## 2 Related Work  
PLS regression is a statistical technique that differs from principal components regression. Instead of identifying hyperplanes that maximize the variance between the response and independent variables, PLS constructs a linear regression model by projecting both the predictor and response variables into a new space. This projection results in a bilinear factor model, as both the \(X\) (predictor) and \(Y\) (response) data are mapped to this new space. A variant of PLS, known as PLS discriminant analysis (PLS-DA), is used when the response variable is categorical. PLS is used to explore the fundamental relationships between two matrices, \(X\) and \(Y\), by modeling the covariance structures in these spaces using a latent variable approach. A PLS model aims to identify the multidimensional direction in the \(X\) space that best explains the maximum multidimensional variance direction in the \(Y\) space. PLS regression is particularly effective when the number of predictor variables exceeds the number of observations, or when there is significant multicollinearity among the \(X\) values. Standard regression would fail under these circumstances unless regularization is applied [4; 5; 6]. Given the predictor and response matrices \(X\in R^{n\times m}\) and \(Y\in R^{n\times P}\), PLS iteratively repeats the following steps \(k\) times to find the directions \(w\) and \(c\) to maximize covariance in input and output space, and then performs least squares regression on the input score. The loadings are chosen so that the scores form an orthogonal basis, which is achieved with deflation process. The model can be written as \(X=TP^{T}+E,Y=UQ^{T}+F\), where \(T,U\) are score matrices and \(P,Q\) are loading matrices. A number of variants of PLS exist for estimating the factor and loading matrices \(T\), \(U\), \(P\) and \(Q\). Some PLS algorithms are only appropriate for the case where \(Y\) is a column vector, while others deal with the general case of a matrix \(Y\). Algorithms also differ in whether they estimate the factor matrix \(T\) as an orthogonal (that is, orthonormal) matrix or not [7; 8; 9; 10]. The final prediction will be the same for all these varieties of PLS, but the components will differ. PLS1 is a widely used algorithm appropriate for the vector \(Y\) case [11]. It estimates \(T\) as an orthonormal matrix. This algorithm features deflation of the matrix \(X\), but deflation of the vector \(y\) is not performed, as it is not necessary. Another extension of PLS regression, named L-PLS for its L-shaped matrices, connects three related data blocks to improve predictability [12]. In brief, a new \(Z\) matrix, with the same number of columns as the \(X\) matrix, is added to the PLS regression analysis and may be suitable for including additional background information on the interdependence of the predictor variables. The performance of PLS is affected by the noise, and the approaches to address this issue mainly include (1) calibrating the signal to remove components orthogonal to the response; (2) preprocessing the model before performing PLS modeling; (3) establishing probabilistic PLS models to estimate the impact of noise on model performance; (4) employing kernel methods for nonlinear modeling of the signal; (5) building PLS models on manifolds; and (6) adopting robust objective functions. The methods for signal correction mainly include OSC [13], OPLS [14], DOSC [15], etc. The core of these methods is to identify and remove the components in the predictor variables \(X\) that are orthogonal to the response variable \(Y\) under certain constraint conditions, in order to enhance the performance of the PLS model. Essentially, they assume that the noise in the predictor variables is orthogonal to \(Y\). If the orthogonality assumption does not hold, then the effectiveness of these methods may not be ideal.