# Air Traffic Trajectory Operation Mode Mining Based on Clustering
## 2 Data Analysis and Processing
### Trajectory Correction Based on Operational Characteristics  
Abnormal time points of trajectory caused by data errors in radar equipment and network transmission, as well as aircraft trajectory under abnormal conditions, will affect the operation law of aircraft under normal conditions of model learning [17; 18]. The noise data need to be filtered before generating a training trajectory set. The former are unreal data caused by sampling error, which should be eliminated. To ensure that there is no lack of timescale, it should be filled by numerical prediction. The latter come from real aircraft operations, and the characteristics are not typical. The data belong to outlier samples that cannot be learned by the model, so they should also be eliminated. For the abnormal time points process, it is usually assumed that the aircraft operates normally in a short period. The speed and heading remain stable. These two variables can determine the reasonable range of three-dimensional coordinates of each time point and mark the time points outside the range as abnormal points. When detecting time point \(\overset{\rightharpoonup}{x}_{t_{0}}\) (a three-dimensional vector of position) in a time series with a time interval of \(\Delta t\), the current moment is denoted by \(t_{0}\), and the nearest time window of length T is chosen, that is, T points before and T points after this point, if the point on one side is selected, and as many points as possible if the number is fewer than T (for example, for the starting point, select T points after it). The series is believed to be full, and the sequence is obtained after selection as:  
\[\overset{\rightharpoonup}{x}_{t_{0}-\Delta t}\,\ \ldots\,\overset{\rightharpoonup}{x}_{t_{0}- \Delta t}\,\ \overset{\rightharpoonup}{x}_{t_{0}}\,\ \overset{\rightharpoonup}{x}_{t_{0}+\Delta t}\,\ \ldots\,\ \overset{ \rightharpoonup}{x}_{t_{0}+\tau\Delta t} \tag{1}\]  
The displacement vector on magnitude and direction in detection is confined in the maximum velocity and the boundary value of the heading angle. The detection time \(t_{0}\) has the following relationship with the previous time point. If this time point is the starting point, the latter time point is used to formulate the relationship:  
\[\overset{\rightharpoonup}{x}_{t_{0}}=\overset{\rightharpoonup}{x}_{t_{0}- \Delta t}+\overset{\rightharpoonup}{\Delta x}=\overset{\rightharpoonup}{x}_{t_{0} -\Delta t}+\int_{t_{0}-\Delta t}^{t_{0}}|\overset{\rightharpoonup}{v}_{t}| \Delta\overset{\rightharpoonup}{\theta_{t}}dt \tag{2}\]  
where \(\overset{\rightharpoonup}{\Delta x}\), \(\overset{\rightharpoonup}{v}_{t}\) are the displacement vector and real-time velocity vector of the aircraft, and \(\overset{\rightharpoonup}{\theta_{t}}\) is the direction vector of the real-time heading. The detection of the point is the detection of the mode and direction of the pair \(\overset{\rightharpoonup}{\Delta x}\). Assuming the velocity maxima of the aircraft in  
Figure 3: Valid trajectory. (**A**) Valid trajectory at higher altitudes. (**B**) Valid trajectory at lower altitudes. this sequence is, respectively, \(v_{max},\ v_{min}\), then the range of displacement distance, \(|\overset{\rightharpoonup}{\Delta x}|\), can be determined:  
\[\left|\int_{t_{0}-\Delta t}^{t_{0}}|\overset{\rightharpoonup}{v}_{min}| \Delta\overset{\theta}{\theta}dt\right|<\left|\int_{t_{0}-\Delta t}^{t_{0}}| \overset{\rightharpoonup}{v}_{t}|\Delta\overset{\theta}{\theta}dt\right|< \left|\int_{t_{0}-\Delta t}^{t_{0}}|\overset{\rightharpoonup}{v}_{max}|\Delta \overset{\theta}{\theta}dt\right| \tag{3}\]  
\[|\overset{\rightharpoonup}{v}_{min}|\Delta t<|\overset{\rightharpoonup}{ \Delta x}|<|\overset{\rightharpoonup}{v}_{max}|\Delta t \tag{4}\]  
The displacement vector is confined by two components, horizontal angle and vertical angle. If the horizontal angle boundaries of this sequence are \(\ \alpha_{1},\ \alpha_{2}\), and the vertical angle boundaries are \(\ \beta_{1},\ \beta_{2}\), then both are within the interval \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\).