基于多批次增长试验的舰空导弹命中概率贝叶斯估计

doi:10.12305/j.issn.1001-506X.2025.03.19

摘要/Abstract

摘要：

现有舰空导弹命中概率估计方法主要从单批次试验角度研究, 未能考虑命中概率多批次增长试验特点的问题, 致使命中概率准确估计较为困难。本文以舰空导弹二维正态射弹散布现象为切入点, 基于贝叶斯方法选取正态-逆伽马分布作为射弹散布参数的先验分布, 融合先验信息弥补小样本试验数据量不足问题, 在各批次试验的射弹散布参数值之间建立顺序约束关系, 并利用马尔可夫链-蒙特卡罗(Markov chain-Monte Carlo, MCMC)方法结合吉布斯抽样进行贝叶斯求解, 从而实现融合多批次增长试验信息的目的。研究结果表明, 该方法相比现有单批次试验命中概率估计方法能够考虑多批次增长试验的特点, 为舰空导弹命中概率估计提供借鉴。

关键词: 舰空导弹, 命中概率, 多批次增长试验, 射弹散布, 贝叶斯, 马尔可夫链-蒙特卡罗

Abstract:

The existing ship-to-air missile hit probability estimation methods are mainly studied from the perspective of single batch tests, but fail to consider the characteristics of multiple batches growth tests, which makes it difficult to accurately estimate the hit probability. In this paper, the two-dimensional normal projectile dispersion phenomenon of ship-to-air missile is taken as the entry point, the normal-inverse Gamma distribution is selected as the prior distribution of projectile dispersion parameters based on Bayesian method, and the prior information is merged to make up for the insufficient data in the small sample tests. The sequential constraint relationship is established between the projectile dispersion parameter values of each batch of tests, and the Markov chain-Monte Carlo (MCMC) method is combined with Gibbs sampling for Bayesian solution, so as to achieve the purpose of fusing multiple batches growth tests information. The research results show that this method can consider the characteristics of multiple batches growth tests compared with the existing single batch tests hit probability estimation method, and provide reference for ship-to-air missile hit probability estimation.

Key words: ship-to-air missile, hit probability, multiple batches growth tests, projectile dispersion, Bayesian, Markov chain-Monte Carlo (MCMC)

中图分类号:

E917

刘昊邦, 陈童, 胡涛, 李明贵, 杜凯. 基于多批次增长试验的舰空导弹命中概率贝叶斯估计[J]. 系统工程与电子技术, 2025, 47(3): 871-882.

Haobang LIU, Tong CHEN, Tao HU, Minggui LI, Kai DU. Bayesian estimation of ship-to-air missile hit probability based on multiple batches growth tests[J]. Systems Engineering and Electronics, 2025, 47(3): 871-882.

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参考文献 32

1	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. doi: 10.3873/j.issn.1000-1328.2010.04.043
	YAN Z Q , JIANG Y J , XIE H W . Statistical evaluation method of hit probability for multi-batch tests based on sequence constraints[J]. Journal of Astronautics, 2010, 31 (4): 1206- 1211. doi: 10.3873/j.issn.1000-1328.2010.04.043
2	ZHAI C L , CHEN X W . Simulation model for studying the effect of function distribution on the evaluation of building da-mage caused by missile attack[J]. Defence Science Journal, 2023, 73 (5): 541- 550. doi: 10.14429/dsj.73.18317
3	HUANG J , WU P F , LI X B . Research on dynamically corrective hit probability model of anti-air missile integrated in war game system[J]. Engineering Letters, 2022, 30 (2): 57- 68.
4	QIU X Q , GAO C S , JING W X . Maneuvering penetration strategies of ballistic missiles based on deep reinforcement learning[J]. Proceedings of the Institution of Mechanical Engineers Part G-Journal of Aerospace Engineering, 2022, 236 (16): 3494- 3504. doi: 10.1177/09544100221088361
5	CORRIVEAU D , RABBATH C , GOUDREAU A . Effect of the firing position on aiming error and probability of hit[J]. Defence Technology, 2019, 15 (5): 111- 126.
6	KHAIKOV L V . Assessment of the single shot hit probability as a function of the horizontal range taking into account different target types and points of aim[J]. Vojnotehnicki Glasnik, 2019, 67 (1): 55- 68.
7	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.
	LI K , SHI X M , LI G N , et al. Estimation method of anti-cruise missile hit probability based on normal-inverse gamma distribution[J]. Systems Engineering and Electronics, 2022, 44 (8): 2621- 2627.
8	刘昊邦, 史宪铭, 赵美, 等. 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 H B , SHI X M , ZHAO M , et al. Bayesian estimation of surface-to-air missile hit probability based on normal-inverse Wishart distribution[J]. Acta Armamentarii, 2024, 45 (1): 339- 348.
9	SUN Y L . Research on ship-to-air missile fire distribution method based on improved genetic algorithm[J]. Computer Informatization and Mechanical System, 2023, 6 (5): 78- 92.
10	WANG Y T , WANG H Y , LIANG H M , et al. Research on horizontal damage zone airspace of ship-to-air missile under cooperative operation[J]. Advances in Computer Science and Ubiquitous Computing, 2018, 7 (2): 654- 660.
11	赵永涛, 焦纲领, 王军生, 等. 舰舰协同制导舰空导弹中末目标交班问题研究[J]. 北京理工大学学报, 2022, 42 (9): 953- 960.
	ZHAO Y T , JIAO G L , WANG J S , et al. Research on midpoint shift of ship-to-air guided missile[J]. Journal of Beijing Institute of Technology, 2022, 42 (9): 953- 960.
12	刘泽乾, 纪义国, 杨林, 等. 制导化改进的常规航空炸弹命中精度分析[J]. 系统工程与电子技术, 2020, 42 (9): 2071- 2076.
	LIU Z Q , JI Y G , YANG L , et al. Analysis of hit accuracy of conventional aerial bomb with improved guidance[J]. Systems Engineering and Electronics, 2020, 42 (9): 2071- 2076.
13	LIU S T , WANG L T , WEI B Y . Modeling and simulation method of flight trajectory for ship-to-air missile based on vector rotation[J]. Journal of Physics: Conference Series, 2018, 10 (1): 60- 78.
14	CIVEK B C , ERTIN E . Bayesian sparse blind deconvolution using MCMC methods based on normal-inverse-gamma prior[J]. IEEE Trans.on Signal Processing, 2022, 70 (6): 1256- 1269.
15	ZHOU Q , GUAN Y T . On the null distribution of Bayes factors in linear regression[J]. Journal of the American Statistical Association, 2018, 13 (23): 112- 126.
16	ZHANG Y Y , RONG T Z , LI M M , et al. The empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-Gamma prior by the moment method and the MLE method[J]. Communications in Statistics-Theory and Methods, 2019, 48 (9): 2286- 2304. doi: 10.1080/03610926.2018.1465081
17	CAI M Y , VAN B S , VINK G . Joint distribution properties of fully conditional specification under the normal linear model with normal inverse-Gamma priors[J]. Scientific Reports, 2023, 13 (1): 33- 46. doi: 10.1038/s41598-022-27202-x
18	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, 12 (3): 44- 56.
19	SHANA S , JOHAANNES K , ANDREAS B , et al. BayesPostEst: an R package to generate postestimation quantities for Bayesian MCMC estimation[J]. Journal of Open Source Software, 2019, 4 (42): 90- 101.
20	KHARAZMI O , HAMEDANI G G , CORDEIRO G M . Log-mean distribution: applications to medical data, survival regression, Bayesian and non-Bayesian discussion with MCMC algorithm[J]. Journal of Applied Statistics, 2023, 5 (5): 63- 72.
21	MINH N N , MINH N T , ROHITASH C . Sequential reversible jump MCMC for dynamic Bayesian neural networks[J]. Neurocomputing, 2023, 7 (3): 156- 168.
22	NGOC N T , HOANG B N . Nexus between tourism and ecological footprint in RCEP: fresh evidence from Bayesian MCMC random-effects sampling[J]. Cogent Business Management, 2023, 10 (1): 34- 45.
23	THEODORE P . Approximate blocked Gibbs sampling for Bayesian neural networks[J]. Statistics and Computing, 2023, 33 (5): 122- 134.
24	JUN K , SHINTARO H . Approximate Gibbs sampler for Bayesian Huberized lasso[J]. Journal of Statistical Computation and Simulation, 2023, 93 (1): 55- 63.
25	PAUL L D M . Bayesian networks: regenerative Gibbs samplings[J]. Communications in Statistics-Simulation and Computation, 2022, 51 (12): 113- 124.
26	BUMENG Z , CHAO G . Mixing time of Metropolis-Hastings for Bayesian community detection[J]. Journal of Machine Learning Research, 2021, 22 (3): 99- 108.
27	REUSCHEN S , XU T , NOWAK W . Bayesian inversion of hie-rarchical geostatistical models using a parallel-tempering sequential Gibbs MCMC[J]. Advances in Water Resources, 2020, 14 (5): 92- 102.
28	SHENG Y . Bayesian estimation of the four-parameter IRT model using Gibbs sampling[J]. Quantitative Research in Education, 2015, 2 (3): 72- 88.
29	THOMPSON J , PALMER T , MORENO S . Bayesian analysis in stata with WinBUGS[J]. Stata Journal, 2006, 6 (4): 530- 549.
30	WILLIAMS M S , EBEL E D , HOETING J A . Bayesian anal-ysis for food-safety risk assessment: evaluation of dose-response functions within WinBUGS[J]. Journal of Statistical Software, 2011, 43 (2): 1- 14.
31	JEVDJEVIC M , YOUN J H , PETERSOHN S , et al. MSR110 a comparison of stan versus WinBUGS software for conducting Bayesian hazard ratio-based network meta-analysis[J]. Value in Health, 2022, 25 (12): 50- 62.
32	PERRAKIS K , NTZOUFRAS I . Bayesian variable selection using the hyper-g prior in WinBUGS[J]. Wiley Interdisciplinary Reviews: Computational Statistics, 2018, 10 (6): 73- 81.

试验批次	舰空导弹射弹散布外场发射试验数据	舰空导弹射弹散布先验信息数据
	(y_i, z_i)	h(u_yj, σ_yj²)	h(u_zj, σ_zj²)
	(y_i, z_i)	N(u_y0, σ_y²/k_y0)· IGa(v_y0/2, v_y0σ_y0²/2)	N(u_z0, σ_z²/k_z0)· IGa(v_z0/2, v_z0σ_z0²/2)
初始批次试验	(8.869, 7.447), (6.338, 4.557), (8.377, 8.556) (10.155, 8.134), (8.773, 7.334), (4.508, 5.667) (8.446, 7.421), (9.883, 6.233), (7.722, 4.584) (9.362, 5.663)	N(8.488, σ_y²/2.135)· IGa(12.247, 40.414)	N(6.363, σ_z²/2.073)· IGa(9.078, 21.812)
第2批次试验	(4.776, 3.586), (6.527, 4.832), (5.324, 4.432) (4.337, 5.253), (3.882, 3.237), (4.118, 5.229) (7.553, 3.793), (5.753, 2.507), (7.133, 1.891) (5.975, 2.736)	N(5.915, σ_y²/1.783)· IGa(8.523, 17.284)	N(3.938, σ_z²/1.926)· IGa(7.883, 13.971)
第3批次试验	(2.569, 2.667), (2.477, 1.556), (3.435, 1.137) (4.932, 3.016), (1.893, 2.669), (2.223, 2.891) (2.464, 1.667), (2.662, 2.349), (3.255, 0.951) (1.875, 1.533)	N(2.753, σ_y²/1.385)· IGa(4.444, 3.548)	N(1.917, σ_z²/1.471)· IGa(4.381, 2.965)

参数	样本数	均值	方差	蒙特卡罗误差	置信区间下限2.5%	中位数	置信区间上限97.5%
u_y1	3 000	8.324	1.763	3.125E-4	5.825	8.346	10.866
σ_y1²	3 000	3.567	1.232	2.306E-4	2.084	3.396	6.155
u_z1	3 000	6.453	1.288	2.663E-4	4.126	6.404	8.631
σ_z1²	3 000	2.663	1.017	2.137E-4	1.643	2.548	5.284
u_y2	3 000	5.783	1.256	2.442E-4	3.675	5.698	8.012
σ_y2²	3 000	2.216	0.812	1.885E-4	0.963	2.032	4.086
u_z2	3 000	3.887	1.013	2.255E-4	2.056	3.882	5.892
σ_z2²	3 000	1.974	0.663	1.866E-4	0.843	1.906	4.175
u_y3	3 000	2.768	0.746	1.623E-4	1.276	2.811	4.453
σ_y3²	3 000	1.027	0.434	1.114E-4	0.127	0.985	2.336
u_z3	3 000	1.973	0.588	1.339E-4	0.878	1.986	3.240
σ_z3²	3 000	0.864	0.323	1.003E-4	0.086	0.873	2.255

参数	样本数	均值	方差	蒙特卡罗误差	置信区间下限2.5%	中位数	置信区间上限97.5%
u_y	3 000	2.953	0.833	1.771E-4	1.355	2.887	4.558
σ_y²	3 000	1.066	0.448	1.157E-4	0.132	0.992	2.388
u_z	3 000	2.008	0.672	1.557E-4	0.922	2.018	3.378
σ_z²	3 000	0.897	0.374	1.083E-4	0.086	0.912	2.305

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