# Air Traffic Trajectory Operation Mode Mining Based on Clustering
## 2 Data Analysis and Processing
### Trajectory Correction Based on Operational Characteristics
If \(\frac{\overset{\rightharpoonup}{\Delta x}}{|\Delta x|}= \begin{pmatrix}n_{1} \\ n_{2}\\ n_{3}\end{pmatrix} \), the horizontal angle is \(\alpha_{t}\) and the vertical angle is \(\beta_{t}\):  
After calculating the angle and distance range, Formula (5) is obtained:  
\[\left\{ \begin{array}{l}|\overset{\rightharpoonup}{v}_{min}|\Delta t<| \overset{\rightharpoonup}{\Delta x}|<|\overset{\rightharpoonup}{v}_{max}| \Delta t\\ \alpha_{1}<\sin^{-1}\frac{n_{2}}{\sqrt[]{1-n_{3}}^{2}}<\alpha_{2}\\ \beta_{1}<\sin^{-1}n_{3}<\beta_{2}\end{array} \right. \tag{5}\]  
Thus, the arc space area that \(\overset{\rightharpoonup}{x}_{t_{0}}\) may fall into is determined, and higher-quality trajectory data can be obtained by correcting the abnormal points outside the range. After eliminating the anomalies that deviate from the trajectory, the corresponding predicted values are estimated to fill in the eliminated moment points by the mean values of velocity and heading in this time period, and this predicted value should also be able to pass the anomaly test described above. For the moment point to be corrected, \(\overset{\rightharpoonup}{x}_{t_{0}}\), the mean values of velocity, horizontal angle, and vertical angle are calculated for each moment point within the proximity time window of its length \(T\). The estimated \(\overset{\rightharpoonup}{\Delta x}^{*}\) value of the displacement vector at that moment point can be expressed as:  
\[\overset{\rightharpoonup}{\Delta x}^{*}=|\overset{\rightharpoonup}{ \Delta x}^{*}|\overset{\rightharpoonup}{n}^{*} \tag{6}\]  
where \(|\overset{\rightharpoonup}{\Delta x}^{*}|\) is the displacement distance and \(\overset{\rightharpoonup}{n}^{*}\) is the displacement direction, which can be expressed by the mean value as:  
\[\left\{\begin{array}{l}|\overset{\rightharpoonup}{\Delta x}^{*}|=\overline {\upsilon}\Delta t\\ \overset{\rightharpoonup}{n}^{*}= \begin{pmatrix}\cos\overline{\beta}\cos \overline{\alpha}\\ \cos\overline{\beta}\sin\overline{\alpha}\\ \sin\overline{\beta}\end{pmatrix} \right. \tag{7}\]  
where \(\overline{\upsilon},\overline{\alpha}\), and \(\overline{\beta}\) are the mean values of aircraft speed, horizontal angle, and vertical angle, respectively. Then, the coordinates of the current time, \(\overset{\rightharpoonup}{x}^{*}_{t_{0}}\), can be determined according to the coordinates of the previous time point. The corrected coordinates are:  
\[\overset{\rightharpoonup}{x}^{*}_{t_{0}}=\overset{\rightharpoonup}{x}_{t_{0 }-\Delta t}+\overline{\upsilon}\Delta t \begin{pmatrix}\cos\overline{\beta} \cos\overline{\alpha}\\ \cos\overline{\beta}\sin\overline{\alpha}\\ \sin\overline{\beta}\end{pmatrix}  \tag{8}\]  
To sum up, the core idea of the methods of trajectory correction and reconstruction is to deduce the possible position range of the next time step according to the possible speed and heading range of the aircraft at the current time and adjust the reconstruction degree by limiting the position range. The default range is from the maximum value, MAX, to the minimum value, MIN. When the motion feature-based trajectory correction and reconstruction technique described in the preceding section is implemented in Section 2.1, the reconstruction parameters \(\lambda\) and time window of length \(T\) are introduced to facilitate model adjustment. The larger the time window of length \(T\) is set, the higher the smoothness during reconstruction. \(\lambda\) is the coefficient limiting the reconstruction range, \(0\leq\lambda\leq 1\), and the larger the \(\lambda\), the smaller the degree of re-configuration. The relationship between the limit range and \(\lambda\) is as follows. Taking the speed range as an example, it refers to the window mean value, AVR:  
\[v_{AVR}+\lambda(v_{AVR}-v_{MIN})\leq v\leq v_{AVR}+\lambda(v_{MAX}-v_{AVR}) \tag{9}\]  
It can be seen that the default range \(\lambda=1\) is used, that is, it is allowed between the maximum values. The smaller the size of \(\lambda\), the stricter the anomaly detection and the greater the degree of reconstruction.