## 3 Integrity Detection Method for Semi-Physical Simulation Data  
### Dimensionality Reduction in Landing Point Data  
The semi-physical simulation landing point coordinates of an aircraft \((x_{i},y_{i},z_{i})\), \(i=1,2,\cdots,m\) include three dimensions. Considering that two-dimensional aircraft data are more convenient to visualize and analyze, the coordinates of the simulated landing point are projected into three, two-dimensional planes \(xoy\), \(yoz\), and \(xoz\), to achieve data dimensionality reduction, as shown in Figure 2. Taking the coordinate data \((x_{i},z_{i}),i=1,2,\cdots,m\) projected on plane \(xoz\) as an example, the coordinate is divided into intervals. Referring to the selection method of the optimal histogram group spacing and the number of groups in histogram theory, the semi-physical simulation data interval \([z_{b},z_{b}]\) is divided into \(n\) equidistant intervals on the \(z\)-axis. Studies have shown that many scholars have not considered the statistical characteristics of the original data in the selection of group distances in histograms and have directly adopted empirical methods according to the sample capacity, lacking emphasis on histogram theory and optimal histogram production methods. Regarding the choice of group distance, there are many methods in existing research results and there is great controversy. The group distance greatly affects the properties and population distribution characteristics of the histogram, and even if an improper group distance is used, the histogram will be \"distorted\". In the same way, in the process of finding the envelope function  
Figure 1: Flow chart of semi-physical simulation and field data fusion evaluation. \(F_{b}(x)\) for the semi-physical simulation data, the different division of the ordinate of the projected scatterplot will also lead to very different or even \"distorted\" envelope functions. Many scholars have discussed the determination of the histogram group distance. [PERSON] adopts an empirical method and believes that when the sample capacity n is large, the number of groups k is between 10 and 20; when the sample capacity n < 50, k is usually between 5 and 6, and the sample frequency in each group of intervals is usually not less than 5. [PERSON] uses [PERSON]'s formula in his work: \(k\approx\zeta\cdot n^{2/5}\zeta=1\sim 3\). [PERSON] gives two suggestions for making histograms: 1 the number of groups can be approximately equal to the square root of the sample capacity and 2 the distances between groups are equal. [PERSON] obtains a rough relationship between the number of groups k and the sample capacity n under the assumption that the sample data satisfy the population distribution symmetry and approximate normality:\(k=1+\log_{2}n\). Many scholars and some statistical software are still based on this formula. [PERSON] modifies the Sturges formula when considering the bias of the sample data. The number of groups \(k=1+\log_{2}n+K_{e}\), where \(K_{e}=\log_{2}\left(1+\hat{\gamma}\sqrt{(n+1)(n+3)/\left[6(n-2)\right]}\right)\), \(n\) is the sample capacity, and \(\hat{\gamma}\) is the estimated skewness. According to the above analysis, when the population distribution of the sample data is symmetrical and approximately normal, the Sturges formula considers both the population distribution characteristics and the sample capacity and is the best method for determining the group distance. The main idea of the Sturges method is to approximate the normal distribution with a symmetric binomial distribution \((p=0.5)\). Consider an idealized grouping with the number of groups as k. Assume that the probability of each sample observation falling in the i-th interval approximately obeys a binomial distribution with a probability of \(p=0.5\). The average sample frequency of the i-th group is \(\gamma_{i}=C_{k-1}^{i},i=0,1,\cdots k-1\), where the number of combinations is \(C_{k-1}^{i}\). Then, when k is large, such an ideal histogram can be approximated as a normal distribution with mean \((k-1)/2\) and variance \((k-1)/4\).