Combining recurrent neural networks with variational mode decomposition and multifractals to predict rainfall time series

Rainfall time series prediction is essential for monitoring urban hydrological systems, but it is challenging and complex due to the extreme variability of rainfall. A hybrid deep learning model (VMD-RNN) is used in order to improve prediction performance. In this study, variational mode decomposition (VMD) is first applied to decompose the original rainfall time series into several sub-sequences according to the frequency domain, where the number of decomposed sub-sequences is determined by power spectral density (PSD) analysis. To prevent the disclosure of forthcoming data, non-Training time series are sequentially appended for generating the decomposed testing samples. Following that, different recurrent neural network (RNN) variant models are used to predict individual sub-sequences, and the final prediction is reconstructed by summing the prediction results of sub-sequences. These RNN variants are long short-Term memory (LSTM), gated recurrent unit (GRU), and bidirectional LSTM (BiLSTM) and bidirectional GRU (BiGRU), which are optimal for sequence prediction. In addition to three common evaluation criteria, mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE), the framework of universal multifractals (UMs) is also introduced to assess the performance of predictions, which enables the extreme variability of predicted rainfall time series to be characterized. The study employs two rainfall time series with daily and hourly resolutions, respectively. The results indicate that the hybrid VMD-RNN model provides a reliable one-step-Ahead prediction, with better performance in predicting high and low values than the pure LSTM model without decomposition.