## 4 Experiment and Analysis
### Mango Dataset  
This dataset contains 58 near-infrared spectroscopic data, which are used to develop prediction models to determine vitamin C (VC) and total acidity (TA) of intact mango fruits ([[https://data.mendeley.com/datasets/ph57](https://data.mendeley.com/datasets/ph57) ynng46/1]([https://data.mendeley.com/datasets/ph57](https://data.mendeley.com/datasets/ph57) ynng46/1) (accessed on 11 November 2023)). The range of the parameter are as follows: VC (189.72-772.77) and TA (28.93-35.66). The figures indicate milligrams per 100 g of fresh mass. The best RMSEP and prediction variance of FPLS, PLS, OSCPLS, VCPLS, PoPLS, LoPLS, DOSC, OPLS, MSC, SNV, SGFilter, and Lasso are listed in Table 4. Meanwhile, this table also provides the boundaries for RMSEP and CS. Based on the RMSEP and variance data presented in Table 4, in TA prediction, FPLS exhibits the lowest RMSEP value of 0.2731 with a variance of 0.0191, suggesting excellent predictive performance and stability. The relatively high number of components (10) may have contributed to its accuracy. On the other hand, models like OSCPLS, VCPLS, DOSC,  
Figure 7: RMSEP values of Octane prediction using FPLS, PLS, OSCPLS, VCPLS, PoPLS, LoPLS, DOSC, OPLS, MSC, SNV, and SGFilter models with 1–16 components. Lasso is different from the previous methods. It is a feature selection method. The horizontal axis in the RMSEP graph represents the number of random experiments for Lasso. and Lasso have higher RMSEP values, indicating poorer predictive accuracy. For instance, Lasso has the highest RMSEP of 0.5035 with a variance of 0.0074, despite using only 1 component. This suggests that the model may be overfitting or not capturing the underlying patterns in the data well. Models such as PLS, PoPLS, LoPLS, OPLS, MSC, SNV, and SGFilter lie somewhere in the middle, with RMSEP values ranging from 0.3825 to 0.4433. 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 of components may depend on the specific characteristics of the data and the model. FPLS stands out as the top-performing model in TA prediction, while Lasso needs further improvement to achieve better predictive accuracy. In VC prediction, FPLS also exhibits a relatively low RMSEP value of 0.6988 with a variance of 0.0196, suggesting good predictive performance and stability. The use of 10 components may have contributed to its ability to capture more of the underlying patterns in the data. Models like OSCPLS, VCPLS, and Lasso have higher RMSEP values, indicating poorer predictive accuracy. For instance, Lasso has the highest RMSEP of 1.2863 with a variance of 0.0488, which may be due to its use of only 1 component, limiting its ability to model complex patterns. Models such as PLS, PoPLS, LoPLS, OPLS, MSC, SNV, and SGFilter lie somewhere in the middle, with RMSEP values ranging from around 1.08 to 1.10. 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. Figure 8a,b show mean RMSEP for the TA and VC varying with the increasing number of components, respectively. 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 4 for TA prediction and 3 for VC. These figures also validate the effectiveness of the FPLS method.