## 4 Experiment and Analysis
### Soil Available Nitrogen Dataset  
At the Yesheng Gongmi planting base in Wuzhong City, Ningxia, 75 soil samples were collected from plot 2 (954 mu, 1 mu \(\approx\) 0.1647 acres) ([[https://www.scidb.cn/en/detail?dataSetId=93899](https://www.scidb.cn/en/detail?dataSetId=93899) ccb63054b5 fb663c010d88892c8&version=V1]([https://www.scidb.cn/en/detail?dataSetId=93899](https://www.scidb.cn/en/detail?dataSetId=93899) ccb63054b5 fb663c010d88892c8&version=V1) (accessed on 2 December 2023)). After removing impurities, the collected soil samples were allowed to air-dry naturally. The dried soil samples were then manually ground and sieved using a 60-mesh sieve, according to the requirements of the analysis project. The soil samples were sealed in self-sealing bags and uniformly numbered [33]. This experiment aims to predict the Soil Available Nitrogen content, which ranges from 39.48 to 86.73 (mg/kg). The best mean RMSEP and corresponding prediction variances of FPLS, PLS, OSCPLS, VCPLS, PoPLS, LoPLS, DOSC, OPLS, MSC, SNV, SGFilter, and Lasso are listed in Table 5. The figure in parentheses indicates the corresponding number of components when the RMSEP reaches its minimum value. According to Table 5, FPLS exhibits a relatively low RMSEP value of 4.2388 with a variance of 0.5872, suggesting good predictive performance and stability. The use of 18 components may have contributed to its ability to capture more of the underlying patterns in the data. On the other hand, models like OSCPLS, VCPLS, DOLS, and Lasso have higher RMSEP values, indicating poorer predictive accuracy. For instance, Lasso has the highest RMSEP of 8.6914 with a variance of 1.5812, which may be due to its use of only 1 component, limiting its ability to model complex patterns in the data. Models such as PLS, PoPLS, LoPLS, OPLS, MSC, SNV, and SGFilter lie somewhere in the middle, with RMSEP values ranging from around 6.06 to 6.16. Their variances are also relatively low, indicating that these models are somewhat stable in their predictions. The number of components used in these models varies, suggesting that the optimal number may depend on the specific characteristics of the data and the problem being addressed. This table also provides the boundaries of RMSEP and CS, which are consistent with our analysis at the beginning of Section 2. Figure 10 shows the nitrogen mean RMSEP varying with the 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, for FPLS, the optimal \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{5}{c}{**Nitrogen**} \\ \cline{2-7}  & **RMSEP** & **Variance** & **CS** & **Lower RMSEP** & **Upper RMSEP** & **Lower CS** \\ \hline FPLS & 4.2388 (18) & 0.5872 & 0.9667 & 2.8807 & 9.5146 & 0.9506 \\ PLS & 6.0781 (7) & 0.7593 & 0.9561 & 3.7772 & 9.5697 & 0.9283 \\ OSCPLS & 7.4131 (1) & 1.0854 & 0.9356 & 5.4830 & 9.6807 & 0.9118 \\ VCPLS & 8.0386 (1) & 0.9987 & 0.9364 & 5.4172 & 9.6763 & 0.9040 \\ PoPLS & 6.4054 (5) & 0.7488 & 0.9410 & 5.0373 & 9.6509 & 0.9243 \\ LoPLS & 5.9168 (6) & 0.7523 & 0.9427 & 4.8964 & 9.6416 & 0.9303 \\ DOLS & 7.4078 (1) & 1.0842 & 0.9306 & 5.8935 & 9.7087 & 0.9119 \\ OPLS & 6.0622 (6) & 1.2427 & 0.9330 & 5.6967 & 9.6953 & 0.9285 \\ MSC & 6.1594 (7) & 0.6804 & 0.9331 & 5.6885 & 9.6947 & 0.9273 \\ SNV & 6.1596 (7) & 0.6798 & 0.9331 & 5.6885 & 9.6947 & 0.9273 \\ SGFilter & 6.0766 (7) & 0.7792 & 0.9212 & 6.6591 & 9.7624 & 0.9284 \\ Lasso & 8.6914 (1) & 1.5812 & 0.9212 & 6.6591 & 9.7624 & 0.8958 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Quantitative analysis results in terms of prediction variance, RMSEP—including its lower and upper bounds—as well as CS and its lower bound. The proposed FPLS is compared to PLS, OSCPLS, VCPLS, PoPLS, LoPLS, DOSC, OPLS, MSC, SNV, SGFilter, and Lasso using the Soil Available Nitrogen dataset. number of components is observed to be 6.