# Modeling Potential Evapotranspiration by Improved Machine Learning Methods Using Limited Climatic Data

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## Abstract

**:**

_{0}) is an important issue for water resources planning and management projects involving droughts and flood hazards. Evapotranspiration, one of the main components of the hydrological cycle, is highly effective in drought monitoring. This study investigates the efficiency of two machine-learning methods, random vector functional link (RVFL) and relevance vector machine (RVM), improved with new metaheuristic algorithms, quantum-based avian navigation optimizer algorithm (QANA), and artificial hummingbird algorithm (AHA) in modeling ET

_{0}using limited climatic data, minimum temperature, maximum temperature, and extraterrestrial radiation. The outcomes of the hybrid RVFL-AHA, RVFL-QANA, RVM-AHA, and RVM-QANA models compared with single RVFL and RVM models. Various input combinations and three data split scenarios were employed. The results revealed that the AHA and QANA considerably improved the efficiency of RVFL and RVM methods in modeling ET

_{0}. Considering the periodicity component and extraterrestrial radiation as inputs improved the prediction accuracy of the applied methods.

## 1. Introduction

_{0}) in a semi-arid region. Their findings show that all tested soft computing models outperformed the empirical models. Among the soft computing models, the artificial neural network (ANN) provided better results than the adaptive neuro-fuzzy inference system (ANFIS) and gene expression programming (GEP). Fan et al. [17] assessed the performance of a new tree-based soft computing model, called light gradient boosting machine (LightGBM), by comparing it with the tree-based M5 Model Tree (M5Tree) and random forests (RF) as well as four empirical models, namely Hargreaves-Samani, Tabari, Makkink and Trabert) using different combinations of daily weather data. They concluded that the proposed model, namely, LightGBM provides good results in terms of accuracy, and it can be used as an alternative model for daily ET

_{0}estimation, especially when long meteorological data are unavailable. Shamshirband et al. [18] estimated ET

_{0}using a combination of the cuckoo search algorithm (CSA) with ANN and ANFIS, respectively, and compared the results with two empirical models. They concluded that both combined soft computing models performed better than empirical models. Aghelpour et al. [19] estimated crop evapotranspiration using multilayer perceptron (MLP), radial basis function (RBF), generalized regression neural network (GRNN), and group method of data handling (GMDH). Their findings show that the GMDH model performed better than other soft computing and empirical models. Mokari, et al. [20] used four machine learning (ML) models, namely extreme learning machine (ELM), genetic programming (GP), random forest (RF), and support vector regression (SVR), for estimating daily ET0 with limited climatic data using a tenfold cross-validation method across different climate zones in New Mexico using different input scenarios of data. Their findings show that SVR and ELM performed better than other soft computing models for all input scenarios. Ferreira, et al. [21] estimated ETo by comparing the FAO56-PM equation with random forest (RF) (ANN), multivariate adaptive regression splines (MARS), and extreme gradient boosting (XGBoost). They concluded that combining the soft computing models with the FAO56-PM equation to estimate ETo performed similarly to using them individually. Sharma et al. [22] estimated ET

_{0}employing Convolution—Long Short-Term Memory (Conv-LSTM) and Convolution Neural Network—LSTM (CNN-LSTM) and compared them with other empirical models such as Hargreaves, Makkink, and Ritchie considering different input combinations of climate data to find out the minimum needed parameters to estimate the ET

_{0}at reasonable accuracy. Their findings show that CNN-LSTM and Conv-LSTM outperform the empirical models.

_{0}at an acceptable accuracy using fewer data than empirical models. Therefore, they suggest that their model can be used as an alternative model to estimate ET

_{0}at a lower cost in terms of required data and computation time. Chia et al. [24] utilized CNN-1D (explainable structure), Long short-term memory (LSTM), and GRU (black-box structure) to estimate monthly ET

_{0}in a humid climate region. They suggest that the LSTM and GRU models would perform better if designed in a hybrid form rather than a simple black-box structure. This suggests that soft computing models perform better than empirical models in estimating ET

_{0}.

## 2. Case Study

## 3. Methods

_{0}. These algorithms were developed using knowledge of hummingbirds’ sophisticated foraging techniques and unique flight abilities. The implemented methods are described in the next sections.

#### 3.1. Random Vector Functional Link Networks (RVFL)

_{i}, is obtained by the RVFL from the Ns sample data (X), which is represented as a pair (x

_{i}, y

_{i}). After that, the middle (hidden) nodes process the input data. The nodes of enhancement are these middle nodes. The output of every middle node is determined as follows Equation (1):

_{2}result and the input sample F

_{1}:

#### 3.2. Relevance Vector Machine (RVM)

_{1},…, t

_{N}) at certain training points x = (x

_{1},……, x

_{N}), RVM begins with the notion of linear models:

_{i},…, w

_{M}) is a vector consisting of the linear combination weights.

_{1}, w

_{2}, …, w

_{N}), the bias w

_{0}, and the kernel function K (x, xi) are all present.

_{1}, …., t

_{N}), w = (w

_{0},…, w

_{N}) and $\Phi $ is the N*(N+1) “design” matrix with ${\Phi}_{nm}=K\left({x}_{n}\xb7{x}_{m-1}\right)$ and ${\Phi}_{n1}=1$. Overfitting frequently occurs when w and ${\sigma}^{2}$ in Equation (6) are estimated with maximum probability. Tipping [33], therefore, advised imposing previous constraints on the parameters by making the likelihood or error function more complex. The learning process’ capacity for generalization is governed by this a priori knowledge. New higher-level parameters are typically used to constrain an explicit zero-mean Gaussian prior probability distribution over the weights (Equation (11)):

#### 3.3. Artificial Hummingbird Algorithm (AHA)

#### 3.4. Quantum-Based Avian Navigation Optimizer Algorithm (QANA)

#### 3.5. Proposed Optimized RVM and RVFL Models

#### 3.6. Assessment of the Developed Methods and Parameters

_{0}, ${Y}_{o}$ is observed ET

_{0}, ${\overline{Y}}_{o}$ is mean FAO PM 56 ET

_{0}$N$ is a number of data. Table 2 gives brief information related to the setting parameters of the used MH algorithms. For each model, 30 populations were set, and 1000 iterations were employed. The algorithms were run 30 times to reach the final solution.

## 4. Results

_{0}using limited input data (Tmin, Tmax, and Ra). The models were assessed using three data split scenarios (50–50%, 60–40%, and 75–25%), and the results were tabulated for the best-split scenario of each method. Training and testing results of the single RVFL models in estimating ET

_{0}of Faisalabad Station are summed up in Table 3. As seen from the table, five input combinations were considered. The fifth combination considers the periodicity component (MN, month number of the output). It is apparent from Table 3 that the RVFL model with full inputs (Tmin, Tmax, Ra, and MN) offers better estimations compared to other input combinations. Table 3 also provides the outcomes of the improved RVFL models, RVFL-AHA and RVFL-QANA, for the Faisalabad Station. It is visible from Table 4 that the RVFL-AHA with input combination (v) acts as the best model. In the case of RVFL-QANA, the models with input combinations (iii) and (iv) perform the best in the estimation of ET

_{0}, respectively.

_{0}than the Tmin, Ra. A comparison of the single RVFL and hybrid versions reveals that the hybrid models improve the accuracy of the single RVFL model in estimating ET

_{0}.

_{0}of Faisalabad Station are reported in Table 4 for the best training-test scenario (75–25%). There is a marginal difference between the input combinations (iv) and (v). Adding MN into the inputs slightly improves the RVM accuracy.

_{0}. Like RVM-based models, here also Tmax, Ra inputs seem more effective on ET

_{0}than the Tmin, Ra inputs. The last performance belongs to the models having Tmin and Tmax inputs. It is apparent from Table 4 that the RVM-AHA and RVM-QANA perform superior to the single RVM models as found for the RVFL-based models. It is visible from the single and hybrid RVFL and RVM models (Table 3 and Table 4) that importing Ra into the model inputs considerably improves the efficiency in estimating ET

_{0}.

_{0}of Peshawar Station for the best training-test scenario. It is visible from the tables that the models with Tmax, Ra inputs perform superior to the models having inputs of Tmin, Ra. The models with Tmin, Tmax, Ra, and MN inputs outperform the other alternatives. In some cases, however, considering MN in inputs deteriorates the models’ accuracy in ET

_{0}estimation; for example, a 75–25% scenario of RVFL-QANA. It should be noted that the models with only temperature inputs (Tmin, Tmax) produce inferior ET

_{0}estimates in all cases and for all RVFL-based models.

_{0}in Peshawar Station for the best training-test scenario (75–25%). It is clear from the table that the models with full inputs (input combination (v)) generally act as the best in estimating ET

_{0}. Here also, Tmax, Ra combination provides the worst outcomes for single and hybrid RVM models. It is understood from the table that tuning RVM parameters with AHA and QANA methods improve its accuracy in estimating ET

_{0}using limited climatic inputs. A comparison of RVM and RVFL models reveals that the RVM-based models generally act better than the RVFL-based models concerning mean statistics of RMSE, MAE, R

^{2}, and NSE.

_{0}. Taylor diagram illustrated in Figure 7 shows the superior accuracy of AHA and QANA-based RVFL and RVM models compared to single models with higher correlation, lower RMSE, and better standard deviation. It is clear from the violin charts in Figure 8 that the RVFL-QANA and/or RVM-QANA models better resemble the distribution of observed ET

_{0}in Faisalabad Station. It can be seen from the scatterplots, Taylor diagram, and violin charts demonstrated in Figure 9, Figure 10 and Figure 11 that the differences between hybrid and single RVFL and RVM models are clearer for the Peshawar Station. These graphs justify hybrid models’ superiority over single models in estimating ET

_{0}.

## 5. Discussion

_{0}using limited inputs. The outcomes were compared with single models considering three data split scenarios. It was observed that the AHA and QANA methods considerably improved the accuracy of single RVFL and RVM models. For example, the mean RMSE of the RVFL model was improved by 7.96% by applying the AHA method (RVFL-AHA) in the test period of Faisalabad Station, respectively.

_{0}estimations. In contrast, the full inputs involving Tmin, Tmax, Ra, and MN generally perform the best. The studies by Dimitriadou and Nikolakopoulos [39,40] also found similar results. They applied multiple linear regression and ANN for predicting ET

_{0}of the Peloponnese Peninsula, Greece. They indicated that the models with only temperature inputs offer inferior outcomes. On the other hand, Importing Ra into the models’ input considerably improves the performance in estimating ET

_{0}. For example. The RMSE of the single RVFL model decreased by 23.94% in the test period of Faisalabad Station. Shiri et al. [41] assessed some machine learning methods in estimating ET

_{0}in humid locations of Iran and found that temperature-based models involving temperature and Ra inputs offer promising results. Adnan et al. [42] investigated the estimation of ET

_{0}by hybrid adaptive fuzzy inferencing coupled with heuristic algorithms and using various climatic inputs. They observed that the Ra is highly effective on models’ accuracy in ET

_{0}estimation.

_{0}in some cases. It was seen from the comparison of input combinations that the Tmax, Ra provided better accuracy than the Tmin, Ra in estimating ET

_{0}in all models in both stations.

_{0}of sub-humid Red River Valley using different climatic input combinations. They obtained the best outcomes (R

^{2}= 0.927) from the RF in the testing period. Adnan et al. [42] investigated the estimation of ET

_{0}by hybrid adaptive fuzzy inferencing (ANFIS) coupled with Moth Flame Optimization (MFO) and Water Cycle Optimization (WCA) heuristic algorithms using different climatic input data. From best to worst, the ANFIS-WCAMFO, ANFIS-MFO, ANFIS-WCA, and ANFIS provided R

^{2}values of 0.950, 0.946, 0.939, and 0.937 in the test stage, respectively. In the presented study, the hybrid RVFL-AHA and RVM-AHA models had R

^{2}values of 0.959 and 0.963 in the test stage. The comparison with previous studies shows the good efficiency of the proposed models. However, the accuracy of the estimations is highly related to the available data. The relationships between climatic input data and ET

_{0}can be more complex in some regions and/or climates.

_{0}) badly affect the accuracy of machine-learning methods. Therefore, different combinations of inputs should be considered to obtain the best one. The provided outcomes of this study assessed and indicated the efficiency of the hybrid RVM and RVFL models for the limited number of stations in the same climatic region. The results cannot apply to other regions with a different climates. To do this, much more data should be used in evaluating the proposed models. On the other hand, the main RVM and RVFL models can be tuned to more recent algorithms and compared with calibrated empirical equations and deep learning approaches to decide the optimal method for any region.

## 6. Conclusions

_{0}) is crucial in drought monitoring and assessment. ET

_{0}calculates the common drought index (standardized precipitation evapotranspiration index). The methods investigated in this study can be used in drought assessment projects by providing accurate ET

_{0}data.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kuriqi, A.; Pinheiro, A.N.; Sordo-Ward, A.; Bejarano, M.D.; Garrote, L. Ecological impacts of run-of-river hydropower plants—Current status and future prospects on the brink of energy transition. Renew. Sustain. Energy Rev.
**2021**, 142, 110833. [Google Scholar] [CrossRef] - Rost, S.; Gerten, D.; Bondeau, A.; Lucht, W.; Rohwer, J.; Schaphoff, S. Agricultural green and blue water consumption and its influence on the global water system. Water Resour. Res.
**2008**, 44, W09405. [Google Scholar] [CrossRef][Green Version] - Adnan, R.M.; Liang, Z.; Heddam, S.; Zounemat-Kermani, M.; Kisi, O.; Li, B. Least square support vector machine and multivariate adaptive regression splines for streamflow prediction in mountainous basin using hydro-meteorological data as inputs. J. Hydrol.
**2020**, 586, 124371. [Google Scholar] [CrossRef] - Li, Z.-L.; Tang, R.; Wan, Z.; Bi, Y.; Zhou, C.; Tang, B.; Yan, G.; Zhang, X. A Review of Current Methodologies for Regional Evapotranspiration Estimation from Remotely Sensed Data. Sensors
**2009**, 9, 3801–3853. [Google Scholar] [CrossRef][Green Version] - Zheng, C.; Jia, L.; Hu, G. Global land surface evapotranspiration monitoring by ETMonitor model driven by multi-source satellite earth observations. J. Hydrol.
**2022**, 613, 128444. [Google Scholar] [CrossRef] - DehghaniSanij, H.; Yamamoto, T.; Rasiah, V. Assessment of evapotranspiration estimation models for use in semi-arid environments. Agric. Water Manag.
**2004**, 64, 91–106. [Google Scholar] [CrossRef] - Alizamir, M.; Kisi, O.; Adnan, R.M.; Kuriqi, A. Modelling reference evapotranspiration by combining neuro-fuzzy and evolutionary strategies. Acta Geophys.
**2020**, 68, 1113–1126. [Google Scholar] [CrossRef] - El-Kenawy, E.-S.M.; Zerouali, B.; Bailek, N.; Bouchouich, K.; Hassan, M.A.; Almorox, J.; Kuriqi, A.; Eid, M.; Ibrahim, A. Improved weighted ensemble learning for predicting the daily reference evapotranspiration under the semi-arid climate conditions. Environ. Sci. Pollut. Res.
**2022**, 29, 81279–81299. [Google Scholar] [CrossRef] - Zhao, L.; Xia, J.; Xu, C.-Y.; Wang, Z.; Sobkowiak, L.; Long, C. Evapotranspiration estimation methods in hydrological models. J. Geogr. Sci.
**2013**, 23, 359–369. [Google Scholar] [CrossRef] - Malik, A.; Saggi, M.K.; Rehman, S.; Sajjad, H.; Inyurt, S.; Bhatia, A.S.; Farooque, A.A.; Oudah, A.Y.; Yaseen, Z.M. Deep learning versus gradient boosting machine for pan evaporation prediction. Eng. Appl. Comput. Fluid Mech.
**2022**, 16, 570–587. [Google Scholar] [CrossRef] - Monteiro, A.F.M.; Martins, F.B.; Torres, R.R.; de Almeida, V.H.M.; Abreu, M.C.; Mattos, E.V. Intercomparison and uncertainty assessment of methods for estimating evapotranspiration using a high-resolution gridded weather dataset over Brazil. Theor. Appl. Clim.
**2021**, 146, 583–597. [Google Scholar] [CrossRef] - Chia, M.Y.; Huang, Y.F.; Koo, C.H. Resolving data-hungry nature of machine learning reference evapotranspiration estimating models using inter-model ensembles with various data management schemes. Agric. Water Manag.
**2021**, 261, 107343. [Google Scholar] [CrossRef] - Aryalekshmi, B.; Biradar, R.C.; Chandrasekar, K.; Ahamed, J.M. Analysis of various surface energy balance models for evapotranspiration estimation using satellite data. Egypt. J. Remote Sens. Space Sci.
**2021**, 24, 1119–1126. [Google Scholar] [CrossRef] - Gocić, M.; Motamedi, S.; Shamshirband, S.; Petković, D.; Ch, S.; Hashim, R.; Arif, M. Soft computing approaches for forecasting reference evapotranspiration. Comput. Electron. Agric.
**2015**, 113, 164–173. [Google Scholar] [CrossRef] - Kaya, Y.Z.; Zelenakova, M.; Üneş, F.; Demirci, M.; Hlavata, H.; Mesaros, P. Estimation of daily evapotranspiration in Košice City (Slovakia) using several soft computing techniques. Theor. Appl. Clim.
**2021**, 144, 287–298. [Google Scholar] [CrossRef] - Gavili, S.; Sanikhani, H.; Kisi, O.; Mahmoudi, M.H. Evaluation of several soft computing methods in monthly evapotranspiration modelling. Meteorol. Appl.
**2017**, 25, 128–138. [Google Scholar] [CrossRef][Green Version] - Fan, J.; Ma, X.; Wu, L.; Zhang, F.; Yu, X.; Zeng, W. Light Gradient Boosting Machine: An efficient soft computing model for estimating daily reference evapotranspiration with local and external meteorological data. Agric. Water Manag.
**2019**, 225, 105758. [Google Scholar] [CrossRef] - Shamshirband, S.; Amirmojahedi, M.; Gocić, M.; Akib, S.; Petković, D.; Piri, J.; Trajkovic, S. Estimation of Reference Evapotranspiration Using Neural Networks and Cuckoo Search Algorithm. J. Irrig. Drain. Eng.
**2016**, 142. [Google Scholar] [CrossRef] - Aghelpour, P.; Bahrami-Pichaghchi, H.; Karimpour, F. Estimating Daily Rice Crop Evapotranspiration in Limited Climatic Data and Utilizing the Soft Computing Algorithms MLP, RBF, GRNN, and GMDH. Complexity
**2022**, 2022, 4534822. [Google Scholar] [CrossRef] - Mokari, E.; DuBois, D.; Samani, Z.; Mohebzadeh, H.; Djaman, K. Estimation of daily reference evapotranspiration with limited climatic data using machine learning approaches across different climate zones in New Mexico. Theor. Appl. Clim.
**2021**, 147, 575–587. [Google Scholar] [CrossRef] - Ferreira, L.B.; da Cunha, F.F.; Filho, E.I.F. Exploring machine learning and multi-task learning to estimate meteorological data and reference evapotranspiration across Brazil. Agric. Water Manag.
**2021**, 259, 107281. [Google Scholar] [CrossRef] - Sharma, G.; Singh, A.; Jain, S. A hybrid deep neural network approach to estimate reference evapotranspiration using limited climate data. Neural Comput. Appl.
**2021**, 34, 4013–4032. [Google Scholar] [CrossRef] - Sharma, G.; Singh, A.; Jain, S. DeepEvap: Deep reinforcement learning based ensemble approach for estimating reference evapotranspiration. Appl. Soft Comput.
**2022**, 125, 109113. [Google Scholar] [CrossRef] - Chia, M.Y.; Huang, Y.F.; Koo, C.H.; Ng, J.L.; Ahmed, A.N.; El-Shafie, A. Long-term forecasting of monthly mean reference evapotranspiration using deep neural network: A comparison of training strategies and approaches. Appl. Soft Comput.
**2022**, 126, 109221. [Google Scholar] [CrossRef] - Thongkao, S.; Ditthakit, P.; Pinthong, S.; Salaeh, N.; Elkhrachy, I.; Linh, N.T.T.; Pham, Q.B. Estimating FAO Blaney-Criddle b-Factor Using Soft Computing Models. Atmosphere
**2022**, 13, 1536. [Google Scholar] [CrossRef] - Zhao, L.; Zhao, X.; Li, Y.; Shi, Y.; Zhou, H.; Li, X.; Wang, X.; Xing, X. Applicability of hybrid bionic optimization models with kernel-based extreme learning machine algorithm for predicting daily reference evapotranspiration: A case study in arid and semi-arid regions, China. Environ. Sci. Pollut. Res.
**2022**, 1–17. [Google Scholar] [CrossRef] [PubMed] - Ikram, R.M.A.; Mostafa, R.R.; Chen, Z.; Islam, A.R.M.T.; Kisi, O.; Kuriqi, A.; Zounemat-Kermani, M. Advanced Hybrid Metaheuristic Machine Learning Models Application for Reference Crop Evapotranspiration Prediction. Agronomy
**2023**, 13, 98. [Google Scholar] [CrossRef] - Elbeltagi, A.; Raza, A.; Hu, Y.; Al-Ansari, N.; Kushwaha, N.L.; Srivastava, A.; Vishwakarma, D.K.; Zubair, M. Data intelligence and hybrid metaheuristic algorithms-based estimation of reference evapotranspiration. Appl. Water Sci.
**2022**, 12, 1–18. [Google Scholar] [CrossRef] - Adnan, R.M.; Mostafa, R.R.; Islam, A.R.M.T.; Gorgij, A.D.; Kuriqi, A.; Kisi, O. Improving Drought Modeling Using Hybrid Random Vector Functional Link Methods. Water
**2021**, 13, 3379. [Google Scholar] [CrossRef] - Caesarendra, W.; Widodo, A.; Yang, B.-S. Application of relevance vector machine and logistic regression for machine degradation assessment. Mech. Syst. Signal Process.
**2010**, 24, 1161–1171. [Google Scholar] [CrossRef] - Pao, Y.-H.; Park, G.-H.; Sobajic, D.J. Learning and generalization characteristics of the random vector functional-link net. Neurocomputing
**1994**, 6, 163–180. [Google Scholar] [CrossRef] - Schölkopf, B.; Smola, A.J. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond; MIT-Press: Cambridge, MA, USA, 2002. [Google Scholar]
- Tipping, M.E. Sparse Bayesian learning and the relevance vector machine. J. Mach. Learn. Res.
**2001**, 1, 211–244. [Google Scholar] - Guo, R.; Wang, J.; Zhang, N.; Dong, J. tate prediction for the actuators of civil aircraft based on a fusion framework of relevance vector machine and autoregressive integrated moving average. Proceedings of the Institution of Mechanical Engineers Part I. J. Syst. Control Eng.
**2018**, 232, 095965181876297. [Google Scholar] - Zhang, Z.; Huang, C.; Ding, D.; Tang, S.; Han, B.; Huang, H. Hummingbirds optimization algorithm-based particle filter for maneuvering target tracking. Nonlinear Dyn.
**2019**, 97, 1227–1243. [Google Scholar] [CrossRef] - Zhao, W.; Wang, L.; Mirjalili, S. Artificial hummingbird algorithm: A new bio-inspired optimizer with its engineering applications. Comput. Methods Appl. Mech. Eng.
**2021**, 388, 114194. [Google Scholar] [CrossRef] - Bajec, I.L.; Heppner, F.H. Organized flight in birds. Anim. Behav.
**2009**, 78, 777–789. [Google Scholar] [CrossRef] - Zamani, H.; Nadimi-Shahraki, M.H.; Gandomi, A.H. QANA: Quantum-based avian navigation optimizer algorithm. Eng. Appl. Artif. Intell.
**2021**, 104, 104314. [Google Scholar] [CrossRef] - Dimitriadou, S.; Nikolakopoulos, K.G. Multiple Linear Regression Models with Limited Data for the Prediction of Reference Evapotranspiration of the Peloponnese, Greece. Hydrology
**2022**, 9, 124. [Google Scholar] [CrossRef] - Dimitriadou, S.; Nikolakopoulos, K.G. Artificial Neural Networks for the Prediction of the Reference Evapotranspiration of the Peloponnese Peninsula, Greece. Water
**2022**, 14, 2027. [Google Scholar] [CrossRef] - Shiri, J.; Zounemat-Kermani, M.; Kisi, O.; Karimi, S.M. Comprehensive assessment of 12 soft computing approaches for modelling reference evapotranspiration in humid locations. Meteorol. Appl.
**2019**, 27, e1841. [Google Scholar] [CrossRef][Green Version] - Adnan, R.M.; Mostafa, R.R.; Islam, A.R.M.T.; Kisi, O.; Kuriqi, A.; Heddam, S. Estimating reference evapotranspiration using hybrid adaptive fuzzy inferencing coupled with heuristic algorithms. Comput. Electron. Agric.
**2021**, 191, 106541. [Google Scholar] [CrossRef] - Niaghi, A.R.; Hassanijalilian, O.; Shiri, J. Estimation of Reference Evapotranspiration Using Spatial and Temporal Machine Learning Approaches. Hydrology
**2021**, 8, 25. [Google Scholar] [CrossRef]

**Figure 2.**Schematic structure of RVFL (adopted from [23]).

**Figure 3.**Three foraging behaviors of AHA [36].

**Figure 6.**Scatterplots of the observed and predicted ET by different RVFL- and RVM-based models in the test period using the best input combination –Station 1 (Faisalabad).

**Figure 7.**Taylor diagrams the predicted ET by different RVFL- and RVM-based models in the test period using the best input combination –Station 1 (Faisalabad).

**Figure 8.**Violin charts of the predicted ET by different RVFL- and RVM-based models in the test period using the best input combination –Station 1 (Faisalabad).

**Figure 9.**Scatterplots of the observed and predicted ET by different RVFL- and RVM-based models in the test period using the best input combination –Station 2 (Peshawar).

**Figure 10.**Taylor diagrams the predicted ET by different RVFL- and RVM-based models in the test period using the best input combination –Station 2 (Peshawar).

**Figure 11.**Violin charts of the predicted ET by different RVFL- and RVM-based models in the test period using the best input combination –Station 2 (Peshawar).

Tmin | Tmax | ET_{0} | Ra | |
---|---|---|---|---|

Faisalabad Station | ||||

Min. | 3.25 | 16.15 | 1.07 | 18.83 |

Max. | 28.8 | 42.2 | 10.53 | 41.34 |

Mean | 17.5 | 31.1 | 4.71 | 31.05 |

Skewness | −0.241 | −0.403 | 0.105 | −0.181 |

Std. dev. | 8.326 | 7.324 | 2.444 | 8.051 |

Peshawar Station | ||||

Min. | −1.45 | 15.95 | 1.41 | 17.23 |

Max. | 26.7 | 40.5 | 10.04 | 41.58 |

Mean | 16.5 | 29.6 | 5.36 | 30.25 |

Skewness | −0.176 | −0.344 | 0.267 | −0.147 |

Std. dev. | 7.936 | 6.824 | 2.127 | 8.697 |

Algorithm | Parameter | Value |
---|---|---|

AHA | Migration coefficient | 2n |

r | ∈ [0,1] | |

QANA | The number of flocks (𝑘) | 10 |

K’ | 9 | |

K’ | 50 | |

Common Settings | Population | 30 |

Number of iterations | 100 | |

Number of runs for each Algorithm | 30 |

**Table 3.**The results of the RFVL-based models for Faisalabad Station using the best training-test partition (75–25% scenario).

Inputs Combinations | Training Period | Test Period | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | R2 | NSE | RMSE | MAE | R2 | NSE | |

RFVL | ||||||||

(i) Tmin, Tmax | 0.9049 | 0.7135 | 0.8012 | 0.8012 | 0.9181 | 0.7261 | 0.7987 | 0.7940 |

(ii) Tmin, Ra | 0.7069 | 0.5387 | 0.9156 | 0.9128 | 0.7173 | 0.5458 | 0.9081 | 0.9058 |

(iii) Tmax, Ra | 0.6849 | 0.5206 | 0.9240 | 0.9210 | 0.6875 | 0.5234 | 0.9180 | 0.9177 |

(iv) Tmin, Tmax, Ra | 0.6919 | 0.5314 | 0.9216 | 0.9210 | 0.6983 | 0.5361 | 0.9166 | 0.9144 |

(v) Tmin, Tmax, Ra, MN | 0.6825 | 0.5183 | 0.9228 | 0.9203 | 0.6866 | 0.5222 | 0.9221 | 0.9186 |

Mean | 0.7342 | 0.5645 | 0.8970 | 0.8953 | 0.7416 | 0.5707 | 0.8927 | 0.8901 |

RFVL-AHA | ||||||||

(i) Tmin, Tmax | 0.8893 | 0.6939 | 0.8116 | 0.8114 | 0.9122 | 0.7140 | 0.8056 | 0.8021 |

(ii) Tmin, Ra | 0.6375 | 0.4868 | 0.9492 | 0.9473 | 0.6447 | 0.4901 | 0.9469 | 0.9456 |

(iii) Tmax, Ra | 0.6182 | 0.4569 | 0.9583 | 0.9567 | 0.6235 | 0.4625 | 0.9558 | 0.9543 |

(iv) Tmin, Tmax, Ra | 0.6081 | 0.4524 | 0.9621 | 0.9596 | 0.6164 | 0.4599 | 0.9587 | 0.9572 |

(v) Tmin, Tmax, Ra, MN | 0.6059 | 0.4507 | 0.9637 | 0.9605 | 0.6158 | 0.4562 | 0.9591 | 0.9580 |

Mean | 0.6718 | 0.5081 | 0.9290 | 0.9271 | 0.6825 | 0.5165 | 0.9252 | 0.9234 |

RFVL-QANA | ||||||||

(i) Tmin, Tmax | 0.8565 | 0.6715 | 0.8340 | 0.8321 | 0.9082 | 0.7051 | 0.8058 | 0.8055 |

(ii) Tmin, Ra | 0.6329 | 0.4816 | 0.9483 | 0.9462 | 0.6466 | 0.4861 | 0.9451 | 0.9448 |

(iii) Tmax, Ra | 0.6034 | 0.4517 | 0.9625 | 0.9617 | 0.6082 | 0.4553 | 0.9608 | 0.9604 |

(iv) Tmin, Tmax, Ra | 0.5958 | 0.4452 | 0.9649 | 0.9632 | 0.6068 | 0.4471 | 0.9625 | 0.9616 |

(v) Tmin, Tmax, Ra, MN | 0.6028 | 0.4459 | 0.9638 | 0.9617 | 0.6079 | 0.4491 | 0.9616 | 0.9609 |

Mean | 0.6583 | 0.4992 | 0.9347 | 0.9330 | 0.6755 | 0.5085 | 0.9272 | 0.9266 |

**Table 4.**The results of the RVM-based models for Faisalabad Station using the best training-test partition (75–25% scenario).

Inputs Combinations | Training Period | Test Period | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | R2 | NSE | RMSE | MAE | R2 | NSE | |

RVM | ||||||||

(i) Tmin, Tmax | 0.9042 | 0.7125 | 0.8074 | 0.8043 | 0.9115 | 0.7134 | 0.8061 | 0.8025 |

(ii) Tmin, Ra | 0.6873 | 0.5397 | 0.9253 | 0.9242 | 0.6960 | 0.5414 | 0.9204 | 0.9200 |

(iii) Tmax, Ra | 0.6827 | 0.5209 | 0.9279 | 0.9269 | 0.6854 | 0.5220 | 0.9259 | 0.9244 |

(iv) Tmin, Tmax, Ra | 0.6809 | 0.5204 | 0.9283 | 0.9274 | 0.6826 | 0.5227 | 0.9269 | 0.9260 |

(v) Tmin, Tmax, Ra, MN | 0.6695 | 0.5106 | 0.9324 | 0.9315 | 0.6734 | 0.5134 | 0.9307 | 0.9294 |

Mean | 0.7249 | 0.5608 | 0.9043 | 0.9029 | 0.7298 | 0.5626 | 0.9020 | 0.9005 |

RVM-AHA | ||||||||

(i) Tmin, Tmax | 0.8615 | 0.6672 | 0.8269 | 0.8257 | 0.9020 | 0.7139 | 0.8125 | 0.8087 |

(ii) Tmin, Ra | 0.6396 | 0.4835 | 0.9503 | 0.9486 | 0.6437 | 0.4890 | 0.9472 | 0.9460 |

(iii) Tmax, Ra | 0.5798 | 0.4348 | 0.9681 | 0.9681 | 0.6229 | 0.4594 | 0.9548 | 0.9546 |

(iv) Tmin, Tmax, Ra | 0.5546 | 0.4149 | 0.9775 | 0.9775 | 0.6068 | 0.4487 | 0.9621 | 0.9610 |

(v) Tmin, Tmax, Ra, MN | 0.5460 | 0.4062 | 0.9805 | 0.9805 | 0.6047 | 0.4462 | 0.9626 | 0.9617 |

Mean | 0.6363 | 0.4813 | 0.9407 | 0.9401 | 0.6760 | 0.5114 | 0.9278 | 0.9264 |

RVM-QANA | ||||||||

(i) Tmin, Tmax | 0.8067 | 0.6214 | 0.8560 | 0.8560 | 0.8900 | 0.7048 | 0.8164 | 0.8163 |

(ii) Tmin, Ra | 0.6119 | 0.4492 | 0.9569 | 0.9569 | 0.6413 | 0.4824 | 0.9484 | 0.9470 |

(iii) Tmax, Ra | 0.6076 | 0.4583 | 0.9612 | 0.9603 | 0.6143 | 0.4547 | 0.9587 | 0.9580 |

(iv) Tmin, Tmax, Ra | 0.5766 | 0.4262 | 0.9706 | 0.9706 | 0.6070 | 0.4542 | 0.9616 | 0.9609 |

(v) Tmin, Tmax, Ra, MN | 0.5604 | 0.4141 | 0.9765 | 0.9765 | 0.6062 | 0.4438 | 0.9639 | 0.9622 |

Mean | 0.6326 | 0.4738 | 0.9442 | 0.9441 | 0.6718 | 0.5080 | 0.9298 | 0.9289 |

**Table 5.**The results of the RFVL-based models for Peshawar Station using the best training-test partition (75–25% scenario).

Inputs Combinations | Training Period | Test Period | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | R2 | NSE | RMSE | MAE | R2 | NSE | |

RFVL | ||||||||

(i) Tmin, Tmax | 0.7186 | 0.5687 | 0.8571 | 0.8571 | 0.8369 | 0.6786 | 0.8525 | 0.8303 |

(ii) Tmin, Ra | 0.7392 | 0.5541 | 0.9220 | 0.9196 | 0.7554 | 0.6083 | 0.9136 | 0.9038 |

(iii) Tmax, Ra | 0.6791 | 0.5034 | 0.9333 | 0.9333 | 0.7144 | 0.5371 | 0.9159 | 0.9098 |

(iv) Tmin, Tmax, Ra | 0.6495 | 0.4879 | 0.9444 | 0.9441 | 0.6961 | 0.5452 | 0.9178 | 0.9128 |

(v) Tmin, Tmax, Ra, MN | 0.6483 | 0.4861 | 0.9446 | 0.9444 | 0.6943 | 0.5360 | 0.9195 | 0.9141 |

Mean | 0.6869 | 0.5200 | 0.9203 | 0.9197 | 0.7394 | 0.5810 | 0.9039 | 0.8942 |

RFVL-AHA | ||||||||

(i) Tmin, Tmax | 0.6998 | 0.5617 | 0.8649 | 0.8645 | 0.7852 | 0.6376 | 0.8342 | 0.8366 |

(ii) Tmin, Ra | 0.5047 | 0.4150 | 0.9531 | 0.9352 | 0.5342 | 0.4101 | 0.9524 | 0.9301 |

(iii) Tmax, Ra | 0.4198 | 0.3594 | 0.9634 | 0.9467 | 0.4905 | 0.3828 | 0.9572 | 0.9447 |

(iv) Tmin, Tmax, Ra | 0.4268 | 0.3360 | 0.9651 | 0.9543 | 0.4367 | 0.3617 | 0.9623 | 0.9502 |

(v) Tmin, Tmax, Ra, MN | 0.3841 | 0.2876 | 0.9724 | 0.9635 | 0.3917 | 0.2904 | 0.9691 | 0.9567 |

Mean | 0.4870 | 0.3919 | 0.9438 | 0.9328 | 0.5277 | 0.4165 | 0.9350 | 0.9237 |

RFVL-QANA | ||||||||

(i) Tmin, Tmax | 0.6519 | 0.5101 | 0.8835 | 0.8824 | 0.7756 | 0.6052 | 0.8535 | 0.8413 |

(ii) Tmin, Ra | 0.4895 | 0.3827 | 0.9564 | 0.9361 | 0.5016 | 0.3921 | 0.9543 | 0.9348 |

(iii) Tmax, Ra | 0.3935 | 0.3060 | 0.9648 | 0.9526 | 0.4408 | 0.3451 | 0.9637 | 0.9514 |

(iv) Tmin, Tmax, Ra | 0.3426 | 0.2690 | 0.9693 | 0.9671 | 0.3754 | 0.2935 | 0.9689 | 0.9632 |

(v) Tmin, Tmax, Ra, MN | 0.3892 | 0.3022 | 0.9704 | 0.9581 | 0.4223 | 0.3435 | 0.9670 | 0.9541 |

Mean | 0.4533 | 0.3540 | 0.9489 | 0.9393 | 0.5031 | 0.3959 | 0.9415 | 0.9290 |

**Table 6.**The results of the RVM-based models for Peshawar Station using the best training-test partition (75–25% scenario).

Inputs Combinations | Training Period | Test Period | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | R2 | NSE | RMSE | MAE | R2 | NSE | |

RVM | ||||||||

(i) Tmin, Tmax | 0.7157 | 0.5647 | 0.8584 | 0.8583 | 0.8346 | 0.6260 | 0.8092 | 0.7915 |

(ii) Tmin, Ra | 0.6678 | 0.5502 | 0.9215 | 0.9136 | 0.6759 | 0.5900 | 0.9199 | 0.8929 |

(iii) Tmax, Ra | 0.6410 | 0.4934 | 0.9363 | 0.9333 | 0.6578 | 0.5414 | 0.9216 | 0.9201 |

(iv) Tmin, Tmax, Ra | 0.6100 | 0.4815 | 0.9328 | 0.9256 | 0.6435 | 0.5362 | 0.9289 | 0.9205 |

(v) Tmin, Tmax, Ra, MN | 0.5951 | 0.4725 | 0.9461 | 0.9452 | 0.6436 | 0.5369 | 0.9297 | 0.9283 |

Mean | 0.6459 | 0.5125 | 0.9190 | 0.9152 | 0.6911 | 0.5661 | 0.9019 | 0.8907 |

RVM-AHA | ||||||||

(i) Tmin, Tmax | 0.7060 | 0.5649 | 0.8629 | 0.8621 | 0.7853 | 0.6096 | 0.8577 | 0.8466 |

(ii) Tmin, Ra | 0.4567 | 0.3540 | 0.9499 | 0.9198 | 0.5385 | 0.4169 | 0.9474 | 0.9346 |

(iii) Tmax, Ra | 0.4284 | 0.3501 | 0.9620 | 0.9418 | 0.4338 | 0.3612 | 0.9575 | 0.9410 |

(iv) Tmin, Tmax, Ra | 0.4110 | 0.3409 | 0.9662 | 0.9560 | 0.4258 | 0.3529 | 0.9650 | 0.9534 |

(v) Tmin, Tmax, Ra, MN | 0.3618 | 0.2988 | 0.9695 | 0.9594 | 0.3688 | 0.3009 | 0.9665 | 0.9573 |

Mean | 0.4728 | 0.3817 | 0.9421 | 0.9278 | 0.5104 | 0.4083 | 0.9388 | 0.9266 |

RVM-QANA | ||||||||

(i) Tmin, Tmax | 0.5643 | 0.4390 | 0.9119 | 0.9119 | 0.7823 | 0.6087 | 0.8537 | 0.8480 |

(ii) Tmin, Ra | 0.4104 | 0.3208 | 0.9618 | 0.9534 | 0.4682 | 0.3880 | 0.9548 | 0.9312 |

(iii) Tmax, Ra | 0.3659 | 0.2810 | 0.9630 | 0.9630 | 0.3999 | 0.3185 | 0.9623 | 0.9498 |

(iv) Tmin, Tmax, Ra | 0.3088 | 0.2328 | 0.9736 | 0.9736 | 0.4347 | 0.3616 | 0.9685 | 0.9407 |

(v) Tmin, Tmax, Ra, MN | 0.3132 | 0.2452 | 0.9729 | 0.9729 | 0.3252 | 0.2696 | 0.9705 | 0.9668 |

Mean | 0.3925 | 0.3038 | 0.9566 | 0.9550 | 0.4821 | 0.3893 | 0.9420 | 0.9273 |

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**MDPI and ACS Style**

Mostafa, R.R.; Kisi, O.; Adnan, R.M.; Sadeghifar, T.; Kuriqi, A. Modeling Potential Evapotranspiration by Improved Machine Learning Methods Using Limited Climatic Data. *Water* **2023**, *15*, 486.
https://doi.org/10.3390/w15030486

**AMA Style**

Mostafa RR, Kisi O, Adnan RM, Sadeghifar T, Kuriqi A. Modeling Potential Evapotranspiration by Improved Machine Learning Methods Using Limited Climatic Data. *Water*. 2023; 15(3):486.
https://doi.org/10.3390/w15030486

**Chicago/Turabian Style**

Mostafa, Reham R., Ozgur Kisi, Rana Muhammad Adnan, Tayeb Sadeghifar, and Alban Kuriqi. 2023. "Modeling Potential Evapotranspiration by Improved Machine Learning Methods Using Limited Climatic Data" *Water* 15, no. 3: 486.
https://doi.org/10.3390/w15030486