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The local predictability of underwater acoustic signal plays an important role in underwater acoustic signal processing, and it is the basis of nonstationary signal detection. Wavelet neural network model, with the advantages of both wavelet analysis and artificial neural network, makes full use of the time-frequency localization characteristics of wavelet analysis and the self-learning ability of artificial neural network; however, this model is prone to fall into local minima or creates convergence. To overcome these disadvantages, a new hybrid model based on fruit fly optimization algorithm (FOA) and wavelet neural network (WNN) is proposed in this paper. The FOA-WNN prediction model is constructed by optimizing the weights and thresholds of wavelet neural network, and the model is applied to underwater acoustic signal prediction. The experimental results show that the FOA-WNN prediction model has higher prediction accuracy and smaller prediction error, compared with wavelet neural network prediction model and BP neural network prediction model.

An important feature of the underwater acoustic signal is local predictability. This feature plays an important role in underwater acoustic signal processing and is the basis for solving nonstationary signal detection [

Wavelet neural network is combined with the characteristics of artificial neural network and wavelet analysis, and it has the advantages of self-learning ability and localization of wavelet transform. Therefore, it is widely used in nonlinear and nonstationary time series prediction and can effectively solve the local minimum problem [

Fruit fly optimization algorithm (FOA) is a method for deriving global optimum based on foraging behavior of fruit flies [

(1) Initialize the population position. Set the initial location of the flies group to (

(2) Set the random direction and distance of individual.

(3) Because the group does not know the location of the optimal solution, the distance between the group and the origin (Dist) is calculated firstly, and then the smell concentration judgment value (S) can be calculated as follows, which is the reciprocal of the distance.

(4) By substituting

(5) Find the individual with the highest Smell among all fly groups.

(6) Keep the best concentration value and position coordinate (

(7) Repeat steps (2)–(5) to enter iterative optimization, and determine whether the taste concentration value at the current moment is better than the iterative flavor concentration value at the previous moment. If yes, execute step (6).

Wavelet neural network (WNN) is a kind of feed-forward neural network which combines BP neural network and wavelet transform. It is network framework is the topological structure of BP neural network, and it takes the wavelet basis function as the hidden layer excitation function [

The structure of wavelet neural network.

In Figure

Calculate the output of the hidden layer

The output of the neural network

By adjusting the parameters of the wavelet neural network by error, the error between the output of the wavelet neural network and the ideal result is

The correction items of the weights of the neural network such as

One training of wavelet neural network refers to the completion of one forward propagation and one reverse error correction. When the neural network is continuously trained and the output parameters meet the specified requirements, the wavelet neural network will stop training. The data is entered into trained wavelet neural network, and then the output signal will be got. The training process of wavelet neural network is shown in Figure

The training process of wavelet neural network.

The weights and thresholds of wavelet neural network are optimized by FOA, and nonlinear time series are normalized firstly. Define the initial position of the fruit fly

Load the data which is divided into training groups and testing groups, and initial processing, respectively.

Initialize the population and iteration count of the fly optimization algorithm, the location of fruit flies, the random direction, and the distance of individual search.

The optimal value is calculated by the fly optimization algorithm.

The optimized weights and thresholds are substituted into the constructed wavelet neural network for training.

The performance of the trained wavelet neural network is tested by using testing groups, and the error is calculated.

The block diagram of FOA-WNN prediction model is shown in Figure

The block diagram of FOA-WNN prediction model.

In this paper, we use normalized pretreatment ship radiated noise signal where the sampling rate is 20 kHz, and there are a total of 2048 data points. 1380 points are randomly selected as experimental data and the time domain waveform is shown in Figure

Waveform of underwater acoustic signal.

The 1380 data points are divided into prediction data and test data. The first four observed values are used as input vector, and the fifth observed value is used as output vector. 1380 data points can be divided into 1372 sets of data, wherein the former 996 sets of data are used as the test data, and the latter 376 sets of data are used as the forecast data. Then the parameters of wavelet neural network are set up. The best hidden layer nodes and the number of layers are determined by trial-and-error method. The number of nodes in the input layer is 4, the number of nodes in the hidden layer is 15, and the number of nodes in the output layer is 1. That is, the structure of the wavelet neural network is 4-15-1. The learning probabilities are 0.01 and 0.001, respectively, and the number of iterations is 200.

Fruit fly population location (

FOA-WNN prediction results.

The red line in Figure

The prediction error of FOA-WNN.

The different number of hidden layers is set. Then the same experiment is done. Lastly, the change of program’s running time in the case of different hidden layers is recorded. The running time for different hidden layers is shown in Table

The running time of different hidden layer number.

The number of hidden layers | Time (s) |
---|---|

1 | 4.93 |

5 | 6.22 |

10 | 7.48 |

15 | 8.65 |

20 | 9.42 |

25 | 10.74 |

30 | 12.07 |

35 | 13.05 |

40 | 14.39 |

45 | 15.45 |

50 | 16.77 |

In order to facilitate comparison, BP neural network and WNN prediction model are used to predict the same time series of underwater acoustic signal. Predicted results of underwater acoustic signal for each model are shown in Figure

Predicted results of underwater acoustic signal for each model.

Local predicted result of underwater acoustic signal for each model.

In order to verify the prediction result, the RMS error (RMSE) and mean absolute error (MAE) are used to estimate the result of prediction model. The RMSE can be used to measure the deviation between the observed value and the true value, which reflects the discrete degree of the data. The smaller the RMSE is, the smaller the deviation is. The mean absolute error is a good reflection of the actual situation of the prediction error. The smaller the MAE is, the more accurate the data fitting is.

The RMS error (RMSE) is

The mean absolute error (MAE) is

Error comparison of RMSE and MAE in three models.

Models | RMSE | MAE |
---|---|---|

BP | 0.0701 | 0.0534 |

WNN | 0.0573 | 0.0468 |

FOA-WNN | 0.0498 | 0.0387 |

As shown in Table

In order to overcome these disadvantages of wavelet neural network model being prone to fall into local minimum or convergence problems, a new hybrid model based on fruit fly optimization algorithm and wavelet neural network is proposed. The FOA-WNN prediction model is constructed by optimizing the weights and thresholds of wavelet neural network, and it is applied to underwater acoustic signal prediction. The experimental results show that the proposed model can improve the prediction precision compared with wavelet neural network prediction model and BP neural network prediction model in predicting the same underwater acoustic signal. It can also be applied to other fields after conducting some modification and has high application value.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (no. 51709228).