# 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
\tag{13}\]  
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 parameter matrix is \(\theta\). The model obtained a feasible optimization objective, but further analysis is needed for the task of reconstructing the track data. The AE-processed data are fed into a density-peak-based clustering model. To make similar traces more easily identifiable, AE is needed to remove traces with the anomalous patterns mentioned earlier and to make the reconstructed trace denser. In this paper, we assumed that there are two types of deviations to be separated: one is the deviation of dimensional track features to be compressed, and the other is the deviation of anomalous tracks. By applying the deviation matrix, \(S\), the penalty term is constrained using the L21-norm, and both requirements can be achieved simultaneously. The L21-norm is defined as follows:  
\[\parallel S\parallel_{2,1}=\sum_{i=1}^{n}\parallel s_{i}\parallel_{2}=\sum_{ i=1}^{n}\left(\sum_{j=1}^{m}s_{ij}\right)^{1/2} \tag{14}\]  
where \(s_{i}\) is the row vector of the i row of the matrix \(S\), and \(s_{ij}\) is the element of the \(i\) row and \(j\) column of the matrix. L21-norm found \(2\) norms for the row and \(1\) norm for the column of the matrix and restricted the sparsity of the row and column at the same time. When processing track data, the time-series characteristics of a track are tiled into rows, and then the sample set is processed with equal length. All tracks are aggregated by columns to obtain the tracking matrix. For track data, the constraint of the L21-norm can be expressed as that the time sequence points in the reconstructed track are as compact as possible and the spacing between different tracks is as short as possible. The elimination of rows and columns in the original track matrix during decoding represents the characteristics of removing deviated tracks and compressing redundancy, respectively. Therefore, it is confirmed that the optimization objectives of the track compression model based on OFAE are:  
\[\left\{ \begin{aligned} \min_{\theta,S}\parallel L& -D_{\theta}(E_{\theta}(L))\parallel_{2}+\lambda\parallel S\parallel_{21}\\ &\text{s.t.}X-L-S=0\end{aligned} \right. \tag{15}\]  
The L2-norm constraint is used for the tiled track feature sequence to make the timing points as compact as possible, and the distance function is defined to calculate the track distance. However, the track characteristics after isometric processing and compression, and the differential correlation characteristics of heading, speed, and other positions will be distorted. Therefore, in the input track characteristics, this variable information needs to be added directly, and the model is difficult to mine directly from the location data. Since the optimization goal has no clear requirements for the codec function, sigmoid is used as the activation function of neurons. The weight matrix of the decoder selects the transpose of the encoder weight matrix, and the bias term is independent. The encoding and decoding functions are as follows:  
\[\left\{ \begin{aligned} & E_{\theta}(x)=E_{W,b}(x)=\text{sigmoid}(Wx+b_{E})\\ & D_{\theta}(x)=D_{W,b}(x)=\text{sigmoid}\big{(}W^{T}E_{W,b}(x) +b_{D}\big{)}\end{aligned} \right. \tag{16}\]  
where \(x\) is the input vector, \(W\) is the weight matrix from the input layer to the hidden layer, \(b_{E}\) is the bias matrix from the input layer to the middle layer, and \(b_{D}\) is the bias matrix from the middle layer to the output layer.