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摘要: 复杂仿真模型一般具有多个不同类型且带有相关性的输出，现有验证方法存在变量信息缺失、变量相关性度量不准确等问题.为此，提出基于变量选择和概率分布差异相结合的多变量仿真结果验证方法，考虑不确定性的影响，对选取到具有相关性的多个变量进行联合验证.首先，引入分形维数和互信息方法对多元异类输出进行相关性分析，提取相关变量子集.而后对相关变量子集中的各变量提取数据特征，进而计算各相关变量子集关于数据特征的联合概率分布，采用概率分布差异法度量仿真输出和参考输出联合概率分布的差异，并将其转化为一致性程度；在此基础上综合多个验证结果得到模型可信度.通过实例应用及对比实验，验证了方法的有效性.Abstract: Complex simulation models often generate the multivariate and different types of output, some problems such as the lacking variables information and the inaccuracy correlation measurement are involved in the existing validation methods. A novel validation method combining variables selection with area metric is proposed, the multiple outputs with correlation are selected for the associated validation under uncertainty. The fractal dimension and mutual information methods are primarily applied to analyze the correlation among multivariate and divise responses and extract the correlated variable subsets. Next, the interesting data characteristics of all variables are extracted in subset and the corresponding joint cumulative distribution function (JCDF) of each subset related to any characteristic is calculated. The area metric is used to measure the difference between the simulation and reference output JCDFs of multivariate characteristics in each subset, and the differences are transformed into the consistency degrees. Then the multiple validation results are integrated to obtain the model credibility. Finally, the method is validated through the application case and comparison experiments.1) 本文责任编委 莫红
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图 2 考虑相关性的多元输出仿真结果验证方法流程
Fig. 2 Procedures of multivariate simulation result validation under correlation
图 3 纵向平面内弹目相对运动几何关系
Fig. 3 Geometrical relationship of relative missile-target movement in longitudinal plane
图 8 飞行器落点X坐标输出散点图
Fig. 8 Scatter diagram of X-direction drop point coordinates of the flight vehicle
图 9 目标终点位置X坐标输出散点图
Fig. 9 Scatter diagram of X-direction terminal point coordinates of the target vehicle
图 10 飞行器落点Z坐标输出散点图
Fig. 10 Scatter diagram of the terminal point of the target in the Z direction
图 11 飞行器落点X坐标与目标终点位置X坐标间的关系
Fig. 11 Relationship of X-direction coordinates between drop point of flight vehicle and terminal point of target
图 12 飞行器落点X方向坐标与Z方向坐标间的关系
Fig. 12 Relationship between X-direction and Z-direction coordinates of the drop point of flight vehicle
图 13 仿真和参考静态输出变量子集Ⅰ的联合CDF对比
Fig. 13 JCDF comparison of variable subset I between static simulation and reference output
图 14 仿真和参考静态输出变量子集Ⅱ的CDF对比
Fig. 14 Comparison of variable subset Ⅱ between static simulation and reference output
表 1 常用变量选择方法对比
Table 1 Comparison of general variable selection methods
变量选择方法 是否为原变量集的子集 是否支持非线性相关关系 个体决策所占比例 是否需要训练样本集 运行速度与变量个数的关系 SVD 否 否 是 否 线性增长 PCA 否 否 是 否 线性增长 KNN 是 是 是 是 指数增长 DT 是 是 是 是 指数增长 BN 是 是 是 是 指数增长 FD 是 是 是 否 线性增长 
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表 2 飞行器末制导过程的不确定参数取值
Table 2 Uncertainty parameters values in the terminal guidance process of flight vehicle
变量名 仿真模型参数分布 参考系统参数分布 大气密度系数${{C}_{{ }\!\!\rho\!\!{ }}}$ $N\left( 0, 0.033 \right)$ $N\left( 0, 0.033 \right)$ 升力系数${{C}_{{D}}}$ $N(0, 0.05)$ $N(0.02, 0.07)$ 阻力系数${{C}_{{L}}}$ $N(0, 0.033)$ $N(0.02, 0.033)$ 初始弹道倾角${{{\theta }_{0}}}~/{{\rm rad}}$ $N\left( 0.17, 0.09 \right)$ $N\left( 0.26, 0.07 \right)$ 初始视线角${{{\lambda }_{0}}}~/{{\rm rad}}$ $N\left( 0.17, 0.09 \right)$ $N\left( 0.17, 0.09 \right)$
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表 3 待验证的模型输出
Table 3 Model outputs to be validated
变量类型 变量名 动态 弹道倾角${\theta }~$(rad) 动态 攻角${\alpha }$ (rad) 动态 视线角${\lambda }~$(rad) 动态 弹目相对距离${{{D}_{{MT}}}}~$(m) 动态 目标速度${{v}_{{T}}}~$(m/s) 静态 飞行器落点X坐标${{{x}_{{f}}}}~$(m) 静态 飞行器落点Z坐标${{{z}_{{f}}}}~$(m) 静态 目标终点位置X坐标${{{x}_{{Tf}}}}~$(m) 静态 目标终点位置Z坐标${{{z}_{{Tf}}}}~$(m)
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表 4 多元输出变量选择结果
Table 4 Variables selection results of multiple outputs
输出类型 变量子集Ⅰ 变量子集Ⅱ 变量子集Ⅲ 动态 $\theta $, $\alpha$, $\lambda$, ${{D}_{{MT}}}$ ${{v}_{{T}}}$ - 静态 ${{x}_{{f}}}$, ${{x}_{{Tf}}}$ ${{z}_{{f}}}$ ${{z}_{{Tf}}}$
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表 5 动态输出均值曲线的一致性分析结果
Table 5 Consistency analysis results of the mean curves of dynamic outputs
变量名 位置特征一致性 形状特征一致性 $\theta $ 0.92 0.74 $\alpha$ 0.63 0.60 $\lambda$ 0.98 0.74 ${{D}_{{MT}}}$ 0.97 0.61
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表 6 仿真和参考输出变量子集的一致性分析结果
Table 6 Consistency analysis results of the variables subset of the simulation and reference outputs
输出变量类型 变量子集标号 累积概率分布差异 可信度结果 动态 变量子集Ⅰ 位置差异: $8.92\times {{10}^{{-8}}}$ 位置特征: 0.99 动态 变量子集Ⅰ 形状差异: $1.1\times {{10}^{{-3}}}$ 形状特征: 0.94 动态 变量子集Ⅱ 0 1 静态 变量子集Ⅰ $1.6\times {{10}^{5}}$ 0.84 静态 变量子集Ⅱ 0.5 0.9 静态 变量子集Ⅲ 0 1
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表 7 验证实验的不确定参数取值
Table 7 Uncertainty parameters values for validation experiments
试验编号 参考系统${{\theta }_{0}}$取值 实验组Ⅰ ${{\theta }_{0}}$取值 实验组Ⅱ ${{\theta }_{0}}$取值 1 $N\left( 0.26, 0.07 \right)$ 0.26 $N\left( 0.15, 0.07 \right)$ 2 $N\left( 0.26, 0.07 \right)$ $N\left( 0.26, 0.04 \right)$ $N\left( 0.21, 0.07 \right)$ 3 $N\left( 0.26, 0.07 \right)$ $N\left( 0.26, 0.07 \right)$ $N\left( 0.26, 0.07 \right)$ 4 $N\left( 0.26, 0.07 \right)$ $N\left( 0.26, 0.1 \right)$ $N\left( 0.31, 0.07 \right)$ 5 $N\left( 0.26, 0.07 \right)$ $N\left( 0.26, 0.13 \right)$ $N\left( 0.37, 0.07 \right)$
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