基于正态-逆威沙特分布的地空导弹命中概率贝叶斯估计

doi:10.12382/bgxb.2022.0547

摘要/Abstract

摘要：

针对现有导弹命中概率贝叶斯估计中主要采用二维正态射弹散布但未考虑X轴、Y轴两向相关导致估计精度不高的问题,采用正态-逆威沙特分布作为射弹散布参数先验分布,对地空导弹命中概率估计方法展开研究。在描述地空导弹脱靶量定义的基础上,以两向相关特点确定相关系数。针对地空导弹外场试验成本高、数据少,经典统计学方法估计命中概率困难问题,采用贝叶斯方法融合先验信息进行研究。基于正态-逆威沙特分布,利用先验信息对先验分布超参数进行求解,并利用贝叶斯公式融合外场实测试验数据,求出射弹散布参数后验分布,最终完成地空导弹命中概率估计。试验结果表明,该方法相比现有正态-逆伽马分布方法,能够考虑地空导弹射弹散布两向相关的实际情况,并充分利用命中概率信息,有利于提升估计准确性,为地空导弹命中概率估计提供理论方法。

关键词: 地空导弹, 命中概率, 射弹散布, 两向相关, 正态-逆威沙特分布, 贝叶斯

Abstract:

The two-dimensional normal projectile dispersion is mainly used in the existing Bayesian estimation of missile hit probability, but the accuracy of estimation is not high enough due to the failure of considering the two-direction correlation of X and Y axes. The normal-inverse Wishart distribution is used as the prior distribution of projectile dispersion parameters, and the hit probability estimation method of surface-to-air missile is studied.Firstly, on the basis of describing the definition of miss distance of surface-to-air missile, the correlation coefficient ρ is determined by two related characteristics.Secondly, in view of the problems of high cost, little dataof the surface-to-air missile field test, and the difficulty of estimating hit probability by classical statistical methods, the Bayesian method is used to integrate the prior informationfor research. Then, based on the normal-inverse Wishart distribution, the prior information is used to solve the hyperparameters of the prior distribution, and the posterior distribution of projectile dispersion parameters are obtained by integrating the field test data with the Bayesian formula, and finally the hit probability of surface-to-air missile is estimated. The results show that, compared with the existing normal-inverse gamma distribution method, the proposed method take into account the actual situation of the projectile dispersion two-direction correlation of surface-to-air missile, and makes full use of the hit probability information, which is conducive to improving the accuracy of estimation,and provide a theoretical method for the estimation of surface-to-air missile hit probability.

Key words: surface-to-air missile, hit probability, projectile dispersion, two-direction correlation, normal-inverse Wishart distribution, Bayesian

中图分类号:

E917

刘昊邦, 史宪铭, 赵美, 张建军. 基于正态-逆威沙特分布的地空导弹命中概率贝叶斯估计[J]. 兵工学报, 2024, 45(1): 339-348.

LIU Haobang, SHI Xianming, ZHAO Mei, ZHANG Jianjun. Bayesian Estimation of Surface-to-Air Missile Hit Probability Based on Normal-inverse Wishart Distribution[J]. Acta Armamentarii, 2024, 45(1): 339-348.

图/表 11

图1 地空导弹脱靶量示意图

Fig.1 Schematic diagram of misse distance of surface-to-air missile

图2 靶平面示意图

Fig.2 Schematic diagram of target plane

表1 先验信息射弹散布参数数据

Table 1 Projectile dispersion parameter data of prior information m

u_xw	u_yw	$σ x w 2$	$σ y w 2$	ρ_w
2.6403	2.4799	7.5723	5.9074	0.132
0.7555	-0.139	6.6879	6.5947	0.347
1.6672	1.1085	6.3770	5.2931	0.293
2.0232	2.1851	7.5895	6.7872	0.281
2.8013	0.4143	7.6679	5.3524	0.306
0.3645	-1.961	7.1918	4.4118	0.211
0.6867	0.1977	8.2224	6.5851	0.471
-0.4182	0.3964	6.4517	5.6393	0.193
-1.6221	-0.1843	7.3581	4.4016	0.207
2.5087	-0.0998	7.2903	5.5550	0.255
0.3648	1.0525	9.1168	6.8629	0.129
2.9984	1.5372	8.2664	6.5618	0.345
3.9850	2.6212	7.1418	5.3819	0.219
2.6754	1.8164	7.3274	4.6665	0.167
2.3072	2.4857	7.2153	5.3975	0.286
2.0955	1.364	6.7565	6.2557	0.108
-0.2608	-1.3818	6.5446	7.1421	0.223
1.0069	-0.621	7.2879	6.4085	0.267
0.5769	0.1977	5.6595	6.2875	0.163
-0.1456	0.4277	6.3870	5.4659	0.275
0.6136	1.9062	9.1106	6.2640	0.233
-1.1208	-0.3237	6.7686	5.3795	0.195
1.0378	0.4674	6.8930	3.8074	0.301
0.9787	1.5857	5.4529	4.3073	0.179
0.0439	0.3098	7.6678	5.2978	0.197

表1 先验信息射弹散布参数数据

Table 1 Projectile dispersion parameter data of prior information m

u_xw	u_yw	$σ x w 2$	$σ y w 2$	ρ_w
2.6403	2.4799	7.5723	5.9074	0.132
0.7555	-0.139	6.6879	6.5947	0.347
1.6672	1.1085	6.3770	5.2931	0.293
2.0232	2.1851	7.5895	6.7872	0.281
2.8013	0.4143	7.6679	5.3524	0.306
0.3645	-1.961	7.1918	4.4118	0.211
0.6867	0.1977	8.2224	6.5851	0.471
-0.4182	0.3964	6.4517	5.6393	0.193
-1.6221	-0.1843	7.3581	4.4016	0.207
2.5087	-0.0998	7.2903	5.5550	0.255
0.3648	1.0525	9.1168	6.8629	0.129
2.9984	1.5372	8.2664	6.5618	0.345
3.9850	2.6212	7.1418	5.3819	0.219
2.6754	1.8164	7.3274	4.6665	0.167
2.3072	2.4857	7.2153	5.3975	0.286
2.0955	1.364	6.7565	6.2557	0.108
-0.2608	-1.3818	6.5446	7.1421	0.223
1.0069	-0.621	7.2879	6.4085	0.267
0.5769	0.1977	5.6595	6.2875	0.163
-0.1456	0.4277	6.3870	5.4659	0.275
0.6136	1.9062	9.1106	6.2640	0.233
-1.1208	-0.3237	6.7686	5.3795	0.195
1.0378	0.4674	6.8930	3.8074	0.301
0.9787	1.5857	5.4529	4.3073	0.179
0.0439	0.3098	7.6678	5.2978	0.197

表2 先验信息射弹散布参数统计

Table 2 Projectile dispersion parameter statistics of prior informationm

统计量	μ	Σ
均值	[1.1426 0.7137]^T	$7.2002 1.5367 1.5367 5.6805$
协方差	$1.8524 1.0020 1.0020 1.3783$

表2 先验信息射弹散布参数统计

Table 2 Projectile dispersion parameter statistics of prior informationm

统计量	μ	Σ
均值	[1.1426 0.7137]^T	$7.2002 1.5367 1.5367 5.6805$
协方差	$1.8524 1.0020 1.0020 1.3783$

图3 参数μ频数直方图

Fig.3 Frequency histogram of parameter μ

图4 参数μ散布图

Fig.4 Scatter plot of parameter μ

表3 射弹散布参数先验分布超参数

Table 3 Prior distribution hyperparameters of projectile dispersion parameters

Σ		μ
Inv-Wishart(ν₀,Ψ₀)		$\mathrm{N}\left(\boldsymbol{\mu}_{0}, \frac{\boldsymbol{\Sigma}}{k_{0}}\right)$
ν₀	Ψ₀	μ₀	k₀
5.3	$38.1611 8.1445 8.1445 30.1067$	[1.1426 0.7137]^T	3.05

表3 射弹散布参数先验分布超参数

Table 3 Prior distribution hyperparameters of projectile dispersion parameters

Σ		μ
Inv-Wishart(ν₀,Ψ₀)		$\mathrm{N}\left(\boldsymbol{\mu}_{0}, \frac{\boldsymbol{\Sigma}}{k_{0}}\right)$
ν₀	Ψ₀	μ₀	k₀
5.3	$38.1611 8.1445 8.1445 30.1067$	[1.1426 0.7137]^T	3.05

表4 射弹散布参数后验分布(导弹数量10)

Table 4 Posterior distribution of projectile dispersion parameters(10 missiles )

Σ			μ
Inv-Wishart(ν_n,Ψ_n)			N(μ_n, $Σ k n$ )
ν_n	Ψ_n	$\hat{\boldsymbol{\Sigma}}_{\mathrm{E}}$	μ_n	k_n	$\hat{\boldsymbol{\mu}}_{\mathrm{E}}$
15.3	$99.604 22.612 22.612 54.653$	$6.5100 1.4779 1.4779 3.5721$	[1.166, 0.802]^T	13.05	[1.166, 0.802]^T

表4 射弹散布参数后验分布(导弹数量10)

Table 4 Posterior distribution of projectile dispersion parameters(10 missiles )

Σ			μ
Inv-Wishart(ν_n,Ψ_n)			N(μ_n, $Σ k n$ )
ν_n	Ψ_n	$\hat{\boldsymbol{\Sigma}}_{\mathrm{E}}$	μ_n	k_n	$\hat{\boldsymbol{\mu}}_{\mathrm{E}}$
15.3	$99.604 22.612 22.612 54.653$	$6.5100 1.4779 1.4779 3.5721$	[1.166, 0.802]^T	13.05	[1.166, 0.802]^T

图5 不同信息地空导弹射弹散布

Fig.5 Surface-to-air missile projectile dispersions of different informations

图6 不同信息地空导弹射弹散布概率密度

Fig.6 Surface-to-air missile projectile dispersion probability densities of different informations

表5 两种方法命中概率计算对比

Table 5 Comparison of hit probability calculations of two methods

计算目标	正态-逆威沙特分布方法 (本文方法)		正态-逆伽马分布方法 (现有研究方法)				外场射击试验样本
	$\hat{\boldsymbol{\mu}}_{\mathrm{E}}$	$\hat{\boldsymbol{\Sigma}}_{\mathrm{E}}$	$\hat{u}_{x \mathrm{E}}$	$\hat{\sigma}_{x \mathrm{E}}$	$\hat{u}_{y \mathrm{E}}$	$\hat{\sigma}_{y E}$	u_x	σ_x	u_y	σ_y	ρ
射弹散布参数	[1.166, 0.802]^T	$6.5100 1.4779 1.4779 3.5721$	1.158	2.513	0.809	1.966	1.173	2.397	0.829	1.468	0.383
命中概率	0.81		0.72				0.84

表5 两种方法命中概率计算对比

Table 5 Comparison of hit probability calculations of two methods

计算目标	正态-逆威沙特分布方法 (本文方法)		正态-逆伽马分布方法 (现有研究方法)				外场射击试验样本
	$\hat{\boldsymbol{\mu}}_{\mathrm{E}}$	$\hat{\boldsymbol{\Sigma}}_{\mathrm{E}}$	$\hat{u}_{x \mathrm{E}}$	$\hat{\sigma}_{x \mathrm{E}}$	$\hat{u}_{y \mathrm{E}}$	$\hat{\sigma}_{y E}$	u_x	σ_x	u_y	σ_y	ρ
射弹散布参数	[1.166, 0.802]^T	$6.5100 1.4779 1.4779 3.5721$	1.158	2.513	0.809	1.966	1.173	2.397	0.829	1.468	0.383
命中概率	0.81		0.72				0.84

参考文献 26

[1]	ANONYMOU S. Dia and Aoc host 2019 modern threats: surface-to-air missile(SAM) systems conference[J]. Journal of Electronic Defense, 2019, 42:56-71.
[2]	闫志强. 装备试验评估中的变动统计方法研究[D]. 长沙: 国防科学技术大学, 2010.
	YAN Z Q. Research on variation statistical method in equipment test evaluation[D]. Changsha: National University of Defense Technology, 2010. (in Chinese)
[3]	VLADISLAV K, VADIM V, OLEKSANDR D, et al. Problematic issues and perspective ways for ensuring documentation of battle work on combat means of surface-to-air missile systems (complexes)[J]. Social Development & Security, 2019, 9:55-67.
[4]	王保华, 裴益轩, 霍勇谋, 等. 一种求解火箭防空武器毁歼概率的简便方法[J]. 兵工学报, 2016, 37(4):751-755. doi: 10.3969/j.issn.1000-1093.2016.04.025
	WANG B H, PEI Y X, HUO Y M, et al. A convenient solution method for kill probability of antiaircraft rocket weapon[J]. Acta Armamentarii, 2016, 37(4):751-755. (in Chinese)
[5]	王树山, 买瑞敏. 集束箭弹命中概率分析的Monte-Carlo方法[J]. 北京理工大学学报, 2005, 25(4):286-288.
	WANG S S, MAI R M. A Monte-Carlo method for the analysis of the hit probability of cluster arrows[J]. Transactions of Beijing Institute of Technology, 2005, 25(4): 286-288. (in Chinese)
[6]	CORRIVEAU D, RABBATH C A, GOUDREAU A. Effect of the firing position on aiming error and probability of hit[J]. Defence Technology, 2019, 15:97-112.
[7]	李宗吉, 程善政, 刘洋. 蒙特卡洛模拟法计算航空自导深弹命中概率[J]. 弹箭与制导学报, 2012, 32(2):22-24.
	LI Z G, CHENG S Z, LIU Y. Calculation of hit probability of aerial self-guided deep missile by Monte Carlo simulation method[J]. Journal of Projectiles, Rockets, Missiles and Guidance, 2012, 32(2):22-24. (in Chinese)
[8]	于涛, 郝永平, 郑斌, 等. 上升扫描式末敏子弹对坦克目标的毁伤效能评估[J]. 兵工学报, 2018, 39(10):1927-1935. doi: 10.3969/j.issn.1000-1093.2018.10.007
	YU T, HAO Y P, ZHENG B, et al. Evaluation of the damage effectiveness of ascending scanning terminal-sensitive bullets to tank targets[J]. Acta Armamentarii, 2018, 39(10): 1927-1935. (in Chinese)
[9]	WANG X R, XIE D X, CHEN Y J. Analysis of the hit rate of a land-based ballistic missile against the aircraft carrier[J]. International Journal of Computational and Engineering, 2018, 3:12-25.
[10]	SEO B G, HONG S J. A study of estimating the hit probability and confidence level considering the characteristic ofprecision guided missile[J]. Journal of the Korea Academia-Industrial cooperation Society, 2016, 17(12):56-72. doi: 10.5762/KAIS.2016.17.5.56 URL
[11]	闫志强, 蒋英杰, 谢红卫. 基于顺序约束的命中概率多批次试验统计评定方法[J]. 宇航学报, 2010, 31(4):1206-1211.
	YAN Z Q, JIANG Y J, XIE H W. Statistical evaluation method of hit probability based on sequence constraint for multiple batch tests[J]. Journal of Astronautics, 2010, 31(4):1206-1211. (in Chinese)
[12]	李康, 史宪铭, 李广宁, 等. 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. doi: 10.12305/j.issn.1001-506X.2022.08.27
	LI K, SHI X M, LI G N, et al. Anti-cruise missile hit probability estimation method based on normal-inverse gamma distribution[J]. Systems Engineering and Electronics, 2022, 44(8):2621-2627. (in Chinese)
[13]	LIU H B, SHI X M, CHEN X J, et al. Bayesian inference of ammunition consumption based on normal-inverse gamma distribution[J]. Discrete Dynamics in Nature and Society, 2022, 4(3):43-53.
[14]	吴越, 刘东升, 孙树国, 等. 岩土强度参数正态-逆伽马分布的最大后验估计[J]. 岩石力学与工程学报, 2019, 38(6):1188-1196.
	WU Y, LIU D S, SUN S G, et al. Maximum a posteriori estimation of normal-inverse gamma distribution of rock and soil strength parameters[J]. Chinese Journal of Rock Mechanics and Engineering, 2019, 38(6):1188-1196. (in Chinese)
[15]	赵永满. 基于威沙特分布的二元过程均值与协方差监控[J]. 统计与决策, 2015(5):71-72.
	ZHAO Y M. Monitoring the mean and covariance of binary processes based on Wishart distribution[J]. Statistics and Decision Making, 2015(5):71-72. (in Chinese)
[16]	许凯, 何道江, 徐兴忠. 正态-逆Wishart先验下多元线性模型中经验Bayes估计的优良性[J]. 数学年刊A辑(中文版), 2014, 35(3):267-284.
	XU K, HE D J, XU X Z. The goodness of empirical bayes estimation in multiple linear models under the normal-inverse Wishart prior[J]. Chinese Annals of Mathematics, Series A, 2014, 35(3):267-284. (in Chinese)
[17]	王润平, 王正向. 地空导弹单发杀伤概率之解析算法[J]. 系统工程与电子技术, 2001, 23(5):39-40.
	WANG R P, WANG Z X. Analytical algorithm of single shot kill probability of surface-to-air missile[J]. Systems Engineering and Electronics, 2001, 23(5):39-40. (in Chinese)
[18]	沈忱, 徐定杰, 沈锋, 等. 基于变分推断的一般噪声自适应卡尔曼滤波[J]. 系统工程与电子技术, 2014, 36(8):1466-1472.
	SHEN C, XU D J, SHEN F, et al. General noise adaptive kalman filter based on variational inference[J]. System Engineering and Electronics, 2014, 36(8):1466-1472. (in Chinese)
[19]	许述文, 王喆祥, 水鹏朗. 海杂波背景下雷达目标贝叶斯检测算法[J]. 西安电子科技大学学报, 2021, 48(2):15-26.
	XU S W, WANG Z X, SHUI P L. Bayesian detection algorithm of radar target under sea clutter background[J]. Journal of Xidian University, 2021, 48(2):15-26. (in Chinese)
[20]	TARAS B, STEPAN M, KRZY P. Singular inverse Wishart distribution and its application to portfolio theory[J]. Journal of Multivariate Analysis, 2016, 143(7):186-200.
[21]	朱光明, 蒋荣欣, 周凡, 等. 带测量偏置估计的鲁棒卡尔曼滤波算法[J]. 浙江大学学报(工学版), 2015, 49(7):1343-1349.
	ZHU G M, JING R X, ZHOU F, et al. Robust Kalman filter algorithm with measurement bias estimation[J]. Journal of Zhejiang University (Engineering Science), 2015, 49(7):1343-1349. (in Chinese)
[22]	TRONARP F, SARKKA S, HENNIG P. BayesianODE solvers:the maximum a posteriori estimate[J]. Statistics and Computing, 2021, 31(3):1-18.
[23]	TOM L, JOHN S. Bayesian inference for a covariance matrix[J]. The Annals of Statistics, 1992, 20(4):23-35.
[24]	IRWIN G, TANW Y. The use of the disguised Wishart distribution in a Bayesian approach to tolerance region construction[J]. Annals of the Institute of Statistical Mathematics, 1973, 25(1):13-25.
[25]	HUI Y, SUN K D, CHEN M H. Bayesian inference for multivariate meta-regression with a partially observed within study sample covariance matrix[J]. Journal of the American Statistical Association, 2015, 31(10):65-87.
[26]	MUNIIA S, CANTET R J C. Bayesian conjugate analysis using a generalized inverted Wishart distribution accounts for differential uncertainty among the genetic parameters-an application to the maternal animal model[J]. Journal of Animal Breeding and Genetics, 2012, 129(3):166-184.