## 3 Results and Discussion
### Comparison of Different PLS Calibration Models  
The performance of the model varied with different preprocessing methods and variable selection methods. Two additional variable selection methods, iPLS and biPLS, were compared to siPLS. To further validate the ability of the D1-MSC-siPLS model to quantify biodiesel, as shown in Table 4, the feature wave numbers were reduced to 87, 261, and 378, selected by the iPLS biPLS and siPLS algorithms, respectively. They were used as input variables to the PLS model to build a quantitative biodiesel model. The siPLS picks out variables that are more included in the location of feature peaks, reflecting the superiority of the algorithms. The cross-validated R\({}^{2}_{\text{CV}}\) of the three feature variable extraction methods was similar, ranging from 0.9738 to 0.9842. However, the RMSECV values of iPLS and biPLS, after variable extraction, decreased from 0.1912% (\(v/v\)) to 0.1692% (\(v/v\)) and 0.1811% (\(v/v\)), respectively, which was not superior to the RMSECV value of siPLS (0.1329% (\(v/v\))). Additionally, both methods had LVs greater than 5, indicating a risk of overfitting. The RMSEP values also did not improve compared to D1 st-MSC-siPLS. Among the models, D1 st-MSC-biPLS (RMSEP = 0.1879% (\(v/v\))) and D1 st-MSC-iPLS  
Figure 4: Feature variables selected using siPLS. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Variable** \\ **Extract** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Interval** \\ **Number** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Variable** \\ **Number** \\ \end{tabular} } & \multicolumn{2}{c}{**LOO-CV**} & \multicolumn{4}{c}{**Prediction Set**} \\ \cline{4-9}  & & & & **R\({}^{2}_{\text{CV}}\)** & **RMSECV\% (v/v)** & **R\({}^{2}_{\text{P}}\)** & **RMSEP\% (v/v)** & **MREP\%** & **RPD\({}_{\text{P}}\)** \\ \hline \multirow{4}{*}{siPLS (2)} & 12 & 189 & 0.9771 & 0.1710 & 0.9771 & 0.2386 & 11.65 & 6.61 \\  & 15 & 174 & 0.9378 & 0.3506 & 0.8837 & 0.3579 & 26.54 & 2.93 \\  & 18 & 161 & 0.9249 & 0.3559 & 0.9053 & 0.3346 & 20.09 & 3.25 \\  & 20 & 149 & 0.9337 & 0.3440 & 0.8207 & 0.3682 & 29.38 & 2.36 \\ \hline \multirow{4}{*}{siPLS (3)} & 12 & 284 & 0.9839 & 0.1355 & 0.9682 & 0.1507 & 8.51 & 5.61 \\  & 15 & 261 & 0.9839 & 0.1355 & 0.9742 & 0.1817 & 14.44 & 6.23 \\  & 18 & 241 & 0.9809 & 0.1568 & 0.9619 & 0.1986 & 13.29 & 5.12 \\  & 20 & 224 & 0.9814 & 0.1586 & 0.9888 & 0.1715 & 7.99 & 9.45 \\ \hline \multirow{4}{*}{siPLS (4)} & 12 & 379 & 0.9842 & 0.1329 & 0.9899 & 0.1367 & 6.31 & 9.95 \\  & 15 & 348 & 0.9839 & 0.1409 & 0.9765 & 0.2652 & 13.47 & 6.52 \\ \cline{1-1}  & 18 & 321 & 0.9811 & 0.1563 & 0.9636 & 0.2916 & 12.76 & 5.24 \\ \cline{1-1}  & 20 & 299 & 0.9806 & 0.1622 & 0.9891 & 0.1780 & 8.34 & 9.58 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cross-validation results of PLS calibration models with different separation intervals and different combinations of interval numbers. (RMSEP = 0.2263% (\(v/v\))) showed marginally improvements in external validation results, but no improvement was observed in cross-validation results. The optimal preprocessing full-spectrum model, D1 st-MSC, had an R2P of 0.9932, which was higher than D1 st-MSC-siPLS (R2P = 0.9899). However, when R2 is within a reasonable range, MRE is the primary consideration. Therefore, D1 st-MSC-siPLS was determined as the optimal model, as it improved the R2 CV of the calibration set from 0.9738 to 0.9842, decreased the RMSECV from 0.1912% (\(v/v\)) to 0.1329% (\(v/v\)), significantly reduced the number of variables from 1044 to 378, decreased the modeling time, and improved the model's predictive performance. Figure 5 clearly shows the correlation between the predicted values from the D1 st-MSC-siPLS calibration model and the sample standard reference values for biodiesel. Clearly, they exhibited a strong linear relationship. Additionally, the MREP value, in particular, significantly decreased after variable selection.