## 2 Methodology  
### Factor Sets Construction Considering Hysteresis Effects  
As an integrated response to the behavior of the dam structure, the dam displacement is a nonlinear function formed by the influence of hydraulic pressure, temperature, and aging effects [9; 56]. The equation for the dam displacement is therefore as follows.  
\[\delta=\delta_{H}+\delta_{T}+\delta_{\theta} \tag{1}\]  
where \(\delta_{H}\) represents the hydraulic pressure factor, \(\delta_{T}\) represents the temperature factor, and \(\delta_{\theta}\) represents the aging factor.  
The hydraulic pressure factor \(\delta_{H}\) reflects the displacement response of a dam under hydraulic pressure. The mathematical expression for this factor can be represented as follows.  
\[\delta_{H}=\sum_{i=1}^{n}a_{i}H^{i} \tag{2}\]  
where \(a_{i}\) represents the regression coefficient, \(H\) represents the water depth in front of the dam, and \(n\) is taken as 3 in gravity dams.  
The temperature factor \(\delta_{T}\) reflects changes in the temperature of the dam and its surroundings. The mathematical expression for this factor can be expressed as follows.  
\[\delta_{T}=b_{1}T_{0}+b_{2}T_{1-2}+b_{3}T_{3-7}+b_{4}T_{8-15}+b_{5}T_{16-30}+b_ {6}T_{31-60}+b_{7}T_{61-90}+b_{8}T_{91-120}+b_{9}T_{121-180} \tag{3}\]  
where \(b_{i}\) represents the regression coefficient, \(T_{0}\) represents the temperature on the day of the measurement moment, and \(T_{p-q}\) represents the average temperature in the \(p\) to \(q\) days before the observation date. According to previous literature studies [35; 36], the last 180 days can be used to study.  
As indicated by earlier studies, the structural behavior of dams frequently exhibits a discrepancy with the changes in environmental factors such as water level and temperature, i.e., this phenomenon can be described as the hysteresis effect in the dam's deformation [37; 40]. The cosine similarity method [41] has been proposed to calculate the specific hysteresis time of the temperature factor \(\delta_{T}\) affecting the displacement and deformation of dams. This method can be used to modify the factor sets of the temperature component. The updated expression for the temperature factor \(\delta^{\prime}{}_{T}\) is as follows.  
\[\delta^{\prime}{}_{T}=b_{1}T_{0}+b_{2}T_{1-2}+b_{3}T_{3-7}+\ldots+b_{n}T_{p-q} \tag{4}\]  
where \(b_{i}\) represents the regression coefficient, \(T_{p-q}\) represents the average air temperature in the \(p\) to \(q\) days before the observation date, and day \(q\) is the specific hysteresis time at which the temperature factor affects the displacement and deformation of the dam.  
The aging factor \(\delta_{\theta}\) reflects the irreversible deformation of the dam body in a certain direction over time. The mathematical expression for this factor can be expressed as follows.  
\[\delta_{\theta}=c_{1}\theta+c_{2}\ln\theta \tag{5}\]  
where \(\theta=t/100\), and \(c_{1}\) and \(c_{2}\) are regression coefficients.  
Thus, according to the expressions of \(\delta^{\prime}{}_{H}\), \(\delta^{\prime}{}_{T}\), and \(\delta_{\theta}\), a new factor set (HT\({}_{\text{H}}\)T) considering the specific hysteresis times of the temperature factor can be represented as follows.  
\[x_{1}=\left\{H^{1},H^{2},H^{3},T_{0},T_{1-2},T_{3-7},\dots,T_{p-\theta},\theta,\ln(\theta+1)\right\} \tag{6}\]  
In this paper, the cosine similarity method is used to calculate the effect of the hydraulic pressure factor and temperature factor on the displacement and deformation of the dam at different measurement points, and the hysteresis time of the temperature factor at each measurement point is derived; then, we can obtain the unique factor sets that reflect the most accurate characteristics of each measurement point for the subsequent displacement prediction study.