ZHAO Rudong,SHI Xianming,WANG Qian,SU Xiaobo,and SONG Xing

1.Department of Equipment Command and Management,Shijiazhuang Campus,Army Engineering University,Shijiazhuang 050003,China;2.Unit 73127 of the PLA,Fuzhou 350503,China;3.The Ninth Comprehensive Training Base of Army,Zhangjiakou 075000,China;4.Army Infantry College of the PLA,Shijiazhuang 050003,China;5.Unit 68303 of the PLA,Golmud 816000,China

Abstract: Aiming at the problem that the consumption data of new ammunition is less and the demand is difficult to predict,combined with the law of ammunition consumption under different damage grades, a Bayesian inference method for ammunition demand based on Gompertz distribution is proposed. The Bayesian inference model based on Gompertz distribution is constructed,and the system contribution degree is introduced to determine the weight of the multi-source information.In the case where the prior distribution is known and the distribution of the field data is unknown, the consistency test is performed on the prior information,and the consistency test problem is transformed into the goodness of the fit test problem.Then the Bayesian inference is solved by the Markov chain-Monte Carlo (MCMC) method, and the ammunition demand under different damage grades is gained. The example verifies the accuracy of this method and solves the problem of ammunition demand prediction in the case of insufficient samples.

Keywords: ammunition demand prediction, Bayesian inference,Gompertz distribution, system contribution, Markov chain-Monte Carlo(MCMC)method.

1.Introduction

Demand prediction is the basic work of ammunition support decision. Accurately predicting the ammunition demand is the key to the precise ammunition supply, and it is also the focus of current ammunition support research.Many experts have researched on ammunition demand prediction and provided guidance for ammunition support work. Wang et al. [1] established an ammunition effectiveness model and a target damage simulation model,and developed an ammunition consumption prediction model and system simulation platform based on damage to enemy firepower. Zhi et al. [2] proposed a method to calculate the average ammunition consumption required for shipto-air missile damage targets. It can calculate the average ammunition consumption required for a ship-to-air missile to damage a single target, an evacuated target, or a dense target. Song et al. [3] established a hybrid target optimal firepower allocation scheme based on the most favorable firepower distribution method of a single target,and established a calculation model of the minimum ammunition consumption. The above methods are based on traditional ammunition, and fully grasp the ammunition consumption data. The relevant information on the new ammunition has not been fully grasped, and the demand for new ammunition can only be predicted based on the existing small amount of data.

In the case of insufficient samples of new ammunition demand, this study decides to use Bayesian inference to solve the demand prediction problem. In other areas, Bayesian inferences about the prior distribution of the Gompertz distribution have made some progress.In the study of Shahrastani [4], the E-Bayesian and hierarchical Bayesian of the scalar parameter of a Gompertz distribution under Type II censoring schemes was estimated based on fuzzy data.Alizadeh et al.[5]used the Bayesian method to obtain estimates of Gompertz distribution parameters under three different loss functions. Bayesian inference requires to calculate the posterior estimators, usually expressed as complex multidimensional integrals,which are difficult to find in most practical applications.The Markov chain-Monte Carlo (MCMC) method proposed by scholars at home and abroad has properly solved this problem.Miguel et al.[6]developed a simulation method based on the Dirichlet process and the Gibbs sampler to estimate the posterior distribution of the main parameters of the model. Martino et al. [7] introduced the adaptive in-dependent sticky MCMC algorithm into the Bayesian inference solution, and effectively extracted samples from the probability density function of any bounded target.The more iterations,the closer the probability density is to the target. Martino et al. [8] also proposed different multiple try metropolis schemes, ensemble MCMC method, particle metropolis-hastings algorithm and delayed rejection metropolis technique to solve the posterior estimator of Bayesian inference.

Based on the research mentioned above,this paper fully considers the distribution rule of ammunition consumption under different damage grades and the lack of new ammunition consumption data. The Bayesian inference model based on Gompertz distribution is established and solved by the MCMC method.And the feasibility and accuracy of this method are verified by examples.It provides methodological guidance for the statistical analysis of data in the case of insufficient samples.

2.Bayesian inference model

2.1 Posterior distribution acquisitions of prior information

In order to ensure that the field test data are not overwhelmed by historical data, this paper uses the credibility of the pre-test information to measure the difference between the two. The degree of consistency between the distribution of simulation data, the distribution of expert experience data and the distribution of parameters to be estimated is reflected by the credibility of prior information.

Suppose there are two subsamples,XandY. To test whether they belong to the same population, a selection hypothesis is given:H0indicates thatXandYbelong to the same population,H1indicates thatXandYdo not belong to the same population,andAindicates the event of adoptingH0.

Credibility of Bayesian inference results is often judged by the credibility of prior information. The higher the credibility is, the more credible the inference result is.The credibility is expressed asP(H0|A). The higher theP(H0|A) value, the higher the credibility of the pre-test information,and the more accurate the Bayesian inference results[9].P(H0|A)is obtained by Bayesian formula:

Before the ammunition demand is determined, the target damage grades must be clarified,and the target damage grades can be determined based on changes in its combat effectiveness.Based on the analysis of target physical damage, functional damage, and combat effectiveness damage, classification criteria of damage grades in [10] are used to classify the target damage grades into five categories,as described in Table 1.

Table 1 Classification criteria of damage grades

Supposeθrepresents the ammunition demand when each damage grade is reached,θirepresents the ammunition demand when theith damage grade is reached,andrepresents the ammunition demand when theith data source reaches each damage grade.

Ammunition demandsθ1,θ2,...,θnat each damage grade are discrete random variables. Multi-source data such as simulation data and expert experience data are used as prior information samples. Under different data sources,the ammunition demands when the target reaches each damage grade areand respectively.

The amount of ammunition consumed at each grade of damage is continuously processed, and the points of discrete distribution are fitted to a distribution curve.The distribution ofcan be obtained by the statistical analysis of the ammunition demandunder each damage grade presented by different source data. Through the simulation data and expert experience data analysis, it is found that the ammunition demand at different damage grades obeys the Gompertz distribution.The growth trend of the ammunition demand when reaching each damage grade is slow in the beginning, fast in the middle,and slow in the later period.obey the Gompertz distribution, with parametersandrespectively, which is expressed as(i=1,2,...,n).

The prior probability density function of the ammuni-tion demandθunder each damage grade is

The prior distribution function of the ammunition demandθis

In the field test,the actual ammunition demand data under each damage grade isAccording to the field test sampleθsand the prior probability densityf(θ;c,λ), the Bayesian formula is used to calculate the posterior probability density of single prior data source[11]:

In order to make the Bayesian inference results more accurate,the sources of prior information are often diversified.How to integrate multi-source information with appropriate methods is a hot topic of current research. The posterior probability density with single prior information has been obtained by using the Bayesian formula above.Extend it to the general case and solve the posterior distribution function based on multi-source prior information. This paper introduces the system contribution to the Bayesian fusion.

2.2 Weight computing model based on system contribution

The degree of system contribution refers to the influence of the combat unit on the comprehensive operational capability of the combat system.The degree of influence of multisource prior information on the prediction system of the ammunition demand is measured by the system contribution.The impact is mainly determined by its reliability.By analyzing the information from different sources,the reliability of ammunition demand data is evaluated.The system contribution degree is the weight of the Bayesian fusion.The higher the reliability is, the greater the system contribution degree is.The intuitionistic fuzzy membership function is used to solve the weights.Ω={Γ1,Γ2,...,Γn}indicates possible sources of ammunition demand data.(m=Γ1,Γ2,...,Γn) indicates the membership degree which expertsPd(d=1,2,...,n)believe the source of ammunition demand dataΓ1,Γ2,...,Γnis reliable.(m=Γ1,Γ2,...,Γn) indicates the membership degree which expertsPd(d=1,2,...,n)believe the source of ammunition demand dataΓ1,Γ2,...,Γnis unreliable.Intuitionistic fuzzy evaluation information from expertsPd(d= 1,2,...,n)on the reliability of ammunition demand dataΓ1,Γ2,...,Γnis expressed[12]as

Intuitive indicators indicate the degree of hesitation that experts believe the source of ammunition demand dataΓ1,Γ2,...,Γnis reliable or not. Half the degree of hesitation is used to revise the weight.The intuition indicator expression is

The membership degree is expressed as

Normalization processing is

Based on the multi-source informationΩ={Γ1,Γ2,...,Γn}, the weight is determined by the contribution degree. The weighted prior probability density function is obtained.

Bayesian fusion formulas are rewritten as

2.3 Bayesian statistical inference output

Combined with(4),(10)and(11),the posterior probability density function is

Equation(12)shows that the fusion posterior density is weighted by the posterior density ofThis process fuses field ammunition demand data with prior information.

In order to simplify the calculation,in (4)can be replaced by the likelihood function.The likelihood function[13]is expressed as

The weighted posterior probability density function is

3.Model solution

3.1 Consistency tests of prior information

In the testing and appraisal of weapons and equipment,due to the limitations of objective conditions such as test costs,the method of the experimental analysis of completely relying on the field test information can no longer meet the actual needs.

With the maturity of the simulation technology, researchers pay more attention to the use of simulation data for scientific statistical decision-making research in the case of insufficient samples.Since the data are simulation of the actual situation,the credibility of the results and the difference with the field test data will directly affect the rational use of subsequent test methods.Therefore,before using the simulation data and empirical data, it is necessary to verify the consistency with the field test data, that is,the consistency test.The prior distributionF(θ)is completely known,and the field test distributionF'(θ')is unknown.At this time,the consistency test problem is transformed into a goodness-of-fit test to verify whether samplesΘ1,Θ2,...,Θnobey distributionF(θ).

Prior information such as simulation data and expert experience data obeys the Gompertz distribution with parameterscandλ. That is, the amount of ammunition used to reach zero damage,mild damage,moderate damage,severe damage and destruction is a group of samples, hereinafter referred to as the amount of ammunition required to reach each damage grade.The distribution function and the probability density function are

In order to judge whether the field test data, that is,the ammunition demand at each damage grade, satisfies the above distribution, the empirical distribution function(EDF)type goodness-of-fit test is used[14].The test model is expressed as

H0indicates that the simulation data and the expert experience data pass the consistency test on the field test data,andH1is the opposite.

SupposeΘ1,Θ2,...,Θnare random samples of the ammunition demand distribution functionF'(θ')drawn from the field test,and its empirical distributionFn(θ)is defined as

whereis an indicative function and#Xis the number of elements in the setX.Fn(θ)is a right continuous step function withnjump points,Fn(?∞) = 0,Fn(∞) = 1.The empirical distribution function ofnordered statisticsis represented[15]by

To measure the difference betweenFn(θ)andF(θ),the Kolmogorov distance is introduced, indicating the maximum distance betweenFn(θ)andF(θ)in the vertical direction,denoted asKnand expressed as

In the actual calculation,Knis expressed as

The significance levelαofis defined as

3.2 Basic principle of the MCMC method and solution process

The MCMC method [16] belongs to the Monte Carlo method.This method introduces the Markov process into the Monte Carlo simulation, directly simulates the posterior probability distribution, and makes the stable sampling distribution approximate the target distribution. It has the advantages of being suitable for high-dimensional distribution functions and convenient implementation.The MCMC method is used to sample the posterior distribution in Bayesian inference. The method first extracts the random samples that converge to the posterior distribution and counts them, and then analyzes the properties of the posterior distribution. The posterior estimation of the model parameters obtained by the MCMC method can simplify the calculation to some extent.

For any given stationary distributionπ(x) of a Markov chain,the MCMC method uses the Metropolis-Hastings algorithm[17–19].The posterior probability density of the ammunition demand at each damage grade is

The MCMC method is applied to the posterior probability density function to obtain sample sequences ofcandλ.

According to the MCMC method, the posterior estimation of the Gompertz ammunition demand prediction model parameters is calculated[20–24].The specific steps are as follows:

Step 1k=0.

Step 2Determine the initial values ofcandλ, and select the estimated valuesof the parameters obtained from the original sample as the initial values.

Step 3Construct the Markov chain transfer function.The parameterscandλare independent of each other.Under the conditions ofc(k)andλ(k), the following transfer functions are constructed forcandλrespectively:

whereεcandελobey the normal distributionN(0,σc)andN(0,σλ)of parametersσcandσλrespectively.Determineσcandσλduring the sampling process by observation.

Extract random numberufrom uniform distribution[0,1]. The transferred value is determined according to(24).

The determination method ofλ(k+1)is the same asc(k+1).

Step 4k=k+1.

Step 5Determine whether the traversal meanandare converged or not. If they do not converge,jump to Step 3,otherwise turn to Step 6.

Step 6Continue to iterateMtimes according to Step 3 and Step 4 to obtain the Markov chain [c(k+1),c(k+2),...,c(k+M)],[λ(k+1),λ(k+2),...,λ(k+M)] corresponding to parameterscandλ.

Step 7On each of the corresponding Markov chains ofcandλ,samples are taken at intervals.Collectnpoints each, which areciandλi,i ∈{1,...,n}. According to(21),since the parameterscandλare independent of each other,the expected value of the parameters can be obtained from the average of thensample points.

wherei=1,...,n.

After obtaining the estimated values of the parameterscandλ,the posterior probability density function and the posterior distribution function of the ammunition demand amount at each damage grade are determined.In turn,the amount of ammunition consumed at each grade of damage is determined.

3.3 Induction of ammunition demand prediction process

In order to facilitate understanding, this section summarizes the above-mentioned ammunition demand prediction process.The prediction process is shown in Fig.1 and the specific prediction process is shown in the following example.

Fig.1 Flow chart of ammunition demand prediction

4.Application example

In recent years, a large number of new ammunition units have been installed. In order to test the tactical and technical performance of ammunition, the army organized a new ammunition field test at a shooting range.In this case,a new type of suppressed weapon ammunition is selected as the research object. The value of the new ammunition is relatively large,and the number of actual tests is small.The amount of ammunition required to reach each grade of damage is not evident. In the case of insufficient samples,in order to further determine the ammunition demand when each damage grade is reached,this example uses the Bayesian inference method.

The following data are collected by consulting experts in the field of ammunition and consulting the literature.Table 2 is the expert experience value obtained from the actual test data of the original ammunition.These data are the amount of ammunition consumed by six pieces of target equipment to achieve mild damageL2, moderate damageL3,severe damageL4,and destructionL5.The mean value is the amount of ammunition consumed by a single target to achieve different grades of damage.

Table 2 Comparison of the damage loss rate of target equipment under different strike strengths

According to the analysis of expert experience data,the prior probability density function of the ammunition demand at each damage grade is

According to the above data,the origin software can be used to fit the ammunition demand curve when the target reaches different damage grades, as shown in Fig. 2. The image is more intuitive to see the trend of the ammunition demand under different damage grades, and it is verified that it obeys the Gompertz distribution.

Fig. 2 Ammunition demand curve under different damage grades obtained from expert experience data

On the basis of the expert data in Table 2,according to the impact of operational conditions such as combat style,combat scale, force comparison, and battlefield environment on the ammunition demand in the current war,combined with fuzzy back propagation (BP) neural network[25],the simulation of ammunition demand under different damage grades of the single object is carried out.The data are shown in Table 3,and the statistical inference study is carried out.

Table 3 Ammunition demand of target equipment at all grades of damage

When the damage grade is zero damageL1, the OEL of the target equipment is less than 5%, and it is determined that there is no ammunition demand at this time.Through the maximum likelihood estimation of the simulation data,ammunition demand of the target equipment to achieve damage at all grades can be obtained,as shown in Table 4. And the ammunition demand curve is shown in Fig.3.

Table 4 Simulation estimates of ammunition demand when equipment achieves all grades of damage

Fig. 3 Ammunition demand curve under different damage grades obtained from simulation data

According to simulation data,the prior probability density function of the ammunition demand at each damage grade is

According to the analysis of ammunition demand data of the field test in Table 5,the probability density function of the ammunition demand at each damage grade is

Table 5 Average value of actual ammunition demand when the target equipment reaches different grades of damage in the field test

The ammunition demand curve when the target reaches different damage grades in the field test is shown in Fig.4.

Fig.4 Ammunition demand curve when the target reaches different damage grades in the field test

Five hundreds sets of ammunition demand data are taken from the field test for the goodness-of-fit test. When it fits with expert experience data, the significance level isα= 0.500 162.When it fits with simulation data,the significance level isα= 0.661 42. Both are more than 0.5,so the null hypothesisH0is accepted, the field test data are subject to the Gompertz distribution, and the sample satisfies the consistency.

The current data on ammunition demand data are of two types: one is the expert experience value obtained from a small amount of the past field test; the other is the simulation data obtained by computer software.Next,use the system contribution to determine the fusion weight of the two. The specific practices are shown in Table 6 and Table 7.

Table 6 Expert rating of empirical data reliability

Table 7 Expert rating of simulation data reliability

According to (13), (14) and normalization, the system contribution of the two can be determined.Fusion weights are?X=0.52,?Y=0.48.

The weighted prior probability density is

According to the Bayesian formula,the probability density of the fusion is obtained by means of the MCMC method:

According to the ammunition demand obtained by the Bayesian inference,the ammunition demand curve in this case can be determined,as shown in Fig.5.

Fig.5 Ammunition demand curve when the target reaches different damage grades under Bayesian inference

In Fig. 6, the circle represents the ammunition demand when the target reaches different damage grades in the field test;the triangle represents the ammunition demand at different damage grades obtained from the simulation data;the square represents the ammunition demand at different damage grades obtained from expert experience data;the five-pointed star represents the ammunition demand when the target reaches different damage grades under the Bayesian inference.

Fig.6 Contrast with actual data

In order to more intuitively compare the similarity between expert experience data, simulation data, Bayesian inference data and field test data, we summarize the contents of Fig.2 to Fig.5 in Fig.6.It can be seen from Fig.6 that compared with the expert experience data and the computer simulation data, the Bayesian inference value of the ammunition demand at different damage grades is closer to the actual value.

It is not difficult to find through Fig.7 that the Bayesian inferred value of the ammunition demand is less distinct from the true value, and the relative error can be within a reasonable range.The relative errors are detailed in Table 8.The Bayesian inference method for the ammunition demand based on Gompertz distribution and MCMC is effective and feasible. The advantage of this method is the fact that it can solve the problem of ammunition demand prediction in the case of insufficient samples.

Fig.7 Comparison of the actual value of ammunition demand with the Bayesian inference value

With a small amount of real ammunition consumption data,the ammunition demand when the target reaches each damage grade can be obtained.This method is particularly suitable for the demand prediction of two types of ammunition. One is the new type of ammunition for the newly installed troops, and the other is the expensive ammunition.

Table 8 Comparison of actual ammunition demand with Bayesian inference

5.Conclusions

The Bayesian inference method based on Gompertz distribution and MCMC for the ammunition demand proposed in this paper has fully considered the ammunition consumption rule under different damage grades and the characteristic of fewer modern ammunition consumption data.It will provide a reference for the statistical analysis of data in the case of deficient samples.The main tasks are as follows:

(i)The consumption rule of new ammunition under different damage grades has been studied.It has been found that the ammunition demand obeys the Gompertz distribution. The growth trend of the ammunition demand when reaching each damage grade is slow in the beginning,fast in the middle, and slow in the later period. From mild damage to moderate damage, moderate damage to severe damage,the ammunition demand increases significantly.

(ii) The fusion weight calculation model based on the system contribution degree has measured the influence of multi-source information on the ammunition demand under different damage grades,and the weight has been given according to the reliability degree of multi-source information.

(iii) Through the goodness-of-fit test of prior information, the field test data has not been overwhelmed by a large amount of prior data. The step has provided a reliable source of data for the next Bayesian inference.

(iv)The Bayesian fusion formula has been used to fuse the ammunition demand information from diverse sources.The MCMC method has been used to determine the posterior probability density of the ammunition demand.Thus,the amount of ammunition consumed when the target reaches each damage grade has been determined.

In the next study of ammunition demand prediction,it is necessary to fully understand the tactical and technical performance of new ammunition and uncertainties affecting the ammunition consumption. The next step is to properly classify the target damage grade based on the consideration of various indicators of the fresh ammunition.It is expected that the proposed method will be more suitable for different ammunitions and achieves the purpose of precise guarantee of ammunition.