# A Completion Method for Missing Concrete Dam Deformation Monitoring Data Pieces
## 2 Method
### Completion Strategy for a Single Missing Value
#### 2.1.1 Traditional Interpolation Completion Methods
\begin{array}{c}\beta_{1}\\ d_{2}\\ \vdots\\ d_{n-1}\\ \beta_{n}\end{array} \right] \tag{10}\]where \(\gamma_{i}=1-\mu_{i}\), \(\mu_{i}=\frac{h_{i-1}}{h_{i-1}+h_{i}}\), \(d_{i}=6\Big{(}\frac{y_{i+1}-y_{i}}{h_{i}}-\frac{y_{i}-y_{i-1}}{h_{i-1}}\Big{)} \frac{1}{h_{i-1}+h_{i}}\), \(\beta_{1}=\frac{6}{h_{1}}\Big{(}\frac{y_{2}-y_{1}}{h_{1}}-y^{\prime}{}_{1} \Big{)}\), \(\beta_{n}=\frac{6}{h_{n-1}}\Big{(}y^{\prime}{}_{n}-\frac{y_{n}-y_{n-1}}{h_{n-1} }\Big{)}\). 4. Cubic Hermite interpolation (4)  
The Hermite interpolation method uses a curve to approximate the objective function, which not only requires that the interpolation curve strictly passes through the data points, but also needs to satisfy that the derivative value of each order at the data points is equal to the original function, to build a smooth interpolation curve. The cubic Hermite interpolation needs to know the function value of two nodes and the first derivative value to complete the construction. The algorithm is simple, and its interpolation results are close to real data, so it has been widely used. Assuming that the two known nodes are \(\big{(}x_{j-1},y_{j-1}\big{)}\) and \(\big{(}x_{j},y_{j}\big{)}\), and the corresponding first derivative values are \(y^{\prime}{}_{j-1}\) and \(y^{\prime}{}_{j}\), the interpolation polynomial \(H_{3}(x)\) can be expressed as:  
\[H_{3}(x)=\alpha_{j-1}(x)y_{j-1}+\alpha_{j}(x)y_{j}+\beta_{j-1}(x)y^{\prime}{}_ {j-1}+\beta_{j}(x)y^{\prime}{}_{j} \tag{11}\]  
where \(\alpha_{j-1}(x),\alpha_{j}(x),\beta_{j-1}(x),\beta_{j}(x)\) are the interpolation basis functions, and their highest degree cannot exceed 3. \(H_{3}(x)\) meets the conditions:  
\[H_{3}\big{(}x_{j-1}\big{)}=y_{j-1},H_{3}\big{(}x_{j}\big{)}=y_{j} \tag{12}\]  
\[H^{\prime}{}_{3}\big{(}x_{j-1}\big{)}=y^{\prime}{}_{j-1},H^{\prime}{}_{3} \big{(}x_{j}\big{)}=y^{\prime}{}_{j} \tag{13}\]  
Therefore, it can be solved by:  
\[\alpha_{j-1}(x)=\left(1+2\frac{x-x_{j-1}}{x_{j}-x_{j-1}}\right)\left(\frac{x- x_{j}}{x_{j}-x_{j-1}}\right)^{2} \tag{14}\]  
\[\alpha_{j}(x)=\left(1-2\frac{x-x_{j}}{x_{j}-x_{j-1}}\right)\left(\frac{x-x_{j -1}}{x_{j}-x_{j-1}}\right)^{2} \tag{15}\]  
\[\beta_{j-1}(x)=\big{(}x-x_{j-1}\big{)}\left(\frac{x-x_{j}}{x_{j}-x_{j-1}} \right)^{2},\beta_{j}(x)=\big{(}x-x_{j}\big{)}\left(\frac{x-x_{j-1}}{x_{j}-x_{ j-1}}\right)^{2} \tag{16}\]  
Thus, the final expression can be obtained as:  
\[ \begin{array}{l}H_{3}(x)=\Big{(}\Big{(}1+2\frac{x-x_{j-1}}{x_{j}-x_{j-1}} \Big{)}y_{j-1}+\big{(}x-x_{j-1}\big{)}y^{\prime}{}_{j-1}\Big{)}\Big{(}\frac{x -x_{j}}{x_{j}-x_{j-1}}\Big{)}^{2}+\\ \Big{(}\Big{(}1-2\frac{x-x_{j}}{x_{j}-x_{j-1}}\Big{)}y_{j}+\big{(}x-x_{j}\big{)} y^{\prime}{}_{j}\Big{)}\Big{(}\frac{x-x_{j-1}}{x_{j}-x_{j-1}}\Big{)}^{2} \end{array}  \tag{17}\]  
The function parameters of the interpolation point can be obtained by substituting the x-coordinate at the interpolation point into Equation (17). The rationality of the traditional interpolation method lies in that the approximation of a small part of missing data does not affect the overall trend and law of deformation time series. When the non-uniform data information is few, this kind of interpolation method can be used to cover up and generate deformation time series with equal intervals.