# Forecasting effective drought index using a wavelet extreme learning machine (W-ELM) model

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## Abstract

A drought forecasting model is a practical tool for drought-risk management. Drought models are used to forecast drought indices (DIs) that quantify drought by its onset, termination, and subsequent properties such as the severity, duration, and peak intensity in order to monitor and evaluate the impacts of future drought. In this study, a wavelet-based drought model using the extreme learning machine (W-ELM) algorithm where the input data are first screened through the wavelet pre-processing technique for better accuracy is developed to forecast the monthly effective DI (EDI). The EDI is an intensive index that considers water accumulation with a weighting function applied to rainfall data with the passage of time in order to analyze the drought-risk. Determined by the autocorrelation function (ACF) and partial ACFs, the lagged EDI signals for the current and past months are used as significant inputs for 1 month lead-time EDI forecasting. For drought model development, 97 years of data for three hydrological stations (Bathurst Agricultural, Wilsons Promontory and Merredin in Australia) are partitioned in approximately 90:5:5 ratios for training, cross-validation and test purposes, respectively. The discrete wavelet transformation (DWT) is applied to the predictor datasets to decompose inputs into their time–frequency components that capture important information on periodicities. DWT sub-series are used to develop new EDI sub-series as inputs for the W-ELM model. The forecasting capability of W-ELM is benchmarked with ELM, artificial neural network (ANN), least squares support vector regression (LSSVR) and their wavelet-equivalent (W-ANN, W-LSSVR) models. Statistical metrics based on agreement between the forecasted and observed EDI, including the coefficient of determination, Willmott’s index, Nash–Sutcliffe coefficient, percentage peak deviation, root-mean-square error, mean absolute error, and model execution time are used to assess the effectiveness of the models. The results demonstrate enhanced forecast skill of the drought models that use wavelet pre-processing of the predictor dataset. Based on statistical measures, W-ELM outperformed traditional ELM, LSSVR, ANN and their wavelet-equivalent counterparts (W-ANN, W-LSSVR). It is found that the W-ELM model is computationally efficient as shown by a faster running time with the majority of forecasting errors in lower frequency bands. The results demonstrate the usefulness of W-ELM over W-ANN and W-LSSVR models and the benefits of wavelet transformation of input data to improve the performance of drought forecasting models.

## Keywords

Wavelet Extreme learning machine Drought model Effective drought index Forecasting## List of symbols

- ACF
Autocorrelation function

- ANN
Artificial neural network

- BOM
Bureau of Meteorology

*c*Y-intercept of linear function

- CWT
Continuous wavelet transformation

- DI
Drought index

- DWC
Discrete wavelet coefficients

- DWT
Discrete wavelet transformation

- EDI
Effective drought index

*EDI*_{o}Observed (calculated) EDI

- EDI
_{p} Forecasted (predicted) EDI

- ELM
Extreme learning machine

- EMD
Empirical mode decomposition

*E*_{NS}Nash–Sutcliffe coefficient

- FFBP
Feed-forward back-propagation

- LSSVR
Least squares support vector regression

*υ*Gradient of linear function

- MAE
Mean absolute error

*MP*_{E}Mean

*P*_{E}- MSE
Mean square error

*P*Precipitation

- PACF
Partial autocorrelation function

*P*_{dv}Peak percentage deviation

*P*_{E}Effective precipitation

- |PE|
Absolute prediction error

*R*^{2}Coefficient of determination

- RBF
Radial basis function

- RCP
Representative concentration pathway

- RDDI
Rainfall-decile drought index

- RMSE
Root mean square error

- SLFN
Single layer feed-forward network

*SP*_{E}Standard deviation of

*P*_{E}- SPI
Standardized precipitation index

- SSA
Singular spectrum analysis

- SVM
Support vector machines

- SVR
Support vector regression

- VC
Vapnik–Chervonenkis

- W-ANN
Wavelet-based ANN

*WI*Willmott’s index

- W-ELM
Wavelet-based ELM

- W-LSSVR
Wavelet-based LSSVR

## Notes

### Acknowledgments

The research paper utilised precipitation data from Australian Bureau of Meteorology. USQ Academic Division funded Dr. RC Deo through a “Research Activation Incentive Scheme (RAIS, July–September 2015)” for collaboration with McGill and Anand Agricultural University. Dr. RC Deo, as Senior Visiting Scholar, also held an Endeavour Executive Fellowship (4293-2015) funded by Australian Government Department of Education. Finally we thank both reviewers, journal Editors and McGill MSc student Sasha Rodrigues whose comments have enhanced the integrity of this paper.

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