The prediction of underwater acoustic signal is the basis of underwater acoustic signal processing, which can be applied to underwater target signal noise reduction, detection, and feature extraction. Therefore, it is of great significance to improve the prediction accuracy of underwater acoustic signal. Aiming at the difficulty in underwater acoustic signal sequence prediction, a new hybrid prediction model for underwater acoustic signal is proposed in this paper, which combines the advantages of variational mode decomposition (VMD), artificial intelligence method, and optimization algorithm. In order to reduce the complexity of underwater acoustic signal sequence and improve operation efficiency, the original signal is decomposed by VMD into intrinsic mode components (IMFs) according to the characteristics of the signal, and dispersion entropy (DE) is used to analyze the complexity of IMF. The subsequences (VMD-DE) are obtained by adding the IMF with similar complexity. Then, extreme learning machine (ELM) is used to predict the low-frequency subsequence obtained by VMD-DE. Support vector regression (SVR) is used to predict the high-frequency subsequence. In addition, an artificial bee colony (ABC) algorithm is used to optimize model performance by adjusting the parameters of SVR. The experimental results show that the proposed new hybrid model can provide enhanced accuracy with the reduction of prediction error compared with other existing prediction methods.

Underwater acoustic signal processing is one of the most active disciplines in the field of ocean and information [

Liang et al. [

Many scholars apply signal features to various problems. Hossen et al. [

As mentioned above, some prediction methods of underwater acoustic signal such as Volterra model [

The main contents of this paper are as follows: in Section

VMD is a typical instantaneous frequency analysis method proposed by Dragomiretskiy and Zosso [

In the following equation, each IMF obtained by VMD is defined as amplitude modulated frequency modulated (AM-FM) signal:

VMD theory assumes that the input signal

Hilbert transform is applied to each mode component, and the unilateral spectrum of the mode function is obtained by Hilbert transform and construction of analytic signal.

The analysis signal of each mode component is mixed with the corresponding center frequency, and the spectrum of each mode component to the baseband is moved.

The bandwidth of each mode component is estimated.

The constrained variational model is constructed by introducing constraints. The concrete structure is as follows:

In equation (

For solving variational problems, the extended Lagrangian function

The variational mode decomposition uses the alternate direction method of multipliers (ADMM) to solve equation (

Given discriminant accuracy

Dispersion entropy (DE) is a new method proposed by Rostaghi and Azami [

Step 1: the normal cumulative distribution function is used to normalize the signal

Step 2: construct the embedding vector

where

Step 3: calculate the relative frequency of

where

Step 4: the formula of DE is defined as follows:

When calculating DE,

Vapnik [

Lagrange functions can be obtained by introducing Lagrange multipliers

By substituting equation (

Substituting equation (

By solving the above problems, the regression function of SVR can be obtained:

In the formula,

It can be seen that penalty factors and kernel function parameters are involved in the calculation process of SVR, which are represented by

Artificial bee colony (ABC) algorithm is an optimization algorithm to simulate bee colony behavior, which was proposed by Karaboga and Basturk [

The working diagram of the bee colony collection.

The food source is defined as a location in the search space. The initial number of food sources is equal to the number of leader bees and follower bees. The main steps are as follows:

Step 1: after initialization, the bee begins to search all the initial solutions circularly. It includes the population number, the maximum iteration number, the control parameter, and the range of the solution.

Step 2: at the beginning of the search process, each leader bee generates a new food source by the following equation:

where

Step 3: after all the leader bees have finished searching, they will share the nectar source information with the follower bees. Then, bees choose the location of the honey source with a certain probability of nectar source

where

Step 4: although all the follower bees have finished the searching process, the follower bees and the leader bees still do not find a more adaptable honey source. Then, the solution falls into the local optimum, and the honey source will be abandoned. At the same time, the leader bee or the follower bee is transformed into a scout bee, and a new honey source is randomly generated according to the following equation:

where

Step 5: fitness is the objective function of optimization problem, and the calculation formula of fitness value is shown in equation (

Extreme learning machine (ELM) is a new algorithm for single hidden layer feed-forward neural network (SLFNs) proposed by Huang et al. [

Structure diagram of extreme learning machine.

In Figure

For

The least square solution of the connection weight

ELM model adopts three-layer structure, in which the number of input layer and output layer corresponds to the number of input and output variables. If the number of neurons in the hidden layer is too large, it will affect the accuracy of model prediction. So in this paper, the number of neurons in the input layer is 5, the number of neurons in the hidden layer is 16, and the number of neurons in the input layer is 1. The activation function of hidden layer is sigmoid.

The flow chart based on the VMD-DE-ELM-ASVR hybrid prediction model for underwater acoustic signal is shown in Figure

The framework of the proposed hybrid VMD-DE-ELM-ASVR model.

VMD decomposes underwater acoustic signal into modes, and each mode have a central frequency, for example, IMF1, IMF2, ... , IMF

In order to solve the problem of overdecomposition and computing burden, DE is used to calculate the entropy value of IMF1, IMF2, … , IMF

In the combined model forecasting process, each new recombination sequence obtained in the DE process is divided into training set and test set. The training set is used for the training model, and the test set is used to verify the effect of prediction. According to the DE value and curve of each mode component, it can be divided into high-frequency sequence and low-frequency sequence. ELM is used to predict the low-frequency subsequence, SVR is used to predict the high-frequency subsequence, and ABC algorithm is used to optimize the model performance by adjusting the parameters of SVR. Figures

Forecasting results of high-frequency sequence of data set A by ELM and ASVR models.

Prediction results of low-frequency sequence of data set A by ELM and ASVR models.

In the hybrid process,

In this paper, the following three error indexes are selected to measure the prediction effect of the proposed prediction model: mean absolute error (MAE), root mean squared error (RMSE), and coefficient of determination (^{2}). Using the performance index MAE and RMSE to quantify the error of prediction value, the smaller the value is, the better the prediction accuracy is. The closer the ^{2} is to 1, the better the prediction performance is. The formulas are as follows:

In order to verify the effectiveness of the proposed prediction model, experiments need to be carried out. A PC with Intel Core i7, 3.6 GHz RAM, 4 GB ROM, 32 GB memory, running the Microsoft Window 8 operating system was used as the platform on which to implement the proposed model. In addition, MATLAB R2014a software platform was also used for the implementation of the proposed model. In this paper, the normalized preprocessing method is applied to the experimental data set [

Prediction error (MSE) and sample number.

It can be seen from Figure

(a) Underwater acoustic signal data set A; (b) underwater acoustic signal data set B.

Before decomposition, the mode number

(a) VMD decomposition results of data set A; (b) VMD decomposition results of data set B; (c) EMD decomposition results of data set A; (d) EMD decomposition results of data set B.

As shown in Figures

IMF with finite complexity approximation is obtained by VMD decomposition. If the prediction model is directly used to predict each IMF, the calculation scale will increase. In order to reduce the computation scale, the complexity of each IMF is analyzed by using the DE algorithm. The calculation results are shown in Figure

DE of each IMF sequences: (a) data set A; (b) data set B.

It can be seen from Figure

Recombined by the DE value of VMD subsequence of data set A.

Sub | DE value | Recombination | New sub |
---|---|---|---|

IMF1 | 0.3476 | IMF1 | DE-IMF1 |

IMF2 | 0.4775 | IMF2 | DE-IMF2 |

IMF3 | 0.5668 | IMF3&IMF4 | DE-IMF3 |

IMF4 | 0.5842 | ||

IMF5 | 0.6192 | IMF5&IMF6 | DE-IMF4 |

IMF6 | 0.6316 | ||

IMF7 | 0.6648 | IMF7 | DE-IMF5 |

IMF8 | 0.7227 | IMF8 | DE-IMF6 |

Recombined by the DE value of VMD subsequence of data set B.

Sub | DE value | Recombination | New sub |
---|---|---|---|

IMF1 | 0.3830 | IMF1 | DE-IMF1 |

IMF2 | 0.4627 | IMF2 | DE-IMF2 |

IMF3 | 0.5439 | IMF3 | DE-IMF3 |

IMF4 | 0.6508 | IMF4&IMF5 | DE-IMF4 |

IMF5 | 0.6681 | ||

IMF6 | 0.7033 | IMF6 | DE-IMF5 |

IMF7 | 0.7471 | IMF7 | DE-IMF6 |

IMF8 | 0.7767 | IMF8&IMF9 | DE-IMF7 |

IMF9 | 0.7640 |

Table

Recombined by the SE value of VMD subsequence of data set A.

Sub | SE value | Recombination | New sub |
---|---|---|---|

IMF1 | 0.1499 | IMF1 | SE-IMF1 |

IMF2 | 0.4962 | IMF2 | SE-IMF2 |

IMF3 | 0.6206 | IMF3&IMF4&IMF6 | SE-IMF3 |

IMF4 | 0.5921 | ||

IMF5 | 0.5409 | IMF5 | SE-IMF4 |

IMF6 | 0.6054 | ||

IMF7 | 0.8079 | IMF7 | SE-IMF5 |

IMF8 | 1.3062 | IMF8 | SE-IMF6 |

For underwater acoustic signal data set B, after VMD, the entropy value of each component and the result of merging are shown in Table

According to the combined results of mode components in Tables

Recombination component processed by the VMD-DE method: (a) data set A; (b) data set B.

The SVR model is a machine learning algorithm, which overcomes some shortcomings of traditional prediction methods. It has obvious advantages in solving nonlinear problems and strong generalization performance. In practical applications, the kernel function

After setting the initial parameters, the training data and test data are input into the ASVR model many times to achieve the best effect. The curve of fitness value and cycle times during training is shown in Figure

ABC-SVR optimal value convergence curve.

In Section

Each high-frequency mode component of data set A after VMD decomposition is superposed into a high-frequency sequence, and the prediction results of ELM model and ASVR model for high-frequency sequence are shown in Figure

The low-frequency sequence often contains the characteristics of the original signal, and its waveform is closer to the real series. Similarly, we use the ELM model and ASVR model to predict the low-frequency series after the superposition of each low-frequency mode component. From Figure

Through the above comparative analysis, this paper uses ELM model to predict the low-frequency sequence of underwater acoustic signal and ASVR model to predict the high-frequency sequence of underwater acoustic signal. The prediction results of ASVR for each component of data set A after VMD-DE decomposition are shown in Figure

(a) VMD-DE-ASVR prediction results of each component; (b) VMD-DE-ELM-ASVR prediction results of each component.

In order to further measure the prediction effect of this method, the VMD-DE-ELM-ASVR model proposed in this paper is compared with the other six models, and the prediction effect of each model is quantitatively analyzed by MAE, RMSE, and ^{2}, so as to verify the superiority of the combined model in the prediction performance of this paper.

Data A are divided into six relatively stable mode components after VMD-DE decomposition. The prediction model is established for each component, and the prediction results are shown in Figure ^{2} are also listed in Figure ^{2} is close to 1, which further explains the effectiveness of the prediction model proposed in this paper.

Prediction results of VMD-DE-ELM-ASVR. MAE = 0.010158; RMSE = 0.012886; ^{2} = 0.99773.

The fitting curves of predicted values and original values of different prediction models are shown in Figure

Prediction results for each model of data set A.

(a) Error box diagram of each model; (b) bar chart of model error index. M1: SVR; M2: ELM; M3: ASVR; M4: EMD-ASVR; M5: VMD-ASVR; M6: VMD-DE-ASVR; M7: VMD-DE-ELM-ASVR.

It can be seen from Figure

The comparison result of different model performances for data set A.

Model | MAE | RMSE | ^{2} | |
---|---|---|---|---|

SVR | 0.0578 | 0.0785 | 0.9147 | 7 |

ELM | 0.0432 | 0.0551 | 0.9602 | 6 |

ASVR | 0.0386 | 0.0504 | 0.9641 | 154 |

EMD-ASVR | 0.0298 | 0.0369 | 0.9832 | 456 |

VMD-ASVR | 0.0140 | 0.0168 | 0.9966 | 460 |

VMD-DE-ASVR | 0.0135 | 0.0167 | 0.9967 | 374 |

VMD-DE-ELM-ASVR |

The values of MAE, RMSE, and ^{2} of all models are shown in Table ^{2} is the closest to 1, indicating that the proposed model has the best prediction accuracy.

By comparing the prediction results of SVR and ASVR model, it can be seen from Table ^{2} of predicted value and real value is 0.9147, while the correlation coefficient of other models is above 0.96. However, the prediction accuracy of the SVR model optimized by the ABC algorithm has been significantly improved. MAE and RMSE are the mean absolute error and root mean square error of predicted value and actual value. Therefore, the smaller the index is, the better the result is. The RMSE value of the optimized SVR model is lower than that of the ASVR model. This shows that the ABC algorithm relies on its strong global and local search ability to find the optimal penalty parameter

Both EMD-ASVR and VMD-ASVR models have improved the prediction accuracy compared with a single ASVR model. It shows that the prediction model with decomposition method can improve prediction accuracy and reduce the prediction error. However, by comparing the results in Table ^{2} are also significantly worse than VMD decomposition. It also shows that the end effect of EMD decomposition and mode mixing will directly affect the prediction results. In addition, the complexity of each mode component is analyzed by DE to reconstruct the subsequence, which improves the prediction accuracy and shortens the training time of the model.

From Table ^{2} is 0.9977. It shows that the combination prediction model proposed in this paper has the highest fitting degree with the original data.

In order to further verify the prediction effect of the VMD-DE-ELM-ASVR model in different underwater acoustic signal sequences, the underwater acoustic signal data set B is selected for discussion and analysis in this section. The prediction results are shown in Figure

Prediction results for each model of data set B.

In addition, Figure ^{2} are shown in Table ^{2} reaches 0.9966, which is similar to the prediction performance of the underwater acoustic signal in data A.

(a) Error box diagram of each model; (b) bar chart of model error index. M1: SVR; M2: ELM; M3: ASVR; M4: EMD-ASVR; M5: VMD-ASVR; M6: VMD-DE-ASVR; M7: VMD-DE-ELM-ASVR.

The comparison result of different model performances for data set B.

Model | MAE | RMSE | ^{2} | |
---|---|---|---|---|

SVR | 0.0420 | 0.0547 | 0.9285 | 10 |

ELM | 0.0394 | 0.0505 | 0.9394 | 8 |

ASVR | 0.0354 | 0.0457 | 0.9498 | 154 |

EMD-ASVR | 0.0259 | 0.0325 | 0.9746 | 470 |

VMD-ASVR | 0.0129 | 0.0159 | 0.9947 | 465 |

VMD-DE-ASVR | 0.0108 | 0.0136 | 0.9961 | 396 |

VMD-DE-ELM-ASVR |

Generally speaking, from the prediction effect of the underwater acoustic signal data sets A and B, the combined prediction model proposed in this paper shows better prediction performance, which can provide a reference for prediction of underwater acoustic signal.

In order to improve the prediction accuracy of underwater acoustic signal, a combined prediction model based on VMD-DE-ELM-ASVR is proposed and applied to the prediction of underwater acoustic signal. The main conclusions are as follows:

The VMD decomposition algorithm can effectively overcome the mode mixing of EMD. The simulation results show that the decomposition effect of VMD is clearer, and the prediction accuracy is higher.

In this paper, DE is used to calculate the entropy of the IMFs of VMD decomposition, and the components with DE approximation are merged and recombined. The simulation results show that eight IMF components obtained by VMD decomposition in data set A are combined into six IMF components, and nine mode components are combined into seven by DE in data set B. In this way, the complexity of calculation is effectively reduced, and the prediction accuracy is improved.

ABC algorithm has few parameter settings and can be used for global and local search. The results show that the optimal penalty parameter

In this paper, ELM model is selected to predict the low-frequency component of underwater acoustic signal, and ASVR is used to predict the high-frequency component of underwater acoustic signal. It is proved by experiments that the combined prediction model can improve the prediction accuracy and reduce the prediction error compared with the single prediction model.

The prediction method proposed in this paper is tested by actual underwater acoustic signal data sets A and B and compares seven kinds of prediction models with three statistical indicators, including SVR, ELM, ASVR, EMD-ASVR, VMD-ASVR, VMD-DE-ASVR, and VMD-DE-ELM-ASVR. The experimental results show that VMD-DE-ELM-ASVR can effectively predict underwater acoustic signal. Compared with other models, the combined model improves the prediction accuracy, reduces the error, and has strong generalization ability and robustness.

Variational mode decomposition

Empirical mode decomposition

Extreme learning machine

Support vector regression

Structural risk minimization

Optimized SVR based on ABC algorithm

^{2}:

Coefficient of determination

Gravitational search algorithm

Particle swarm optimization

Sample entropy

Genetic algorithm

Dispersion entropy

Intrinsic mode function

Artificial bee colony

Support vector machine

Root mean squared error

Mean square error

Maximum cycle number

Index of orthogonality.

The data used to support the findings of this study are currently under embargo, while the research findings are commercialized. Requests for data, 12 months after publication of this article, will be considered by the corresponding author.

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).