## 4 Experiment and Analysis
### Octane Dataset Analysis  
This dataset contains 39 near-infrared spectra, which need to be modeled to predict the content of Octane. The wavelength range is 1100-1550 nm with 2 nm intervals. The range of Octane is 86.7-92.4%. We use the original spectra from the dataset for modeling and analysis [31; 32]. The best mean RMSE values for FPLS, PLS, OSCPLS, VCPLS, PoPLS, LoPLS, DOSC, OPLS, MSC, SNV, SGFilter, and Lasso are listed in Table 3. The numbers in parentheses indicate the number of components. Table 3 also presents the upper and lower bounds of RMSE, as well as the lower bound of CS. Based on the RMSE and variance values presented in Table 3, we found that FPLS stands out with the lowest RMSE value of 0.0645, indicating its exceptional predictive accuracy. The moderate variance suggests stability in its predictions, despite the relatively high number of components (15). PLS follows closely with a competitive RMSEP and variance, making it a viable alternative. OSCPLS and VCPLS, while offering comparable RMSE values, exhibit higher variance, potentially indicating less stability. On the other hand, methods like PoPLS and LoPLS offer lower variance but at a cost of higher RMSEP values, suggesting a trade-off between stability and accuracy. DOSC  
Figure 6: (a) \(w^{T}(X*h)^{T}Yc\) converges as iteration increases. (b) Correlation between the projection of the test samples of PLS, FPLS models on vector \(w\), and the moisture content. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{6}{c}{**Octane**} \\ \cline{2-7}  & **RMSEP** & **Variance** & **CS** & **Lower RMSEP** & **Upper RMSEP** & **Lower CS** \\ \hline FPLS & 0.0645 (16) & 0.0165 & 0.9998 & 0.0015 & 2.7065 & 0.9994 \\ PLS & 0.0949 (3) & 0.0166 & 0.9911 & 0.0796 & 2.2716 & 0.9894 \\ OSCPLS & 0.0883 (2) & 0.0045 & 0.9910 & 0.0766 & 2.2711 & 0.9898 \\ VCPLS & 0.4435 (9) & 0.0413 & 0.9798 & 0.4752 & 2.4115 & 0.9736 \\ PoPLS & 0.3036 (3) & 0.0296 & 0.9477 & 0.4735 & 2.8059 & 0.9352 \\ LoPLS & 0.0950 (3) & 0.0167 & 0.9899 & 0.0811 & 2.7219 & 0.9892 \\ DOSC & 0.0885 (2) & 0.0043 & 0.9990 & 0.0766 & 2.2711 & 0.9898 \\ OPLS & 0.0949 (2) & 0.0049 & 0.9988 & 0.0766 & 2.2711 & 0.9898 \\ MSC & 0.0843 (2) & 0.0049 & 0.9995 & 0.0038 & 2.7065 & 0.9991 \\ SNV & 0.0945 (2) & 0.0049 & 0.9994 & 0.0045 & 2.7065 & 0.9988 \\ SGFilter & 0.0853 (2) & 0.0049 & 0.9990 & 0.0796 & 2.2716 & 0.9894 \\ Lasso & 0.4313 (1) & 0.0354 & 0.9894 & 0.0796 & 2.2716 & 0.9750 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Quantitative analysis results in terms of RMSEP, prediction variance, Lower and Upper boundaries of RMSEP and lower boundary of CS. The proposed FPLS is compared to PLS, OSCPLS, VCPLS, PoPLS, LoPLS, DOSC, OPLS, MSC, SNV, SGFilter, and Lasso using the Octane dataset. and OPLS present intermediate RMSEP values with moderate variance, balancing both aspects. MSC, SNV, and SGFilter, despite their simplicity, demonstrate competitive RMSEP values with varying degrees of variance. Lasso, with its significantly higher RMSEP and moderate variance, may not be the optimal choice for predictive accuracy. The number of components for each method also plays a role, as more components can potentially increase complexity and computational cost. The boundaries of RMSEP and CS are consistent with Equations (33) and (37). Figure 7 shows that the mean RMSEP of Octane prediction varies with an increasing number of components. Among these methods, FPLS exhibits a faster decline in RMSEP compared to other approaches. Furthermore, beyond a certain number of components, the RMSEP of FPLS becomes significantly smaller than that of other methods. Specifically, the number of components is observed to be 3 for Octane prediction. This figure also validates the effectiveness of the proposed FPLS model. Prediction variance of FPLS is essentially consistent with that of PLS, which is consistent with our analysis in Section 3.3.