Estimation of Ammunition Hit Probability in Multiple-Batch Tests Using Multinomial Distribution

doi:10.12382/bgxb.2022.0490

Abstract

Abstract:

Ammunition hit probability tests are usually carried out in multiple batches. With each test batch, the physical properties of ammunition evolve, influencing hit probabilities dynamically. Before each batch, simulated ammunition hit probability information is obtained. Effectively utilizing the shooting tests information and simulation data of each batch is key to hit probability estimation. In addition, considering the different degrees of damage caused by ammunition hitting different areas of the target, the original binomial distribution hit probability test is improved and described by multinomial distribution. System contribution is introduced to measure the impact of each batch of tests on the estimation of the ammunition hit probability, and the Bayesian method is used to fuse the test data of each batch. The ammunition hit property improvement is identified by constructing sequence constraint relation. Finally, the estimation of ammunition hit probability in multiple batch tests based on multinomial distribution is proposed. The example shows that the method can describe the hit properties of ammunition in detail compared with the binomial distribution method. It can also scientifically integrate the information of each batch shooting tests and simulation tests, providing reference for the estimation of the hit probability.

Key words: ammunition, hit probability, multiple batches tests, multinomial distribution, Bayesian inference, system contribution

CLC Number:

E917

LIU Haobang, SHI Xianming, HAO Bing, JIANG Yongchao. Estimation of Ammunition Hit Probability in Multiple-Batch Tests Using Multinomial Distribution[J]. Acta Armamentarii, 2023, 44(10): 3156-3164.

Figures/Tables 8

Fig.1 Information fusion process of multiple batches hit probability trials

Table 1 Ammunition hit probabilities in multiple-batch tests

批次	x⁽ⁱ⁾	α⁽ⁱ⁾
初始批次	( $3322$ )	( $1.2 1 0.8 0.75$ )
第2批次	( $2422$ )	( $1.5 2.8 2 1.8$ )
第3批次	( $1243$ )	( $1 2.4 3.5 2.8$ )

Table 1 Ammunition hit probabilities in multiple-batch tests

批次	x⁽ⁱ⁾	α⁽ⁱ⁾
初始批次	( $3322$ )	( $1.2 1 0.8 0.75$ )
第2批次	( $2422$ )	( $1.5 2.8 2 1.8$ )
第3批次	( $1243$ )	( $1 2.4 3.5 2.8$ )

Table 2 Prior information membership score in expert interview

专家序号	$ω δ 1 d$	$υ δ 1 d$	$ω δ 2 d$	$υ δ 2 d$
1	0.5	0.3	0.4	0.3
2	0.6	0.3	0.7	0.2
3	0.8	0.2	0.5	0.4
4	0.4	0.4	0.5	0.3
5	0.6	0.4	0.6	0.3
6	0.7	0.3	0.6	0.4

Table 2 Prior information membership score in expert interview

专家序号	$ω δ 1 d$	$υ δ 1 d$	$ω δ 2 d$	$υ δ 2 d$
1	0.5	0.3	0.4	0.3
2	0.6	0.3	0.7	0.2
3	0.8	0.2	0.5	0.4
4	0.4	0.4	0.5	0.3
5	0.6	0.4	0.6	0.3
6	0.7	0.3	0.6	0.4

Fig.2 Iteration trace graph of initial batch hit probability parameters

Fig.3 Iteration trace graph of the second-batch hit probability parameters

Fig.4 Iteration trace graph of the third-batch hit probability parameters

Fig.5 Probability density function diagram of ammunition hit probability parameters of each batch

Table 3 Bayesian statistical analysis of ammunition hit probability parameters for each batch

P⁽ⁱ⁾	初始批次试验贝叶斯估计		第2批次试验贝叶斯估计		第3批次试验贝叶斯估计
P⁽ⁱ⁾	均值	方差	均值	方差	均值	方差
$P 1 (i)$	0.305	0.2256	0.231	0.1832	0.124	0.0847
$P 2 (i)$	0.291	0.2197	0.344	0.2447	0.251	0.1993
$P 3 (i)$	0.204	0.1556	0.218	0.1646	0.343	0.2443
$P 4 (i)$	0.20	0.1544	0.207	0.1573	0.282	0.2123

Table 3 Bayesian statistical analysis of ammunition hit probability parameters for each batch

P⁽ⁱ⁾	初始批次试验贝叶斯估计		第2批次试验贝叶斯估计		第3批次试验贝叶斯估计
P⁽ⁱ⁾	均值	方差	均值	方差	均值	方差
$P 1 (i)$	0.305	0.2256	0.231	0.1832	0.124	0.0847
$P 2 (i)$	0.291	0.2197	0.344	0.2447	0.251	0.1993
$P 3 (i)$	0.204	0.1556	0.218	0.1646	0.343	0.2443
$P 4 (i)$	0.20	0.1544	0.207	0.1573	0.282	0.2123

References 24

[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.
	YAN Z Q, JIANG Y J, XIE H W. Evaluation of hitting probability for multi-batch tests based on ordinal constraints[J]. Journal of Astronautics, 2010, 31(4): 1206-1211. (in Chinese)
[2]	TRZUN Z, VRDOLJAK M. Probability of hitting the target with an imperfectly manufactured projectile[J]. Proceedings in Applied Mathematics and Mechanics, 2021, 21 (1): 321-334.
[3]	CORRIVEAU D, RABBATH C A, GOUDREAU A. Effect of the firing position on aiming error and probability of hit[J]. Defence Technology, 2019, 15 (5): 78-95.
[4]	KHAIKOV V L. 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]. Vojnotehnički Glasnik, 2019, 67(1): 108-121.
[5]	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)
[6]	SEO B G, HONG S J. A study of estimating the hit probability and confidence level considering the characteristic of precision guided missile[J]. Journal of the Korea Academia-Industrial cooperation Society, 2016, 17(12): 211-223. doi: 10.5762/KAIS.2016.17.7.211 URL
[7]	KANG H I, KIM H S, KIM G. A study on prediction of probability of hit for an anti-aircraft artillery[J]. Indian Journal of Science and Technology, 2016, 9(S1): 53-69.
[8]	LI K, SHI X M, LI J, et al. Bayesian estimation of ammunition demand based on multinomial distribution[J]. Discrete Dynamics in Nature and Society, 2021, 14(2): 56-67.
[9]	GUO J Y, LI Y F, PENG W W, et al. Bayesian information fusion method for reliability analysis with failure-time data and degradation data[J]. Quality and Reliability Engineering International, 2022, 38(4): 99-117.
[10]	PAPANANIAS M, MCLEAY T E, MAHFOUF M, et al. A Bayesian information fusion approach for end product quality estimation using machine learning and on-machine probing[J]. Journal of Manufacturing Processes, 2022, 6(3): 37-54.
[11]	ZHANG X Y, ZHANG Q, ZHENG J F. Research on optimization method of weapon equipment system contribution rate evaluation index based on meta-evaluation[J]. Scientific Journal of Intelligent Systems Research, 2021, 3(11): 107-121.
[12]	于录, 曲宝忠, 时维科. 反舰导弹单发命中概率检验方法[J]. 海军航空工程学院学报, 2012, 27(1): 23-26.
	YU L, QU B Z, SHI W K. Test method for single-shot hit probability of anti-ship missile[J]. Journal of Naval Aeronautical Engineering College, 2012, 27(1): 23-26. (in Chinese)
[13]	LIU H B, SHI X M. Damage effectiveness calculation of hitting targets with ammunition based on Bayesian multinomial distribution[J]. Symmetry, 2022, 14(5): 57-70. doi: 10.3390/sym14010057 URL
[14]	LIGTVOET R. Exact bayes factors for the comparison of multinomial distributions[J]. The American Statistician, 2021, 75(1): 201-215.
[15]	CHANDRA N, RATHAUR V K. Bayes estimation of augmenting gamma strength reliability of a system under non-informative prior distributions[J]. Calcutta Statistical Association Bulletin, 2017, 69(1): 67-82.
[16]	RABOULI S, SERRE M, DUBOIS V, et al. Spatialization of saturated hydraulic conductivity using the Bayesian maximum entropy method: application to wastewater infiltration areas[J]. Water Research, 2021, 4(20): 156-169
[17]	NAJAR F, BOUGUILA N. Exact fisher information of generalized Dirichlet multinomial distribution for count data modeling[J]. Information Sciences, 2022, 56(8): 42-58.
[18]	闫志强. 装备试验评估中的变动统计方法研究[D]. 长沙: 国防科技大学, 2010.
	YAN Z Q. Research on variation statistical methods in equipment test evaluation[D]. Changsha: National University of Defense Technology, 2010. (in Chinese)
[19]	殷小静, 胡晓峰, 荣明, 等. 体系贡献率评估方法研究综述与展望[J]. 系统仿真学报, 2019, 31(6): 1027-1038. doi: 10.16182/j.issn1004731x.joss.19-0240
	YIN X J, HU X F, RONG M, et al. Review of evaluation methods of contribution rate to system of systems[J]. Journal of System Simulation, 2019, 31(6): 1027-1038. (in Chinese) doi: 10.16182/j.issn1004731x.joss.19-0240
[20]	ZHAO R D, SHI X M, WANG Q, et al. Bayesian inference for ammunition demand based on gompertz distribution[J]. Journal of Systems Engineering and Electronics, 2020, 31(3): 567-577. doi: 10.23919/JSEE.2020.000035
[21]	陈立新, 徐仲祥, 王从容, 等. 面向相互比较的装备体系建设贡献率评估[J]. 兵工学报, 2022, 43(5): 1208-1214. doi: 10.12382/bgxb.2021.0052
	CHEN L X, XU Z X, WANG C R, et al. Intercomparison-oriented evaluation of equipment’s contribution rate to armament system of systems construction[J]. Acta Armamentarii, 2022, 43(5): 1208-1214. (in Chinese)
[22]	赵汝东, 史宪铭, 苏小波, 等. 基于Bayesian体系融合的新型弹药消耗预计方法[J]. 兵器装备工程学报, 2020, 41(2): 75-80.
	ZHAO R D, SHI X M, SU X B, et al. New ammunition consumption prediction method based on bayesian system fusion[J]. Journal of Ordnance Equipment Engineering, 2020, 41(2): 75-80. (in Chinese)
[23]	VAJARGAH K F, BENIS S G, GOLSHAN H M. Detection of the quality of vital signals by the Monte Carlo Markov Chain(MCMC) method and noise deleting[J]. Health Information Science and Systems, 2021, 9(1): 55-71.
[24]	HU Q, SHEN J J, WANG K, et al. A web service clustering method based on topic enhanced Gibbs sampling algorithm for the dirichlet multinomial mixture model and service collaboration graph[J]. Information Sciences, 2022, 6(2): 99-113.