# Dam Deformation Monitoring Model Based on Deep Learning and Split Conformal Quantile Prediction
## 2 Methodologies
### Split Conformal Quantile Prediction  
SCQP is an interval prediction method proposed by [PERSON] et al. (2022) based on Split conformal prediction (SCP) [34; 35]. The method inherits the advantages of conventional SCP and is able to construct PIs without any prior distribution [36]. Moreover, the addition of the pinball loss function also solves the defect of SCP that relies too much on the accuracy of the point prediction model. This approach requires splitting the training samples into a training subset indexed by \(\mathcal{L}_{1}\) and a calibration subset indexed by \(\mathcal{L}_{2}\). The regression algorithm \(\mathcal{M}_{q}\) integrates pinball loss function and sets the minimization function loss as the objective. The two curves \(\hat{q}_{\alpha UB}\) and \(\hat{q}_{\alpha LB}\) that represent the upper and lower bounds of the PIs are fitted thus:  
\[\begin{array}{l}\{\hat{q}_{\alpha LB}=q_{a}(x;\hat{\theta}),\hat{q}_{\alpha UB }=q_{a}(x;\hat{\theta})\}\leftarrow\mathcal{M}_{q}(\{(X_{i},Y_{i}):i\in \mathcal{L}_{1}\})\\ \hat{\theta}=\underset{\theta}{\text{argmin}}=\frac{1}{n}\sum\limits_{i=1}^{n }\rho_{a}(Y_{i},q_{a}(X_{i};\hat{\theta}))+L(\theta)\end{array} \tag{10}\]  
where \(\rho_{a}(Y_{i},q_{a}(X_{i};\hat{\theta}))\) represents the pinball loss function, which can also be written as:  
\[\rho_{a}(y;z)=\left\{\begin{array}{l}(1-\alpha)(y-z),y\geq z\\ \alpha(y-z),z<y\end{array}\right. \tag{11}\]  
In the next key step, the consistency score \(E_{i}\) can be calculated using the calibration set \(\mathcal{L}_{2}\) and the fit function of the upper and lower bounds, by which the error between the prediction interval \(PI^{a}(x)=\left[\hat{q}_{\alpha LB}=q_{a}(x;\hat{\theta}),\hat{q}_{\alpha UB}=q _{a}(x;\hat{\theta})\right]\) and the actual observed value \(Y_{i}\) can be quantified thus:  
\[E_{i}=\max\{\hat{q}_{\alpha LB}(X_{i})-Y_{i},Y_{i}-\hat{q}_{\alpha UB}\} \tag{12}\]  
From Equation (14), it is not difficult to explore the meaning of the consistency score \(E_{i}\). When the observed value \(Y_{i}\) lies below the PIs, \(Y_{i}<\hat{q}_{\alpha LB}(X_{i})\), \(E_{i}=|\hat{q}_{\alpha LB}(X_{i})-Y_{i}|\) can be expressed as the error between the PI and the observed value. Similarly, if the observed value \(Y_{i}\) lies above the PIs, \(Y_{i}>\hat{q}_{\alpha UB}(X_{i})\), then the error between PIs and the observed value \(Y_{i}\) can be expressed as \(E_{i}=|\hat{q}_{\alpha LB}(X_{i})-Y_{i}|\). In addition, if \(Y_{i}\) always lies within the PI, \(\hat{q}_{\alpha LB}(X_{i})<Y_{i}<\hat{q}_{\alpha UB}(X_{i})\), then \(E_{i}\) will be chosen from the largest non-positive number.  
So far, given the new input variable \(X_{n+1}\), the PIs of the response variable \(Y_{n+1}\) can be described:  
\[PI^{a}(X_{n+1})=[\hat{q}_{\alpha LB}(X_{n+1})-Q_{1-\alpha}(R,\mathcal{L}_{2} ),\hat{q}_{\alpha LB}(X_{n+1})+Q_{1-\alpha}(R,\mathcal{L}_{2})] \tag{13}\]  
\[Q_{1-\alpha}(E,\mathcal{L}_{2}):=(1-\alpha)(1+1/|\mathcal{L}_{2}|)-\text{the empirical quantile of }\{E_{i}:i\in\mathcal{L}_{2}\} \tag{14}\]  
Figure 3: The structure of Bi-LSTM.