## 2 Related Work
Signal preprocessing methods encompass low-pass filtering, MSC (Multiplicative Scatter Correction) [16], SNV (Standard Normal Variate) [17], and SG Filter (Savitzky-Golay Filter) [18], among others. These methods, being unsupervised (non-data-driven), have limited potential for enhancing model performance. Motivated by probabilistic PCA and probabilistic curve-fitting, a new Probabilistic-PLSR (PPLSR) model is proposed and the Estimation Maximization (EM) algorithm is employed to estimate the parameters [19]. By comparing the results with those from traditional least squares (LS) methods and PLS, PPLSR is shown to be more robust and accurate. This method assumes that the parameters are all Gaussian-distributed, which makes it easier to solve. Data collected in modern industrial processes often exhibit complex non-Gaussian and multimodal characteristics. In order to address these problems, a robust mixture probabilistic PLS (RMPPLS) model-based soft sensor is developed [20], where two different kinds of hidden variables are introduced in the formulated model structure. The multivariate Laplace distribution is employed for robust modeling, and a hybrid form of the probabilistic PLS model is adopted for multimodal description. When the assumed distribution is consistent with the true distribution of the data, the model will achieve good results; when it is not, the effect may not be good. [PERSON] extends regularized least squares regression models by incorporating the KPLS method [21]. The KPLS approach maps input data into a high-dimensional feature space where a linear PLS model is constructed. Good generalization properties are achieved by estimating regression coefficients appropriately and selecting a suitable kernel function. [PERSON] et al. [22] have analyzed the limitations of PLS in Euclidean spaces and employ Riemannian optimization on manifolds for better numerical properties. The study focuses on two types of manifolds: the generalized Stiefel manifold and the generalized Grassmann manifold. Algorithms are developed for optimizing PLS models on these manifolds, including SIMPLS with the generalized Grassmann manifolds (PLSRGGr) and SIMPLS with product manifolds (PLSRGStO). Experimental results show that the proposed models and algorithms are more robust and have better performance than traditional methods in Euclidean space. [PERSON] [23] have proposed Partial Least Trimmed Squares Regression (PLTS) as a robust regression technique, focusing on addressing outliers in the dataset. The PLTS model incorporates trimming methods to mitigate the influence of outliers. The study demonstrates the application of PLTS on datasets such as fish and biscuit, highlighting its effectiveness through comparisons with other methods. The results showcase reduced error rates, particularly in the presence of outliers, demonstrating the PLTS model's robustness and accuracy in predictive modeling. The existing methods for handling Partial Least Squares (PLS) regression exhibit several limitations, including the following:  
1. Static Preprocessing Filtering: Traditional filtering approaches are often applied as a preprocessing step before modeling. The limitation of this method lies in its reliance on preset filter coefficients, which may not be optimal for all datasets or specific data characteristics, potentially leading to suboptimal model performance. 2. Sensitivity to Noise: Standard PLS models are highly sensitive to noise, especially when dealing with complex, non-Gaussian, or multimodal data distributions. While probabilistic PLS models attempt to mitigate this issue by assuming Gaussian-distributed parameters, their effectiveness can diminish significantly if the actual data distribution does not align with these assumptions. 3. Lack of Adaptability: Many existing PLS extensions (e.g., kernel PLS, manifold-based regression) enhance model robustness but lack flexibility in dynamically adapting to different types of noise and selecting features that best represent the underlying data structure.