## 3 Manifold Dimension Transformation of Raman Spectra of Oil-Paper Insulation
### Isometric Mapping Feature-Extraction Method  
Regarding the specific classification of oil-paper insulation samples, the 175 aging samples are defined as follows:  
\[\mathbf{x}_{i},\;i=1,2,3,\ldots,n, \tag{1}\]  
where \(\mathbf{x}_{i}\) represents the Raman spectrum vector of the \(i\)-th aging sample. The total Raman spectrum samples are denoted as \(\mathbf{X}=\left[\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{x}_{3},\ldots,\mathbf{x}_{n}\right]^{T} \in\mathbf{R}^{n\times n}\), where \(n\) is the number of samples, and \(m\) is the number of data points collected for each Raman spectrum sample, representing the data dimension. As shown in Figure 5, the manifold distance between samples \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\) is longer than the Euclidean distance. If the distance between Raman spectrum samples of the same  
Figure 4: Raman spectrum preprocessing. (**a**) Single Raman spectrum preprocessing; (**b**) Raman spectrum preprocessed images of different aging days. aging state is determined as the Euclidean distance, and the distance between samples of different aging states is determined as the manifold distance, it can increase the relative distance between different class samples after feature extraction, thereby highlighting the discrimination between samples of different aging states. The formula for calculating the Euclidean distance between two Raman spectrum samples is as follows:  
\[d=\sqrt{\sum\limits_{j=1}^{m}{(\mathbf{x}_{ij}-\mathbf{x}_{kj})^{2}}}. \tag{2}\]  
Let \(\mathbf{D}\) be the adjacency matrix that represents the Euclidean distances between each pair of Raman spectrum samples, which has the following form:  
\[\mathbf{D}= \begin{bmatrix}d_{11}&d_{12}&\infty&\cdots&\infty\\ d_{21}&d_{22}&\infty&\cdots&\infty\\ \infty&d_{32}&d_{33}&\cdots&\infty\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \infty&d_{n2}&\infty&\cdots&d_{nm}\end{bmatrix} . \tag{3}\]  
Let \(k\) define the number of neighbors, where the high-dimensional distances between the \(i\)-th Raman spectrum and its \(k\) nearest-neighboring spectra are set as Euclidean distances. The distances between the remaining samples and the \(i\)-th spectrum are defined as \(\infty\). Based on the four or ten categories of aging states, the value of \(k\) is determined as shown in Equation (4). \[k=\frac{\sum\limits_{I=1}^{S}{N(\mathbf{X}_{J})}}{S},\ J=1,2,\ldots,S, \tag{4}\]  
where \(N(\mathbf{X}_{J})\) represents the number of Raman spectrum samples in the \(J\)-th class, and \(S\) refers to the number of aging states. Recalculate the shortest distance matrix for all Raman spectral data points, and redefine the manifold distance between any two Raman spectra \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\). If:  
\[d_{ij}>d_{ik}+d_{kj}, \tag{5}\]  
then:  
\[d^{\prime}_{ij}=d_{ik}+d_{kj}. \tag{6}\]  
Figure 5: Aging sample distance diagram. Reconstructing the Raman spectra sample manifold distance matrix by substituting the manifold distance \(d^{\prime}_{ij}\) into the Euclidean distance adjacency matrix results in:  
\[\mathbf{D^{\prime}}= \begin{bmatrix}d_{11}&d_{12}&d^{\prime}_{13}&\cdots&d^{\prime}_{ 1n}\\ d_{21}&d_{22}&d^{\prime}_{23}&\cdots&d^{\prime}_{2n}\\ d^{\prime}_{31}&d_{32}&d_{33}&\cdots&d^{\prime}_{3n}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ d^{\prime}_{n1}&d_{n2}&d^{\prime}_{n3}&\cdots&d_{nn}\\ \end{bmatrix} . \tag{7}\]  
The Raman spectral data in a low-dimensional space can be defined as:  
\[\mathbf{Z}=\{\mathbf{z}_{1},\mathbf{z}_{2},\ldots,\mathbf{z}_{s}\}\in\mathbf{R}^{d^{\prime}\times n}, \tag{8}\]  
where \(d^{\prime}\) is the spatial dimension after feature extraction.