2.9883 s. However, compared with the 279.4006 s required by the Kriging method, the actual time they took is not worth mentioning. (3) In terms of error, it is obvious that as the dimensionality of the low-dimensional space increases, the number of errors produced by the KMDS model becomes lower and lower. The error score of KMDS-1 and KMDS-2 is very large compared with the Kriging model, but when \(d^{\prime}\) is 3, the error score of KMDS-3 is very close to that of the Kriging model. Even though the RMSE will become lower and lower as the dimension \(d^{\prime}\) increases, the rate of decline is almost invisible. In addition, these figures also show that KMDS-1 and KMDS-2 not only have large number of errors, but also have poor stability. However, KMDS-3, KMDS-4 and KMDS-5 not only produce fewer errors, but they are also more stable in comparison to KMDS-1 and KMDS-2. Based on the above examples and analysis, and balancing the requirements of accuracy and efficiency, it is reasonable to reduce the high-dimensional problem to a three-dimensional problem. The reason is that when \(d^{\prime}\) is 3, the model error score of KMDS-3 is very close to that of the Kriging model, which can meet the accuracy requirements, and the time taken is dozens of times faster than the Kriging model. When \(d^{\prime}\) is greater than 3, the KMDS-4 and KMDS-5 methods do have fewer errors, but compared to KMDS-3, the error score is only reduced a little, while the time has doubled. In addition, Sammon mapping [36] was used to further verify the dimensionality reduction effect of the KMDS method in this work. Sammon mapping achieves dimensionality reduction by minimizing the distance difference between corresponding points in the two spaces of the original space and the low-dimensional space; that is, by minimizing the error function of Equation (27):  
\[E=\frac{1}{\sum\limits_{i<j}dist(\mathbf{x}_{i},\mathbf{x}_{j})}\times\sum\limits_{i< j}\frac{\left[dist(\mathbf{x}_{i},\mathbf{x}_{j})-dist(\overline{\mathbf{x}}_{i},\overline{ \mathbf{x}}_{j})\right]^{2}}{dist(\mathbf{x}_{i},\mathbf{x}_{j})} \tag{27}\]  
The above error is calculated for the original data and low-dimensional data of each modeling process, and the average error is taken as the final error. Accordingly, Table 2 shows the results of the error given by Equation (27). It can be seen from Table 2 that when \(d^{\prime}\) is 3, the range of the distance error score between the two spaces is 0.4 to 0.5, which is very small. It also proves the effectiveness of the KMDS dimensionality reduction method.