## 3 Manifold Dimension Transformation of Raman Spectra of Oil-Paper Insulation
### Feature-Extraction Results on the Raman Spectra after Oil-Paper Insulation Aging  
To validate the correctness of the chosen value of the neighboring parameter \(k\) in the Isomap feature-extraction method mentioned earlier, the value of the \(k\) was sequentially changed within the range of 25 to 85. The corresponding two-dimensional Raman spectrum feature maps are shown in Figure 6.  
From Figure 6, it can be observed that when the neighboring parameter \(k\) is set to 45, the differences between different classes of samples are most pronounced. According to Equation (4), the calculated value of \(k\) is 44, and the difference between the two neighboring numbers is almost negligible. This proves the feasibility of using the method described in Equation (4) to determine the Isomap neighboring parameter value. Similarly, to validate the correctness of selecting the Isomap nearest-neighbor parameter \(k\) for the classification of ten aging states in Raman spectra, we conducted verification with different values of \(k\). Due to the relatively large number of aging state categories, the spatial map is sensitive to changes in the nearest neighbor parameter. Therefore, we limited the range of \(k\) to a small interval of 12-27 and varied it sequentially. The results are shown in Figure 7.  
From Figure 7, it can be observed that when the nearest neighbor parameter \(k\) is set to 17, the Raman spectra samples of different aging states in the three-dimensional space are most widely dispersed. This closely matches the nearest neighbor parameter value of 18 obtained from Equation (4), demonstrating the applicability of the nearest  
Figure 6: Isomap feature-extraction maps for classifying the four aging states with different values of the neighboring parameter. (**a**) \(k=25\); (**b**) \(k=45\); (**c**) \(k=65\); (**d**) \(k=85\). neighbor parameter selection method described in Equation (4) for the classification of ten aging states. Using the aforementioned manifold feature-extraction method, the Raman spectrum samples were categorized into ten groups (see Table 1). Taking into account the number of samples in each aging stage for the ten-state classification, the neighbor parameter was set to 18 according to Equation (4). Through the Isomap feature extraction, the sample dimensionality was reduced from 953 to 3, as depicted in Figure 8c. Similarly, MDS and PCA techniques were employed to decrease the sample dimensionality to 3, as shown in Figure 8a,b, respectively. Figure 8: Three-dimensional renderings of ten categories extracted via (**a**) MDS; (**b**) PCA; (**c**) Isomap method. Figure 7: Isomap feature extraction-maps for classifying the ten aging states with different values of the neighboring parameter. (**a**) \(k=12\); (**b**) \(k=17\); (**c**) \(k=22\); (**d**) \(k=27\). A comparison of the results obtained using three feature-extraction algorithms revealed that, after applying the Isomap method, the samples belonging to ten categories were distinctly separated from each other. Except for the third and fourth aging stage samples, the remaining samples within the same category exhibited a high level of clustering. Conversely, MDS and PCA techniques yielded lower degrees of sample aggregation within the same category and inferior discrimination between samples of different categories compared to the Isomap tool. Similarly, the Isomap, MDS, and PCA algorithms were employed to extract features for the four-state classification of Raman spectrum samples, reducing the data to two dimensions. The results are presented in Figure 9. As seen from the figure, the Isomap method allowed for the highest discrimination among different sample categories. In spite of a significant range between samples in the pre-aging and middle aging states, no overlap was observed between different categories of samples. In contrast, the extraction results of PCA and MDS algorithms showed interconnected and overlapping regions among samples before aging, in the middle aging states, and at the end of aging.