## 2 Deformation Zoning Model of Ultra-High Arch Dam
### Matrix Expression of Spatiotemporal Characteristic Level of Panel Data  
Panel data is the direct embodiment of the dynamic clustering of dam deformation, and its Spatio-temporal characteristic attributes are mainly described from three aspects [35]: (a) absolute index, which is the original value of panel data, (b) growing index, which is the embodiment of the growth rate of sample objects, and (c) fluctuation index, which is the embodiment of the growth rate and change rate of the sample. When analyzing the dam deformation monitoring value, the similarity of the deformation sequence is reflected by effectively integrating the three indexes of \"absolute quantity\", \"growing\", and \"fluctuation\" of the dam deformation value [23]. The third index is very important to describe the deformation of an ultra-high arch dam under complex water levels, but many studies have not been carried out. For the deformation data set \(X_{it}\), \(i=1,2,\cdots,N;t=1,2,\cdots,T\), the following definitions are given [38]:  
**Definition 1**. : _The mean function \(\overline{x}_{i}\) and variance function \(S_{i}\) of dam deformation measuring point \(i\) in period \(T\) are_  
\[\overline{x}_{i}=\frac{1}{T}\overline{\sum_{i=1}^{T}x_{it}};\;\;S_{i}=\frac{1} {T-1}\sum_{t=1}^{T}\left(x_{it}-\overline{x}_{i}\right)^{2},\;0\leq t\leq T \tag{2}\]  
_where all symbols are the same as Equation (1)._  
**Definition 2**. : _The \"absolute\" distance \(d_{ij}(AQED)\) between the measuring point \(i\) and the measuring point \(j\) as_  
\[d_{ij}(AQED)=\left[\sum_{t=1}^{T}\left(x_{it}-x_{it}\right)^{2}\right]^{0.5} \tag{3}\]  
_where \(x_{it}\) and \(x_{it}\) are the deformation values of measuring points \(i\) and \(j\) in period \(t\) respectively._  
**Definition 3**. : _The \"growing\" distance \(d_{ij}(ISED)\) between measuring point \(i\) and measuring point \(j\) as:_  
\[d_{ij}(ISED)=\left[\sum_{t=1}^{T}\left(\frac{\Delta x_{it}}{x_{it-1}}-\frac{ \Delta x_{it}}{x_{it-1}}\right)^{2}\right]^{0.5} \tag{4}\]  
_where \(\Delta x_{it}=x_{it}-x_{it-1}\); \(\Delta x_{it}=x_{it}-x_{it-1}\); \(\Delta x_{it}\) and \(\Delta x_{it}\) are the difference of absolute quantity between two adjacent periods of deformation respectively._  
**Definition 4**. : _The \"fluctuation\" distance \(d_{ij}(VCED)\) between measuring point \(i\) and measuring point \(j\) as:_  
\[d_{ij}(VCED)=\left[\sum_{t=1}^{T}\left(\frac{\overline{x}_{i}}{S_{i}}-\frac{ \overline{x}_{j}}{S_{j}}\right)^{2}\right]^{0.5} \tag{5}\]  
_where all symbols are the same as Equation (2)._  
In addition, to prevent large differences in the order of magnitude between the three distances, the above distances can also be standardized, so as to ensure that each distance plays a role. For example, the standardized transformation formula of absolute distance \(d_{ij}(AQED)\) is:  
\[zd_{ij}(AQED)=\frac{d_{ij}(AQED)-\text{min}d_{ij}(AQED)}{\text{max}d_{ij}(AQED )-\text{min}d_{ij}(AQED)} \tag{6}\]  
where \(zd_{ij}(AQED)\) is the standardized treatment of absolute distance and \(\text{max}d_{ij}(AQED)\) and \(\text{min}d_{ij}(AQED)\) are the maximum and minimum values of absolute distance, respectively. The standardized formula of other distances is the same. During a specific calculation, whether to standardize or not can be selected according to the order of magnitude difference of the three distances. Generally, when the order of magnitude difference is greater than or equal to \(10^{2}\), it is recommended to standardize.