# 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 spatial adjacent point regression interpolation method establishes the correlation between the deformation value of the target measurement point and the deformation value of the adjacent points, which can effectively reveal the relationships among the deformation values of the spatial adjacent measurement points. However, the measuring points that are located on the same deformation body, such as the measuring points on the same section of the concrete gravity dam and concrete arch dam, have integrity, correlations, mutual influences, and correlations in deformation, so the specific relationships between the deformation of these measuring points are complex. However, the spatial adjacent point regression method, which is based on the modeling idea of the statistical model, only regresses the power series expansion of finite integer terms of variables, so it is difficult to fully describe the unknown relationships between the deformation of measurement points, and therefore this regression method has limitations. It is difficult to represent the complex and unknown relationship between the deformation of spatial measurement points by specific mathematical expressions. But the BP neural network, with strong nonlinear mapping ability, can delineate the complex information relationship behind the data through learning and training of samples. Meanwhile, the BP neural network also has strong generalization ability, so that the trained network can effectively process new input samples and give appropriate output results. Therefore, in order to improve the accuracy of the missing value completion and find the true value of deformation closest to the missing time, the BP neural network is introduced in this section to deal with the unknown relationship between the deformation of spatial measurement points. A corresponding missing value completion method is proposed. The BP network is a kind of multi-layer feedforward neural network, which realizes operation through forward signal propagation and back error propagation. It has three layers: input layer, hidden layer, and output layer. Each layer is composed of nodes (namely neurons). The upper and lower nodes are connected by weight, and the nodes of the same layer are independent of each other. Through the connection weights between the upper and lower neurons, the network transforms the output of the upper neuron to the input of the lower neuron, thus realizing the learning calculation of the samples. Assume that there are n monitoring points that are spatially adjacent and structurally related at a concrete dam body, such as points of a concrete gravity dam that are on the same vertical line or points within the same deformation zone of a concrete arch dam (partition method is not explained in detail in this paper), when the deformation information of the \(i\)th measuring point is missing due to some reasons, the known information of other m = n - 1 measuring points can be used to estimate the information of point \(i\). The steps to establish a multi-value data missing completion method based on BP neural network mapping are as follows:  
Suppose that the sample set contains \(Z\) pattern pairs between the input vector and the output vector, randomly select a pattern pair \(k\), while the input pattern vector is \(A_{k}=\left(a_{1}^{k},a_{2}^{k},\cdots,a_{m}^{k}\right)\), and the expected output vector is \(Y_{k}=\left(y_{1}^{k}\right)\). The input vector of the middle layer element is \(S=\left(s_{1},s_{2},\cdots,s_{p}\right)\) (\(p\) is the number of hidden layer nodes, the same below), and the output vector is \(B_{k}=\left(b_{1},b_{2},\cdots,b_{p}\right)\). The input vector of the output layer element is \(L_{k}=\left(l_{1},l_{2},\cdots,l_{p}\right)\), and the output vector is \(C=\left(c\right)\). The connection weight between the input layer and the hidden layer is \(w\left(w=w_{ij};i=1,2,\cdots,m;j=1,2,\cdots,p\right)\).