TOPSIS missile target selection method supported by the posterior probability of target recognition

Zhang Jianqiang; Li Jialin; Li Yan

Dziennik i szczegóły wydania

Applied Mathematics and Nonlinear Sciences

Informacje o czasopiśmie

Format: Czasopismo
eISSN: 2444-8656
Pierwsze wydanie: 01 Jan 2016
Częstotliwość wydawania: 2 razy w roku
Języki: Angielski

Addressing the problem of being easily interfered by false targets such as chaff and corner reflector when selecting missile targets, a multi-feature multi-model target selection method based on the technique for order preference by similarity to ideal solution (TOPSIS) is proposed if only single-target characteristics such as target position or radar cross section (RCS) size is used in the selection of missile targets. The target selection problem under the comprehensive utilisation of multiple features is regarded as a problem in decision-making based on multiple attributes. The BP network based on the target RCS frequency domain statistical features, the radial basis network based on the target polarisation feature and the radial size feature based on high-resolution range profile (HRRP) are realised. The comprehensive utilisation of C-support vector machine (SVM) and other target recognition methods and the comprehensive sorting of target selection, to a certain extent, enable anti-ship missiles to make the correct target selection more accurately. The simulation results display that the effective selection of ship targets can be achieved in the case of passive interference by surface ships.

Keywords

vector machines
neural networks
TOPSIS
target recognition

Artykuł

Introduction

Given a certain capability of target selection, the force-distribute-code method could effectively perform target selection and force distribution during the collective attack of anti-ship missiles at the enemy formation, which prevents the simultaneous attack of several missiles at the target with larger radar cross section (RCS). Modern naval battle involves a complicated electromagnetic environment, causing a noticeable role of passive interferences including chaff and corner reflector in the protection against missile attacks. The problem of being easily interfered by false targets such as chaff and corner reflector is encountered if only single-target characteristics such as target position or RCS size are used in the selection of missile targets. Because of the rapid development, the terminal guidance radar technology has not only revealed the RCS frequency sequence of targets, but also collected the high-resolution features of targets including polarisation and radial length of high-resolution range profile (HRRP). Therefore, a variety of classifiers (e.g. BP neural network, RBF, C-support vector machine (SVM), etc.) could be established for different target features in the process of recognition. The recognition results could be further converted into the posterior probability under maximum likelihood estimation by employing sigmoid function mapping and other methods [1]. Subsequently, the problem of anti-ship missile target recognition and selection under the background of real and false targets is equal to a multi-attribute decision-making problem. The posterior probabilities generated by diverse classifiers with different features are utilised to construct the decision-making matrix. Eventually, the technique for order preference by similarity to ideal solution (TOPSIS) is employed for decision-making with multiple attributes to comprehensively sequence the types of target and support for decision-making on the preset targets of anti-ship missiles.

Technique for order preference by similarity to ideal solution

TOPSIS is a method for making a decision involving multiple attributes [2]. It is mainly employed to resolve the problem of decision-making on the best alternative with multiple attributes or indexes [3]. Multiple attribute indexes may restrict or contradict each other, so that there is normally not the best solution, but it is possible to obtain ideal solution and negative ideal solution. TOPSIS utilises such two solutions simultaneously to sequence the alternatives in terms of quality. In the meanwhile, these two solutions are used for the following reason: when the distance of two alternatives to the ideal solution is the same, priority is given to the alternative that is farthest away from the negative ideal solution [4].

TOPSIS has the basic principle as shown in Figure 1. Among them, f₁ and f₂ are two attribute indexes of alternatives. There are six alternatives from x₁ to x₆. For these alternatives, ideal solution is x^* and negative ideal solution is x⁰. As shown in Figure 1, alternatives x₄ and x₅ have the same distance to the ideal solution x^*, so that their distance from the negative ideal solution x⁰ should be further calculated. Alternative x₄ is better than alternative x₅ since the former has the longest distance from the negative ideal solution x⁰ than the latter.

Ideal solution and negative ideal solution schemes

Calculation methods for posterior probability of recognition results from classifiers

3.1

Calculation method for posterior probability of recognition results from feed-forward neural network

Ruck, Richard and Ken-ichi et al. [5, 6] have proved that, if the class coding of ‘1 out of C’ is adopted by network output, and minimum mean-square error is taken as the training target, the outputs of all feed-forward neural networks correspond to the posterior probabilities of training sample categories. In this case, ‘1 out of C’ means that the expected output of feed-forward neural network must be: (1) $d_{i} (x) = {\begin{matrix} 1 & x \in w_{i} \\ 0 & x \notin w_{i} \end{matrix} i = 1, 2, \dots, c$ {d_i}(x) = \left\{{\matrix{1 & {x \in {w_i}} \cr 0 & {x \notin {w_i}} \cr}} \right.\quad i = 1,2, \cdots,c

If neural network estimation is quite accurate, the sum of its outputs for each class should be 1. If inaccuracy is taken into account, the sum will not be 1. In this case, normalisation should be conducted as follows: (2) $d_{i}^{'} (x) = \frac{d_{i} (x)}{\sum_{i = 1}^{c} d_{i} (x)} i = 1, 2, \dots, c$ d_i^{'}(x) = {{{d_i}(x)} \over {\sum\limits_{i = 1}^c {d_i}(x)}}\quad i = 1,2, \cdots,c

3.2

Calculation method for posterior probability of recognition results from support vector machine (SVM)

For the training sample (z_i,y_i), i = 1,2,3,…, l, z_i ∈ R^d, y ∈ {−1,1}, SVM has the judgement function as follows: (3) $sgn (\sum_{i = 1}^{l} y_{i} a_{i} (K (z_{i}, z) + b)) = sgn (f)$ {\rm sgn} \left({\sum\limits_{i = 1}^l {y_i}{a_i}(K({z_i},z) + b)} \right) = {\rm sgn}(f)

Clearly, SVM could not output the posterior probability, but the classification result of sample data, that is, 1 or −1. Platt [7, 8] utilised the sigmoid function to realise the mapping between SVM output and probability. The transformation is in the form of following equation: (4) $P (y = 1 | f) = \frac{1}{1 + exp (A f + B)}$ P\left({y = 1|f} \right) = {1 \over {1 + \exp (Af + B)}} where f represents the classification output result of SVM, and P(y = 1| f) indicates the classification correctness probability under the output f. Moreover, A and B represent the parameters of the function. The optimisation could be achieved by solving the maximum likelihood: (5) $F (z) = min_{z = (A, B)} (- \sum_{i = 1}^{l} t_{i} log (p_{i}) + (1 - t_{i}) log (1 - p_{i}))$ F(z) = \mathop {\min}\limits_{z = (A,B)} \left({- \sum\limits_{i = 1}^l {t_i}\log \left({{p_i}} \right) + \left({1 - {t_i}} \right)\log \left({1 - {p_i}} \right)} \right) where (6) $p_{i} = \frac{1}{1 + exp (A f (z_{i}) + B)}$ {p_i} = {1 \over {1 + \exp \left({Af({z_i}) + B} \right)}} (7) $t_{i} = {\begin{matrix} \frac{N_{+} + 1}{N_{+} + 2} & if y_{i} = 1 \\ \frac{1}{N_{-} + 2} & if y_{i} = - 1 \end{matrix}$ {t_i} = \left\{{\matrix{{{{{N_ +} + 1} \over {{N_ +} + 2}}} & {{\rm{if}}\;{y_i} = 1} \cr {{1 \over {{N_ -} + 2}}} & {{\rm{if}}\;{y_i} = - 1} \cr}} \right. where i = 1,2,3,…, l N₊,N₋; represents the number of positive samples and negative samples respectively in the model.

TOPSIS target selection method with support of target recognition posterior probability

The principles of TOPSIS target selection method with support of target recognition posterior probability is presented in the Figure 2. RCS frequency statistical feature, polarisation feature and the HRRP mirror size are selected for classified target recognition by employing BP neural network, RBF network, and SVM, respectively. The recognition results are transformed into the posterior probability as detailed in Sections 3.1 and 3.2. Subsequently, TOPSIS is adopted to comprehensively evaluate and determine the sequence of target types. The steps are detailed as follows:

Step 1. Select the suitable classifier based on different target features for the target recognition, and obtain the respective recognition results.

Step 2. Calculate the posterior probability for each type of model recognition result as detailed in Section 3.

Step 3. Build the weighted standard matrix X_c = {x_ij}.

Let w_c = [w₁, w₂,…, w_n]^T, so that

x_ij = w_j·p_ij, i = 1,…, m, j = 1,…, n

where p_ij stands for the posterior probability of the target with the jth feature recognised into the ith class.

Step 4. Obtain the ideal solution x^* and the negative ideal solution x⁰.

The posterior probability of target recognition classification is a benefit attribute, so that the ideal solution $x_{j}^{*} = max_{i} (x_{i j})$ x_j^* = \mathop {\max}\limits_i \left({{x_{ij}}} \right) and the negative ideal solution $x_{j}^{0} = min_{i} (x_{i j})$ x_j^0 = \mathop {\min}\limits_i \left({{x_{ij}}} \right) .

Step 5. Calculate the distances of sequences alternative for each class to the ideal solution and each class to the negative ideal solution.

The distance of target class x_i from the ideal solution is given by $d_{i}^{*} = \sqrt{\sum_{j = 1}^{n} {(x_{ij} - x_{j}^{*})}^{2}}, i = 1, \dots, m target class, j = 1, \dots, 6 attribute$ d_i^* = \sqrt {\sum\limits_{j = 1}^n {{({x_{ij}} - x_j^*)}^2}},\;i = 1, \ldots,m\;{\rm{target}}\;{\rm{class}},\;j = 1, \ldots,6\;{\rm{attribute}}

The distance from target class x_i to the negative ideal solution is given by $d_{i}^{0} = \sqrt{\sum_{j = 1}^{n} {(x_{ij} - x_{j}^{0})}^{2}}, i = 1, \dots, m$ d_i^0 = \sqrt {\sum\limits_{j = 1}^n {{({x_{ij}} - x_j^0)}^2}},\;i = 1, \ldots,m

Step 6. Calculate and determine the comprehensive evaluation index by using $C_{i}^{*} = d_{i}^{0} / (d_{i}^{0} + d_{i}^{*}), i = 1, \dots, m$ C_i^* = d_i^0/\left({d_i^0 + d_i^*} \right),\;i = 1, \ldots,m

Step 7. Sequence the alternatives of target class based on $C_{i}^{*}$ C_i^* in a descending order.

Principle diagram of TOPSIS selection method with support of recognition posterior probability. TOPSIS, technique for order preference by similarity to ideal solution

Simulation experiment

Based on the analysis in Reference [9], a warship normally places chaffs in the form of missile axis symmetric ‘four targets’ when it detects an incoming anti-ship missile, as shown in Figure 3 [10]. In the meantime, it also rapidly deploys corner reflector, and then moves fast to create a complicated interference situation involving many types of false targets, which confuses the incoming missile and makes it select a false target. Under such situation, if the target selection method of the anti-ship missile is defined as ‘selecting the right side’, the anti-ship missile is very likely to select the chaffer or corner reflector when only RCS size is taken as the basis for type judgement.

It is assumed that the terminal guidance radar could gather the RCS frequency statistical feature, polarisation feature and HRRP radial size feature of targets. TOPSIS is employed to comprehensively sequence the classification results of these features, in a bid to rule out the interferences by chaff and corner reflector. As shown in Table 1, the decision matrix reveals that if single RCS frequency domain feature is employed, false target 1 (corner reflector) is mistakenly considered as the real warship; If single BP neural network mode is employed, false target 5 (chaff cloud) is mistaken as the real warship; if single polarisation feature or single RBF radial basis network model is employed, false target 1 (corner reflector) is mistakenly considered as the real warship; if single radial size feature is employed, false target 2 (chaff cloud) is mistakenly considered as the real warship, and if single SVM model is employed, false target 3 (chaff cloud) is mistakenly identified as the real warship. When the multi-feature and multi-model comprehensive classification model based on TOPSIS is adopted, and the weighted standard matrix is set as w = [0.4 0.4 0.2]^T, the distance of sequence alternatives for each class to the ideal spot and the negative ideal spot is calculated, respectively as presented in Table 2. The sequence for decision on the target type is completely accurate, revealing the effectiveness of the proposed method.

Table 1

Decision matrix

Object	Target type	RCS frequency domain feature (BP)	Polarisation feature (RBF)	Radial size (SVM)
Real warship	Warship	0.4642037	0.607021	0.401691
	Chaff cloud	0.1075953	0.1135784	0.379228
	Corner reflector	0.428201	0.2794006	0.219081
False target 1 (corner reflector)	Warship	0.412431	0.4401453	0.1364145
	Chaff cloud	0.120652	0.146565	0.1177154
	Corner reflector	0.466917	0.4132897	0.7458701
False target 2 (chaff cloud)	Warship	0.312082	0.2197485	0.438642
	Chaff cloud	0.561797	0.6395	0.340408
	Corner reflector	0.126121	0.1407515	0.22095
False target 3 (chaff cloud)	Warship	0.317219	0.2038501	0.486033
	Chaff cloud	0.559468	0.664808	0.302723
	Corner reflector	0.123313	0.1313419	0.211244
False target 4 (chaff cloud)	Warship	0.316303	0.2121808	0.330903
	Chaff cloud	0.520689	0.647954	0.454883
	Corner reflector	0.163008	0.1398652	0.214214
False target 5 (chaff cloud)	Warship	0.477049	0.2406688	0.379228
	Chaff cloud	0.40926	0.63095	0.401691
	Corner reflector	0.113691	0.1283812	0.219081

RCS, radar cross section; SVM, support vector machine.

Table 2

Distance of each class to the ideal spot and negative ideal spot

Object	Target type	$d_{i}^{}$ d_i^	$d_{i}^{0}$ d_i^0	$C_{i}^{}$ C_i^
Real warship	Warship	0	0.6356	1.0000
	Chaff cloud	0.6092	0.1601	0.2082
	Corner reflector	0.3768	0.3610	0.4893
False target 1 (corner reflector)	Warship	0.6119	0.4143	0.4037
	Chaff cloud	0.7750	0.0000	0
	Corner reflector	0.0269	0.7653	0.9661
False target 2 (chaff cloud)	Warship	0.4884	0.2970	0.3781
	Chaff cloud	0.0982	0.6729	0.8726
	Corner reflector	0.6971	0.0000	0.0000
False target 3 (chaff cloud)	Warship	0.5207	0.3440	0.3978
	Chaff cloud	0.1833	0.6951	0.7913
	Corner reflector	0.7418	0.0000	0.0000
False target 4 (chaff cloud)	Warship	0.4970	0.2058	0.2928
	Chaff cloud	0.0000	0.6663	1.0000
	Corner reflector	0.6663	0.0000	0.0000
False target 5 (chaff cloud)	Warship	0.3909	0.4127	0.5135
	Chaff cloud	0.0678	0.6110	0.9001
	Corner reflector	0.6465	0.0000	0.0000

CST computing software is used with the CPU/GPU heterogeneous parallel computing method to determine the scattering characteristics of extremely large electric size targets including warship and corner reflection in the high-performance computing cluster. The calculation process in this paper involves eight computing nodes. Each computing node is provided with two 16-core and 64 GB CPUs, and the IB network with transmission speed up to 56 GB/s for data exchange. The parameters of CST simulation software are as follows: incident horizontal angle 30°, pitch angle 5.1°, central frequency 10 GHz, bandwidth 400 MHz, and 10,001 sampling points. After all, 178 pieces of data (including 89 pieces of training data and 89 pieces of test data) were obtained for the RCS frequency statistical feature, polarisation feature and radial size feature of warship, corner reflector and four chaff clouds. The classification recognition outcome is as shown in Table 3. TOPSIS has significantly improved the recognition accuracy up to 97.75%.

Table 3

The classification recognition outcome

Classification method	RCS frequency domain statistical feature	Polarisation feature	HRRP radial size feature	Comprehensive feature
BP neural network	69.66%	75.28%	82.02%	84.27%
Radial basis network (RBF)	74.16%	78.65%	86.52%	88.76%
SVM	80.09%	78.65%	87.64%	92.75%
TOPSIS comprehensive decision-making	–	–	–	97.75%

HRRP, high-resolution range profile; RCS, radar cross section; SVM, support vector machine; TOPSIS, technique for order preference by similarity to ideal solution.

In general, this method includes the recognition results of multiple target features for the comprehensive evaluation and decision-making, so that it could overcome the defects to some extent, that is, the target selection of missile is very easily interfered by false targets including chaff and corner reflector if the missile relies only on a single target feature such as target position or RCS size.

Conclusion

To address the problem of being easily interfered by false targets such as chaff and corner reflector when selecting missile targets, a TOPSIS missile target selection method with support of target recognition posterior probability is proposed if only a single target feature such as target position or RCS size is used for the selection of missile targets. In the proposed method, the problem of target selection with the comprehensive utilisation of multiple features is regarded as a decision-making problem involving multiple attributes. Three types of features, i.e. RCS frequency statistical feature, polarisation feature and HRRP mirror size are selected to conduct the classified target recognition with BP neural network, RBF network and SVM, respectively. Subsequently, the calculation methods for posterior probability are employed to further transform the recognition results into posterior probabilities. In the end, TOPSIS is adopted for the comprehensive evaluation to determine the sequence of target types. As revealed in the simulation results, the ship targets can be effectively selected in the case of passive interference by surface ships.

Ilustracje i tabele

The classification recognition outcome

Classification method	RCS frequency domain statistical feature	Polarisation feature	HRRP radial size feature	Comprehensive feature
BP neural network	69.66%	75.28%	82.02%	84.27%
Radial basis network (RBF)	74.16%	78.65%	86.52%	88.76%
SVM	80.09%	78.65%	87.64%	92.75%
TOPSIS comprehensive decision-making	–	–	–	97.75%

Distance of each class to the ideal spot and negative ideal spot

Object	Target type	di* d_i^*	di0 d_i^0	Ci* C_i^*
Real warship	Warship	0	0.6356	1.0000
	Chaff cloud	0.6092	0.1601	0.2082
	Corner reflector	0.3768	0.3610	0.4893
False target 1 (corner reflector)	Warship	0.6119	0.4143	0.4037
	Chaff cloud	0.7750	0.0000	0
	Corner reflector	0.0269	0.7653	0.9661
False target 2 (chaff cloud)	Warship	0.4884	0.2970	0.3781
	Chaff cloud	0.0982	0.6729	0.8726
	Corner reflector	0.6971	0.0000	0.0000
False target 3 (chaff cloud)	Warship	0.5207	0.3440	0.3978
	Chaff cloud	0.1833	0.6951	0.7913
	Corner reflector	0.7418	0.0000	0.0000
False target 4 (chaff cloud)	Warship	0.4970	0.2058	0.2928
	Chaff cloud	0.0000	0.6663	1.0000
	Corner reflector	0.6663	0.0000	0.0000
False target 5 (chaff cloud)	Warship	0.3909	0.4127	0.5135
	Chaff cloud	0.0678	0.6110	0.9001
	Corner reflector	0.6465	0.0000	0.0000

Decision matrix

Object	Target type	RCS frequency domain feature (BP)	Polarisation feature (RBF)	Radial size (SVM)
Real warship	Warship	0.4642037	0.607021	0.401691
	Chaff cloud	0.1075953	0.1135784	0.379228
	Corner reflector	0.428201	0.2794006	0.219081
False target 1 (corner reflector)	Warship	0.412431	0.4401453	0.1364145
	Chaff cloud	0.120652	0.146565	0.1177154
	Corner reflector	0.466917	0.4132897	0.7458701
False target 2 (chaff cloud)	Warship	0.312082	0.2197485	0.438642
	Chaff cloud	0.561797	0.6395	0.340408
	Corner reflector	0.126121	0.1407515	0.22095
False target 3 (chaff cloud)	Warship	0.317219	0.2038501	0.486033
	Chaff cloud	0.559468	0.664808	0.302723
	Corner reflector	0.123313	0.1313419	0.211244
False target 4 (chaff cloud)	Warship	0.316303	0.2121808	0.330903
	Chaff cloud	0.520689	0.647954	0.454883
	Corner reflector	0.163008	0.1398652	0.214214
False target 5 (chaff cloud)	Warship	0.477049	0.2406688	0.379228
	Chaff cloud	0.40926	0.63095	0.401691
	Corner reflector	0.113691	0.1283812	0.219081

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