## 3 Results and Discussion
### PLS Calibration Model Based on siPLS Feature Variables Selection  
The PLS calibration model based on preprocessed spectral data showed a slight improvement in prediction performance over the original spectra. However, there were still some irrelevant variables in the spectra, as well as multicollinearity among the variables, which not only increased the modelling time and complexity, but also reduced the robustness and accuracy of the model [44]. Therefore, a model is constructed by extracting the relevant variables of the characteristic biodiesel peaks from the whole spectrum as a way to improve the predictive performance of the model [46]. In this study, the method of siPLS was employed to divide the preprocessed spectral matrix (D1-MSC) into 10-20 equal intervals. Since the siPLS algorithm requires selecting combinations of spectral bands and the computational process is complicated, the randomly selected number of sub-interval combinations (j) was typically less than 5. In this experiment, j was set to 2, 3, and 4. Table 3 presents the results of feature subinterval selection by siPLS for different values of k and j. It can be observed that as the quantity of separation intervals decreases and the combination intervals rises, the RMSECV value decreases, indicating the selection of more effective spectral regions. Considering that the RMSEP value is based on the prediction error calculated from the test set data, the high RMSEP value of (siPLS (2)) indicates that the model tends to over-fit under this parameter. However, this problem is overcome as the number of combined intervals increases and the stability of the proposed method is improved. When k = 12 and j = 4, the corresponding R\({}^{2}\)CV was 0.9842, and the minimum RMSECV was achieved at 0.1329% (\(v/v\)).  
Figure 4 illustrates the feature variable intervals selected by siPLS, which closely corresponded to the characteristic peaks of C=C stretching and C-H bending in the original spectra. This confirms the feasibility of the siPLS-feature variable selection method. The calculation error expresses the mathematical significance of the model, while the screening of characteristic peaks better corresponds to the physicochemical significance of the study. The optimization of LVs was carried out simultaneously with siPLS variable selection. When LVs were set to 5, the RMSECV of the optimal combination model reached a stable state. The calibration model was then established using the optimal combination (k = 12, j = 4) and the best LVs. The ability of the model to capture variations in measurement data was evaluated using R\({}^{2}\), which represents the ability of the independent variables to explain the induced variables. A higher R\({}^{2}\) value indicates better fitting performance. The RMSE, which represents the predictive accuracy of the model, is the primary indicator for evaluating the performance of regression models. A lower RMSE value is desirable. RPD is a measure of the predictive power of the model, with higher RPD values indicating better regression performance. Models with RPD values greater than 6 are considered to have good regression performance. The variable selection methods in Table 3 (siPLS (4)) all have better performance on RMSECV compared to PLS (0.9812) after full spectral  
Figure 3: Predictive performance of PLS models with different parameters ((**a**): the effect of latent variables; (**b**): the effect of different smoothing points of D1 st).  
preprocessing, indicating the necessity of using variable selection as a preprocessing step in spectral detection, and the D1 st-MSC-siPLS model showed a significant improvement in predictive performance. The R\({}^{2}_{\text{CV}}\) and R\({}^{2}_{\text{P}}\) values were both above 0.98, and the RMSEP decreased from 0.1413% (\(v/v\)) to 0.1367% (\(v/v\)). The MREP decreased from 7.10% to 6.31%, while the RPDp was 9.95, indicating the successful modeling of the model.