# The Current Development of Structural Health Monitoring for Bridges: A Review
## 3 Data Processing Methods
### Data Reconstruction
[101] proposed a new conditional generative adversarial network (CGAN) to improve the generative adversarial interpolation network (GAIN) [99]. They extracted the implicit category information contained in the data set and used it to generate a classifier to evaluate and optimize the interpolated data. The experimental results showed that this method was superior to the GAIN algorithm, even in the case of a high data miss rate. GAN contains a generative model \(G\) and a discriminant model \(D\), and the two models constantly update each other's parameters until they reach the optimum in training. In fact, the operation of the GAN model is a minimax iterative process about \(G\) and \(D\). CGAN is a common extension of GAN. It adds certain conditional information \(y\) before the generative model and discriminant model in GAN. Figure 7 shows the structure of a simple conditional adversarial network. In the study by [PERSON] et al. [101], \(y\) is set as a category label. Under such conditions, the objective function of the optimization problem is as follows:  
\[\underset{G}{\text{minmax}}V(D,G)=E_{X\sim P_{data(x)}}[logD(x|y)]+E_{X\sim P_ {z}(z)}\Big{[}\log\Big{(}1-D\big{(}G(x|y)\big{)}\Big{)}\Big{]} \tag{2}\]  
where \(V(D,G)\) is the objective function of the CGAN; \(x\) and \(y\) represent the training data and the input condition data, respectively; and \(P_{z}(z)\) is the prior noise variable. \(G\big{(}z;\theta_{g}\big{)}\) is used to learn the distribution \(p_{g}\) of training data \(x\), ranging from \(P_{z}(z)\) to data space. The output of discriminator \(D(x;\theta_{d})\) is used to represent the probability that \(x\) came from the training data, from which we can judge whether \(x\) can be identified as \(p_{g}\). In 2019, [PERSON] et al. [103] studied temporal correlations among the sensor data. Before this, most sensor data recovery methods only focused on the spatial correlation among data. They proposed a data-driven bidirectional recursive neural network for data reconstruction. Sensor data from the Telegraph Road Bridge in Michigan were used to verify the accuracy of the proposed method in data reconstruction. [PERSON] et al. [104] developed a response recovery method for heterogeneous structures. They reported that this approach took into account not only the temporal and spatial dependencies of the data but also the dependencies between heterogeneous structural responses. Furthermore, they proposed a parallel optimization method to optimize the parameters of the network. Three months of monitoring data on a bridge were collected to train and test the proposed method. The results showed that this method can interpolate the missing data accurately, especially when the data loss rate was high. [PERSON] et al. [105] introduced a recursive neural network when studying the temporal correlation of data, and built a framework for abnormal data recovery. The monitoring data of the Bund Bridge in Ningbo, China were used to verify the accuracy of their framework. [PERSON] et al. [106] developed a spatiotemporal graph attention network for restoring missing data. The network uses the inherent temporal and spatial dependencies of sensor networks for modeling to obtain temporal and spatial features to reconstruct missing signals. They also discussed the recovery accuracy of multi-source and single-source data fusion. In the application of the cable force data recovery of long-span cable-stayed bridges, the model showed satisfactory accuracy. [PERSON] et al. [107] proposed two matrix decomposition (MF) methods based on autoregression (AR) for data interpolation and structural response prediction. In the first approach, the authors developed a time regularizer and combined it with the standard MF formula. In the second approach, they introduced an additional AR-based matrix. The accuracy and reliability of the proposed method were verified by the SHM data set collected in the field.