A hybrid prediction method based on empirical mode decomposition and multiple model fusion for chaotic time series

Abstract

Chaotic time series exist in nature, such as in the field of meteorology or physics, with unpredictable features caused by their inherent high complexity and nonstationary motion. To improve the prediction effect of chaotic time series, a hybrid prediction method on the basis of the empirical mode decomposition (EMD) and neural networks (NN) is proposed. First, the original chaotic time series is decomposed into several intrinsic mode functions (IMFs) and one residual by EMD, whose components are divided into high, medium and low components by using the runs test. The high IMFs change dramatically, and the correlation among data is not high; therefore, we use the fuzzy first-order transition rules trained neural network (NN-FFOTR) model to predict the high frequency components. Autoregressive integrated moving average (ARIMA) models are applied according to the random characteristic exhibited by the medium frequency IMFs and residual. Finally, the prediction results of the two models are adaptively superimposed together to obtain the final prediction results. The proposed hybrid method is named EMD-ARIMA-NN-FFOTR for simplicity. The simulation results are verified through two typical chaotic time series including the Lorenz-63 system and Mackey-Glass time series. For the x-axis of the Lorenz-63 system, the experimental results indicates that the symmetric mean absolute percentage error (SMAPE) of the ARIMA, NN-FFOTR, ARIMA-NN-FFOTR and EMD-ARIMA-NN-FFOTR are 1.0657, 1.5281, 1.0657 and 0.0041;the root mean square error (RMSE) are 2.2591,0.6830,0.4231 and 0.0012; the mean absolute error (MAE) are 50.5149,15.2130,9.4610 and 0.0019; the sum of error squares (SSE) are 101.4,96.32,65.8300 and 21.6351; and the Theil inequality coefficient are 0.0674, 0.0741, 0.0213 and 0.0162, the proposed model's TIC are much lower than those of the comparison models. The values of the R square of the developed prediction approach utilizing ARIMA, NN-FFPTR, ARIMA-NN-FFOTR and EMD-ARIMA-NN-FFOTR method are 0.9674,0.9741, 0.9799 and 0.9963, and the index of agreement are 0.8992,0.9921,0.9845 and 0.9991, the value of the R-square and the index of agreement of the proposed model's are much higher than those of the comparison models. The Pearson's test results show that the association strength between the actual value and the predicted values of the proposed model is stronger. These results show the following: (a) compared with other related, recent studies, the prediction accuracy of the hybrid system EMD-ARIMA-NN-FFOTR proposed in this article is higher; (b) the proposed hybrid system attains superior performance compared with single models; and (c) the proposed hybrid system balances the forecast accuracy and convergence speed simultaneously during prediction. Therefore, it is feasible to apply the hybrid model to the prediction of chaotic time series.