# Intelligent Trajectory Prediction Algorithm for Reentry Glide Target Based on Intention Inference
## 3 Parameter Value Prediction Network
### Importance Map of Strategic Places  
Usually, strategic places occupy an area. To accurately describe this area, it is assumed that the mapping area of a strategic place is elliptic in several directions, and its probability distribution conforms to the joint two-dimensional Gaussian distribution, which can be expressed as  
\[f_{k}(x,y)\ \sim N(x_{c}^{k},\delta_{1k^{\prime}}^{2},y_{c}^{k},\delta_{2k^{ \prime}}^{2},\rho) \tag{4}\]  
where \(f_{k}(x,y)\) represents the probability density of position \((x,y)\) for the \(k-th\) strategic place, \(\left(x_{c}^{k},y_{c}^{k}\right)\) is the central position of the \(k-th\) strategic place's ellipse, \(\delta_{1},\delta_{2}\) represent the variance of the distribution, and \(\rho\) is the correlation coefficient between \(x,y\).  
According to the theory of two-dimensional normal distribution, assuming that the central position of the distribution is \((x_{c},y_{c})\), the rotation angle of mapping ellipse is \(\delta\), and the radius of the major and minor axis of the ellipse's mapping standard deviation are \(t_{1},t_{2}\), respectively. Then, the distribution density can be expressed as  
\[f(x,y)\ \sim N\big{(}x_{c},\sigma_{1},y_{c},\sigma_{2},\rho_{xy}\big{)}=- \exp\left(-\frac{1}{2(1-\rho_{xy}^{2})}t_{xy}\right)/2\pi\sigma_{1}\sigma_{2} \sqrt{1-\rho_{xy}^{2}} \tag{5}\]where, \(\rho_{xy}=(t_{2}^{2}-t_{1}^{2})\cos\delta\sin\delta/\sqrt{\sigma_{1}\sigma_{2}}\ \sigma_{1}=\sqrt{t_{1}^{2}\cos^{2}\delta+t_{2}^{2}\sin^{2}\delta}\), \(\sigma_{2}=\sqrt{t_{2}^{2}\cos^{2}\delta+t_{1}^{2}\sin^{2}\delta}\), \(t_{xy}=(x-x_{c})^{2}/\sigma_{1}^{2}-2\rho_{xy}(x-x_{c})(y-y_{c})/\sigma_{1} \sigma_{2}+(y-y_{c})^{2}/\sigma_{2}^{2}\).  
According to the absolute importance of each strategic place \(\pi_{k}\), considering the combined effect for the location's importance in the map, the importance of each position in the map is calculated based on the Gaussian mixture model (GMM),  
\[p(x,y)=\sum_{i=1}^{K}\pi_{i}f_{i}(x,y) \tag{6}\]  
The extreme points of mixing probability are considered the key points attacked by the target. The distribution of probability density can be shown through the heat map, which means the absolute importance of each position and is named as importance map. Compared with the representation by simple points, the importance map can describe the importance of strategic places in more detail and rationality. What is more, based on the proposed model, the comprehensive influence of all strategic places on each position can be obtained that may generate a new key point. As shown in the following three examples, the importance distributions of the strategic places calculated by the importance map are shown in Figure 3.  
As can be seen from Figure 3, for the importance map with three strategic places, there are three kinds of cases. The first is that no new key points are generated and the attack positions are the centre of each strategic place, as shown in Figure 3a. The second is that a new key point is generated due to the influence of the two strategic places, as shown in Figure 3b. What is more, the new generated key point should also be considered when inferring the intention of the target. The third is that a new key point is generated by the joint influence of all the strategic places, as shown in Figure 3c.