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Echo state network based on improved fruit fly optimization algorithm for chaotic time series prediction

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Abstract

Chaos is a common phenomenon in nature and society. Chaotic system affects many fields. It is of great significance to find out the regularity of chaotic time series from chaotic system. Chaotic system has extremely complex dynamic characteristics and unpredictability. The traditional prediction methods for chaotic time series have some problems, such as low accuracy, slow convergence speed and complex model structure. In this paper, an echo state network prediction method based on improved fruit fly optimization algorithm for chaotic time series is proposed. The phase space reconstruction is introduced for the prediction of chaotic time series. The C–C method is used to determine the delay time. The embedding dimension is obtained by the G–P method. After reconstructing the phase space of the chaotic time series, an improved echo state network is proposed as the prediction model. In order to improve the prediction accuracy, an improved fruit fly optimization algorithm is proposed to optimize the parameters of the prediction model. Three typical chaotic time series, including Lorenz, Mackey–Glass, and short-term wind speed, are selected as simulation objects. The simulation results show that the prediction method proposed in this paper has good prediction indicators. At the same time, the results of the reliability and Pearson's test also show the better predictive effect.

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Abbreviations

ARMA:

Auto regressive moving average

ARIMA:

Auto regressive integrated moving average

ELM:

Extreme learning machine

SVM:

Support vector machine

LSSVM:

Least squares support vector machine

ESN:

Echo state network

FOA:

Fruit fly optimization algorithm

C–C:

Correlation–correlation

GP:

Grassberger–Procaccia

DR:

Dynamic reservoir

SR:

Spectral radius

SD:

Sparse degree

IS:

Input scale

RMSE:

Root mean square error

MAE:

Mean absolute error

MAPE:

Mean absolute percentile error

\({R^{2}}\) :

R square

EMD:

Empirical mode decomposition

\(\tau\) :

Delay time

m :

Embedding dimension

\(x(k)\) :

Time series at sampling time k

\(\overline{x} (k)\) :

Predictive value of time series at sampling time k

\(\overline{x}\) :

The mean value of chaotic time series

\(N\) :

The length of time series

M :

The amount of m-dimensional phase points

\(\mathop S\limits^{\_} (t)\) :

Average test statistics

\(\Delta \mathop S\limits^{\_} (t)\) :

Mean deviation

\(\varepsilon\) :

The standard deviation of time series

r :

The multiple of deviation

\(n_{m}\) :

The number of possible values of m

\(n_{k}\) :

The number of possible values of r

\(D_{m}\) :

Searching radius

\(C_{n} (r)\) :

Correlation integral

\(\theta ()\) :

Heaviside unit function

\({\mathbf{W}}_{in}\) :

Input weight matrix

\({\mathbf{W}}_{out}\) :

Output weight matrix

\({\mathbf{u}}(k)\) :

Input of ESN

K :

Input layers of ESN

B :

Internal processing units of ESN

L :

Output layers of ESN

U :

Input matrix of ESN

Y :

Output matrix of ESN

\(c\) :

Compensation signal

K :

Input layers of ESN

\(\alpha\) :

Ratio coefficient of reservoir

β :

Output feedback coefficient of reservoir

\(\lambda _{{\max }}\) :

The absolute value of the largest eigenvalue of connection weight matrix of ESN

\(X_{i}^{best}\) :

Abscissa of the position coordinates of the optimal individual

\(Y_{i}^{best}\) :

Ordinate of the position coordinates of the optimal individual

\(X_{i}^{{{\text{Secbest}}}}\) :

Abscissa of the position coordinates of the sub-optimal individual

\(Y_{i}^{{{\text{Secbest}}}}\) :

Ordinate of the position coordinates of the sub-optimal individual

\(X\_{\text{axis}}\) :

Abscissa of the coordinates of the position after the movement of the ith fruit fly

\(Y\_{\text{axis}}\) :

Ordinate of the position coordinates of the sub-optimal individual

DIST:

The optimal individual is the minimum distance

\(S_{i}\) :

The determination of the smell concentration of each fruit fly individual in populations

\({\text{Dist}}_{{{\text{best}}}}\) :

The distance of the fruit fly individual and the origin

t :

The iterative algebra

\(\eta\) :

The threshold

NI :

The amount of all individuals

\({\rm smell}_{{best}}^{r}\) :

The determination of the smell concentration of the optimal individual at the tth iteration

\({\rm smell}_{{best}}^{t}\) :

The sum of the determination of the smell concentration of all individuals at the tth Iteration

pos_up:

The lower limit of the fruit fly's search range

pos_low:

The upper limit of the fruit fly's search range

NI :

The amount of all individuals

maxgen:

Maximum amount of iterations

sizepop:

The size of the fruit fly population

\(\xi ^{{(1 - a)}}\) :

The amount of actual values falling into the confidence interval under the confidence level 1−α

\(\Delta T\) :

Sampling interval of Lorenz time series

\(\Delta\) :

Delay parameter of Mackey-Glass time series

γ :

Regularization parameters of LSSVM

\(\delta^{2}\) :

Radial basis function width of LSSVM

λ :

Sigmoid function parameter

μ :

Scale factor of the improved ELM

a, b:

Parameters to be estimated of grey model

\(c_{1} ,c_{2}\) :

Weight coefficient of hybrid prediction model

ρ :

Discard probability

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Acknowledgements

This paper is supported by the Science Research Project of Liaoning Education Department (No. LGD2016009), Natural Science Foundation of Liaoning Province of China (No. 20170540686).

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  1. College of Information Science and Engineering, Shenyang University of Technology, Shenyang, 110870, China

    Zhongda Tian

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Tian, Z. Echo state network based on improved fruit fly optimization algorithm for chaotic time series prediction. J Ambient Intell Human Comput 13, 3483–3502 (2022). https://doi.org/10.1007/s12652-020-01920-4

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