Xin-hua HE，Hao CHENG，*，Rui-tao ZOU，Xing-gang ZHAO，Yun-long LI

（1Department of Information Engineering，Academy of Army Armored Force，Beijing 100072，China）

（2 The66136 troops of the Chinese People’s Liberation Army，Beijing 100042，China）

Abstract：There are many components and complex structure relationships in weapon system.The ability model can be reduced and optimized by effectively judging the weight of the relationship，and the computational complexity of the subsequent simulation evaluation can be effectively reduced by reducing the complexity of the model.In order to solve this problem，nonlinear thinking is introduced into the theory of structural equation modeling（SEM）.Considering the factors of synergy and secondary effects between capability subjects，a method for solving the relationship between combat capabilities based on nonlinear SEM is proposed.Finally，taking the tactical information distribution capability as an example，the effectiveness of the method is verified by comparing the evaluation results of linear and nonlinear SEM.

Key words：Weapon SoS，Nonlinear SEM，Relationship judging

1 Introduction

There are a lot of information exchanges among the main units in the structure model of combat capability of weapon and equipment system，which makes the relationship between them present non-linear characteristics，which is also the fundamental reason for the emergence of combat capability of system.Therefore，considering the non-linear relationship among the main bodies of combat capability and making effective and accurate judgment on it becomes an important part of the evaluation of combat capability of the system.The traditional method of judging association relationship is usually based on expert evaluation or linear mathematical statistics，which is time-consuming，laborious and lack of scientific objective reality.Therefore，a nonlinear correlation determination method is needed to make up for the shortcomings of traditional methods.

2 Structural equation modeling

SEM is a method used to analyze multivariate data association in the field of applied statistics［1］.Usually it is used to analyze the relationship model with implicit variables，and ultimately solve the complex relationship in the model.SEM mainly combines the ideas of multiple regression calculation，exploratory factor analysis and path analysis.

2.1 Basic ideas

Structural equation model mainly uses the factors that can be observed directly to characterize the hidden factors，and then through the analysis of the complex correlation between various factors，regression calculates the hidden relationship between the latentvariables.

2.2 Basic concepts

A.Variable

There are two classifications of variables in SEM.If the variable can be observed directly，it is called explicit variable；otherwise，it is called implicit variable.If the variable is affected by external factors，it is called exogenous variable；otherwise，it is called endogenous variable.As shown in the Fig.1，the combination of exogenous explicit variable X，exogenous implicit variable E，endogenous implicit variable N and endogenous explicit variable Y.

B.Equation

SEM is mainly composed of measurement equation and structure equation.

The measurement equation is mainly used to express the relationship between explicit and implicit variables.The general form is：

Structural equation is mainly used to express the relationship between hidden variables.

C.RAM diagram

We use graphical method to visually express the relationship between variables in the estimation model，namely RAM graph.It is currently the most effective way to express the results of structural modeling.The schematic diagram is shown in the Fig.1.

Fig.1 RAM diagram

As shown in the figure above，the theory of weapon equipment system is studied.Tactical communication capability is taken as an example.ξ1represents integrated communication capability，ξ2represents network coverage capability，η1represents tactical communication capability，η2represents command and control capability.The explicit variables mainly represent the underlying parameters of the competent subjects，while the path is used to represent the relationship between the subjects.The relationship equation is expressed as follows：

Among them，the four coefficient matrices are：

On this basis，the concept of quadratic term and cross term of latent variable is introduced，considering the influence of non-linear correlation among subjects.The RAM diagram is shown in the Fig.2.

Fig.2 Non linear RAM diagram

The non-linear structural equation is：

The non-inear measurement equation is：

Based on the model estimation of the non-linear SEM and the reasonable relationship parameters，the quantitative model of operational capability index of weapon and equipment system is obtained.

2.3 Solution idea

According to statistical knowledge，in reality，there is a certain gap between the experimental observations and the ideal actual values，that is，residual e.This feature can be used to estimate model parameters based on the minimum residual e.

This idea is also applicable to the solution of nonlinear SEM.Firstly，the estimated covariance matrix is obtained by using the drawn RAM diagram.Assuming that the difference between the estimated covariance matrix and the sample covariance matrix is very small，the estimation model is reliable and has the following relationship：

Because the two matrices of sample covariance and estimated covariance are identical，the corresponding positions of the two matrices are equal，so the estimation parameter matrixθcan be solved.

Further deduction can be made from the above formula：

After calculating the relationship between the estimated covariance and the sample covariance，the model estimation parameters with the largest similarity and the smallest error are obtained by the mathematical statistics method of Iterative Regression calculation.Finally，a series of evaluation criteria are set to evaluate the estimated parameters.

3 Solution method of main unit association relation of weapons SoS operational capability based on nonlinear SEM

According to the above solution idea，the procedure of solving the main body association relation of combat capability of weapon and equipment system based on non-linear SEM is as follows：

1）Data acquisition and preprocessing

The bottom measurement data of the system’s combat capability can be obtained through a variety of actual combat drill data，equipment performance index parameters and laboratory simulation experiments.At the same time，it is necessary to standardize the acquisition of data，examine and screen abnormal data，fill in missing data，and ensure high value and predictability of sample data.

2）Model setting

In-depth study of weapon and equipment system expertise，based on the known prior knowledge，assume the relationship between the combat capability subjects，and draw the RAM chart.Then the corresponding non-linear SEM model is established according to the topological structure of RAM.

3）Model recognition

In order to ensure that the model is estimable，the model must be identified first.If the model is identifiable，the unknown parameters in the model can be estimated.T rule is usually used as recognition rule.

Rule t：t＜（p＋o＋q）（p＋o＋q＋1）／2，which has（p＋o）exogenous explicit variables，q endogenous explicit variables，t is the number of unknown parameters.If the t rule is satisfied，the non-linear SEM can be identified.

4）Model estimation

The most common methods of model estimation are maximum likelihood estimation，least squares，moment estimation and Bayesian methods.In this paper，the maximum likelihood estimation method is used.

The idea of maximum likelihood estimation is to list the probability density function of multivariate normal distribution and iterate a set of model parameters to maximize the probability density function.That is to say，the concept of minimum difference between sample covariance and estimated covariance is the largest.

The concrete method is to list the probability density function of multivariate normal distribution.

Among them，∑ is the total covariance matrix，the number of endogenous index variable X，derivative index variable XiXjand exogenous index variable Y are p，O and Q respectively，and Z is a vector of（p＋o＋q）×1.

The probability function of a single set of observations Ziis：

Assuming that the observed values of each sampleare independent of each other，the joint density function of all the observed values appears as follows：

Namely：

After a series of simplified deductions，it is concluded：

When the likelihood function FMLgets the minimum value，the estimated parameterθis the final result.Likelihood function is very complex，and it usually needs to be solved by computer.

4 Case analysis

Based on Amos platform and taking tactical information distribution ability as an example，this paper analyzes and shows the method of association relationship determination.It is mainly divided into the following steps：

1）Establish the tactical information distribution capability index system，as shown in the Fig.3.

Fig.3 Tactical distribution capability index system chart

2）According to the above index system，establishing the SEM of tactical information distribution ability.As shown in the Fig.4.

Fig.4 SEM of tactical information dissemination capability

The indicators corresponding to the variables in figure 4 are shown in Table 1.

Table 1 Variable name table

The structural equation of the model is：

The measurement equation of the model is：

In the linear model，the number of exogenous explicitvariables（P）is 6，the number of endogenous explicit variables（q）is 3，and the number of unknown parameters is 24.According to the t rule，t＝24≤（p＋q）（p＋q＋1）／2＝45，as the nonlinear SEM can be identified.

3）Assuming that there is a nonlinear relationship betweenξ1andξ2，only considering the interaction effect between them，establishing the nonlinear SEM of tactical information distribution capability of tactical communication system，as shown in Fig.5.

Fig.5 Nonlinear SEM of tactical information dissemination capability

The structural equation of the model is：

The nonlinear measurement equation is：

In the nonlinear model，the number of exogenous explicit variables（p）is 14，the number of endogenous explicit variables（q）is 3，and the number of unknown parameters is42.According to the t rule，t＝42≤（p＋q）（p＋q＋1）／2＝153，as the nonlinear SEM can be identified.

4）Combat experiment simulation data

In this paper，based on a digital equipment simulation test platform in the army，Monte Carlo simulation is used to calculate the index values of each capability subject under different schemes.

5）Analysis of experimental results

Calculate the structural equation model of SEM and nonlinear SEM respectively，measure the capacity value of different schemes，compare the gap between the two，draw a histogram，and verify the emergence of the system caused by nonlinearity.

Amos is used to solve the nonlinear SEM parameter estimates.The correlation model is as follows Fig.6：

Fig.6 Judgment and estimation experiment

According to the estimated values of SEM parameters，the correlation model is：

According to the estimated values of nonlinear SEM parameters，the correlation model is：

The comparison of linear and non-linear SEMevaluation results is shown in Fig.7：

Fig.7 Comparison chart of evaluation results

In the above evaluation experiments，considering the interaction between the capabilities of weapons and equipment，a nonlinear SEM regression calculation method is designed to determine the relationship between combat capabilities.Compared with the results of linear SEM，it can be proved that the nonlinear interaction between combat capabilities is a cause of emergence of weapon system.

5 Conclusion

In the research on the method of determining the correlation between the combat capabilities of the weapon equipment system，how to consider the complex interaction between various capabilities is the key to whether a reasonable evaluation model can be constructed.This paper studies in detail the basic ideas，principles and solutions of linear and non-linear SEM，and proposes a method for solving the association relationship of non-linear SEM，and uses the tactical information distribution capability as an example to verify the effectiveness of the method.Finally，by comparing the evaluation results of linear and non-linear SEM capabilities，it is shown that the non-linear characteristics of the system characterize the overall emergence of the system’s combat capabilities to a certain extent.