## 2 Methodology
### Feature Selection by the Relieff Method
#### 2.3.2 Multi-Output Least Squares Support Vector Machines  
The majority of existing SVM-based dam displacement prediction models are single-objective regression models, where the target variables are modeled and analyzed independently by input factors. However, such models are not applicable to multi-objective regression situations, especially in terms of how to capture potential correlation information among various target variables [53; 55]. The MIMO-LSSVM model proposed in this paper is a generalization of the LSSVM model, which follows the idea of multi-objective regression to model the mutual correlation between the outputs by decomposing each weight vector \(\omega_{i}\) (\(\omega_{i}=\omega_{0}+v_{i}\)). In this model, each output weight vector fluctuates around a certain mean vector \(\omega_{0}\) and a range of variations of \(v_{i}\) that carry the correlation and discrepancy information, respectively. In the case of MIMO regression, the general functional form of the MIMO-LSSVM is as follows. \[f_{i}(x)=(\omega_{0}+v_{i})^{T}\phi(x)+b_{i} \tag{16}\]  
where \(f_{i}(x)\), \(\omega_{0}+v_{i_{i}}\) and \(b_{i}\) represent the predicted values, weight vectors, and bias terms under i-dimensional outputs, respectively. Similar to the single-output support vector machine, minimizing the following objective function under the structural risk minimiza  
Figure 1: General architecture of single-output support vector machine. tion principle, multi-output support vector regression can be defined as the following optimization problem. \[\min\limits_{\omega_{0},V,\xi}J(\omega_{0},V,\xi)=\frac{1}{2}\|\omega_{0}\|^{2}+ \frac{1}{2}\frac{\lambda}{m}\sum\limits_{i=1}^{m}v_{i}^{T}v_{i}+\frac{1}{2} \gamma\sum\limits_{i=1}^{m}\xi_{i}^{T}\xi_{i} \tag{17}\]  
\[\text{s.t.}\ Y_{i,j}=\big{(}\omega_{0}+v_{j}\big{)}^{T}\phi(x_{i})+b_{j}+\xi_{ i,j} \tag{18}\]  
where \(\xi\) is the slack variable, \(y_{i}\) is the i-dimensional output vector of the sample set, \(\lambda\) and \(\gamma\) are the two regularization parameters. The Lagrangian function for solving the above equation can be expressed as follows. \[L=J(\omega_{0},V,\xi)-\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{l}A_{i,j}^{T} \Big{[}\big{(}\omega_{0}+v_{i}\big{)}\phi\big{(}x_{j}\big{)}^{T}+b_{i}+\xi_{j, i}-Y_{j,i}\Big{]} \tag{19}\]  
where \(A=(a_{1},a_{2},\ldots,a_{m})\), \(\alpha_{i}\) is the Lagrange multiplier. Taking partial derivatives with respect to \(\omega_{0}\), \(V\), \(b\), \(\xi\), and \(A\) for \(L\), respectively, and making them equal to 0, the following system of linear equations is obtained as follows. \[\left[ \begin{array}{cc}0_{ml\times m}&P^{T}\\ P&\Omega+\gamma^{-1}I_{ml}+\frac{m}{\lambda}Q\end{array} \right]\left[ \begin{array} []{c}b\\ a\end{array} \right]=\left[ \begin{array}{c}0_{m}\\ y\end{array} \right] \tag{20}\]  
where \(P\) is a block matrix consisting of m all-ones vectors arranged in diagonal positions with zeros in the remaining positions, \(\Omega\) is a block matrix consisting of matrix \(K\) repeated in m rows and m columns, and \(Q\) is a block matrix consisting of matrices \(K\) arranged in diagonal positions with zeros in the remaining positions. \(Z=\big{(}\phi(x_{1}),\phi(x_{2}),\ldots,\phi(x_{l})\big{)}\), \(y=\big{(}y_{1}^{T},y_{2}^{T},\ldots,y_{m}^{T}\big{)}^{T}\), \(a=\big{(}a_{1}^{T},a_{2}^{T},\ldots,a_{m}^{T}\big{)}^{T}\), \(K=Z^{T}Z\), \(K_{i,j}=\phi(x_{i})\phi\big{(}x_{j}\big{)}=k\big{(}x_{i},x_{j}\big{)}\) is the element of the matrix \(K\), and \(k\big{(}x_{i},x_{j}\big{)}\) is the kernel function. The regression equation for MIMO-LSSVM can be expressed as follows. \[\left\{ \begin{array}{l}f(x)=\big{(}\omega_{0}^{*}+v_{i}^{*}\big{)}^{T}\phi (x)+b_{i}^{*}=\sum\limits_{i^{\prime}=1}^{m}\sum\limits_{j=1}^{l}a_{i^{\prime },j}^{*}k\big{(}x,x_{j}\big{)}+\frac{m}{\lambda}\sum\limits_{j=1}^{l}a_{j,i}^{* }k\big{(}x,x_{j}\big{)}+b_{i}^{*}\\ k(x,x^{\prime})=\exp\Bigl{(}-p\|x-x_{i}\|^{2}\Bigr{)}\,\ p>0\end{array} \right.