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Long-lead streamflow forecasting using computational intelligence methods while considering uncertainty issue

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Abstract

While some robust artificial intelligence (AI) techniques such as Gene-Expression Programming (GEP), Model Tree (MT), and Multivariate Adaptive Regression Spline (MARS) have been frequently employed in the field of water resources, documents aimed to explore their uncertainty levels are few and far between. Meanwhile, uncertainty determination of these AI models in practical applications is highly important especially when we aimed to use the AI models for streamflow forecast due to the repercussions of poorly managed water resources. With the aid of a global daily streamflow dataset, understanding the uncertainty of GEP, MT, and MARS for forecasting streamflow of natural rivers was studied. The efficiency of uncertainty analysis was quantified by two statistical indicators: 95% Percent Prediction Uncertainty (95%PPU) and R-factor. The results demonstrated that MT had lower uncertainty (95%PPU=0.59 and R-factor=1.67) in comparison with MARS (95%PPU=0.61 and R-factor=1.92) and GEP (95%PPU=0.64 and R-factor=2.03). Overall, although the confidence interval bands of uncertainty for the AI models almost captured the mean streamflow measurements, wide bands of uncertainty were obtained and consequently remarkable uncertainty in the calculation of monthly streamflow values was met.

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Data Availability

Some or all data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

a 0 :

Bias term of multivariate linear equation

a 1, a 2, a 3 :

Weighting coefficients of multivariate linear equation

ACF :

Auto-Correlation Function

ANFIS :

Adaptive Neuro-Fuzzy Inference System

AI :

Artifiucuial Intelligence

AR :

Auto-Regression

ARMA :

Auto-Regression Moving Average

ARIMA :

Auto-Regression Integrated Moving Average

BF :

Basis Function

CC :

Correlation of Coefficient, R.

CM :

Conventional Model

\(\overline d\) :

Average distance between the lower and upper 95PPU band

EPR :

Evolutionary Polynomial Regression

FFNN :

Feed Forward Neural Network

GA :

Genetic Algorithm

GEP :

Gene-Expression Programming

GLUE :

Genetalized Likelihood Uncertainty Estimation

GMDH :

Group Method of Data Handling

KNN :

K-Nearest Neighbor

KELM :

Kernel Extreme Learning Machine

MARS :

Multivariate Adaptive Regression Spline

MCS :

Monte-Carlo Simulation

MSE :

Mean Square Error

MT :

Model Tree

NBF :

Number of Basis Function

P-factor :

The key element for quantifying the uncertainty of AI models performance

PACF :

Partial Auto-Correlation Function

PSO :

Particle Swarm Optimization

Q :

Monthly or dialy streamflow discharge

RE :

Relative Error

R-factor :

The key elements for quantifying the uncertainty of AI models performance

RC :

Coefficient of Correlation

RMSE :

Root Mean Square Error

SF :

Streamflow

SHDI :

Standardized Hydrological Drought Index

SOI :

Southern Oscillation Index

STD :

Standard Deviation

SVM :

Support Vector Machine

t :

Time

t−1,  t−2,  t−3:

Lag times of streamflow discharge

TF :

Thomas-Fiering method

WRM :

Water Resources Management

x U :

Upper limit of 95PPU

x L :

Lower limit of 95PPU

ψ 0 :

Bias term of MARS model

ψ i :

Weigthing coefficients of MARS model

95PPU :

95 Percent Prediction Uncertainty

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Authors and Affiliations

  1. Department of Water Engineering, Faculty of Civil and Surveying Engineering, Graduate University of Advanced Technology, P.O. Box 76315117, Kerman, Iran

    Mohammad Najafzadeh

  2. Department of Ecology, Institute of Science and High Technology and Environmental Sciences, Graduate University of Advanced Technology, P.O. Box 76315117, Kerman, Iran

    Sedigheh Anvari

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Contributions

Mohammad Najafzadeh: providing computer programming codes for AI models, performing Artificial Intelligence models for stream flow forecast, writing analysis of AI models results, improving introduction, and writing literature review, abstract, conclusion, and comparisons sections. Sedigheh Anvari: writing introduction, data analysis, case study, describing uncertainty descriptions, and performing uncertainty analysis of AI models, and providing figures and tables.

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Correspondence to Mohammad Najafzadeh.

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Najafzadeh, M., Anvari, S. Long-lead streamflow forecasting using computational intelligence methods while considering uncertainty issue. Environ Sci Pollut Res 30, 84474–84490 (2023). https://doi.org/10.1007/s11356-023-28236-y

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