## 3 Integrity Detection Method for Semi-Physical Simulation Data
### Data Expansion and Regularization Based on the Bootstrap Method  
Considering that the coordinates of the landing point of an aircraft are small sample data, the data need to be expanded and regularized. At present, many methods have been accumulated in engineering to address the problem of small sample data expansion. These methods can be roughly divided into two categories according to whether they utilize prior information. One is traditional estimation methods represented by the Bayesian method, which uses historical information as prior information to estimate the parameters of small sample information. The other is represented by the bootstrap method, which does not require historical prior information and obtains new samples by resampling the small sample data, thereby transforming the small sample problem into a large sample. Since the focus of this paper is to expand the sample capacity based on the existing small sample data information, there is no prior information available, and a Bayesian-like method is not applicable. In recent years, some scholars have proposed data expansion methods based on neural networks, but the process of unsupervised learning in these methods requires a large sample capacity and is not suitable for the expansion of small sample data. To satisfy the needs of small sample data expansion, this paper presents a data expansion method based on bootstrapping. This method requires less calculation and is suitable for a situation in which the number of dynamic data sequence calculation calls is high, and the calculation amount is large. The observed sample \(X=(x_{1},x_{2},\cdots,x_{n})\) is the overall sample, and its sample capacity is limited. The sample is called the original sample. If \(x_{i}\sim F(x)\), \(i=1,2,\cdots,n,F(x)\) is unknown, it is sorted to obtain sequential samples \(x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)}\). It is assumed that the probability of datapoints falling into each cell interval \(\left[x_{(k)},x_{(k+1)}\right],k=1,2,\cdots,n-1\) is equal to \(\frac{1}{n}\), that is, it is uniformly distributed in each cell. The empirical distribution function constructed from these original samples is:  
\[F_{n}(x)= \begin{cases}0&x<x_{(1)}\\ k/n&x_{(k)}\leq x<x_{(k+1)}\\ 1&x\geq x_{(n)}\end{cases}  \tag{3}\]  
Let \(\theta=\theta(F)\) be the unknown parameter of population distribution F, and \(\hat{\theta}=\hat{\theta}(F_{n})\) is an estimate of \(\theta\). Then, the estimated error is:  
\[R(X,F)=\hat{\theta}(F_{n})-\theta(F)\triangleq T_{n} \tag{4}\]  
In the formula, \(T_{n}\) is a function of \(X\) and \(F\). Let \(X_{1}^{*},X_{2}^{*},\cdots,X_{n}^{*}\) be another set of samples resampled from \(F_{n}\), and let  
\[R^{*}(X^{*},F_{n})=\hat{\theta}(F_{n}^{*})-\hat{\theta}(F_{n})\triangleq R_{n} \tag{5}\]  
In the formula, \(F_{n}^{*}\) is the empirical distribution of \(X_{1}^{*},X_{2}^{*},\cdots,X_{n}^{*}\) and \(R_{n}\) is the bootstrap statistic of \(T_{n}\). Therefore, the mean and variance of \(R(X,F)\) characterize the mean and variance of the estimation error of \(\theta(F)\). Using a computer to perform \(N\) rounds of sample resampling, \(N\) bootstrap statistics \(R^{*}1,R^{*}2,\cdots,R^{*}N\) can be obtained. Thus, the probability distribution of \(R_{n}\) can be obtained from the frequency curve of \(R^{*}(i)\), and the distribution of \(T_{n}\) with probability distribution \(R_{n}\) can be modelled. From Formula (5), \(\theta(F)\approx\hat{\theta}(F_{n})-R_{n}\). \(N\) samples and \(\theta(F)\) can be obtained through computer simulation (usually \(N\) is more than 1000). The distribution and eigenvalues of the unknown parameter \(\theta\) can be obtained via corresponding statistical methods. The most critical aspect of the above process is the sampling method. At present, there are three kinds of sampling methods, including sampling methods that generaterandom numbers between 0 and 1 and the cumulative sum is one, methods that directly use the original data for replacement sampling, and empirical distribution sampling methods.