基于正态-逆伽马分布的反巡航导弹命中概率估计方法

doi:10.12305/j.issn.1001-506X.2022.08.27

摘要/Abstract

摘要：

有效估计反巡航导弹武器系统命中概率是确定反巡航导弹武器系统弹药消耗的基础, 反巡航导弹试验成本高, 导致数据不足, 给估计反巡航导弹武器系统的命中概率带来困难。本文以防空试验中靶面落点服从二维联合正态分布的现象为切入点, 结合巡航导弹等效矩形域的特点, 提出了基于正态-逆伽马分布的反巡航导弹武器系统命中概率估计方法。选取正态分布和逆伽马分布作为反巡航导弹武器系统命中概率特征参数的先验分布, 并通过贝叶斯分析方法得到相应的后验分布函数。采用参数估计得到先验分布中的超参数, 确定命中概率的最大后验估值, 示例表明了方法的可操作性, 具有广阔应用前景。

关键词: 命中概率, 贝叶斯推断, 正态逆伽马分布, 最大后验估计

Abstract:

Effective estimation of hit probability of anti-cruise missile weapon system is the basis of determining ammunition consumption of anti-cruise missile weapon system. The problem of insufficient data caused by the high cost in the anti-cruise missile test makes it difficult to estimate the hit probability of the anti-cruise missile weapon system. Considering that the target surface's landing point obeys the two-dimensional joint normal distribution in the air defense test, and with the characteristics of equivalent rectangular domain for cruise missile, a hit probability estimation method of anti-cruise missile weapon system based on normal-inverse Gamma distribution is proposed. Normal distribution and inverse Gamma distribution are selected as the prior distribution of hit probability characteristic parameters of the anti-cruise missile weapon system, and the corresponding posterior distribution function is obtained by the Bayesian analysis method. Parameter estimation is used to obtain the hyper-parameter in the prior distribution, and the maximum posterior estimate of the hit probability is determined. The example shows that the method is operable and has broad application prospects.

Key words: hit probability, Bayesian inference, normal-inverse Gamma distribution, maximum posterior estimation

中图分类号:

E917

李康, 史宪铭, 李广宁, 刘昊邦. 基于正态-逆伽马分布的反巡航导弹命中概率估计方法[J]. 系统工程与电子技术, 2022, 44(8): 2621-2627.

Kang LI, Xianming SHI, Guangning LI, Haobang LIU. Estimation method of hit probability of anti-cruise missile weapon based on normal-inverse Gamma distribution[J]. Systems Engineering and Electronics, 2022, 44(8): 2621-2627.

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m_y_i	σ_y_i²	m_z_i	σ_z_i²
1.377 490	0.172 372	1.515 220	0.907 454
0.708 472	2.187 920	1.550 930	0.594 746
0.347 654	0.377 026	1.707 720	0.293 170
1.075 980	1.189 570	1.870 490	1.187 260
0.278 324	0.167 917	1.293 320	0.352 415
0.743 739	1.191 890	1.039 020	0.411 809
0.926 964	0.222 406	1.197 160	0.585 176
2.022 300	0.451 792	1.309 380	0.639 399
1.565 120	0.358 150	2.163 270	0.401 670
0.808 005	0.190 375	1.029 120	0.555 096
0.462 536	0.116 871	1.214 530	0.262 952
1.041 060	0.227 478	1.946 280	0.379 593
1.463 190	0.141 815	1.755 720	0.381 999
0.847 913	0.768 632	1.780 910	0.266 508
1.435 770	0.452 978	0.912 044	0.397 532
-0.525 189	0.256 578	1.309 400	0.255 707
0.160 629	0.667 809	1.250 100	0.142 136
0.047 482	0.287 915	1.434 490	0.408 504
0.967 343	0.159 587	0.256 380	0.287 542
0.813 588	0.387 056	1.082 690	0.165 968
0.984 163	0.110 569	0.909 952	0.264 000
1.552 070	0.266 407	0.748 825	0.561 851
0.735 931	0.193 000	1.326 590	0.607 474
0.573 856	0.215 326	1.136 700	0.207 383
0.815 197	0.144 584	1.478 740	0.297 837
0.863 712	0.500 342	0.465 459	0.619 345
0.525 191	0.290 132	0.849 605	0.365 040
0.973 616	0.268 516	1.138 420	0.104 562
0.582 212	0.860 912	1.744 190	2.386 830
1.440 790	0.107 284	1.437 070	0.464 996

统计量	m_y	σ_y²	m_z	σ_z²
均值	0.853 837	0.431 107	1.295 124	0.491 865
方差	0.272 658	0.195 300	0.183 653	0.179 561

坐标轴	n	IGa(α, β)超参数				N(u₀, σ²/k₀)超参数
		$\widehat{\alpha}\left(\frac{v_{0}}{2}\right)$	$\hat{\beta}\left(\frac{v_{0} \sigma_{0}^{2}}{2}\right)$	K-S检验		$\hat{u}_{0}$	(σ²/k₀)′	$\hat{\sigma}^{2}$	k₀	K-S检验
		$\widehat{\alpha}\left(\frac{v_{0}}{2}\right)$	$\hat{\beta}\left(\frac{v_{0} \sigma_{0}^{2}}{2}\right)$	statistic	p-value	$\hat{u}_{0}$	(σ²/k₀)′	$\hat{\sigma}^{2}$	k₀	statistic	p-value
OY	30	2.311 1	0.569 590	0.233 3	0.171 9	0.853 837	0.272 658	0.431 107	1.581 125	0.166 7	0.407 2
OZ	30	3.006	1.002 365	0.066 7	0.866 1	1.295 124	0.183 653	0.491 865	2.678 227	0.133 3	0.562 7

m_yi	σ_yi²	m_zi	σ_zi²
1.055 9	0.258 8	0.165 1	0.507 4
1.461 1	0.594 1	1.037 6	0.414 3
1.568 9	0.163 6	0.347 8	0.604 2
0.872 6	0.239 6	1.139 9	0.301 9
1.414 0	0.395 9	0.690 2	0.308 1
1.535 5	0.328 5	0.963 4	1.032 6
0.627 4	0.702 4	1.739 4	0.172 6
0.349 4	0.319 1	1.527 0	0.553 9
1.007 2	0.399 7	1.234 5	0.380 2
1.264 8	0.171 3	1.290 2	0.233 0

坐标轴	参数
坐标轴	n	${\bar x}$	s²	k₀	k_n	v₀	v_n	u₀	u_n	υ₀ σ₀²	υ_n σ_n²	m_x_MD	σ_x_MD²
OY	10	1.115 7	0.168 0	1.581 1	11.581 1	4.622 2	14.622 2	0.853 8	1.079 9	1.139 2	2.744 8	1.079 9	0.311 5
OZ	10	1.013 5	0.244 7	2.678 2	12.678 2	6.012 0	16.012 0	1.295 1	1.073 0	2.004 7	4.374 5	1.073 0	0.460 2