Long-Term Forecasts of Droughts Using a Conjunction of Artificial Neural Networks and Wavelet Transforms

The accurate prediction of drought conditions plays a vital role in developing a management policy for water supply systems. This study presents a conjunction model to forecast droughts in the Conchos River Basin in Mexico, which supplies approximately 70-80% of the Lower Bravo/Grande River flows. The model uses wavelet transforms (WT) in order to improve forecast accuracy of artificial neural networks (ANNs) for a regional drought index.

In recent decades, ANNs have shown great ability in the modeling and forecasting of nonlinear and nonstationary time series in various fields of study due to their innate nonlinear property and flexibility for modeling. Wavelet analysis has become a common tool for analyzing local variations in a time series. The “á trous” algorithm for the dyadic wavelet transform, which performs successive convolutions with the discrete low-pass filter, has been used to forecast a time series (Aussem and Murtagh, 1997; Aussem et al., 1998; Zhang and Dong, 2001). However, the inconsistency of the decomposed sub-signal remains problematic in a forecasting model. We took an intuitively acceptable approach to this issue by taking a convolution value in the “á trous” algorithm fixed to the beginning and the end of the signal. Mallat’s quadratic spline (Mallat, 1998) was used as a low-pass filter.

The proposed conjunction model has two phases. In the training/forecasting phase, the “á trous” wavelet transform is performed twice to obtain input values and target values of ANNs in order not to contain any information about the target values. In forecasting models using a transforming preprocess, very careful attention must be given to the end of the signal during transforming and reconstructing. After training the network, ANNs predict sub-signals with a specific lead time. Then, in a reconstruction phase, the forecasted sub-signals are reconstructed. The decomposed sub-signals can be reconstructed by a routine that inverts the dyadic wavelet transform. In forecasting models, however, it is difficult to reconstruct the forecasted sub-signals leaped over a lead time, because successive convolutions can not be carried out using the forecasted sub-signals. We used ANNs once more to reconstruct signals in the reconstruction phase. The ANNs used in the reconstruction phase have different architecture and weights from the ANNs used in the training/forecasting phases. The schematic representation for the proposed forecasting model is given in Fig. 1. Wavelet transforms separate a real world signal into sub-signals at different resolution levels, which improve the ability of ANNs to capture valuable information in the sub-signals and successfully forecast the original signal.

The monthly Palmer Drought Severity Index (PDSI) was used to represent regional drought severity in the Conchos River Basin. Improved forecasts of the PDSI allow water resources decision makers to develop drought preparedness plans far in advance to mitigate the social, environmental, and economic costs of drought. The 1957-1990 period was used to find efficient architectures of ANNs available for decomposition levels and train the networks. The 1991-2000 period was used for validation. The three-layered ANNs with a back-propagation algorithm were examined for four wavelet decomposition levels and four long-term lead times up to 12 months. Their architectures were determined by empirical experiments.

Fig. 2 shows the time series of the observed and one-month ahead forecasted values of the Conchos regional PDSI by the conjunction model (ANN-DD). The one-month ahead forecasts captured the interannual variability and turning points in the time series, which represent the end of dry or wet spells. In Fig. 3, the normalized root mean square errors (NRMSE) and the forecasting skill score (SS) referred to climatological average values were compared with simple ANNs (ANN). Fig. 3 shows that the proposed model (ANN-DD) has forecast skills up to six months and gives better results than using ANN alone. Wavelet transform is a useful tool for forecasting time series capturing the dynamics of observed signals with a multi-resolution. The proposed conjunction model is real-time and may produce valuable forecasts for the indexed regional drought through wavelet decompositions.

Acknowledgments:

This study is supported by SAHRA (Sustainability of semi-Arid Hydrology and Riparian Areas) at the University of Arizona under the STC Program of the National Science Foundation, Agreement No. EAR-9876800.

References:

Aussem, A., J., and F. Murtagh, 1997: Combining neural network forecasts on wavelet-transformed time series. Connection Science, 9(1), 113-121.

Aussem, A., J. Campbell, and F. Murtagh, 1998: Wavelet-based feature extraction and decomposition strategies for financial forecasting. Journal of Computational Intelligence in Finance, 6(2), 5-12.

Mallat, S., 1998: A Wavelet Tour of Signal Processing, Academic Press, 577pp.

Zhang, B.L. and Z.Y. Dong, 2001: An adaptive neural-wavelet model for short term load forecasting. Electric Power Systems Research, 59, 121-129.