## 4 Calculus Analysis
### Model Parameters and Hyperparameter Selection  
The residual cascade smoothing and MSSAN proposed in this paper have some model parameters that need to be determined, and the parameter values are determined by grid-seeking parameters, as shown in Tables 1 and 2. In Table 1, the MSSAN employs multiscale convolution with convolution kernels of 3, 7, and 11, and different levels of padding to maintain the same number of features for easy splicing, respectively. The GRU maps the convolved feature dimensions to 32 to obtain the bridge deflection correlation between different periods. Then, SA linearly projects the input data to obtain the query matrix Q, the keyword matrix K, and the value matrix V, all set to 16. Finally, the predicted values are output by linear mapping. In Table 2, the Adam optimizer is used in this paper, and the learning rate is set to 0.001, the number of training and batches are 500 and 256, respectively, the history step is 5, and the prediction step is 1.  
Figure 10: Xin’anjiang Bridge data collection.  
The Mean Absolute Error (MAE), Root Mean Square Error (RMSE) and coefficient of determination (\(R^{2}\)) are used as the evaluation indexes of the model in this experiment, and their calculation formulas are shown in Equations (10)-(12):  
\[MAE=\frac{1}{n}\sum_{i=1}^{n}\left|\hat{y}-y_{i}\right| \tag{10}\]  
\[RMSE=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}\left(\hat{y}-y_{i}\right)^{2}} \tag{11}\]  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}\left(y-\hat{y}_{i}\right)^{2}}{\sum\limits _{i=1}^{n}\left(y-\overline{y}\right)^{2}} \tag{12}\]  
where \(\hat{y}=\left\{\hat{y}_{1},\hat{y}_{2},\ldots,\hat{y}_{n}\right\}\) is the predicted value, \(y=\left\{y_{1},y_{2},\ldots,y_{n}\right\}\) is the measured value, and \(n\) is the number of variables. The RMSE and MAE are susceptible to the range of predicted values, and if the raw data are large, they may produce large metrics. \(R^{2}\), on the other hand, is a value ranging from 0 to 1, with values close to 1 indicating that the model fits the data well and values close to 0 indicating that the model fits poorly. In the cascade residual smoothing method proposed in this paper, there are Gaussian smoothing standard deviation \(\sigma\), smoothing intensity coefficient \(\alpha\) and trend coefficient \(\beta\), and these three hyperparameters are selected by averaging the coefficient of determination, \(R^{2}\), over five experiments, as shown in Figure 11.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Module** & **Kernels** & **Channel** & **Strides** & **Dimension** & **Padding** \\ \hline Conv1 & 3 & 16 & 1 & / & 1 \\ Conv2 & 7 & 16 & 1 & / & 3 \\ Conv3 & 11 & 16 & 1 & / & 5 \\ GRU & / & / & / & 32 & / \\ SA-\(Q\) & / & / & / & 16 & / \\ SA-\(K\) & / & / & / & 16 & / \\ SA-\(V\) & / & / & / & 16 & / \\ Linear & / & / & / & 1 & / \\ \hline \end{tabular}
\end{table}
Table 1: MSSAN parameters.  
\begin{table}
\begin{tabular}{c c} \hline
**Hyperparameter** & **Value** \\ \hline Optimizer & Adam \\ Learning rate & 0.001 \\ epoch & 500 \\ Batch size & 256 \\ time step & 5 \\ \(\sigma\) & 1.25 \\ \(\alpha\) & 0.5 \\ \(\beta\) & 0.01 \\ \hline \end{tabular}
\end{table}
Table 2: Hyperparameter selection.  
In Figure 11, it can be seen that cascade residual smoothing with Gaussian smoothing standard deviation \(\sigma\) works best at 1.25. In addition, the smoothing intensity coefficient \(\alpha\) in quadratic exponential smoothing is not as large as possible, reaching better results at 0.5, and the trend coefficient \(\beta\) is optimal at 0.01.