# A Completion Method for Missing Concrete Dam Deformation Monitoring Data Pieces
## 2 Method
### Completion Strategy for Multi Missing Values
#### 2.2.2 Spatial Adjacent Point Regression  
When the fitting accuracy of the regression model to the deformation sequence is low or the environmental variables of the missing segment are unknown, the accuracy of the above completion method is low. Since the single section of concrete gravity dam and the whole dam body of concrete arch dam can be regarded as a whole, the deformation is naturally integrated and coherent, so the deformation in local areas is correlated to a certain extent. In other words, the missing information of a target measurement point can be estimated according to the deformation value of its adjacent measurement point.  
Assume that there are three monitoring points, A, B, and C, with similar locations and structures in the local area of a specific concrete dam section. The deformation sequences of measuring points A and C are complete, and a partial sequence of measuring point B is missing, as shown in Figure 2.  
Figure 2: Schematic diagram of partial sequence deletion.  
Considering the correlation of dam deformation at measuring points A, B, and C, there is a certain correlation between the deformation value of measuring point B and the deformation value of measuring points A and C. Therefore, according to the modeling idea of the statistical model, this paper takes the deformation value of measuring points A and C as the influence factors, and the deformation value of measuring point B as the target output to establish the correlation between measuring point B and measuring points A and C. The expression is:  
\[\delta_{B}=f(\delta_{A},\delta_{C})+\varepsilon \tag{23}\]  
where \(f(\delta_{A},\delta_{C})\) represents the general function between \(\delta_{B}\) and two influence factors, \(\delta_{A}\), \(\delta_{C}\), and the function relation can be expressed by polynomial as:  
\[\delta_{B}=\sum_{i=1}^{K_{A}}\lambda_{Ai}\delta_{A}^{i}+\sum_{i=1}^{K_{C}} \lambda_{Ci}\delta_{C}^{i}+\beta_{B}+\varepsilon \tag{24}\]  
where \(\lambda_{Ai}\) and \(\lambda_{Ci}\) represent the coefficients of each polynomial of \(\delta_{A}\) and \(\delta_{C}\) respectively, \(K_{A}\) and \(K_{C}\) represent the highest order of \(\delta_{A}\) and \(\delta_{C}\), and \(\beta_{B}\) is the translation term.  
If Equation (24) is expanded, let the number of adjacent measurement points in the local area of the target measurement point be abstracted as \(L\), then:  
\[\delta_{it}=\sum_{j=1}^{L}\lambda_{ij}f\big{(}\delta_{ji}\big{)}+\beta_{i}+\varepsilon \tag{25}\]  
where \(\delta_{it}\) and \(\delta_{it}\) represent the deformation values of the measuring point \(i\) and the adjacent measuring point \(j\) at time \(t\) respectively, and \(\lambda_{ij}\) represents the influence coefficient of each factor.  
From the above analysis, with the known deformation information of the target measurement point and the adjacent measurement points, the least square method can also estimate the influence coefficients, and the expression of the model is thus established. By substituting the hypothesis missing information of the adjacent measurement point into Equation (25), the missing information of the target measurement point can be estimated.