# Air Traffic Trajectory Operation Mode Mining Based on Clustering
## 3 Clustering Analysis-Based Aircraft Operation Pattern Mining
### Historical Track Clustering  
The density peak clustering algorithm is a clustering algorithm proposed in recent years. Compared with the traditional DBSCAN clustering algorithm, it has a faster solution speed, and can extract the points that are most likely to become the class center in the global data through the two indicators of local density and similarity measurement,  
Figure 7: The flow chart of the algorithm based on ADMM. which solves the problem that it is difficult for the traditional algorithm to distinguish low-density clusters. Although DPCA can effectively identify some tactical traffic flows that form a much lower density than normal flight patterns due to control or regional reasons, DPCA cannot identify abnormal data, and abnormal trajectories that deviate significantly from normal data are also directly classified into nearby cluster classes, so this paper used OFAE in combination with DPCA, and removed the deviation points from the original data by OFAE, reconstructing the trajectory. At the same time, the normal trajectory was condensed to improve the local density, which makes up for the defect that DPCA cannot denoise. After AE was used to compress the track, the time characteristics of the track were erased, leaving only the spatial characteristics. Therefore, the Euclidean distance was used to measure the similarity between the two tracks. The hard threshold was selected as the distance threshold of the model parameter, and the step count function was selected as the density statistical function, \(\chi(\cdot)\). Then, the local density of the track \(i\) was calculated as follows:  
\[\rho_{i}=\sum_{j}\chi\big{(}d_{ij}-d_{c}\big{)} \tag{18}\]  
where \(\chi(\cdot)\) is the density statistical function, which specifies the calculation of density when the condition is satisfied, and here the step counting function was chosen. \(d_{ij}\) is the distance between the points \(i\) and \(j\), and \(d_{c}\) is the distance threshold, which specifies the condition for density counting and can be determined by hard or soft thresholding (e.g., Gaussian kernel function). Next, the similarity distance, \(\delta_{i}\), of point \(i\) was calculated, and the rule is quite simple: if point \(i\) is the point with the largest local density, then \(\delta_{i}\) is the distance between that point and the point farthest away from it; conversely, the shortest distance between that point and all points with local density higher than itself is calculated. Then, we have:  
\[\delta_{i}= \begin{cases}\underset{\begin{subarray}{c}\rho_{j}>\rho_{i}\\ \text{max}\end{subarray}}{\min}\exists\rho_{j}>\rho_{i}\\ \text{max}\quad\quad\text{{other}}\end{cases}  \tag{19}\]  
where \(\rho_{j}\) and \(\rho_{i}\) are the local densities at points \(j\) and \(i\), respectively, \(\min\) is the shortest distance between point \(i\) and all points with a local density higher than \(i\), and \(\max\) is the distance between point \(i\) and the farthest point from \(i\) when \(\rho_{i}\) is the local maximum. As shown in Figure 8, the dataset contains two cluster classes, which are labeled in red and blue in Figure 8A. The local density, \(\rho_{i}\), and similarity distance, \(\delta_{i}\), were calculated for all data points to obtain Figure 8B. Figure 8: Dataset to be classified. (**A**) Dataset cluster class. (**B**) The figure of similarity distance. In Figure 8B, points 10 and 1 have the maximum local density, \(\rho_{i}\), and similarity distance, \(\delta_{i}\), which are the cluster centers of the two cluster classes, respectively, so the cluster centers can be determined directly from these two metrics without iterations. However, the more obvious drawback of DPCA is the high empirical component, which relies on the subjective tendency of the user in the problem of cluster center selection. In this example in Figure 8B, the classification result is clear, and points 1 and 10 can be clearly distinguished from other points in Figure 8B.