Long-Term Forecasts of Droughts Using a Conjunction of Wavelet

sciencediscussionAI and Robotics

Oct 20, 2013 (4 years and 6 months ago)



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

Woong Kim

Juan B. Valdés
, and
Javier Aparicio

1. Department of Civil Engineering and Engineering Mechanics, and Center for
Sustainability of

Arid Hydrology and Riparian Areas (SAHRA), The University of
Arizona, Tucson, Arizona.

2. Hydrologic Technology Division, Mexican Institute of Water Technology (IMTA),
Morelos, Mexico

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
80% of the Lower Bravo/Grande River flows. The model uses wavelet

(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 succes
sive convolutions with the discrete low
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
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 propose
d 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 model
s 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 fo
recasted 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 tim
e, because successive convolutions can not be carried out using the forecasted
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
e 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 o
f ANNs
to capture valuable information in the sub
signals and successfully forecast the original

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 decom
position levels and
train the networks. The 1991
2000 period was used for validation. The three
ANNs with a back
propagation algorithm were examined for four wavelet decomposition
levels and four long
term lead times up to 12 months. Their architec
tures 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
ahead forecasts captured the interannual var
iability 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
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
resolution. The proposed conjunction model is
time and may produce valuable forecasts for the indexed regional drought through
wavelet decompositions.


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


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

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

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

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


Figure. 1. Schematic representation of the forecasti
ng model with a conjunction of
wavelet transform and artificial neural networks.

Figure 2. Time series of the observed and one
month ahead forecasted values of the
Conchos regional PDSI.


Figure 3. Forecast skills measured by normalized RMSE (NRMSE)
and skill score (SS).