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Estimation of monthly evaporative loss using relevance vector machine, extreme learning machine and multivariate adaptive regression spline models

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Abstract

The forecasting of evaporative loss (E) is vital for water resource management and understanding of hydrological process for farming practices, ecosystem management and hydrologic engineering. This study has developed three machine learning algorithms, namely the relevance vector machine (RVM), extreme learning machine (ELM) and multivariate adaptive regression spline (MARS) for the prediction of E using five predictor variables, incident solar radiation (S), maximum temperature (T max), minimum temperature (T min), atmospheric vapor pressure (VP) and precipitation (P). The RVM model is based on the Bayesian formulation of a linear model with appropriate prior that results in sparse representations. The ELM model is computationally efficient algorithm based on Single Layer Feedforward Neural Network with hidden neurons that randomly choose input weights and the MARS model is built on flexible regression algorithm that generally divides solution space into intervals of predictor variables and fits splines (basis functions) to each interval. By utilizing random sampling process, the predictor data were partitioned into the training phase (70 % of data) and testing phase (remainder 30 %). The equations for the prediction of monthly E were formulated. The RVM model was devised using the radial basis function, while the ELM model comprised of 5 inputs and 10 hidden neurons and used the radial basis activation function, and the MARS model utilized 15 basis functions. The decomposition of variance among the predictor dataset of the MARS model yielded the largest magnitude of the Generalized Cross Validation statistic (≈0.03) when the T max was used as an input, followed by the relatively lower value (≈0.028, 0.019) for inputs defined by the S and VP. This confirmed that the prediction of E utilized the largest contributions of the predictive features from the T max, verified emphatically by sensitivity analysis test. The model performance statistics yielded correlation coefficients of 0.979 (RVM), 0.977 (ELM) and 0.974 (MARS), Root-Mean-Square-Errors of 9.306, 9.714 and 10.457 and Mean-Absolute-Error of 0.034, 0.035 and 0.038. Despite the small differences in the overall prediction skill, the RVM model appeared to be more accurate in prediction of E. It is therefore advocated that the RVM model can be employed as a promising machine learning tool for the prediction of evaporative loss.

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Abbreviations

ANN:

Artificial neural network

ARMA:

Autoregressive moving average

ELM:

Extreme learning machine

GCV:

Generalized cross validation

MAE:

Mean absolute error

MARS:

Multivariate adaptive regression spline

MLR:

Multiple linear regression

RMSE:

Root-mean-square-error

RVM:

Relevance vector machine

SLFN:

Single layer feedforward neural network

SRMP:

Structural risk minimisation principle

SVM:

Support vector machine

SWAT:

Soil water assessment tool

VP:

Vapor pressure

B(x):

Basis functions of MARS model

c 0, c m :

Constant or coefficients of BF(x)

E :

Evaporative loss

\( \overline{E} \) :

Mean predicted E

E ai :

Actual evaporative loss

E flat :

Flatness of predicted E

E p25 :

25th percentile predicted E

E p50 :

Median of predicted E

E p75 :

75th percentile predicted E

E pi :

Predicted evaporative loss

E range :

Range of predicted E

E skew :

Skewness of predicted E

N :

Number of predictor datum

ρ :

Performance index

P :

Precipitation

R :

Coefficient of correlation

S :

Incident solar radiation

σE :

Standard deviation of E

T max :

Maximum temperature

T min :

Minimum temperature

w :

Weight

x :

Predictor data (≡ S, T max, T min, etc.)

y :

Objective variable (≡ E)

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Acknowledgment

The data were acquired from Science Delivery Division, Department of Science, Information Technology, Innovation and Arts. Assistance by Torben Marcussen for data extraction is very much appreciated. School of Agricultural, Computational and Environmental Sciences supported Dr R.C. Deo to collaborate with Profs. P. Samui and D Kim. We thank the reviewers and Associate Editor whose comments have improved our manuscript. Author contributions were as follows: Prof. P. Samui performed the experimental and modeling tasks, assisted by Prof. D. Kim and Dr R. C. Deo completed the manuscript write-up. Authors declare no conflict of interest.

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Authors and Affiliations

  1. School of Agricultural, Computational and Environmental Sciences, International Centre for Applied Climate Sciences (ICACS), University of Southern Queensland, Springfield, QLD, 4300, Australia

    Ravinesh C. Deo

  2. Centre for Disaster Mitigation and Management, VIT University, Vellore, 632014, India

    Pijush Samui

  3. Department of Civil Engineering, Kunsan National University, Jeonbuk, Republic of Korea

    Dookie Kim

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

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Deo, R.C., Samui, P. & Kim, D. Estimation of monthly evaporative loss using relevance vector machine, extreme learning machine and multivariate adaptive regression spline models. Stoch Environ Res Risk Assess 30, 1769–1784 (2016). https://doi.org/10.1007/s00477-015-1153-y

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