# A Completion Method for Missing Concrete Dam Deformation Monitoring Data Pieces
## 2 Method
### Completion Strategy for Multi Missing Values
#### 2.2.3 BP Mapping of Spatial Adjacent Points
The connection weight between the hidden layer and the output layer is \(\upsilon\left(v=v_{j};j=1,2,\cdots,p\right)\). The output threshold of each unit in the hidden layer is \(\theta\left(\theta=\theta_{j};j=1,2,\cdots,p\right)\). The output threshold of the output layer unit is \(\gamma=\left(\gamma\right)\). (1) Network parameters are initialized by using random assignment functions to assign \(w\), \(v\), \(\theta\), and \(\gamma\) small random values between (\(-1\),\(1\)). (2) Input vector \(A_{k}\), connection weight \(w\), and threshold \(\theta\) are used to calculate the input \(S\) of the hidden layer. Calculate the output \(B_{k}\) of the hidden layer through Sigmoid function with \(S\), namely:  
\[s_{j}=\sum\limits_{i=1}^{m}w_{ij}a_{i}-\theta_{j}\quad j=1,2,\cdots,p \tag{26}\]  
\[b_{j}=f(s_{j})\quad j=1,2,\cdots,p \tag{27}\]  
\[f(x)=\frac{1}{1+e^{-x}} \tag{28}\]  
(3) The output \(B_{k}\), connection weight \(v\), and threshold \(\gamma\) of the hidden layer are used to calculate the input \(L_{k}\) of the output layer element, and then the output vector \(c\) of the output layer element is calculated with \(L_{k}\), namely:  
\[l=\sum\limits_{j=1}^{p}v_{j}b_{j}-\gamma \tag{29}\]  
\[c=f(l) \tag{30}\]  
(4) The expected output vector \(Y_{k}\) and the actual network output \(c\) are used to calculate the generalized error \(d^{k}\) of the output layer element, namely:  
\[d^{k}=\left(y^{k}-c\right)c(1-c) \tag{31}\](5) The connection weight \(v\), the generalization error \(d^{k}\) of the output layer, and the output \(B_{k}\) of the hidden layer are used to calculate the generalization error \(e^{k}\) of each element of the hidden layer, namely:  
\[e_{j}^{k}=\Big{(}d^{k}v_{j}\Big{)}b_{j}\big{(}1-b_{j}\big{)} \tag{32}\]  
(6) Use the generalized error \(d^{k}\) of the output layer element and the output \(B_{k}\) of each element in the middle layer to correct the connection weight \(v\) and threshold \(\gamma\), that is:  
\[v_{j}(N+1)=v_{j}(N)+\eta d^{k}b_{j}+\alpha\big{(}v_{j}(N)-v_{j}(N-1)\big{)} \tag{33}\]  
\[\gamma(N+1)=\gamma(N)+\eta d^{k} \tag{34}\]  
where \(\eta\) stands for learning efficiency and take \(\eta=0.01\sim 0.8\). \(\alpha\) is the momentum factor and take \(\alpha=0.9\). (7) The connection weight \(w\) and threshold \(\theta\) are modified by the generalized error \(e_{j}^{k}\) and input mode vector \(A_{k}\) of each element of the hidden layer, namely:  
\[w_{ij}(N+1)=w_{ij}(N)+\eta e_{j}^{k}a_{i}^{k}+\alpha\big{(}w_{ij}(N)-w_{ij}(N- 1)\big{)} \tag{35}\]  
\[b_{j}=f(s_{j})\quad j=1,2,\cdots,p \tag{27}\]0  
(8) Randomly select another learning pattern pair in the training sample set and repeat steps (3)-(6) until all pattern pairs are trained. (9) Calculate the global error function \(E\) of the network, and its formula is:  
\[b_{j}=f(s_{j})\quad j=1,2,\cdots,p \tag{27}\]1  
If \(E\) is less than a preset error value, the network stops learning; otherwise, repeat steps (3)-(8) for the next round of learning and training of the sample set. (10) The trained network is saved, while new samples are input, and the output result of missing information completion is obtained. Take \(n=5\) as an example, the input layer is the deformation sequence of four relevant measurement points, and the output layer is the deformation sequence of the target measurement points. The network structure is shown in \[b_{j}=f(s_{j})\quad j=1,2,\cdots,p \tag{27}\]2