## 6 Application  
Take the landing point data of a 534A aircraft as an example. The coordinates of the landing point where the aircraft hits the air for the first time are marked as the origin (0, 0, 0) and a space Cartesian coordinate system is established. \(x,y,\) and \(z\) represent the coordinate axis data in the three directions. The front, right, and top are taken as the positive directions of the three coordinate axes and the unit of distance is meters. The sample size of the semi-physical simulation data of the landing point is \(N_{A}=50\) and the sample size of the field test data landing point is \(N_{B}=10\). These parameters are shown in Table 1. All the following calculations are based on the software MATLAB 2021b implementation. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**No.** & \(x\)**-Axis** & \(y\)**-Axis** & \(z\)**-Axis** & **No.** & \(x\)**-Axis** & \(y\)**-Axis** & \(z\)**-Axis** \\ \hline
1 & 0.788436 & \(-0.84452\) & \(-0.03967\) & 6 & \(-0.11351\) & \(-1.35334\) & 0.201431 \\
2 & \(-1.25091\) & 0.620047 & \(-0.10734\) & 7 & \(-0.12106\) & \(-0.12984\) & \(-0.70126\) \\
3 & \(-1.06563\) & \(-0.18271\) & 0.204729 & 8 & \(0.911668\) & \(0.005804\) & \(0.972401\) \\
4 & \(-0.15802\) & 0.921861 & \(-0.49578\) & 9 & \(-1.22589\) & 0.262081 & \(-0.68309\) \\
5 & 1.02129 & \(-0.39676\) & 0.606478 & 10 & \(-0.53463\) & 0.827411 & \(-0.39213\) \\ \hline \end{tabular}
\end{table} 
Table 1: Field landing data. Figure 5: Diagram of the area measurement method based on the probability distribution distance. Through the introduction and analysis of the mainstream group distance determination methods, we conducted experiments on the \(z\)-axis data projected from the semi-physical simulation data on the plane \(xoz\) to find the best group distance determination method. The methods proposed by [PERSON], [PERSON], [PERSON], [PERSON], [PERSON], and [PERSON] are used to determine the group distance, and the grouping situation is shown in Figure 6. The number of groups is \(k=6,5,10,7,11,6\). Clearly, the histograms based on the methods proposed by [PERSON], [PERSON], and [PERSON] are too flat, and the original data information is largely lost. The results of the normal distribution hypothesis test on the \(z\)-axis data showed that the \(p\)-values of the Jarque-Bera test and the Lilliefors test (modified Kolmogorov-Smirnov test) at a confidence level of 95% were 0.4093 and 0.1176, respectively. All are greater than 0.05, so the null hypothesis is accepted; that is, the \(z\)-axis data are considered to obey a normal distribution, so it is effective and meaningful to use the methods proposed by [PERSON], [PERSON], and [PERSON] to determine the group distance. The skewness of the data is tested as follows and the skewness coefficient is calculated:  
\[S_{k}=\frac{\sum\limits_{i=1}^{n}{(x_{i}-\overline{x})}^{3}}{\sigma^{3}\cdot N }=0.1330\]  
Therefore, the sample data are unbiased and [PERSON]'s correction method is not considered. We choose to use the Sturges method to determine the group distance. According to the projection map of the landing point data of the semi-physical simulation on the plane, the interval is divided according to the Sturges method of determining the group distance. The sample size is 50, so the number of groups k and the group distance h can be calculated as follows:  
\[k=1+\log_{2}50=1+\frac{\text{lg50}}{\text{lg2}}=6.64386\approx 7\]  
\[h=\frac{\text{max}(y)-\text{min}(y)}{k}=\frac{0.8157-(-0.9091)}{7}=0.2464\]  
Taking  
each cell is expanding to 100 as the standard, and a scatterplot of the sample data after the expansion is shown in Figure 8. Similarly, the sample data expansion map projected on the planesandcan be obtained by the same method.