In target estimating sea clutter or actual mechanical fault diagnosis, useful signal is often submerged in strong chaotic noise, and the targeted signal data are difficult to recover. Traditional schemes, such as Elman neural network (ENN), backpropagation neural network (BPNN), support vector machine (SVM), and multilayer perceptron- (MLP-) based model, are insufficient to extract the weak signal embedded in a chaotic background. To improve the estimating accuracy, a novel estimating method for aiming at extracting problem of weak pulse signal buried in a strong chaotic background is presented. Firstly, the proposed method obtains the vector sequence signal by reconstructing higher-dimensional phase space data matrix according to the Takens theorem. Then, a Jordan neural network- (JNN-) based model is designed, which can minimize the error squared sum by mixing the single-point jump model for targeting signal. Finally, based on short-term predictability of chaotic background, estimation of weak pulse signal from the chaotic background is achieved by a profile least square method for optimizing the proposed model parameters. The data generated by the Lorenz system are used as chaotic background noise for the simulation experiment. The simulation results show that Jordan neural network and profile least square algorithm are effective in estimating weak pulse signal from chaotic background. Compared with the traditional method, (1) the presented method can estimate the weak pulse signal in strong chaotic noise under lower error than ENN-based, BPNN-based, SVM-based, and -ased models and (2) the proposed method can extract the weak pulse signal under a higher output SNR than BPNN-based model.

As early as the end of the 19th century, the French Poincare found that the solution of the three-body problem could be random within a certain range when he studied the three-body problem, but this discovery did not arouse widespread concern among scholars. It was not until 1963 that Lorenz, an American meteorologist, published a paper [

Pulse signal is a kind of discrete signal, which is usually submerged by noise and not easy to detect. It is widely existed in communication engineering, biomedicine, and industrial automatic control. Therefore, the prediction of the pulse signal from chaotic noise has certain practical significance. Experts and scholars at home and abroad have also studied the prediction of the pulse signal in chaotic noise and obtained a lot of research results, such as Volterra filter, correlation detection in time domain, spectral analysis in frequency domain, chaotic Duffing oscillator [

At present, experts and scholars have proposed many methods to predict chaotic time series and extract impulse signal from chaotic noise [

Neural network is a model that imitates the information processing mode of the human nervous system. It is the abstraction and simplification of human neural network. Neural networks can approximate and model the processes of nonlinear dynamic systems [

In this paper, the Jordan neural network is used to fit the chaotic background and estimate the pulse signal from its residuals. Because the pulse signal is very weak, the characteristics of chaotic time series are mainly reflected in the chaotic background. Firstly, the observed signal was reconstructed, then the Jordan neural network and the single-point pulse signal model were established, the weight and bias of the Jordan neural network were optimized by the gradient descent algorithm, and then the amplitude of the pulse signal was estimated by the profile least square method.

The remainder of the paper is organized as follows. Section

Our work mainly involves two aspects: the application of feedback neural network and the application of feedback neural network in time series prediction.

Chaotic time series is a kind of irregular motion in deterministic system and a special time series. However, the traditional time series prediction models—MA, ARMA, and ARIMA—and other models are not ideal for chaotic time series prediction.

The applications for chaotic time series prediction are wide and range from financial prediction to weather prediction [

Recursive neural networks are used in a wide range of applications, such as weather prediction [

Liu et al. combined the dual stage two-phase model with the feedback neural network to make a long-term prediction of multiple time series in [

This paper analyzes the weak pulse signal which is fixed chaotic noise. Let

Pulse signal can be boiled down by periodic signal:

Chaotic time series is a univariate or multivariable time series produced by a dynamical system. Phase space reconstruction theorem is mapping a chaotic time series to high-dimensional space. According to the Takens theorem of embedding [

For an observed chaotic signal

In mathematics, Takens embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes, but it does not preserve the geometric shape of structures in phase space. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function.

Cao’s method is a practical method which determines the minimum embedding dimension from a scalar time series. The average ratio of Euclidean distance of a vector with embedded dimension _{0} if the time series comes from an attractor. Then _{0} + 1 is the minimum embedding dimension we look for.

Jordan combined the Hopfield network storage concept and distributed parallel processing theory to create a new circular neural network in 1986 [

Jordan neural network topology.

According to Figure

In order to express clearly and write easily, the following signs will be introduced:

So the Jordan neural network is obtained:

To better understand equation (

Details of Jordan neural network recurrent connections.

The following equation is obtained by the Takens theorem of embedding:

According to the Takens theorem of embedding, the model of pulse signal, and Jordan neural network, we can obtain the following equation for estimating the pulse signal:

Then, we can deduce the following equation with unknown parameters involving only

Profile least-square method [

Profile least square algorithm is usually divided into two steps: first, estimating the nonparametric function with a given

_{.} From equation (

Gradient descent algorithm is used to update the weights of Jordan neural network to achieve the purpose of optimization. The basic idea of gradient descent algorithm is the linear approximation by first-order Taylor expansion. Therefore, the partial derivatives of the weights and the bias that need to be optimized in this paper are, respectively, calculated as follows:

The updates of weights and the bias are equal to the sum of the corresponding old one and the product of the partial derivative and the learning rate.

The method of estimating

The first derivative and the second derivative of

You obtain the following equation from Sections

It can be seen that equation (

The component form of the first derivative in equation (

The second derivative of

Similarly, the component form of the second derivative in equation (

We now outline the Algorithm

Input: Observation sequence

Output:

Begin

While (

Step 1: For given

Step 2: The phase point data set

Step 3: Based on (A) in Section

Step 4:

End while

Obtain the optimal

End

Because the mathematical expression of the Jordan neural network is very complex, the difference is used to solve the approximate first and second derivatives for simplifying the calculation when the profile least square is used for estimation. The solution process of difference is as follows:

We take 3 points (

In order to verify the validity and feasibility of the proposed prediction model, three simulation experiments were carried out. The experimental data in this paper were generated by Lorenz dynamic system and realized by

The equation of Lorenz dynamic system is in the following form:

We normalize the original data sets and there are two kinds of normalized equations. Different neural network structures correspond to different normalization equations in this paper:

Absolute error (AE) and absolute percentage error (APE) are used to measure the error, and they are

Chaotic time series

Predicted value of

AE | APE | MSE | |||||
---|---|---|---|---|---|---|---|

(1) | 130 | 0.1 | −59.18164 | 0.0969737 | 0.0030263 | 0.030263 | 6.9 × 10^{−8} |

(2) | 130 | 0.12 | −57.59802 | 0.1161813 | 0.0038187 | 0.031823 | 1.1 × 10^{−7} |

(3) | 140 | 0.15 | −55.95726 | 0.1527467 | 0.0027467 | 0.018311 | 5.3 × 10^{−8} |

(4) | 140 | 0.22 | −52.63063 | 0.2215733 | 0.0015733 | 0.007151 | 1.7 × 10^{−8} |

(5) | 150 | 0.3 | −50.25632 | 0.3005346 | 0.0005346 | 0.001782 | 1.9 × 10^{−9} |

(6) | 150 | 0.4 | −47.75755 | 0.3982338 | 0.0017662 | 0.004415 | 2.0 × 10^{−8} |

(7) | 150 | 0.5 | −45.81935 | 0.4988764 | 0.0011236 | 0.002247 | 8.2 × 10^{−9} |

(8) | 180 | 0.8 | −42.65903 | 0.7941931 | 0.0058069 | 0.007259 | 1.9 × 10^{−7} |

(9) | 200 | 1.1 | −40.32545 | 1.1067487 | 0.0067487 | 0.006135 | 2.2 × 10^{−7} |

(10) | 200 | 1.5 | −37.63148 | 1.5022624 | 0.0022624 | 0.001508 | 2.4 × 10^{−8} |

(11) | 200 | 1.9 | −35.57823 | 1.9066362 | 0.0066362 | 0.003493 | 2.1 × 10^{−7} |

(12) | 210 | 3.6 | −30.26097 | 3.5552166 | 0.0447834 | 0.012440 | 9.0 × 10^{−6} |

To demonstrate the stability of the model presented in this paper, we performed Monte Carlo simulations consisting of 50 runs on the same data and the experimental results were averaged.

From Table ^{−5}. As can be seen from Table

Iteration results of the proposed algorithm for Example 1. (1) The iteration result at

Predicted value of

Mean of | Standard deviation (std) | |||
---|---|---|---|---|

(1) | 130 | 0.1 | 0.09665155 | 0.001731646 |

(2) | 130 | 0.12 | 0.1161727 | 0.002553445 |

(3) | 140 | 0.15 | 0.1497219 | 0.001991941 |

(4) | 140 | 0.22 | 0.2191495 | 0.001827229 |

(5) | 150 | 0.3 | 0.3030018 | 0.001105854 |

(6) | 150 | 0.4 | 0.4028839 | 0.001321245 |

(7) | 150 | 0.5 | 0.5030558 | 0.001121901 |

(8) | 180 | 0.8 | 0.7990241 | 0.001515569 |

(9) | 200 | 1.1 | 1.102412 | 0.001636155 |

(10) | 200 | 1.5 | 1.502059 | 0.002687494 |

(11) | 200 | 1.9 | 1.901625 | 0.004017169 |

(12) | 210 | 3.6 | 3.563963 | 0.011684620 |

In this experiment, different models were used to fit the same set of observed data, which had an amplitude of 2.5 and a period of 400. The SNR of the observed data is −36.42872. The chaotic noise and original observation signal are shown in Figure

Data display diagram. (a) Chaotic noise

From Table

Predicted value of

Model | Mean of | AE | APE | std | |
---|---|---|---|---|---|

JNN | 2.5 | 2.501952 | 0.001952 | 0.0007808 | 0.004211562 |

BPNN | 2.5 | 2.503297 | 0.003297 | 0.0013188 | 0.006069684 |

ENN | 2.5 | 2.503410 | 0.003410 | 0.0013640 | 0.005308634 |

SVM | 2.5 | 2.387866 | 0.112134 | 0.0448536 | 0.010575260 |

MLP | 2.5 | 2.969629 | 0.469629 | 0.1878516 | 0.091993390 |

Comparison diagram of different models.

Comparison diagram of original signal and predicted signal and the predicted error diagram: (a) original signal vs. the result of Jordan neural network; (b) Jordan neural network prediction error; (c) original signal vs. the result of support vector machine; (d) support vector machine prediction error; (e) original signal vs. the result of multilayer perceptron; (f) multilayer perceptron prediction error.

Figure

Output SNR vs. input SNR.

This section reports the performance of the profile method combined with Jordan neural network in estimation of pulse signals against chaotic background. The purpose of the experiments was to evaluate if JNN can maintain quality in prediction performance when compared to conventional methods.

The results have shown that the method proposed in this paper not only solves the weak pulse signal in chaotic signal well, but also has good stability. We evaluate the application breadth of the proposed model by comparing the experimental data of different SNR. Table

The results report the estimation of pulse signals by different neural networks under chaotic noise background, as well as the mean amplitude and standard deviation (std) of 50 test runs in Table

A major innovation of this method is to use Jordan neural network to fit the chaotic noise background and then estimate the impulse signal in the chaotic noise by combining the profile method. Compared with other neural networks, this method is more accurate and the overall training time is shorter.

In this paper, we are interested in weak pulse signal in chaotic background. Based on the short-term predictability and Takens theorem, we provide an algorithm for directly estimating the problems; that is, the Jordan neural network is used to fit the chaotic time series, and the one-step prediction error was obtained. Then, starting from the error, the single-point jump signal model was connected, and the amplitude of the pulse signal was estimated by the profile least square method, so as to achieve the prediction effect of the pulse signal under the chaotic background. The following conclusions can be drawn from the experimental results: the model proposed in this paper can predict the weak pulse signal in the chaotic background, and it can be seen from the results of experiment 1 that the prediction accuracy is high. It can be seen from the results of experiment 2 that the Jordan neural network is significantly better than other feed-forward neural networks in fitting the nonlinear dynamic system, and the absolute percentage error of the Jordan neural network is the smallest. In future work, we can try to improve in two aspects, so that it has the best generalization performance in most cases, and apply the method to solve the relevant practical problems. Firstly, we can try to estimate the amplitude of the impulse signal without knowing the period of the impulse signal. Secondly, the Jordan neural network can be trained by other optimization methods, and the Jordan neural network can be improved.

No data were used to support this study.

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

This work was supported by the Fundamental and Advanced Research Project of CQ CSTC of China (Grant no. cstc2018jcyjAX0464).