# Air Traffic Trajectory Operation Mode Mining Based on Clustering
## 3 Clustering Analysis-Based Aircraft Operation Pattern Mining  
### Trace Feature Compression  
#### 3.1.1 Autoencoder of Trajectory Based on L21-Norm Constraints  
The L1 regularization and relative entropy constraint method can effectively suppress the overfitting of the autoencoder [19]. The RSPCA (robust principal component analysis) method attempts to learn and express rather than resist the part of the original input that cannot be learned by the model, which is called the deviation. The ability of the autoencoder to restore the rest of the input data is improved by removing the deviation. Inspired by this, this paper applied a similar constraint method to the AE model, removed the abnormal trajectory, extracted the main features of the track data, and further reduced the reconstruction error while constraining the fitting ability of the autoencoder. The optimization objectives of the conventional AE model are:  
\[\min_{\theta}\parallel X-D_{\theta}(E_{\theta}(X))\parallel_{2} \tag{10}\]  
where \(\theta\) is the parameter matrix of the model (including two parts of encoding and decoding), \(E_{\theta}(\cdot)\) is the encoder whose parameter matrix is \(\theta\), and \(D_{\theta}(\cdot)\) is the decoder whose  
Figure 5: Reconstruction trajectory. (**A**) Reconstruction trajectory with \(\lambda\) = 0.5, T = 51. (**B**) Reconstruction trajectory with \(\lambda\) = 0.2, T = 91. Figure 6: Trajectory comparison. parameter matrix is \(\theta\). The coding network and decoding network of the default model are multi-layer depth. The input feature \(X\) consists of two parts: low-rank information matrix \(L\) and sparse deviation matrix \(S\), as follows:  
\[X=L+S \tag{11}\]  
where \(L\) is a low rank with the same dimension of \(X\) and represents a low dimensional manifold that can be well-restored in the encoder in the input data, and \(S\) is the deviation that is difficult for the model to learn, which is usually caused by the characteristics of the data itself or noise. Traditional AE models are prone to overfitting problems when learning the deviation in the input. Some optimized AE models such as DAE (denoising autoencoder) resist the deviation, \(S\), through noise learning. The denoising autoencoder adds noise to the input training data and makes the autoencoder learn to remove this noise to obtain the real input that is not contaminated with noise, so that the most important features can be extracted, and more robust representations of the input data can be learned. The generalization capability is stronger than that of the general autoencoder. Here, convex relaxation technology was used to learn and express the deviation, so that when the model restores the information matrix, \(L\), it can better extract the main traffic flow in the airspace, non-linearly. In this paper, the coding model using this deviation extraction method is called the filtered autoencoder (OFAE). The optimization objectives of OFAE are as follows [20]:  
\[\left\{ \begin{aligned} &\underset{L,S}{\text{min}}(L)+\lambda\parallel S \parallel_{0}\\ &\text{s.t.}\parallel X-L-S\parallel_{F}^{2}=0\end{aligned} \right. \tag{12}\]  
where \(\lambda\) is the control parameter of medium sparsity, \(S\), \(\rho(\cdot)\) is the rank of the matrix, \(\parallel\cdot\parallel_{0}\) represents the number of non-zero elements, and \(\parallel\cdot\parallel_{F}\) represents F-norm. Then, the kernel norm and \(1\) norm were used to replace the rank operator and \(0\) norms for optimization, to solve the NP-hard problem of the matrix rank operator and \(0\) norm optimization. The compressed vector can obtain the linear mapping of the original input, and the following optimization objectives were obtained:  
\[\left\{ \begin{aligned} &\underset{\theta,S}{\text{min}}\parallel L-D_{\theta}(E_{ \theta}(L))\parallel_{2}+\lambda\parallel S\parallel_{1}\\ &\text{s.t.}X-L-S=0\end{aligned} \right.