Bayesian Estimation of Surface-to-Air Missile Hit Probability Based on Normal-inverse Wishart Distribution

doi:10.12382/bgxb.2022.0547

Abstract

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

CLC Number:

E917

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.

Figures/Tables 11

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

Fig.2 Schematic diagram of target plane

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

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

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	$1.8524 1.0020 1.0020 1.3783$

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	$1.8524 1.0020 1.0020 1.3783$

Fig.3 Frequency histogram of parameter μ

Fig.4 Scatter plot of parameter μ

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

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

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

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

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

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

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

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

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