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Target threat assessment is a key issue in the collaborative attack. To improve the accuracy and usefulness of target threat assessment in the aerial combat, we propose a variant of wavelet neural networks, MWFWNN network, to solve threat assessment. How to select the appropriate wavelet function is difficult when constructing wavelet neural network. This paper proposes a wavelet mother function selection algorithm with minimum mean squared error and then constructs MWFWNN network using the above algorithm. Firstly, it needs to establish wavelet function library; secondly, wavelet neural network is constructed with each wavelet mother function in the library and wavelet function parameters and the network weights are updated according to the relevant modifying formula. The constructed wavelet neural network is detected with training set, and then optimal wavelet function with minimum mean squared error is chosen to build MWFWNN network. Experimental results show that the mean squared error is

With the development of science and technology, the requirement of information is increasingly improving in modern warfare. To adapt to this change, many countries have begun the research of multisensor information fusion from the 1970s. After years of research, the United States, Britain, and other military powers have developed a number of information fusion systems which can be used for combat. Target threat assessment belongs to the third level in information fusion model and is a kind of high-level information fusion. The target threat assessment is the essential basis for the allocation of force and fire in C4ISR system.

The traditional methods to solve threat assessment are Bayesian inference [

Firstly proposed by Grossman and Morlet in the 1980s, wavelet theory [

Wavelet function is constructed through a series of basic transformation with a mother wavelet function. Not all functions can be used as wavelet mother function if a wavelet function is to be available and then develop into a good wavelet transform function, it must satisfy many conditions. Therefore, it is difficult to find the practical wavelet function. In the practical wavelet functions, some of them do not have expressions.

Let

As the translation factor

Wavelet transform calculates the inner product between the signal

Equivalent expression in time domain is given as

The conclusion can be drawn from (

Wavelet transform has time-frequency localization property and focal features and neural network (NN) has self-adaptive, fault tolerance, robustness, and strong inference ability. How to combine the advantages of wavelet transform and NN to solve practical problems has been one of the hot spots. So-called wavelet neural network (WNN) or wavelet network (WN) is a variety of two techniques and inherits the advantages of the neural network and wavelet transformation. Proposed by Q. Zhang in 1992 [

For WNN, its topology is based on BP network; the transfer function of hidden layer nodes is the mother wavelet function; and the network signal is prior to transmission while error is backpropagation in the training process. The network topology is shown in Figure

Topology of wavelet neural network.

For the input signal sequence

Currently, the choice of mother wavelet functions has not yet formed a standard theory; commonly used wavelet functions are Morlet, Haar, Daubechies (dbN), Symlet (symN), Meryer, Coiflet, Biorthogonal wavelets, and so on.

The output of the output layer is calculated as

For WNN, the updating weight algorithm is similar to BP network; the gradient method is used to update mother wavelet function parameters and connection weights between the layers, making the prediction output closer and closer to the desired output. The weights of WNN and the parameters of wavelet function are updated as follows.

(1) Calculating the prediction error of WNN

(2) Updating the weights of WNN and the parameters of wavelet function according to the prediction error

The process of training WNN is as follows

Data preprocessing: first, the original data is quantified and normalized, and then the data is divided into training set and testing set for network training and testing, respectively.

Initializing WNN: connection weights

Training network: input the training set into WNN, compute network predicted output values, and calculate the error

Updating the weights: update mother wavelet function parameters and network weights according to the prediction error

If the results satisfy the given conditions, use the testing set to test the network, otherwise, return to Step

MWFWNN will be provided in this section.

Initializing: initialize mother wavelet function library

Choosing the best mother wavelet function: for each mother wavelet function,

Update the weights and parameters of wavelet function

if

Constructing MWFWNN using

Testing constructed MWFWNN network in Step

Analyzing results.

Strictly speaking, threat assessment is an NP-hard problem, belonging to the third level in the JDL information fusion model. Target threat assessment needs to consider many factors (such as geography, weather, enemy, etc.), and the relation among the various factors is not a simple linear combination and it is difficult to determine a function between the target threat value and various factors. Therefore, we must consider various factors and their relationships when studying the threat assessment. However, we consider the following six factors in general: target type, target speed, target heading angle, target height, and target distance. We will test the performance of MWFWNN using these factors in this paper.

We mainly consider the following six key factors when studying the target threat assessment in the paper:

Target Type: large targets (such as fighter-bombers), small targets (such as stealth aircraft, cruise missiles), and helicopters;

Target heading angle: such as 22°, 26°, and 6°;

Target speed: such as 100 m/s, 500 m/s, and 220 m/s;

Target height: such as very low, low, medium, and high;

Target interference: such as strong, medium, and weak;

Target distance: such as 100 km, 110 km, and 220 km.

We design MWFWNN model according to the data characteristics. Because the data is 6-dimensional, and the output is 1-dimensional, the structure of WNN is 6-12-1. Firstly, we input six indicators that are the target type, target speed, target heading angle, target interference, target height, and the distance to the input layer. The hidden layer nodes are formed by the wavelet function, and the output layer outputs predicted target threat assessment value under the current indicators. On the basis of the above analysis, we construct the target threat assessment model based on MWFWNN with these six selected indicators, and its architecture is shown in Figure

Architecture for the model of target threat assessment based on MWFWNN.

In this section, we will test the target threat assessment model using MWFWNN proposed in Section

Part of the data used in our work is shown in Table

Target Type: helicopter, large target (such as fighter-bombers), and small targets (such as stealth aircraft, cruise missiles), are quantified by 3, 5, and 8, respectively;

Target interference: strong, medium, weak, and no are quantified by 8, 6, 4, and 2, respectively;

Target height: very low, low, medium, and high are quantified 8, 6, 4, and 2, respectively;

Target speed: 0 m/s~1800 m/s equal interval (200 m/s) is quantified by 9 to 1.

Target heading angle:

Target distance: 0 km~450 km equal interval (50 km) is quantified by 9 to 1.

Determining the target output: firstly normalize the various factors from air combat situation and then put them into the WMF_WNN proposed in Section

Part of data.

No. | Type | Velocity (m/s) | Heading angle (°) | Inference | Height | Distance (km) | Threat value |
---|---|---|---|---|---|---|---|

1 | Large | 450 | 8 | Medium | Low | 300 | 0.5843 |

2 | Large | 400 | 3 | Strong | High | 100 | 0.5707 |

3 | Large | 450 | 16 | Medium | Low | 200 | 0.5333 |

4 | Large | 800 | 4 | Strong | High | 100 | 0.6895 |

5 | Large | 800 | 12 | Strong | Low | 320 | 0.6896 |

6 | Small | 530 | 6 | Strong | Medium | 230 | 0.6056 |

7 | Small | 650 | 8 | Strong | Medium | 200 | 0.7425 |

8 | Small | 700 | 12 | Strong | Low | 320 | 0.7336 |

9 | Small | 750 | 15 | Medium | Very low | 400 | 0.7541 |

10 | Small | 640 | 18 | Strong | Medium | 280 | 0.6764 |

11 | Helicopter | 90 | 12 | Weak | Very low | 320 | 0.3937 |

12 | Helicopter | 110 | 3 | No | Medium | 100 | 0.3927 |

13 | Helicopter | 100 | 9 | No | Medium | 260 | 0.3351 |

14 | Helicopter | 120 | 15 | No | Low | 160 | 0.3586 |

15 | Helicopter | 80 | 6 | Weak | High | 180 | 0.3471 |

After quantifying the data, we can normalize the training set and testing set using the following expression:

In this paper, we implement the MWFWNN algorithm by MATLAB R2009a with the CPU Pentium (R) 4 3.06 GHz, 1 G memory (

Because many mother wavelet functions have no specific expression, we cannot work out their derivatives. Therefore, in this work, we only use the following seven mother wavelet functions to bulid a library waveFunciton. Their expressions are as follows:

Haar wavelet function:

Gaussian wavelet function:

Morlet wavelet function:

Mexican Hat (Mexihat) wavelet function:

Shannon wavelet function:

Meyer wavelet function (approximate formula):

Wavelet function GGW constructed by the authors:

We use the following parameters to initialize the network: the input layer nodes

We construct wavelet neural network using each wavelet function in wavelet function library and then input training set into network and train the network. The MSEs are as follows (from small to large): Morlet < Mexihat < GGW < Haar < Gaussian < Shannon < Meyer, as shown in Table

Predicting results of different mother wavelet function.

Wavelet function | Haar | Gaussian | Morlet | Mexihat | Shannon | Meyer | GGW |
---|---|---|---|---|---|---|---|

MSE | 2.10 × 10^{−2} |
1.03 × 10^{54} |
1.23 × 10^{−3} |
1.27 × 10^{−3} |
1.11 × 10^{55} |
3.90 × 10^{57} |
1.60 × 10^{−2} |

Running time (s) | 4.74 | 4.76 | 4.85 | 4.88 | 4.84 | 5.24 | 4.88 |

Similar to MWFWNN, WNN, BP network, and support vector machine (SVM) can be used to solve the target assessment. As the choice of the support vector machine parameters

The structure of wavelet neural network and BP neural network is 6-12-1 according to the characteristics of the data used. Where the WNN and other parameters are setting as shown in Section

Threat assessment is predicted by the trained WNN, BP network, and PSO_SVM and their MSE are

Predicting results of BP, PSO_SVM, MWFWNN, and WNN.

Neural network | BP | PSO_SVM | MWFWNN | WNN |
---|---|---|---|---|

MSE | 9.39 × 10^{−3} |
4.80 × 10^{−3} |
1.23 × 10^{−3} |
4.01 × 10^{−3} |

Running time (s) | 1.86 | 10.6 | 4.88 | 5.10 |

Prediction error of BP, PSO_SVM, WNN, and MWFWNN is shown in Figure

Result of target threat assessment based on WNN, BP, MWFWNN and PSO_SVM.

Based on requirements for quickly processing information in the modern information war, aiming to the characteristics of threat assessment in data fusion functional model, we adopt MWFWNN to solve the threat assessment under the comprehensive consideration of various factors which influence the threat degree. After constructing wavelet function library with 7 wavelet functions, we get the best performance wavelet function Morlet as mother wavelet function to construct MWFWNN network and its result is compared with the WNN, BP, and PSO_SVM networks. Simulation results show that, the MSE of MWFWNN is 1.23 × 10^{−3}, which is far better than WNN (4.01 × 10^{−3}), BP network (9.39 × 10^{−3}), and PSO_SVM (4.80 × 10^{−3}), achieving the desired goal. In our future work, we will further expand the scale of wavelet function library to find more suitable wavelet function to solve the threat assessment and other problems.