1.

INTRODUCTION

Air combat target threat assessment^1-3 is a typical multi attribute decision-making problem. Decision-makers need to assess the threat from potential targets and attack the higher-threat targets.

At present, the methods used for threat assessment are mainly divided into two categories, one is to build a specific mathematical model, the other is based on artificial intelligence assessment methods. Due to measurement error, environmental noise, different equipment, enemy non-cooperation and other factors, the detection data of the incoming target is often missing, inaccurate and fuzzy, while the current threat assessment methods mostly deal with the same type of characteristic attributes. For example, multi-attribute decision making based on interval number^4-7 and intuitionistic fuzzy set^{2, 7-10} is the current research hotspot. Han⁴ proposed a multi-attribute decision making method for quantum bee colonies with target attribute values of interval number and unknown weights. Dong⁵ proposed a GRA-TOPSIS radiation source threat assessment model based on game theory under interval conditions. Studies on different types of data generally convert them into a certain type of data and then process them. For example, Kong⁸ puts forward a processing method of converting the quantitative results of indicators into a unified intuitionistic fuzzy set for evaluation, aiming at the problem of diverse and difficult quantification of target threat indicators in ground combat. Chen³ proposed a new intuitionistic fuzzy multi attribute decision making model based on intuitionistic fuzzy number, right-angle triangle fuzzy number and the proposed intuitionistic fuzzy number, which transformed intuitionistic fuzzy set average operator. The research status shows that there are still technical difficulties in threat assessment processing of multi-type heterogeneous data directly at present.

At the same time, the current threat assessment methods are mostly single moment, and a few methods with time series values is to use Poisson inverse distribution^{7, 11, 12} to generate time series weights to fuse decision information of multiple moments, involving less factors, did not consider the possible influence of numerical change of characteristic attribute and internal law on weight at different time.

Based on the above, this paper proposes an exponential correlation dynamic threat assessment method for multi-type heterogeneous data. Firstly, referring to the theory of set pair, the D-U space mapping method for multi-type heterogeneous data is defined to keep the uncertainty of information while isomorphizing multi-type heterogeneous data from different sensors. Then the attribute weight was calculated based on the method of exponential correlation degree, and the weight of time series was calculated based on the method of combining deviation and grey entropy. Finally, the comprehensive evaluation value and proximity degree of targets were calculated, and the threat ranking results were obtained.

2.

AIR COMBAT TARGET THREAT ASSESSMENT INDEX SYSTEM

The factors affecting target threat assessment mainly include space posture, tactical intention and combat capability, which must be considered comprehensively. Space posture includes Angle, velocity, distance and altitude threat, which reflects the dynamic property of the target. Tactical intention is divided into attack, cover, interference and evasion, which reflects the behavioral characteristics of the target. Combat capability is measured by maneuvering, attack, detection, range, electronic countermeasures and so on. It reflects the static attributes of the target. The threat assessment system constructed in this paper is shown in Figure 1. The specific indicators and calculation formula are elaborated¹³.

Figure 1.

Two-dimensional D-U space.

3.

HETEROGENEOUS DATA AND ISOMERIZATION

3.2

Isomer method

In the threat assessment problem, the use of heterogeneous data such as accurate number, interval number and intuitionistic fuzzy set as attribute values is more consistent with the operational reality. It is difficult to directly evaluate heterogeneous data, which usually requires isomerization. Currently, the commonly used method for isomerization of heterogeneous data is to directly transform different types of data into a certain type of data for aggregation evaluation¹⁴, but the uncertainty of data will be lost. In order to retain the uncertain information of data, this paper refers to set pair theory¹⁵ and comprehensively analyses the certainty and uncertainty of heterogeneous data as a whole, so as to realize the isomerization processing.

3.2.1

Set Pair Theory and D-U Space.

As a theory to solve the problem of uncertainty, the core idea of set pair theory is to construct set pair from two associated sets in the uncertain system, and to construct the set pair connection degree by analyzing the characteristics of set pair from three aspects of sameness, difference and opposition, and its mathematical expression is

where H is the generated set pair, Q is the total number of set pair features, R is the common feature of two sets, F is neither opposite nor identical feature of two sets, P is the opposite feature of two sets, satisfying P = Q – R – F. R / Q, R / Q and P / Q are respectively represent the degree of identity, difference and opposition of the two sets; τ and ε are the coefficient of difference and the coefficient of opposition respectively.

D-U space is a kind of mapping space of set pair theory. Among the n coordinate axes that constitute n-dimensional space, there are at least 1 dimensional coordinate to describe uncertainty. Take the two-dimensional space shown in Figure 2 as an example, the certainty of variables is represented by the X-axis and the uncertainty by the Y-axis, so the variable θ is called the complex quantity of deterministic uncertainty, and its two-dimensional coordinate system is called the two-dimensional D-U space.

Figure 2.

The target threat assessment values of the three methods

3.2.2

Interval Number Mapping in D-U Space.

The two endpoints of interval data are determined, but the value is uncertain within the interval. In this section, uncertainty information is reserved by mapping it to D-U space.

It is assumed that the attribute value of the evaluation object S_i(i ∈ n) under the attribute a_j(j ∈ m) is the normalized interval type value , that is, meets , , and , when , . degrades to a real number.

The expression of interval number mapping to D-U space is defined as: represents the degree of certainty about , represents the degree of uncertainty about , τ is the uncertainty coefficient, represents the degree of uncertainty, this paper takes , therefore, the binary form of interval number mapping to D-U space is

3.2.3

Intuitionistic Fuzzy Number Mapping in D-U Space.

Intuitionistic fuzzy number’s expression of complex uncertainty through membership degree, non-membership degree and hesitation degree is similar to the description of uncertainty in set pair theory to a certain extent. Therefore, this paper maps intuitionistic fuzzy number to d-u space by referring to the method^{13, 16}.

It is assumed that the attribute value of the evaluation object s_i(i ∈ n) under the attribute is the intuitionistic fuzzy value after normalization, μ_ij, v_ij is respectively the membership degree and non-membership degree of , satisfies μ_ij, v_ij ∈ [0,1] and 0 ≤ μ_ij + v_ij ≤ 1, hesitation degree μ_ij = 1 – μ_ij – v_ij, obviously 0 ≤ π_ij ≤ 1.

The expression of intuitionistic fuzzy number mapping to D-U space is defined as: θ_Hij, =a_ij + b_ijτ, a_ij = μ_ij is the degree of certainty of representation , b_ij =1 – μ_ij – v_ij is the degree of uncertainty of representation , and the uncertainty coefficient τ = a_ij/(a_ij + b_ij) = μ_ij/(1 – v), then the form of intuitionistic fuzzy number mapping to D-U space is:

4.

WEIGHT DETERMINATION METHOD

To determine the attribute weight and time weight, the dynamic multi-attribute decision problem D needs to be described first. Set the evaluation objection set of D as S = (s₁, s₂, ⋯, s_n), the attribute set as A = (a₁, a₂, ⋯, a_m), and time series set as T = (t₁, t₂, ⋯, t_p). It can be known that at moment t_k, the value of the object s_i on the attribute a_j. is , i = 1,2, ⋯, n, j = 1,2, ⋯·, m, k = 1,2, ⋯·, p, that is, the threat assessment matrix at moment t_k is , is the weight of the attribute a_j at moment t_k, meets ω_k is the time weight at moment t_k, meets .

4.1

Attribute weight determination method

This section refers to the attribute weight determination method based on exponential function correlation degree¹⁷ to improve the low resolution accuracy of traditional methods. The specific calculation method is as follows:

Firstly, the positive ideal solution and negative ideal solution of the threat assessment matrix are calculated.

Positive ideal solution:

Benefit indicator: ; Cost indicator: .

Negative ideal solution:

Benefit indicator: ; Cost indicator: .

The exponential correlation degree between the attribute values of the number i evaluation object and the positive ideal solution at the moment t_k is defined as

where:i = 1,2, ⋯, n, j = 1,2, ⋯, m, k = 1,2, ⋯, p.

Finally, we get the attribute weight vector at moment .

4.2

Time weight determination method

In dynamic multi-attribute decision making problem, the importance of data at different time to the final threat assessment is different, so how to determine the time weight scientifically and reasonably is the key to the assessment. At present commonly used method for the Poisson distribution inverse form⁹ way to generate time weighting fusion decision-making information, but this method doesn’t consider attribute values and interconnectedness, the lack of a certain theoretical basis, this article uses the deviation combined with grey entropy weight method to determine the time, the concrete calculation method is as follows:

Define the comprehensive positive and negative deviation of the object s_i at time t_k as

Then we know that the positive and negative comprehensive deviation of all objects is

The ideal time weight should minimize the positive ideal synthesis deviation and maximize the negative ideal synthesis deviation. Accordingly, the weight determination problem can be transformed into the following multi-objective programming problem:

At the same time, because the weight of decision system with incomplete information has certain uncertainty, the uncertainty of time weight sequence should be reduced as much as possible, which can be obtained by maximum entropy principle:

By introducing the coordinated equilibrium coefficient μ, the three optimization problems above are transformed into a single minimization problem:

In this paper, three optimization problems are considered equally important and the equilibrium coefficient is μ = 1/3.

Construct Lagrange function, according to the existence condition of extreme value, solve the above equation

Considering , we can get the optimal solution of the time weight vector W = ω₁, ω₂, ω₃ ⋯ ω_p.

On this basis, the comprehensive evaluation value of the evaluation object s_i is calculated as

The closeness degree of each evaluation object was solved. Finally, the threat ranking was completed according to the closeness degree of each evaluation object.

5.

THREAT ASSESSMENT ALGORITHM

According to the above, the dynamic threat assessment model of multi-type heterogeneous data is established, and the specific process is as follows.

Step 1. Obtain the original threat assessment matrix. According to the battlefield situation, m enemy targets constitute the combat unit set X = {X₁, X₂, ⋯ X_m} and n target attributes constitute the attribute set N = {N₁, N₂, ⋯, N_n}, where the j-th attribute value of the i-th target at the moment t_k is , and the numerical structure of different attributes of is different, and the original target threat assessment matrix at the moment t_k is obtained.

Step 2. Isomerization and normalization. According to the isomorphic method in Section 2, the isomer threat assessment matrix is obtained by processing the original threat assessment matrix U^k at the moment t_k; Due to the different attributes and dimensions of evaluation indicators, they can be divided into benefit and cost type. The value of benefit index is positively correlated with the degree of threat, while the value of cost index is negatively correlated with the degree of threat. The following is a specification for the normalization of different attributes.

Process the normalized target threat assessment matrix at the time t_k obtained.

Step 3. Determine the positive and negative ideal solutions , of each target attribute, and calculate the exponential correlation matrix.

Step 4. Calculate the optimal attribute weight of t_k time based on the method of exponential function correlation degree. Step 5. Integrate attribute weights of multiple moments to solve the optimal time weight vector W.

Step 6. Calculate the comprehensive evaluation value of each target according to attribute weight and time weight vector. Step 7. Calculate the proximity of each target and sort the threats.

6.

EXAMPLE ANALYSIS

In an air battle, our side was attacked by five enemy targets, and the enemy target set X = {x₁, x₂, x₃, x₄, x₅} was set. This paper selected five threat attributes for evaluation, among which, cost index: [combat ability, anti-interference ability], and benefit index: [target speed, target height and detection distance], and the corresponding threat attribute values of the three moments t₁, t₂, t₃ given by comprehensive combat experience are shown in Table 1. Among them, combat ability and anti-interference ability are intuitionistic fuzzy numbers, target speed is accurate number, and target height and detection distance are interval numbers.

Table 1.

Threat attribute value of target.

6.1

Calculation process of threat assessment

Step 1. Isomerization: The data in Table 1 have been normalized, so it only needs to be isomerized. The original threat assessment matrix is processed according to the D-U space mapping rule, and the isomerized threat assessment matrix is

Step 2. According to Z₁, the positive and negative ideal solutions of time t₁ are respectively:

Step 3. According to Formula (5), calculate the exponential correlation degree matrix at time t₁.

Step 4. According to Formula (6), the optimal attribute weight of time t₁ is

Step 5. In the same way, the optimal attribute weight of t₁ and t₃ is respectively calculated as

Step 6. The optimal time weight vector W

Step 7. The comprehensive evaluation value of each target is calculated by combining the optimal time weight vector and the attribute weight of each moment

Step 8. The proximity degree of each target was calculated and the threat ranking was carried out

It can be concluded that the threat ordering result of targets is as follows: 3>1>2>4>5. The objective classification result of target threats is priority attack target 3. Targets 1, 2 and 4 delay decision-making due to the uncertainty of battlefield information, and sufficient data need to be obtained by further reconnaissance for confirmation. It can be seen that the threat ranking results of the method in this paper are consistent with the objective classification results, which proves the effectiveness of the method in the decision-making process of air combat.

6.2

Experimental comparison and analysis

• (1) Method validity

  Figure 2 shows the comparison results of the method proposed in this paper, the current time t₃ and the multi-criteria compromise solution sequencing method (VIKOR)¹⁸ respectively. TOPSIS is used for static evaluation at time t₃, and VIKOR is used to sort multi-time target threats.

  As can be seen from Figure 2b, the result of ordering at time t₃ is 3>2>1>4>5, and the closeness degree of each target is {0.498 0.499 0.684 0.492 0.351} respectively, which is roughly the same as the evaluation result of the method proposed in this paper. From Figure 2c, the ranking result using VIKOR is 3>4>2>1>5, which is quite different from the result of the method proposed in this paper.

  All three methods listed target 3 as the highest threat and target 5 as the lowest threat. By analyzing the parameters at each time, it can be seen that the speed of target 3 keeps increasing and its height keeps decreasing, that is, it is in the accelerated subduction state towards us. The combat ability and anti-interference ability of target 5 are both minimum, and they are in the state of accelerating and climbing away from our side. The ranking of the two is consistent with objective cognition.

  Target 1 has outstanding combat capability, the strongest anti-interference capability, and its speed keeps increasing. Although it is far away from our side, it does not leave the combat radius, so the possibility of launching a surprise attack cannot be ruled out.

  Target 2 has the strongest combat capability and has a trend of approaching our side obviously. Although the speed is gradually decreasing, it is possible to launch a surprise attack, and the threat value should be constantly increased.

  Although target 4 tends to approach our side, its height remains unchanged, so the attack intention is unknown. Its threat level should be lower than target 1 and 2. Obviously, there are some problems in the ranking result of VIKOR.

  At the same time, only according to the ranking results obtained at the current time, the difference between the threat values of targets 1, 2 and 4 is too small, especially that targets 1 and 2 are approximately equal, while the difference between targets 1, 2 and 4 in this method is larger than that in (b), so the differentiation degree is better. In addition, the threat value of target 2 keeps increasing and the combat capability is the strongest, the combat intention of target 4 is not obvious, and the threat value of target 4 should be significantly smaller than that of Target 2 by objective analysis. Therefore, there are also certain defects in the threat ranking only by time.

  The results show that the proposed method is consistent with objective analysis and general cognition, which verifies the effectiveness of the proposed method. At the same time, the method presented in this paper is more distinguishable than the time-only threat ranking method and more conducive to decision execution.

• (2) Comparison of different time weight determination methods: Poisson inverse distribution

  In order to verify the validity of the time weight determination method in this paper, it is compared with poisson inverse distribution method⁷. Poisson inverse distribution method selects the observation data at the current (p moment) and the previous p-1 moment for comprehensive evaluation, and assigns weights to the time series, where the weight of time t_k is

  

  where ϕ=1.5, then η_k = {0.200,0.267,0.533}, other parameters are unchanged, and the target closeness degree under this time weight is calculated as {0.536 0.510 0.660 0.475 0.374}, the ranking result of target threat is 3>1>2>4>5. The comparison result with the method in this paper is shown in Figure 3.

  It can be seen from Figure 3 that although the threat ranking results under the two time weight determination methods are consistent, the method presented in this paper has good discrimination. Taking target 1 and target 2 as examples, it can be seen from Figure 3 that the threat value of target 1 at moment t₁ is much greater than that of target 2, and the threat value of target 1 and target 2 at moment t₂, t₃ is not much different from each other. However, due to the large deviation and entropy value of the threat assessment matrix at moment t₁, the weight of time t₁ in this paper increases when calculating the time weight. As a result, the influence of the threat value of each target at the time t₁ increases, which indirectly improves the differentiation of target 1 and 2 and improves the effect of threat ranking.

• (3) Dynamic multi-attribute threat assessment is compared with single-moment threat assessment: single-moment & multimoment.

Figure 3.

The target threat assessment value of different time weight determination methods.

The attribute weight is determined by using the exponential correlation degree method, and the threat is sorted at different times respectively. The results are shown in Table 2.

Table 2.

Ranking results of target threats at each time.

Time	Target threat value	Ranking result
target1	target 2	target 3	target 4	target 5
t1	0.720	0.508	0.522	0.532	0.501	1>4>3>2>5
t2	0.530	0.497	0.686	0.454	0.406	3>1>2>4>5
t3	0.489	0.550	0.778	0.480	0.397	3>2>1>4>5
Multi time	0.572	0.521	0.672	0.488	0.432	3>1>2>4>5

The analysis results show that the ranking results of target threats at different times are different, and the difference between target and other times is particularly significant. By analyzing each target separately, it can be seen that the threat value of target 1 and target 5 decreases continuously over time. The threat value of target 3 keeps increasing and the increase is significant, indicating that the threat is getting bigger and bigger. The threat value of target 2 and 4 firstly decreased and then increased with time. At the same time, it should be noted that the threat value of target 2 eventually exceeds that of target 1, which results in different results when dynamic assessment and threat sequencing are adopted.

7.

CONCLUSION

In this paper, an exponential correlation dynamic threat assessment method is proposed for multi-type heterogeneous data. Firstly, the multi-type heterogeneous data is mapped to D-U space based on set pair theory to achieve isomerization while preserving the uncertainty of information. Secondly, aiming at the problem that the resolution of correlation degree is not high when the traditional method is used to determine the attribute weight, the method of exponential function correlation degree is proposed, which can effectively improve the existing problem and improve the distinction between attributes. Finally, the time weight determination method based on deviation and grey entropy is adopted to optimize the existing Poisson inverse distribution method, and the effectiveness of the proposed method is verified by an example.