In view of the small sample size of combat ammunition trial data and the difficulty of forecasting the demand for combat ammunition, a Bayesian inference method based on multinomial distribution is proposed. Firstly, considering the different damage grades of ammunition hitting targets, the damage results are approximated as multinomial distribution, and a Bayesian inference model of ammunition demand based on multinomial distribution is established, which provides a theoretical basis for forecasting the ammunition demand of multigrade damage under the condition of small samples. Secondly, the conjugate Dirichlet distribution of multinomial distribution is selected as a prior distribution, and Dempster–Shafer evidence theory (D-S theory) is introduced to fuse multisource previous information. Bayesian inference is made through the Markov chain Monte Carlo method based on Gibbs sampling, and ammunition demand at different damage grades is obtained by referring to cumulative damage probability. The study result shows that the Bayesian inference method based on multinomial distribution is highly maneuverable and can be used to predict ammunition demand of different damage grades under the condition of small samples.

Ammunition supply support is a key element in the formation of combat effectiveness [

Traditionally, there are three ammunition demand prediction methods: ammunition consumption standard correction method, empirical deduction algorithm, and theoretical calculation method. The ammunition consumption standard correction method is based on the ammunition consumption standard, which is modified according to actual combat conditions to predict the ammunition demand, but with the rapid development of weapons and ammunition, the formulation of ammunition consumption standards cannot be followed up in time. In an empirical deduction algorithm, the law of ammunition consumption is summarized by analyzing typical war cases, and ammunition demand is predicted according to the future war development trend and ammunition consumption characteristics. Considering historical limitations in respect to the complexities of the future combat mode and the diversity of the combat means, the empirical deduction algorithm is difficult to apply. The theoretical calculation method is a mathematical method for calculating ammunition consumption based on military indicators, such as the number of weapons and equipment. It can be further classified into the task quantity method and the battle scenario method. When the theoretical calculation method is used for ammunition demand estimation, large amounts of accurate and reliable trial data are required to be as a basis [

With the development of ammunition demand prediction theory, the simulation prediction method based on military operations, the prediction method based on an intelligent algorithm, and the combination forecasting method [

In order to solve the problem of a small amount of test data caused by economic problems in ammunition demand prediction, the Bayesian method is included in the research scope [

The prior information comes from expert information and field test information in the process of using the Bayes method. To make better use of the prior information, it is necessary to fuse them. The D-S theory is widely used in multisource information fusion because of its effectiveness in data fusion. Tang et al. [

Ammunition demand refers to the amount of ammunition needed for a target to reach a certain degree of damage. Without considering the cumulative effect of ammunition damage, the calculation of ammunition demand comes from the single-shot damage probability of ammunition. In fact, the probability that the target reaches a certain degree of damage is stable in the process of ammunition damage. Therefore, this paper assumes that the damage result of the target obeys multinomial distribution. Obtaining multinomial distribution parameters becomes a key link, and its value depends on expert experience and field test data.

Based on the above analysis, a Bayesian inference method of ammunition demand based on multinomial distribution is proposed. It considers the different damage grades in combat ammunition hitting and the actual demand of ammunition prediction under the condition of a small sample size. The multinomial distribution is used for describing the damage result of the target and the D-S theory is used for fusing expert information and field test information. And the ammunition demand as reaching different damage grades is forecasted based on Bayesian statistics, which provide a method for solving the ammunition demand prediction under the condition of multiple damage grades and small samples.

Compared with the classical statistical inference method, the Bayesian inference method integrates general information, sample information, and prior information and regards the parameters to be estimated as random variables. Assuming that the overall distribution is

It can be seen from formula (

The methods to determine the prior distribution function include the maximum entropy method commonly used in engineering, the conjugate distribution method with known overall distribution, and the expert scoring method. Assuming that the conditions of each shooting test are basically the same and each shooting is independent, each shooting test has

To facilitate the calculation and reasonably explain the parameters, the conjugate distribution method is used to determine prior distribution in this paper. Given that the conjugate prior distribution of multinomial distribution is Dirichlet distribution, the previous probability density function of ammunition demand

The prior distribution function of ammunition demand

To obtain more accurate Bayesian inference results, the fusion of different prior information should be attended to in the application of Bayesian methods. Expert information comes from the damage data of different similar targets, which has a certain credibility. The field test data may also have certain deviations because of the small number of samples. The D-S theory can effectively fuse and correct the deviation. Therefore, the D-S theory is introduced in this paper for the fusion of prior information.

Before the firing trial, experts could have a specific prior understanding of the ammunition demand required to reach the damage grade. To utilize the information and the field trial data simultaneously, multisource Bayes prior information is fused based on the D-S theory, and the fusion formula is [

In this paper, the D-S theory is used to fuse expert information and field test data, as shown in Table

Basic probability assignment of expert information and field test information.

Recognition frameworks | Expert information | Field test information |
---|---|---|

According to the data in Table

After fusing expert information and field test information, the prior information of damage grade is as follows:

According to the analysis of the experimental data of ammunition effectiveness, the identification framework

Assume that the number of damage grades is

Zhang et al. [

Based on the Bayesian theorem, the posterior distribution density function of the parameter

Let

It can be seen that the posterior distribution density function obeys Dirichlet distribution.

After using the Dirichlet distribution of conjugate prior distribution of multinomial distribution to obtain the estimated value of posterior distribution parameters, the updated ammunition demand

Combining the fused prior information in the light of D-S theory with the Bayesian theorem, the formula of fusion density can be obtained as follows:

The fusion posterior density is

According to formula (

The parameter means

After

Equation (

The damage of ammunition to the target can be regarded as a random event. When calculating the ammunition consumption, it is necessary to determine the probability critical value

According to the above formula, the best ammunition demand

The premise of Bayes statistical inference method using prior information is that previous information can reflect the statistical characteristics of parameters; that is, advance information and outfield trial data approximately obey the same distribution [

Compatibility test consists of two methods: a parametric test and a nonparametric test [

It is assumed that the observed values of samples from populations

If

In the formula,

When

Up to this point, the compatibility test of the prior information of ammunition demand estimation is completed, and further solutions can be made.

According to the fusion posterior distribution density function (15), to get posterior integral and posterior estimation values, complex high-dimensional calculus needs to be calculated, which is difficult to realize. In this paper, Markov chain Monte Carlo (MCMC) is adopted to realize the numerical calculation of high-dimensional integral. Its basic principle is as follows:

Constructing Markov chain: start from the initial state

Generating samples: starting from a particular point

Monte Carlo integration: the marginal distribution of each state cannot be considered as a stable distribution after

The Metropolis–Hastings method and the Gibbs sampling method are two widely used MCMC methods in Bayesian analysis. The Markov chains constructed by which are reversible [

Step 1: determine the posterior conditional distribution of parameters. Generating simulation sample data from posterior conditional distributions is easier than from parameter vectors. [

Step 2: determine the initial value, select an initial state point

Step 3: sampling iteration. Extract sample values of parameters

……

A new sample is

Step 4: repeat the iteration of Step 3 and stop when the maximum iteration number

Step 5: calculate the parameters. In this paper, sample mean serves as the parameter point estimation value in this paper:

So far, the estimated values of parameters corresponding to each damage grade are obtained, which can be used for Bayes inference of ammunition demand.

It is assumed that five damage grades are caused after ammunition hitting, and the conditions of each shooting trial are nearly the same. Sixty shooting trials are conducted in advance and five essential events occurred in each strike, corresponding to five different damage grades, namely, zero damage, mild damage, moderate damage, severe damage, and scrapping. To test the technical performance of high-tech weapons and ammunition, an individual unit has carried out a specific actual installation and firing trial in the shooting range, out of which a certain kind of ammunition of high-tech weapons is selected as the research object in this study. Given the low frequency of firing and the indefinite amount of ammunition needed to reach the corresponding damage grade, in this study, the Bayesian statistical inference method is used to predict the amount of ammunition required to reach different damage grades while applying a small sample statistical analysis method to data analysis.

Combining expert information with field trial data, the frequency distribution for reaching different damage grades under the condition of striking the same target is shown in Table

Target frequency distribution for reaching different damage grades.

Serial number | Number of trials | Zero damage | Mild damage | Moderate damage | Severe damage | Scrapping |
---|---|---|---|---|---|---|

1 | 60 | 5 | 10 | 12 | 18 | 15 |

2 | 60 | 3 | 8 | 12 | 15 | 22 |

3 | 60 | 7 | 8 | 15 | 14 | 16 |

4 | 60 | 8 | 12 | 13 | 14 | 13 |

5 | 60 | 11 | 12 | 10 | 12 | 15 |

6 | 60 | 6 | 12 | 12 | 12 | 18 |

7 | 60 | 15 | 12 | 13 | 10 | 10 |

Python software is used to count the number of trials, and visual processing is shown in Figure

Frequency distribution of targets with different damage grades.

In this paper, through the Markov chain Monte Carlo method, the statistical inference in the WinBUGS software environment is made. With the Gibbs sampling algorithm, three Markov chains are generated by sampling from the complete conditional probability distribution. To estimate the posterior for the parameters

Iterative tracer diagram for parameters

Autocorrelation function diagram of parameters

We found all three Markov chains converge by observing the iterative trace graph and autocorrelation function graph. Then, the posterior estimation of parameters can be performed, which is shown in Figure

Nuclear density diagram of parameters

After 1500 iterations of each chain, the posterior estimation results of parameters

Posterior statistics of parameters

Parameter | Mean | SD | MC error | Median | 97.5% CrI | |
---|---|---|---|---|---|---|

theta_1 | 4500 | 0.1997 | 0.1632 | 0.0026 | 0.1603 | (0.006, 0.609) |

theta_2 | 4500 | 0.2009 | 0.1635 | 0.0023 | 0.1607 | (0.005, 0.608) |

theta_3 | 4500 | 0.1991 | 0.1637 | 0.0023 | 0.1591 | (0.006, 0.600) |

theta_4 | 4500 | 0.2017 | 0.1645 | 0.0025 | 0.1591 | (0.007, 0.601) |

theta_5 | 4500 | 0.1986 | 0.1631 | 0.0024 | 0.1575 | (0.006, 0.595) |

In the actual attack, the damage grade of the target cannot be determined. Assuming that the threshold value

Optimal ammunition demand for each damage grade.

Damage grade | Ammunition amount | Cumulative destruction injury probability | Whether optimal |
---|---|---|---|

Mild damage | 1 | 0.8003 | √ |

Mild damage | 2 | 0.9601 | |

Moderate damage | 1 | 0.5994 | |

Moderate damage | 2 | 0.8395 | √ |

Moderate damage | 3 | 0.9357 | |

Severe damage | 1 | 0.4003 | |

Severe damage | 2 | 0.6404 | |

Severe damage | 3 | 0.7843 | |

Severe damage | 4 | 0.8707 | √ |

Severe damage | 5 | 0.9224 | |

Scrapping | 1 | 0.1986 | |

Scrapping | 2 | 0.3578 | |

Scrapping | 3 | 0.4853 | |

Scrapping | 4 | 0.5875 | |

Scrapping | 5 | 0.6694 | |

Scrapping | 6 | 0.7351 | |

Scrapping | 7 | 0.7877 | |

Scrapping | 8 | 0.8299 | √ |

From the calculation results in Table

To predict the ammunition demand for reaching different damage grades, the Bayesian inference method based on multinomial distribution is proposed by fully considering the current situation of little data in ammunition strike trials, which provides a reference for the prediction of ammunition demand under the condition of small samples in the trial of high-tech weapons and ammunition. The main work accomplished includes the following:

To provide a prerequisite for Bayesian inference, multinomial distribution assumptions are made in the process of ammunition hitting.

With the fusion of multisource prior information based on the D-S theory, multiple information sources can be taken full advantage of. This paper adopts the classical fusion rule, but when there is a large amount of data, there may be conflicts between evidence, so this rule will no longer be applicable. Therefore, a new basic probability assignment method proposed by Jing et al. can be considered to adopt [

On the basis of analyzing the prior information, this paper chooses the Dirichlet distribution, which is the conjugate distribution of multinomial distributions, as the prior distribution of the Bayesian inference model of ammunition demand, that not only does it simplify the calculation but also the posterior distribution parameters are better explained.

By using the MCMC method based on Gibbs sampling to solve the Bayesian inference model of ammunition demand based on multinomial distribution and with the application of WinBUGS software to simulation, the ammunition demand for different damage grades is obtained with the idea that the cumulative probability exceeds the given probability value.

To sum up, the ammunition demand forecasting method in this paper makes the demand forecasting of a new type of ammunition feasible, which has practical guidance to the ammunition supply support, yet it is necessary to consider the damage efficiency of new ammunition in further study.

All the data supporting this study are included within this paper.

The authors declare that they have no conflicts of interest.

Our work was supported by army scientific research (LJ20202C050369).