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Accurate prediction of hydrological processes is key for optimal allocation of water resources. In this study, two novel hybrid models are developed to improve the prediction precision of hydrological time series data based on the principal of three stages as denoising, decomposition, and decomposed component prediction and summation. The proposed architecture is applied on daily rivers inflow time series data of Indus Basin System. The performances of the proposed models are compared with traditional single-stage model (without denoised and decomposed), the hybrid two-stage model (with denoised), and existing three-stage hybrid model (with denoised and decomposition). Three evaluation measures are used to assess the prediction accuracy of all models such as Mean Relative Error (MRE), Mean Absolute Error (MAE), and Mean Square Error (MSE). The proposed, three-stage hybrid models have shown improvement in prediction accuracy with minimum MRE, MAE, and MSE for all case studies as compared to other existing one-stage and two-stage models. In summary, the accuracy of prediction is improved by reducing the complexity of hydrological time series data by incorporating the denoising and decomposition.

Accurate prediction of hydrological processes is key for optimal allocation of water resources. It is challenging because of its nonstationary and multiscale stochastic characteristics of hydrological process which are affected not only by climate change but also by other socioeconomic development projects. The human activities also effected the climate change through contributing in Earth’s atmosphere by burning of fossil fuels which release carbon dioxide in atmosphere. Instead of these, greenhouse and aerosols have made effect on Earth’s atmosphere by altering in-out coming solar radiations which is the part of Earth’s energy balance. This makes the prediction of hydrological time series data challenging. To predict such hydrological processes, two broad types of models are commonly used, one is the process-based models which further included the lumped conceptual models, hydrological model, and one-two-dimensional hydrodynamic models [

To conquer the limitations of existing single models, some hybrid algorithms such as data preprocessing methods are utilized with data-driven models with the hope to enhance the prediction performance of complex hydrological time series data by extracting time varying components with noise reduction. These preprocess based hybrid models have already been applied in hydrology [

This study aimed to develop a robust hybrid model to decompose the hydrological time varying characteristics using CEEMDAN [

In this study, two novel approaches are proposed to enhance the prediction accuracy of the hydrological time series. Both models have the same layout except in stage of denoising, where two different approaches have been used to remove noises from hydrological time series data. In both models, at decomposition stage, an improved version of EEMD, i.e., CEEMDAN, is used to find oscillations, i.e., the high to low frequencies in terms of IMF. At prediction stage, multimodels are used to accurately predict the extracted IMFs by considering the nature of IMFs instead of using only single stochastic model. The purpose of using multimodel is two-way: one is for accurately predicting the IMFs by considering the nature of IMFs and the other is to assess the performance of simple and complex models after reducing the complexity of hydrological time series data through decomposition. Predicted IMFs are added to get the final prediction of hydrological time series. The proposed three stages involve denoising (D-step), decomposition (Decompose-step), and component prediction (P-step), which are briefly described below:

For convenient, two proposed methods as named as EMD (denoising), CEEMDAN (decomposing), MM (multi-models) i.e. EMD-CEEMDAN-MM and WA (denoising), CEEMDAN (denoising) and MM (multi-models) i.e. WA-CEEMDAN-MM. The proposed architecture of WA/EMD-CEEMDAN-MM is given in Figure

The proposed WA/EMD-CEEMDAN-MM structure to predict hydrological time series data.

In hydrology time series data, noises or stochastic volatiles are inevitable component which ultimately reduced the performance of prediction. To reduce the noise from data, many algorithms have been proposed in literature such as Fourier analysis, spectral analysis, WA, and EMD [

The EMD structure is defined as follows:

Identify all local maxima and minima from time series

Find the mean of upper and lower envelope

Check the properties defined in

Repeat the process of (1-3), until the residue

Finally, original series can be written as the sum of all extracted IMFs and residue as

Add a white Gaussian noise series to the original data set.

Decompose the signals with added white noise into IMFs using conventional EMD method.

Repeat steps (a) and (b)

Obtain the ensemble means of all IMFs

In CEEMDAN, extracted modes are defined as

Then replace

The resulting

More details of EMD, EEMD, and CEEMDAN are given in [

The IMF prediction with group method of data handling type neural network: ANN has been proved to be a powerful tool to model complex nonlinear system. One of the submodels of NN, which is constructed to improve explicit polynomial model by self-organizing, is Group Method of Data Handling-type Neural Network (GMDH-NN) [

Transfer functions of GMDH-NN algorithms.

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Sigmoid function | |

Tangent function | |

Polynomial function | |

Radial basis function | |

Architecture of GMDH-type neural network (NN) algorithms.

Topological structure of radial basis function.

^{2} area in which estimated 13% of the areas are covered by Upper Indus Basin (UIB) [

Rivers and irrigation network of Pakistan.

Without denoising and decomposing, only single statistical model is selected, i.e., ARIMA (for convenience, we call one-stage model 1-S) as used in [

Only denoised based models: in this stage, the noise removal capabilities of WA and EMD are assessed. The wavelet based models are WA-ARIMA, WA-RBFNN, and WA-RGMDH whereas the empirical mode decomposition based models are EMD-ARIMA, EMD-RBFNN, and EMD-RGMDH. The different prediction models are chosen for the comparison of traditional statistical models with artificial intelligence based models as RBFN and RGMDH (for convenience, we call two-stage model 2-S). The 2-S selected models for comparison are used from [

With denoising and decomposition (existing method): for that purpose, three-stage EMD-EEMD-MM model is used from [

The WA and EMD based denoised Indus and Jhelum rivers inflow are shown in Figure

Statistical measures of WA- and EMD-based denoised rivers inflow of four hydrological time series data sets.

| | | | | | |
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| Original series | 80.2194 | 87.5044 | |||

EMD | 80.5931 | 87.3925 | 3.9118 | 0.1275 | 36.7636 | |

Wavelet | 80.2267 | 86.1632 | 3.8188 | 0.0987 | 22.9626 | |

| Original series | 30.2001 | 23.6743 | |||

EMD | 30.1412 | 23.1641 | 2.7118 | 0.1666 | 16.4864 | |

Wavelet | 30.2023 | 22.7799 | 2.5579 | 0.1418 | 10.8837 | |

| Original series | 29.1746 | 25.2352 | |||

EMD | 25.23524 | 25.1181 | 2.5474 | 0.2036 | 12.5216 | |

Wavelet | 29.18118 | 24.29148 | 2.7386 | 0.1615 | 12.2447 | |

| Original series | 31.9557 | 29.4916 | |||

EMD | 32.0024 | 29.2734 | 2.271784 | 0.1470 | 10.6797 | |

Wavelet | 31.9585 | 28.2591 | 3.1958 | 0.17228 | 17.8353 |

The denoised series of the two hydrological time series of Indus and Jhelum rivers inflow. The figure shows the denoised results obtained through the EMD-based threshold method (in red color) and the wavelet analysis-based threshold method (in blue color).

The EMD-CEEMDAN decomposition of Indus (left) and Jhelum rivers inflow (right). The two series are decomposed into nine IMFs and one residue.

The WA-CEEMDAN decomposition of Indus (left) and Jhelum rivers inflow (right). The two series are decomposed into nine IMFs and one residue.

Evaluation index of testing prediction error of proposed models (EMD-CEEMDAN-MM and WA-CEEMDAN-MM) with all selected models for all four case studies.

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Indus Inflow | 1-S | ARIMA | 4.2347 | 0.0685 | 64.7141 |

2-S | WA-ARMA | 3.2862 | 0.0430 | 53.4782 | |

WA-RGMDH | 3.2548 | 0.0393 | 46.7382 | ||

WA-RBFN | 20.1949 | 0.2598 | 2301.772 | ||

EMD-ARMA | 4.9898 | 0.0960 | 76.1440 | ||

EMD-RGMDH | 4.9653 | 0.0915 | 76.0884 | ||

EMD-RBFN | 34.3741 | 0.7762 | 3931.601 | ||

3-S | EMD-EEMD-MM | 5.2710 | 0.1721 | 44.0115 | |

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Jhelum Inflow | 1-S | ARMA | 3.5224 | 0.1201 | 47.5529 |

2-S | WA-ARMA | 2.6129 | 0.0748 | 37.1441 | |

WA-RGMDH | 2.6208 | 0.0773 | 37.7954 | ||

WA-RBFN | 9.8608 | 0.7714 | 180.7443 | ||

EMD-ARMA | 3.7354 | 0.1551 | 48.3164 | ||

EMD-RGMDH | 3.7357 | 0.1620 | 48.3606 | ||

EMD-RBFN | 2.8822 | 0.2506 | 51.9916 | ||

3-S | EMD-EEMD-MM | 2.0096 | 0.1269 | 7.3565 | |

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Kabul Inflow | 1-S | ARMA | 2.4910 | 0.0883 | 25.0136 |

2-S | WA-ARMA | 1.9999 | 0.0592 | 20.6874 | |

WA-RGMDH | 2.0794 | 0.0729 | 21.0612 | ||

WA-RBFN | 1.6565 | 0.0997 | 13.3554 | ||

EMD-ARMA | 2.9538 | 0.1484 | 28.5767 | ||

EMD-RGMDH | 3.0114 | 0.2280 | 28.9351 | ||

EMD-RBFN | 4.9355 | 0.7613 | 69.9346 | ||

3-S | EMD-EEMD-MM | 1.8758 | 0.3166 | 5.8020 | |

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Chenab Inflow | 1-S | ARMA | 5.4157 | 0.4646 | 108.185 |

2-S | WA-ARMA | 3.9652 | 0.1087 | 84.2359 | |

WA-RGMDH | 3.6147 | 0.0943 | 81.6493 | ||

WA-RBFN | 4.1424 | 0.2757 | 47.6184 | ||

EMD-ARMA | 4.7971 | 0.2721 | 100.7013 | ||

EMD-RGMDHA | 4.4812 | 0.1865 | 95.6680 | ||

EMD-RBFN | 10.8228 | 2.1666 | 284.5627 | ||

3-S | EMD-EEMD-MM | 2.7172 | 0.2298 | 14.5191 | |

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Prediction results of Indus and Jhelum rivers inflow using proposed EMD-CEEMDAN-MM with comparison to other EMD based denoised and predicted models.

Prediction results of Indus and Jhelum river inflow using proposed WA-CEEMDAN-MM with comparison to WA based denoised predicted models.

To improve the prediction accuracy of complex hydrological time series data from simple time series models one can take the advantage from three principals of “denoising,” “decomposition,” and “ensembling the predicted results.” The 2-S model, with simple ARIMA and GMDH, can perform well as compared to 2-S models with complex models and 1-S models by optimal decomposition methods. Moreover, with addition to extracting time varying frequencies from denoised series, one can get the more precise results over 2-S models. However, from Table

The following conclusions are drawn based on the testing error presented in Table

The accurate prediction of hydrological time series data is essential for water supply and water resources purposes. Considering the instability and complexity of hydrological time series, some data preprocessing methods are adopted with the aim to enhance the prediction of such stochastic data by decomposing the complexity of hydrological time series data in an effective way. This research proposed two new methods with three stages as “denoised,” decomposition, and prediction and summation, named as WA-CEEMDAN-MM and EMD-CEEMDAN-MM, for efficiently predicting the hydrological time series. For the verification of proposed methods, four cases of rivers inflow data from Indus Basin System are utilized. The overall results show that the proposed hybrid prediction model improves the prediction performance significantly and outperforms some other popular prediction methods. Our two proposed, three-stage hybrid models show improvement in prediction accuracy with minimum MRE, MAE, and MSE for all four rivers as compared to other existing one-stage [

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this article.

The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group no. RG-1437-027.