## 2 Methodology
### The Procedure of the Proposed Approach for Dam Health Monitoring  
The procedure of the proposed approach for dam health monitoring is summarized in the flowchart in Figure 3, with the specific steps outlined as follows.  
Figure 3: Flowchart of dam displacement prediction.  
Figure 2: General architecture of multi-output support vector machine.  
Step 1: Data collection and processing. The monitoring data of environmental variables and dam displacements are collected from the dam monitoring system, and the data are normalized.  
Step 2: Determination of the hysteresis effects of environmental factors and the spatial correlation of measurement points. The sliding matching method and cosine similarity calculation method are utilized to determine the specific hysteresis times at which air temperature affects displacement at different measurement points. The correlation analysis of the displacements at different measurement points is conducted by using the maximum information coefficient method.  
Step 3: Training the model for prediction after determining the model dataset. Based on the calculated specific hysteresis effects of environmental factors, the datasets of different measurement points are constructed. The first 80% of the samples are used as the training set according to the time series, and the remaining 20% are used as the testing set. Regression prediction based on PSO-LSSVM is used to select the best parameters to construct the prediction model, and the output is obtained as the preliminary prediction results.  
Step 4: Screening factor sets with the feature selection method. The ReliefF algorithm is utilized to calculate the feature importance of each factor and sort the factors according to their importance. The importance of different locations of measurement points affected by the feature factors is then analyzed, and the features are added to the model for prediction according to the order of importance from the largest to the smallest. The root mean square error (RMSE) is selected as the evaluation index, and, when it reaches the minimum value, it can be considered that we have obtained the optimal factor set.  
Step 5: Establishing a MIMO model for synchronous prediction of dam displacement in multiple blocks. After filtering by feature selection, a uniform factor set of measurement points in the same area is determined. The synchronous prediction of dam displacements at multiple measurement points is then realized based on the MIMO-PSO-LSSVM model.  
Step 6: Comparative assessment of model performance. Three single-output prediction models, SVM, Back Propagation Neural Network (BPNN), and LSSVM optimized by PSO (PSO-LSSVM), are selected for comparative analysis with the multi-output prediction model MIMO-PSO-LSSVM. The RMSE, mean absolute error (MAE), and coefficient of determination (R\({}^{2}\)) are used to compare and evaluate the performance of the single-output prediction model and the multi-output prediction model. The mathematical expressions for these metrics are shown below.  
\[RMSE=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}\left(y_{m}^{(i)}-y_{p}^{(i)}\right) ^{2}} \tag{22}\]  
\[MAE=\frac{1}{n}\sum\limits_{i=1}^{n}\left|y_{m}^{(i)}-y_{p}^{(i)}\right| \tag{23}\]  
\[R^{2}=\frac{\left(\frac{n}{\sum\limits_{i=1}^{n}\left(y_{m}^{(i)}-\overline{y }_{m}\right)\left(y_{p}^{(i)}-\overline{y}_{p}\right)}\right)^{2}}{\sum\limits _{i=1}^{n}\left(y_{m}^{(i)}-\overline{y}_{m}\right)^{2}\sum\limits_{i=1}^{n} \left(y_{p}^{(i)}-\overline{y}_{p}\right)^{2}} \tag{24}\]  
where \(n\) is the total number of data samples, \(y_{m}^{(i)}\) and \(y_{p}^{(i)}\) denote the measured and predicted values of the ith data, respectively, \(\overline{y}_{m}\) is the average of the measured values, and \(\overline{y}_{p}\) is the average of the predicted values. Among these metrics, RMSE denotes the squared mean deviation of the predicted values from the measured values, MAE denotes the mean of the central absolute errors, and R\({}^{2}\) is a common metric for assessing the goodness of fit of the model.