## 3 Manifold Dimension Transformation of Raman Spectra of Oil-Paper Insulation
### Processing the Raman Data on Oil-Paper Insulation Aging Via XGBoost Discrimination Method  
XGBoost uses decision trees (CART) as base classifiers, and the aging stages determination of the previous \(t\) trees provides the error source for the \(t\)+1 tree. Figure 9: Four categories of two-dimensional renderings after (**a**) MDS; (**b**) PCA; (**c**) Isomap feature extraction. Its objective function is expressed as:  
\[obj^{t}=\sum\limits_{i=1}^{n}l(\mathbf{y}_{i},\overline{\mathbf{y}}_{i})+\sum\limits_{i= 1}^{t}\Omega(f_{i}), \tag{16}\]  
where \(l(\mathbf{y}_{i},\overline{\mathbf{y}}_{i})\) is the loss function of each Raman sample, representing the error value between the actual label \(\mathbf{y}_{i}\) of the spectral sample and the predicted label \(\overline{\mathbf{y}}_{i}\) of the \(t-1\) decision tree; \(\sum\limits_{i=1}^{t}\Omega(f_{i})\) serves as the regularization term, indicating the complexity of the decision tree model, aiming to prevent overfitting caused by a small number of Raman spectroscopy samples. Among them, the loss function and regularization term are defined as follows:  
\[l(\mathbf{y}_{i},\overline{\mathbf{y}}_{i})=\sum\limits_{i=1}^{n}(\mathbf{y}_{i}-\overline {\mathbf{y}}_{i})^{2} \tag{17}\]  
\[\Omega(f_{k})=\gamma T+\frac{1}{2}\lambda\sum\limits_{j=1}^{T}w_{j}^{2}, \tag{18}\]  
where \(T\) is the number of leaves; \(w_{j}\) is the weight of the \(j\)-th leaf; \(\gamma\) is the contraction coefficient; and \(\lambda\) is the norm coefficient. The second-order [PERSON] expansion of the objective function is as follows:  
\[obj=\sum\limits_{i=1}^{n}\left[g_{i}f_{t}(x_{i})+\frac{1}{2}h_{i}f_{t}^{2}(x) \right]+\gamma T+\frac{1}{2}\lambda\sum\limits_{j=1}^{T}w_{j}^{2}, \tag{19}\]  
where \(g_{i}\) and \(h_{i}\) are the first and second derivatives of the loss function, respectively. The above formula can be simplified by the quadratic matrix optimization theory:  
\[obj=\sum\limits_{j=1}^{T}\left[G_{j}w_{j}+\frac{1}{2}(H_{j}+\lambda)w_{j}^{2} \right]+\gamma T, \tag{20}\]  
where \(G_{i}=\sum\limits_{i=1}^{n}g_{i}\), \(H_{i}=\sum\limits_{i=1}^{n}h_{i}\). Based on these expressions, the optimal weights can be obtained as Since the only difference between the four-class and ten-class classification is the number of categories, they exhibit consistent sensitivity to hyperparameters. Therefore, the ten-class aging state classification, which has a higher number of categories, was chosen to study the sensitivity of hyperparameters. After applying Isomap feature extraction to the Raman spectroscopy samples to transform them into a three-dimensional space while keeping other hyperparameters at their default values, the accuracy of XGBoost for classifying the ten aging states of oil-paper insulation is evaluated. The hyperparameter values are systematically varied one by one, and for each change, ten-fold cross-validation is performed. The average classification accuracy of XGBoost and its relationship with the hyperparameters are shown in Figure 10.  
According to Figure 8, it can be observed that, as the value of n_estimators increases, the average classification accuracy gradually improves. Conversely, as the gamma value increases, the average classification accuracy shows a declining trend. The values of max_depth, subsample, learning_rate, and reg_alpha do not exhibit a clear relationship with the accuracy. The values of colsample and reg_lambda have minimal impact on the accuracy. To further quantify the sensitivity of the XGBoost model's classification accuracy to hyperparameters, we calculated the corresponding standard deviation of accuracy for each hyperparameter change. A larger standard deviation of accuracy indicates a higher