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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.

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Notes

  1. For any given leap year, the P value was added to the P value for 1st March.

Abbreviations

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

EDIp :

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

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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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  1. School of Agricultural Computational and Environmental Sciences, International Centre of Applied Climate Sciences (ICACS), University of Southern Queensland, Springfield, Australia

    Ravinesh C. Deo

  2. Department of Soil and Water Engineering, College of Agricultural and Technology, Anand Agricultural University, Anand, Gujarat, India

    Mukesh K. Tiwari

  3. Department of Bioresource Engineering, Faculty of Agricultural and Environmental Science, McGill University, Montreal, QC, H9X 3V9, Canada

    Jan F. Adamowski & John M. Quilty

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  1. Ravinesh C. Deo
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Correspondence to Ravinesh C. Deo.

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Deo, R.C., Tiwari, M.K., Adamowski, J.F. et al. Forecasting effective drought index using a wavelet extreme learning machine (W-ELM) model. Stoch Environ Res Risk Assess 31, 1211–1240 (2017). https://doi.org/10.1007/s00477-016-1265-z

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