## 3 Methodology
### Data Augmentation
#### 3.3.1 TimeGAN Construction  
TimeGAN consists of an autoencoder network, a supervised network, and an adversarial network. The specific structure is shown in Figure 3.  
Autoencoder network construction  
The embedding function and the recovery function establish a reversible mapping between the feature space and the latent space, thereby facilitating the adversarial network's capacity to learn the interrelationships inherent within time-series data. The embedding function is articulated as follows:  
\[h_{t}=e_{p}(h_{t-1},p_{t}) \tag{3}\]  
In Formula (3), \(e_{p}\) denotes the embedding function, which comprises three layers of gated recurrent units (GRUs) followed by a fully connected layer. The hidden layer comprises 24 neurons, which utilize the sigmoid activation function. The variable \(p_{t}\) denotes the code corresponding to the processed fault rate data of PSDs in the potential  
Figure 3: Fault data generation model for platform screen door systems based on TimeGAN. space. The subscript \(t\) denotes temporal information. Let \(h_{t}\) represent a low-dimensional temporal feature at time \(t\). The recovery function is characterized as follows:  
\[\widetilde{p}_{t}=r_{p}(h_{t}) \tag{4}\]  
\[\hat{p}_{t}=r_{p}\big{(}\hat{E}_{sup,t}\big{)} \tag{5}\]  
In Formulas (4) and (5), \(r_{p}\) denotes the recovery function, which serves the purpose of reconstructing \(h_{t}\) with respect to the data \(\widetilde{p}_{t}\), ensuring that it maintains the same dimensionality as the processed PSD fault rate data. It further re-establishes the supervisory space \(\hat{E}_{sup,t}\) of the generated data to correspond with the data \(\hat{p}_{t}\), maintaining dimensional congruence with the processed PSD fault rate data. * Supervised network construction  
The potential space coding \(h_{t}\) corresponding to the fault data of screen doors after processing and the fault data \(E_{t}\) generated by the generator are expressed in the supervised space by the following formulas:  
\[\hat{h}_{sup,t}=s_{p}(h_{t}) \tag{6}\]  
\[\hat{E}_{sup,t}=s_{p}(E_{t}) \tag{7}\]  
where \(s_{p}\) represents the supervision function, which is composed of three layers of GRUs and a fully connected layer, and _sigmoid_ is used as the activation function. \(\hat{h}_{sup,t}\) and \(\hat{E}_{sup,t}\) are the representations in the supervised space of \(h_{t}\) and \(E_{t}\). * Adversarial network construction  
The data produced by the generator from the stochastic time series are initially generated within the latent space. The formula associated with the generator network is delineated as follows:  
\[E_{t}=g_{p}(E_{t-1},z_{t}) \tag{8}\]  
In Formula (8), \(g_{p}\) denotes the generating function, which comprises a three-layer architecture of gated recurrent units (GRUs) in conjunction with a fully connected layer. Let \(E_{t}\) denote the PSD fault data generated by the generator at time \(t\), while \(E_{t-1}\) signifies the data produced in the preceding time step. The variable \(z_{t}\) represents a stochastic time series that possesses the same dimensionality as the preprocessed PSD fault rate data. The discriminator operates within the potential space to differentiate between three specific domains: the potential space \(h_{t}\) representing the fault data generated by screen doors after input processing, the potential space \(E_{t}\) associated with the generation of fault data, and the supervised space \(\hat{E}_{sup,t}\), which is also concerned with the generation of fault data. The formula pertinent to the discriminator network is delineated as follows:  
\[c_{real,t}=d_{p}(h_{t}) \tag{9}\]  
\[c_{fake\_e,t}=d_{p}(E_{t}) \tag{10}\]  
\[h_{t}=e_{p}(h_{t-1},p_{t}) \tag{3}\]0  
In Formulas (9)-(11), \(d_{p}\) denotes the discriminant function, which exhibits a structural similarity to the generating function. The discriminant function utilizes the processed fault data alongside the generated data as input variables for classification purposes.