## 3 Filter Learning-Based Partial Least Square (FPLS) Regression and Algorithm
### Model Performance Analysis  
It is difficult to directly analyze the model because it involves the eigenvalues and eigenvectors of \(X^{T}\gamma Y^{T}X\). In this section, for simplicity without losing generality, we consider the PLS-one model. Suppose \(X=[x_{ij}]\in R^{n\times m}\), \(y=[y_{1},\cdots,y_{n}]^{T}\), \(w=[w_{1},\cdots,w_{m}]^{T}\). The Lagrangian function is then constructed as  
\[\mathcal{L}=w^{T}X^{T}Y-\lambda(w^{T}w-1)=\sum_{i=1}^{n}y_{i}\sum_{j=1}^{m}x_{ ij}w_{j}-\lambda(w^{T}w-1) \tag{24}\]  
Calculating the partial derivative of \(\mathcal{L}\) with respect to \(w_{i}\), we have  
\[\frac{\partial\mathcal{L}}{\partial w_{1}}=\sum_{i=1}^{n}x_{i1}y_{i}-2\lambda w _{1}=0,\cdots\frac{\partial\mathcal{L}}{\partial w_{m}}=\sum_{i=1}^{n}x_{im}y _{i}-2\lambda w_{m}=0 \tag{25}\]  
Thus,  
\[w^{T}X^{T}Y\propto Y^{T}XX^{T}Y=\sum_{i=1}^{m}\left(\sum_{j=1}^{n}y_{j}x_{ji} \right)^{2} \tag{26}\]  
If a filter with length \(N\) is employed to filter \(X\), without loss of generality, the first term of \(\sum_{i=1}^{m}\left(\sum_{j=1}^{n}y_{j}x_{ji}\right)^{2}\) is equal to \((y_{1}(h_{1}x_{11}+\cdots\ +h_{N}x_{1N})+\cdots+y_{n}(h_{1}x_{n1}+\cdots+h_{N}x_{nN}))^{2}\). The spectrum is usually smooth in a small neighborhood. If \(h_{i}\) has the same sign for all \(i=1,\cdots,N\), then  
\[\max y_{1}(h_{1}x_{11}+\cdots+h_{N}x_{1N})=\max y_{1}\sum_{i=1}^{N}h_{i}x_{11} =\sqrt{N}y_{1}x_{11} \tag{27}\]  
when \(h_{i}=\frac{1}{\sqrt{N}}\) for all \(i=1,2,\cdots,N\). By appropriately choosing \(N\), we can approximate \(\max w^{T}(X*h)^{T}Y\) as \(Nw^{T}X^{T}Y\). Assuming \(\max w^{T}X^{T}Y>0\), we have \(0<w^{T}(X*h)^{T}Y<Nw^{T}X^{T}Y\), which can be rewritten as  
\[0<\frac{w^{T}(X*h)^{T}Y}{w^{T}X^{T}Y}<N \tag{28}\]  
The closer the filter \(h\) coefficients are, the closer \(\frac{w^{T}(X*h)^{T}Y}{w^{T}X^{T}Y}\) is to \(N\). For the proposed model shown in Equation (2), we adopt a similar approach for analysis. The filtered \(X\) is defined as \(X^{\prime}\), where the term \(x^{\prime}_{ij}\) is given by \(\sum_{k=i}^{N}h_{k}x_{(i+k)j}\). When \(N\) is relatively small, considering the smoothness in the neighborhood, we approximately assume that \(\sum_{k=i}^{N}h_{k}x_{(i+k)j}\approx x_{ij}\sum_{k=i}^{N}h_{k}\). When \(h_{k}=\frac{1}{\sqrt{N}}\) for \(k=1,2,\cdots,N\), \(x_{ij}\sum_{k=i}^{N}h_{k}\) obtains the maximum value \(\sqrt{N}x_{ij}\). Thus, \(X^{\prime}=\sqrt{N}X\), \(X^{\prime T}YY^{T}X^{\prime}w=\left(\sqrt{N}X\right)^{T}YY^{T}\left(\sqrt{N}X \right)w=XX^{T}\gamma Y^{T}Xw=\lambda_{filtered}w\). So, we have \(\frac{\lambda_{filtered}}{\lambda}=N\) and, similarly, we have  
\[0<\frac{w^{T}(X*h)^{T}Yc}{w^{T}X^{T}Yc}<N \tag{29}\]  
Obviously, the vector \(w\) must differ before and after filtering, and the above analysis is also approximate to a certain extent. The prediction variance can be analyzed using probabilistic methods. Let \(u\) represent a hidden variable; the relation between \(x\) and \(u\) is \(x=Pu+\mu_{x}+\xi\) and \(y=Cu+\mu_{y}+\eta\), where \(\xi\) and \(\eta\) represent Gaussian noise, satisfying \(\xi\sim N(0,\sigma_{2}^{2}I)\) and \(\eta\sim N(0,\sigma_{y}^{2}I)\), respectively; \(\mu_{x}\) and \(\mu_{y}\) represent the mean vectors of \(X\) and \(Y\), respectively; and \(P\) andrepresent the transformation matrices for \(X\) and \(Y\), respectively. The variance of prediction distribution is \(C\sigma_{x}^{2}(P^{T}P+\sigma_{x}^{2})^{-1}C^{T}+\sigma_{y}^{2}I\), which is related to the inverse of matrix \((P^{T}P+\sigma_{x}^{2}I)\)[19]. Assuming \(x=[x_{1},x_{2},\cdots,x_{m}]\), the filtered vector \(x^{\prime}\) can be approximated as \(x^{\prime}=\left[\sum_{i=1}^{N}h_{i}x_{i},\sum_{i=1}^{N}h_{i}x_{i+1},\cdots, \sum_{i=1}^{N}h_{i}x_{i+m-1}\right]\approx\left[x_{1}\sum_{i=1}^{N}h_{i},x_{2 }\sum_{i=1}^{N}h_{i},\cdots,\ x_{m}\sum_{i=1}^{N}h_{i}\right]\), and the approximate variance of \(x^{\prime}\) is given by \(\sum_{i=1}^{N}h_{i}^{2}\sigma_{x}^{2}I\). For example, in the two-dimensional case, suppose \(P=[p_{1},p_{2},p_{3},p_{4}]\) and \(C=[c_{1},c_{2}]\).