## 3 Case Study
### Implementation Details  
For the dataset of the vehicle-induced cable tension \(D=\{T_{v,i}(t)\}\), the vehicle loads are the same; therefore, the relations among them are related to their influence surface and their relative stiffness only. Therefore, the model learned using the dataset of a specified period forms an agent model of that period; once the model changes in another period, the structural condition can be inferred to have changed. And the reconstruction error predicted by the pretrained baseline model for a specific period (i.e., the early days of bridge operation) can be used as the damage indicator.  
The proposed model learns the spatiotemporal relationships among the monitored data of all channels with the input and output nodes:  
\[T_{v,i}(t)=AE_{\theta}\Big{(}T_{v,j}(t+\tau)\Big{)} \tag{14}\]  
Algorithm 1 is the pseudo-code for the training procedure: the length of the LSTM is the sequence of length \(T\): \(\left\{\mathbf{x}^{(t_{n}+1)},\mathbf{x}^{(t_{n}+2)},\cdots,\mathbf{x}^{(t_{n}+\mathrm{T}) }\right\}\), inputs \(\mathbf{x}^{(t_{n})}:\left\{T_{v,i}^{b}(t_{n})\right\}\in R^{batch\_size\times M}\) represent the training batch of the vehicle-induced cable tension of M cables at \(t_{n}\) step, outputs \(\mathbf{y}^{(t)}\) represent the corresponding predictions in the unsupervised learning way, and target outputs are equal to inputs \(\mathbf{y}^{(t)}=\mathbf{x}^{(t)}\). [PERSON]'s optimization algorithm is employed to train the model.  
```
1. Initialize parameters \(\mathbf{\theta}\);
2. Specifies batch_size, the length of the input \(T\), learning rate \(\alpha\);
3. For \(k=1,\cdots,max\_iteration\)
4. Randomly generates batch_size integers from \([1,N-T]\);
5. Generates normalized training set \(\mathbf{x}=\left\{T_{v}^{b,i}(t_{n}:t_{n}+T)\right\}\in R^{batch\_size\times T \times M}\), target_y = \(\mathbf{x}\), randomly sets dimensions \(M_{1}\in[1,M]\) in \(\mathbf{x}\) to \(0\) to obtain train_x = \(\mathbf{\overline{x}}\);
6. Updating model parameters using AdamOptimizer.
```  
**Algorithm 1**LSTM-structured autoencoder training  
In this task, the units for the inputs and outputs are 14, corresponding to the 14 channels of the monitoring dataset. The LSTM-structured encoder and the decoder cell have both 3 layers and 32 units in each layer. Linear layers of _layer_\(1:\mathbb{R}^{32}\rightarrow\mathbb{R}^{5}\) and _layer_\(2:\mathbb{R}^{32}\rightarrow\mathbb{R}^{14}\) are added to the encoder and the decoder, respectively, to satisfy the shapes of the hidden layer and the output layer. Other hyperparameters are set as: \(T=2400\), \(max\_iteration=100\),000, \(batch\_size=30\) and \(\alpha=0.005\), and MSE error is employed. The whole model is built on TensorFlow, and the optimizer is named Adam Optimizer. An autoencoder structured by a 3-layer DNN with [64; 32; 5] hidden units and a decoder structured by a 3-layer DNN with [5; 32; 14] hidden units are also developed as comparison models.  
The dataset obtained from 13 May 2015 to 19 May 2006 is used to generate the training dataset, and the dataset of the other 3 days in 2007, 2009, and 2011 is used to generate the test dataset. Segmentations with 100 points of overlap are adopted for the data augmentation. The total length of the cable tension dataset during the 7-day monitoring period in 2006 is \(7\times 3600\times 24\times 2=1\),209,600, therefore, with the above overlap and the segmentation length, the size of the training dataset is 12,073.