## 4 Experiment and Analysis  
In this section, we compare the accuracy of the proposed and alternative methods with four public datasets: Corn, Octane, Mango, and Soil Infrared Spectra datasets. These databases are commonly employed to evaluate regression-based analysis. In our experiments, the accuracy is calculated using Root Mean Squared Error of Prediction (RMSEP). The experiments were implemented with a laptop with a 2.80 GHz i7 processor, 16.0 GB RAM and Win10 system. In our experiments, we compared our FPLS with eight other methods: PLS [27], Orthogonal Signal Correction + PLS (OSCPLS) [13], Variance Constrained PLS (VCPLS) [28], Powered PLS (PoPLS) [29], Low-pass Filter + PLS (LoPLS), Direct Orthogonal Signal Correction + PLS (DOSC) [15], Orthogonal Projections to Latent Structures (OPLS) [14], Multiplicative Signal Correction (MSC), Standard Normal Variate (SNV), Savitzky-Golay (SG) Filtering, and Least Absolute Shrinkage and Selection Operator (Lasso) [30]. In our experiments, we randomly selected 70% of the sample as the training set, and the remaining 30% as the test set. We repeated the experiment 300 times. In our experiment, OSCPLS, OPLS, DOSC, and VCPLS settings are accordance with reference [28]. LoPLS is an unsupervised method, and thus, we specify the low-pass filter as \(h=\left[1/\sqrt{2},1/\sqrt{2}\right]\). The coefficients \(1/\sqrt{2}\) ensure that the filter preserves signal energy without excessive amplification or attenuation. It smooths out high-frequency noise while retaining low-frequency components of the signal, making it effective for denoising. The filter is computationally efficient, using just two coefficients, which is suitable for real-time processing. The equal weighting of current and previous samples minimizes phase distortion. In the theoretical analysis of FPLS, the local smoothness of the spectra was considered, which means that the length of the filter could not be too long. As \(N\) increases, \(\frac{w^{T}(Xsh)^{T}\gamma_{c}}{w^{T}X^{T}\gamma_{c}}\) gradually deviates from \(N\), because smoothness of spectra cannot be guaranteed after \(N\) increases, as shown in Figure 2. Thus, in our experiments, the range of length of filter was set to \([2,3,4]\). The conclusions drawn from filter length selection on the Octane, Mango, and soil datasets are consistent and will not be repeated here. The optimal number of components for the FPLS was determined through 10-fold cross-validation. For the SGFilter method, the order and frame length were also determined with 10-fold cross-validation. We compared the performance of different models in terms of \(RMSE\) on the testing set. \(RMSE\) is defined as the square root of the mean squared prediction error, which measures the average magnitude of the errors between predicted and actual values. \[RMSE=\sqrt{\frac{1}{n}\frac{n}{i}(y_{i}-\hat{y}_{i})^{2}}=\sqrt{\frac{1}{n}\frac {n}{i}(\hat{y}_{i}-y_{i})^{2}} \tag{31}\]  
The prediction and response of the \(i\)-th observation are denoted as \(\hat{y}_{i}\) and \(y_{i}\), respectively. The number of the test set is \(n\). The variance of prediction is \(\frac{1}{n}\sum_{i=1}^{n}\left(\hat{y}_{i}-\frac{1}{n}\sum_{i=1}^{n}\hat{y}_{i }\right)^{2}\). We also compare the performance of different models in terms of Cosine Similarity (\(CS\)) on the testing set. It is a commonly used method to measure the similarity between two non-zero vectors. \[CS(\hat{y},\mathbf{y})=\frac{\mathbf{y}^{T}\hat{\mathbf{y}}}{\|\mathbf{y}\|\|\hat{\mathbf{y}}\|}= \frac{\hat{\mathbf{y}}^{T}\mathbf{y}}{\|\mathbf{y}\|\|\hat{\mathbf{y}}\|} \tag{32}\]  
\(\hat{\mathbf{y}}\in R^{n\times 1}\) and \(\mathbf{y}\in R^{n\times 1}\) represent the predicted result and the reference, respectively. We then discussed the range of \(CS\) when \(RMSE\) is given, as well as the range of \(RMSE\) when \(CS\) is provided. For convenience of description, let \(RMSE\) be denoted as \(\alpha\). According to Equation (31), \(\hat{\mathbf{y}}\) lies on a hypersphere centered at \(\mathbf{y}\) with radius \(\sqrt{n}\alpha\).