A Neural Network Scheme for Long-Term Forecasting of Chaotic Time Series

The accuracy of a model to forecast a time series diminishes as the prediction horizon increases, in particular when the prediction is carried out recursively. Such decay is faster when the model is built using data generated by highly dynamic or chaotic systems. This paper presents a topology and training scheme for a novel artificial neural network, named “Hybrid-connected Complex Neural Network” (HCNN), which is able to capture the dynamics embedded in chaotic time series and to predict long horizons of such series. HCNN is composed of small recurrent neural networks, inserted in a structure made of feed-forward and recurrent connections and trained in several stages using the algorithm back-propagation through time (BPTT). In experiments using a Mackey-Glass time series and an electrocardiogram (ECG) as training signals, HCNN was able to output stable chaotic signals, oscillating for periods as long as four times the size of the training signals. The largest local Lyapunov Exponent (LE) of predicted signals was positive (an evidence of chaos), and similar to the LE calculated over the training signals. The magnitudes of peaks in the ECG signal were not accurately predicted, but the predicted signal was similar to the ECG in the rest of its structure.