# Hybridized Adaptive Neuro-Fuzzy Inference System with Metaheuristic Algorithms for Modeling Monthly Pan Evaporation

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

^{2}), Taylor diagram, and Violin plot. The results showed that maximum temperature was the most influential variable for evaporation estimation compared to the other input variables. The effect of periodicity input was investigated, demonstrating the efficacy of this variable in improving the models’ predictive accuracy. Among the models developed, the ANFIS-HHO and ANFIS-WOA models outperformed the other models, predicting Epan in the study stations with different combinations of input variables. Between these two models, ANFIS-WOA performed better than ANFIS-HHO. The results also proved the capability of the models when they were used for the prediction of Epan when given a study station using the data obtained for another station. Our study can provide insights into the development of predictive hybrid models when the analysis is conducted in data-scare regions.

## 1. Introduction

## 2. Case Study

^{2}, the basin is located in the south of the Yangtze River. It covers 12% of the Yangtze River Basin and is ranked as the world’s second-largest freshwater lake. The DLB is drained by four main rivers, i.e., Li River, Yuan River, Zi River, and Xiang River. The DLB has a humid climate with many variations (mean annual =1380 mm) in precipitation that result in drought periods and flood events. The mean annual temperature of the basin is 17 °C, with the lowest temperature of 4.2 °C in January and the highest temperature of 28.9 °C in July. This region is mainly known for rice production, although production has been reduced due to drought events in recent years. The basin has an altitude gradient ranging from 30 m in plain areas to 2500 m in mountainous areas. Monthly minimum and maximum temperature data, as well as evaporation data, for the period between 1962 and 2001 (480 months) at the three selected stations were collected from the China Meteorological Administration (CMA). A summary of the statistical characteristics of the data is listed in Table 2. For the application of the machine learning models, data were divided into two sets, with 75% (360 months) of the data for model training and the remaining 25% (120 months) for model validation [13,28,33].

## 3. Methods

#### 3.1. Adaptive Neuro-Fuzzy Inference System (ANFIS)

_{1}, x

_{2}, and x

_{3}are the input variables; and A

_{i}, B

_{i}, and C

_{i}are the fuzzy sets [36]. ANFIS consists of five layers which can be summarized as follows. In the first layer, all nodes are adaptive and their parameters (i.e., the premise parameters) are updated during the training process. For each node in the first layer, the linguistic label is calculated using the corresponding membership functions (MFs), as follows:

_{i}) of the fuzzy rule.

_{i}, β

_{i}, γ

_{i}, and δ

_{i}are the linear parameters (i.e., the consequent parameters) of the fuzzy rules. For the fifth layer, only one node is available and it provides the final response by calculating the overall output as a summation:

#### 3.2. Particle Swarm Optimization (PSO)

_{i}), and position, i.e., location (X

_{i}):

_{best}the optimal position of each particle and G

_{best}the cluster (i.e., entire population) historical optimal position (i.e., location) as follows:

_{1}and θ

_{2}are random numbers ranging from 0 to 1; δ

_{1}and δ

_{2}are the cognitive and social acceleration coefficients; ω is an inertia weight parameter; ${p}_{id}^{k}$ is the best position of the particle; and ${p}_{id}^{k}$ is the swarm best position [39].

#### 3.3. Whale Optimization Algorithm (WOA)

#### 3.4. Harris Hawk Optimization (HHO)

_{1}, α

_{2}, α

_{3}, and α

_{4}are random numbers ranging from 0 to 1; β and δ are the upper and lower bounds of variables; and ${X}_{rand}$ is a randomly selected Harris Hawk among all available individuals. The average position of the population is obtained using Equation (28):

_{0}is the prey energy before starting the flight forward; and T is the total number of iterations. From a mathematical point of view, the absolute value of E corresponds to two different cases: (i) if |E|≥1, we are in the exploration phase, and (ii) if |E|< 1, we are in the exploitation phase. We logically assume that the value of E

_{0}ranges in the interval from −1 to +1, and that its value should be updated at each iteration in decreasing or increasing ways. If the value of E

_{0}decreases from 0 to −1, we assume that the prey is weak, and if E

_{0}is higher than 0, we assume that the prey is strong [41].

_{5}is a random number ranging from 0 to 1. The function of the hard besiege can be formulated as follows (r ≥ 0.5 and |E| < 0.5):

#### 3.5. ANFIS Optimization Using PSO, WOA, and HHO

## 4. Results and Discussion

^{2}(0.8956) and NSE (0.8922) compared to ANFIS, with only Tmin or Ra as input. Among the double input combinations, Tmin and Tmax had the highest accuracy (see combinations 4–6 in Table 4). Of the all input combinations, three inputs (Tmin, Tmax, and Ra) provided the lowest RMSE and MAE, and the highest R

^{2}and NSE in estimating monthly Epan for all four models. The last input combination comprised optimum inputs (Tmin, Tmax, and Ra) and α (Opt inputs, α). Table 4 reveals that the periodicity positively affected the accuracy of the ANFIS-HHO and ANFIS-WOA models, while involving α decreased the performance of the ANFIS and ANFIS-PSO models in the estimation of monthly Epan. Among the models developed here, the ANFIS-WOA provided the best accuracy, and the model with Opt inputs, α (Tmin, Tmax, Ra, and α), had the lowest RMSE (0.3127 mm) and MAE (0.2561 mm), and the highest R

^{2}(0.9726) and NSE (0.9704), followed by the ANFIS-HHO and ANFIS-PSO models. Table 4 shows that metaheuristic algorithms improved the performance of ANFIS in both the training and testing phases; the modeling error (RMSE) decreased by about 1.1%, 24.7%, and 29.1% in the testing phases of ANFIS-PSO, ANFIS-HHO, and ANFIS-WOA, respectively. The dominance of ANFIS-WOA can be clearly examined from the mean values of the statistics, which improved from 0.6661 to 0.5032, 0.5260 to 0.4030, 0.8622 to 0.9082, and 0.8592 to 9054 for the RMSE, MAE, R2, and NSE for the split scenarios from ANFIS to ANFIS-WOA, respectively.

^{2}and NSE both in the training and testing phases of all models. Periodicity improved the estimation accuracy of the ANFIS-HHO and ANFIS-WOA models. Improvements in testing accuracy of ANFIS-WOA were 11.4%, 10.6%, 0.6, and 0.5% in terms of RMSE, MAE, R

^{2}, and NSE, respectively. It is clear from Table 5 that the metaheuristic algorithms improved the efficiency of ANFIS, and that the hybridized models performed much better in estimating monthly Epan using limited climatic input. The ANFIS-WOA model with three parameters and periodicity (Tmin, Tmax, Ra, and α) achieved the lowest RMSE (0.5886 mm) and MAE (0.4629 mm) and the highest R

^{2}(0.9526) and NSE (0.9507) in the testing phase. The WOA improved the accuracy of ANFIS with input Tmin, Tmax, and Ra by 23.9%, 20.7%, 4.5%, and 4.6% with respect to RMSE, MAE, R

^{2}, and NSE, respectively. The mean values of the comparison of RMSE, MAE, R2, and NSE statistics also endorsed the outperformed performance of ANFIS-WOA by improving these statistical values from 0.9029 to 0.7712, 0.6843 to 0.5893, 0.8484 to 0.9079, and 0.8461 to 0.9057 for the models ANFIS to ANFIS-WOA, respectively.

^{2}and NSE of the ANFIS-WOA model in the testing phase ranged from 0.8152 and 0.8126 to 0.9281 and 0.9286 for the worst and best input combinations. Again, the results demonstrated the efficiency of the metaheuristic algorithms for the improvement of the accuracy of ANFIS. The ANFIS-WOA performed the best in estimating monthly Epan using external input data.

^{2}and NSE of the ANFIS-WOA model in the testing phase ranged from 0.8153 and 0.8125 to 0.9330 and 0.9283. The improvements in the accuracy of ANFIS when the metaheuristic algorithms were used were evident. A comparison between the results presented in Table 7 and Table 8 reveals that the models performed better in estimating Epan at the Nanxian Station using the climatic input data of Jinzhou Station. The main reason for this might be related to the climatic characteristics of the stations. The Yueyang Station is located very near to Dongting Lake (Figure 1), and relative humidity may affect Epan. This study used the temperature input without considering this information.

## 5. Conclusions

^{2}≥ 0.81 and RMSE ≤ 1.04 mm performed the best using data from the all three stations, and successfully outperformed the other hybridized models in modeling Epan with local external temperature data. Our study can provide insights into the development of predictive models when analysis is based upon a narrow range of climatic variables.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The Dongting Lake basin and the study stations (Jingzhou, Nanxian, and Yueyang) in southern China.

**Figure 3.**Scatterplots of the observed and predicted Epan using the hybridized ANFIS models in the testing phase using the best input combination—Jingzhou Station.

**Figure 4.**Scatterplots of the observed and predicted Epan using the hybridized ANFIS models in the testing phase using the best input combination—Nanxian Station.

**Figure 5.**Scatterplots of the observed and predicted Epan using the hybridized ANFIS models in the testing phase using the best input combination—Yueyang Station.

**Figure 6.**Taylor diagram of the predicted Epan using the hybridized ANFIS models in the testing phase using the best input combination—Nanxian Station.

**Figure 7.**Violin plots of the predicted Epan using the hybridized ANFIS models in the testing phase using the best input combination—Nanxian Station.

**Table 1.**Comparing the performance of standalone ANFIS, hybridized ANFIS with metaheuristic algorithms, and other ML models in modeling Epan based on the previous reports.

Study | Developed Model(s) | Performance Comparison |
---|---|---|

Jasmin et al. [2] | ANFIS and hybridized ANFIS with FFA, GA, and PSO | The ANFIS-PSO model with R^{2} = 0.99 and RMSE = 9.73 performed the best. |

Wang et al. [9] | Multi-layer perceptron (MLP), generalized regression neural network (GRNN), fuzzy genetic (FG), LSSVM, MARS, ANFIS with grid partition (ANFIS-GP) | The ANFIS-GP model did not perform better than MLP and GRNN. It provided less accurate results than SVM. Therefore, the use of metaheuristic algorithms was recommended for improving ANFIS. |

Malik et al. [33] | MLP, co-active ANFIS (CANFIS), radial basis neural network (RBNN), and self-organizing map neural network (SOMNN) | The hybridized CANFIS model with RMSE = 0.627 was ranked among the most accurate models. |

Arya Azar et al. [34] | Least-squares support vector regression (LS-SVR), ANFIS, and ANFIS-HHO | The hybridized ANFIS-HHO (RMSE = 2.35 and NSE = 0.95) model successfully outperformed the other models. |

Seifi et al. [35] | Copula-based Bayesian Model Averaging (CBMA) and hybridized ANFIS with seagull optimization algorithm (SOA), crow search algorithm (CA), FA, and PSO | The hybridized models improved prediction accuracy by 20.35–64.36%. Thus, solidifying ANFIS with metaheuristic algorithms was recommended. |

Jingzhou Station | Nanxian Station | Yueyang Station | |||||||
---|---|---|---|---|---|---|---|---|---|

Whole Data | Training | Testing | Whole Data | Training | Testing | Whole Data | Training | Testing | |

Tmin | |||||||||

Mean | 13.336 | 13.016 | 13.639 | 13.562 | 13.444 | 13.918 | 14.384 | 14.193 | 14.960 |

Min. | −2.360 | −2.360 | 0.742 | −1.303 | −1.303 | 0.761 | −0.935 | −0.935 | 1.426 |

Max. | 26.039 | 26.039 | 25.165 | 27.148 | 26.706 | 27.148 | 28.165 | 27.952 | 28.165 |

Skewness | −0.102 | −0.850 | −0.071 | −0.051 | −0.050 | −0.047 | −0.051 | −0.046 | −0.056 |

Std. dev. | 7.928 | 8.743 | 7.671 | 8.325 | 8.373 | 8.170 | 8.375 | 8.444 | 8.138 |

Tmax | |||||||||

Mean | 21.397 | 21.301 | 21.685 | 20.774 | 20.681 | 21.053 | 20.769 | 20.716 | 20.929 |

Min. | 4.448 | 4.448 | 6.448 | 3.162 | 3.162 | 5.706 | 2.852 | 2.852 | 5.677 |

Max. | 36.284 | 36.284 | 34.726 | 35.084 | 35.084 | 34.445 | 35.174 | 35.174 | 34.116 |

Skewness | −0.138 | −0.125 | −0.176 | −0.139 | −0.130 | −0.162 | −0.122 | −0.111 | −0.154 |

Std. dev. | 8.446 | 8.514 | 8.232 | 8.534 | 8.614 | 8.283 | 8.511 | 8.600 | 8.236 |

Extraterrestrial radiation | |||||||||

Mean | 31.398 | 31.398 | 31.397 | 31.696 | 31.696 | 31.695 | 31.888 | 31.888 | 31.887 |

Min. | 19.753 | 19.753 | 19.753 | 20.382 | 20.382 | 20.382 | 20.797 | 20.797 | 20.797 |

Max. | 41.133 | 41.133 | 41.133 | 41.016 | 41.016 | 41.016 | 40.934 | 40.934 | 40.934 |

Skewness | −0.185 | −0.185 | −0.187 | −0.199 | −0.200 | −0.201 | −0.210 | −0.210 | −0.212 |

Std. dev. | 7.639 | 7.639 | 7.640 | 7.377 | 7.377 | 7.378 | 7.202 | 7.202 | 7.203 |

Evaporation | |||||||||

Mean | 3.630 | 3.653 | 3.562 | 3.385 | 3.256 | 3.773 | 3.956 | 3.872 | 4.207 |

Min. | 0.884 | 0.961 | 0.884 | 0.803 | 0.803 | 0.997 | 0.911 | 0.911 | 1.116 |

Max. | 10.619 | 10.619 | 7.861 | 9.087 | 9.087 | 9.045 | 11.119 | 11.119 | 11.029 |

Skewness | 0.605 | 0.666 | 0.332 | 0.706 | 0.753 | 0.543 | 0.846 | 0.894 | 0.729 |

Std. dev. | 1.816 | 1.857 | 1.683 | 1.810 | 1.751 | 1.926 | 2.185 | 2.177 | 2.189 |

Method/Algorithm | Parameter | Value |
---|---|---|

ANFIS | Error goal | 0 |

Increase rate | 1.1 | |

Initial step | 0.01 | |

ANFIS-DEcrease rate | 0.9 | |

Maximum epochs | 100 | |

PSO | Cognitive component (${c}_{1}$) | 2 |

Social component (${c}_{2}$) | 2 | |

inertia weight | 0.2–0.9 | |

HHO | $\beta $ | 1.5 |

${E}_{0}$ | $\in \left[-1.-1\right]$ | |

WOA | $a$ | $\in \left[0.2\right]$ |

${a}_{2}$ | $\in \left[-1.-2\right]$ | |

All algorithms | Population | 30 |

Number of iterations | 150 | |

Number of runs for each algorithm | 10 |

**Table 4.**Training and testing performances of the models for monthly Epan prediction—Jingzhou Station.

Model Inputs | Training | Test | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | NSE | RMSE | MAE | R^{2} | NSE | |

ANFIS | ||||||||

Tmin | 0.8500 | 0.6285 | 0.7907 | 0.7884 | 0.8846 | 0.6902 | 0.7464 | 0.7436 |

Tmax | 0.6340 | 0.4837 | 0.8836 | 0.8817 | 0.6508 | 0.5210 | 0.8956 | 0.8922 |

Ra | 1.0604 | 0.7956 | 0.6740 | 0.6712 | 0.8408 | 0.6658 | 0.7545 | 0.7521 |

Tmin, Tmax | 0.5162 | 0.3991 | 0.9246 | 0.9193 | 0.5135 | 0.4124 | 0.9185 | 0.9163 |

Tmin, Ra | 0.7856 | 0.5680 | 0.8211 | 0.8184 | 0.9212 | 0.7285 | 0.7845 | 0.7816 |

Tmax, Ra | 0.5744 | 0.4378 | 0.9044 | 0.9017 | 0.6186 | 0.4853 | 0.9109 | 0.9070 |

Tmin, Tmax, Ra | 0.4593 | 0.3562 | 0.9388 | 0.9362 | 0.4412 | 0.3510 | 0.9455 | 0.9426 |

Opt inputs, α | 0.4698 | 0.3587 | 0.9360 | 0.9321 | 0.4582 | 0.3534 | 0.9417 | 0.9378 |

Mean | 0.6687 | 0.5035 | 0.8592 | 0.8561 | 0.6661 | 0.5260 | 0.8622 | 0.8592 |

ANFIS-PSO | ||||||||

Tmin | 0.8111 | 0.6063 | 0.8093 | 0.8074 | 0.7851 | 0.6249 | 0.7798 | 0.7752 |

Tmax | 0.6182 | 0.4753 | 0.8892 | 0.8876 | 0.5628 | 0.4577 | 0.9020 | 0.8996 |

Ra | 1.0502 | 0.7867 | 0.6802 | 0.6775 | 0.8327 | 0.6560 | 0.7597 | 0.7564 |

Tmin, Tmax | 0.5121 | 0.3917 | 0.9269 | 0.9237 | 0.5180 | 0.4244 | 0.9236 | 0.9215 |

Tmin, Ra | 0.7493 | 0.5501 | 0.8312 | 0.8284 | 0.7706 | 0.5989 | 0.8105 | 0.7905 |

Tmax, Ra | 0.5305 | 0.4110 | 0.9184 | 0.9167 | 0.5326 | 0.4419 | 0.9183 | 0.9158 |

Tmin, Tmax, Ra | 0.4590 | 0.3529 | 0.9389 | 0.9358 | 0.4365 | 0.3218 | 0.9561 | 0.9542 |

Opt inputs, α | 0.4627 | 0.3565 | 0.9382 | 0.9362 | 0.4476 | 0.3327 | 0.9521 | 0.9493 |

Mean | 0.6491 | 0.4913 | 0.8665 | 0.8642 | 0.6107 | 0.4823 | 0.8753 | 0.8703 |

ANFIS-HHO | ||||||||

Tmin | 0.8029 | 0.5987 | 0.8131 | 0.8115 | 0.7368 | 0.5781 | 0.8168 | 0.8143 |

Tmax | 0.6163 | 0.4658 | 0.8899 | 0.8862 | 0.5268 | 0.4247 | 0.9094 | 0.9067 |

Ra | 0.8419 | 0.6159 | 0.7794 | 0.7754 | 0.7185 | 0.5647 | 0.8263 | 0.8242 |

Tmin, Tmax | 0.5051 | 0.3865 | 0.9290 | 0.9268 | 0.3749 | 0.2993 | 0.9609 | 0.9573 |

Tmin, Ra | 0.7498 | 0.5445 | 0.8370 | 0.8341 | 0.6813 | 0.5383 | 0.8439 | 0.8422 |

Tmax, Ra | 0.5132 | 0.3903 | 0.9236 | 0.9205 | 0.4892 | 0.3815 | 0.9261 | 0.9236 |

Tmin, Tmax, Ra | 0.4569 | 0.3528 | 0.9395 | 0.9372 | 0.3646 | 0.2958 | 0.9623 | 0.9604 |

Opt inputs, α | 0.4439 | 0.3405 | 0.9429 | 0.9404 | 0.3322 | 0.2746 | 0.9691 | 0.9675 |

Mean | 0.6163 | 0.4619 | 0.8818 | 0.8790 | 0.5280 | 0.4196 | 0.9019 | 0.8995 |

ANFIS-WOA | ||||||||

Tmin | 0.6970 | 0.4977 | 0.8590 | 0.8563 | 0.7155 | 0.5646 | 0.8274 | 0.8243 |

Tmax | 0.5722 | 0.4326 | 0.9051 | 0.9026 | 0.5148 | 0.4160 | 0.9125 | 0.9103 |

Ra | 0.8342 | 0.6083 | 0.7945 | 0.7918 | 0.7119 | 0.5585 | 0.8294 | 0.8261 |

Tmin, Tmax | 0.4385 | 0.3181 | 0.9443 | 0.9421 | 0.3643 | 0.2961 | 0.9618 | 0.9595 |

Tmin, Ra | 0.6943 | 0.4914 | 0.8602 | 0.8579 | 0.6706 | 0.5291 | 0.8480 | 0.8452 |

Tmax, Ra | 0.4947 | 0.3565 | 0.9291 | 0.9274 | 0.4147 | 0.3435 | 0.9447 | 0.9417 |

Tmin, Tmax, Ra | 0.4203 | 0.3122 | 0.9488 | 0.9457 | 0.3214 | 0.2604 | 0.9691 | 0.9658 |

Opt inputs, α | 0.4098 | 0.3050 | 0.9513 | 0.9492 | 0.3127 | 0.2561 | 0.9726 | 0.9704 |

Mean | 0.5701 | 0.4152 | 0.8990 | 0.8966 | 0.5032 | 0.4030 | 0.9082 | 0.9054 |

**Table 5.**Training and testing performances of the models for monthly Epan prediction—Nanxian Station.

Model Inputs | Training | Test | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | NSE | RMSE | MAE | R^{2} | NSE | |

ANFIS | ||||||||

Tmin | 0.6732 | 0.5114 | 0.8523 | 0.8503 | 1.0000 | 0.7716 | 0.8126 | 0.8103 |

Tmax | 0.5479 | 0.4118 | 0.9023 | 0.9014 | 0.8130 | 0.6330 | 0.9092 | 0.9067 |

Ra | 1.0960 | 0.8547 | 0.6082 | 0.6058 | 1.2974 | 0.9336 | 0.6258 | 0.6225 |

Tmin, Tmax | 0.4819 | 0.3566 | 0.9243 | 0.9221 | 0.7961 | 0.5879 | 0.9087 | 0.9064 |

Tmin, Ra | 0.6509 | 0.4962 | 0.8602 | 0.8583 | 0.9712 | 0.7511 | 0.8053 | 0.8032 |

Tmax, Ra | 0.5272 | 0.3971 | 0.9094 | 0.9067 | 0.7928 | 0.6259 | 0.9053 | 0.9027 |

Tmin, Tmax, Ra | 0.4395 | 0.3318 | 0.9370 | 0.9342 | 0.7738 | 0.5837 | 0.9114 | 0.9093 |

Opt inputs, α | 0.4548 | 0.3340 | 0.9328 | 0.9301 | 0.7792 | 0.5877 | 0.9091 | 0.9075 |

0.6089 | 0.4617 | 0.8658 | 0.8636 | 0.9029 | 0.6843 | 0.8484 | 0.8461 | |

ANFIS-PSO | ||||||||

Tmin | 0.6548 | 0.4979 | 0.8602 | 0.8583 | 0.9631 | 0.7320 | 0.8312 | 0.8283 |

Tmax | 0.5338 | 0.4048 | 0.9061 | 0.9042 | 0.7775 | 0.5987 | 0.9183 | 0.9156 |

Ra | 1.0342 | 0.7981 | 0.6512 | 0.6503 | 1.2263 | 0.8799 | 0.6744 | 0.6717 |

Tmin, Tmax | 0.4781 | 0.3500 | 0.9255 | 0.9228 | 0.7829 | 0.5947 | 0.9384 | 0.9352 |

Tmin, Ra | 0.6389 | 0.4878 | 0.8627 | 0.8601 | 0.9518 | 0.7155 | 0.8202 | 0.8179 |

Tmax, Ra | 0.5236 | 0.3903 | 0.9106 | 0.9084 | 0.7841 | 0.6032 | 0.9185 | 0.9156 |

Tmin, Tmax, Ra | 0.4333 | 0.3174 | 0.9388 | 0.9356 | 0.7635 | 0.5672 | 0.9321 | 0.9305 |

Opt inputs, α | 0.4395 | 0.3318 | 0.9370 | 0.9342 | 0.7692 | 0.5742 | 0.9286 | 0.9253 |

0.5920 | 0.4473 | 0.8740 | 0.8717 | 0.8773 | 0.6582 | 0.8702 | 0.8675 | |

ANFIS-HHO | ||||||||

Tmin | 0.6382 | 0.4943 | 0.8672 | 0.8647 | 0.8859 | 0.6745 | 0.8565 | 0.8537 |

Tmax | 0.5318 | 0.4014 | 0.9078 | 0.9053 | 0.7757 | 0.5975 | 0.9226 | 0.9205 |

Ra | 0.6944 | 0.5117 | 0.8428 | 0.8402 | 0.9909 | 0.7659 | 0.8126 | 0.8103 |

Tmin, Tmax | 0.4557 | 0.3478 | 0.9323 | 0.9304 | 0.7624 | 0.5782 | 0.9418 | 0.9402 |

Tmin, Ra | 0.6390 | 0.4950 | 0.8669 | 0.8652 | 0.8689 | 0.6570 | 0.8583 | 0.8563 |

Tmax, Ra | 0.5130 | 0.3831 | 0.9142 | 0.9127 | 0.7660 | 0.5912 | 0.9251 | 0.9227 |

Tmin, Tmax, Ra | 0.4273 | 0.3159 | 0.9405 | 0.9387 | 0.7484 | 0.5689 | 0.9413 | 0.9394 |

Opt inputs, α | 0.4197 | 0.3094 | 0.9426 | 0.9403 | 0.7243 | 0.5637 | 0.9432 | 0.9408 |

0.5399 | 0.4073 | 0.9018 | 0.8997 | 0.8153 | 0.6246 | 0.9002 | 0.8980 | |

ANFIS-WOA | ||||||||

Tmin | 0.5540 | 0.4073 | 0.8999 | 0.8973 | 0.8437 | 0.6499 | 0.8715 | 0.8702 |

Tmax | 0.5241 | 0.3850 | 0.9104 | 0.9082 | 0.7726 | 0.5955 | 0.9233 | 0.9214 |

Ra | 0.6925 | 0.5103 | 0.8436 | 0.8407 | 0.9853 | 0.7563 | 0.8153 | 0.8127 |

Tmin, Tmax | 0.3991 | 0.2846 | 0.9481 | 0.9456 | 0.7367 | 0.5131 | 0.9516 | 0.9493 |

Tmin, Ra | 0.5474 | 0.3982 | 0.9023 | 0.9007 | 0.8201 | 0.6345 | 0.8748 | 0.8721 |

Tmax, Ra | 0.4485 | 0.3396 | 0.9344 | 0.9324 | 0.7581 | 0.5842 | 0.9266 | 0.9234 |

Tmin, Tmax, Ra | 0.3687 | 0.2709 | 0.9557 | 0.9531 | 0.6643 | 0.5177 | 0.9472 | 0.9456 |

Opt inputs, α | 0.3643 | 0.2690 | 0.9567 | 0.9548 | 0.5886 | 0.4629 | 0.9526 | 0.9507 |

0.4873 | 0.3581 | 0.9189 | 0.9166 | 0.7712 | 0.5893 | 0.9079 | 0.9057 |

**Table 6.**Training and testing performances of the models for monthly Epan prediction—Yueyang Station.

Model Inputs | Training | Test | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | NSE | RMSE | MAE | R^{2} | NSE | |

ANFIS | ||||||||

Tmin | 0.7807 | 0.5907 | 0.8716 | 0.8694 | 0.9335 | 0.7200 | 0.8545 | 0.8524 |

Tmax | 0.7081 | 0.5217 | 0.8944 | 0.8917 | 0.8843 | 0.6562 | 0.8663 | 0.8638 |

Ra | 1.3904 | 1.0628 | 0.5920 | 0.5906 | 1.3849 | 1.0313 | 0.6239 | 0.6207 |

Tmin, Tmax | 0.5888 | 0.4411 | 0.9269 | 0.9243 | 0.8178 | 0.6095 | 0.8911 | 0.8893 |

Tmin, Ra | 0.7277 | 0.5562 | 0.8883 | 0.8856 | 0.8831 | 0.6875 | 0.8548 | 0.8524 |

Tmax, Ra | 0.6508 | 0.4872 | 0.9107 | 0.9082 | 0.8210 | 0.6139 | 0.8816 | 0.8793 |

Tmin, Tmax, Ra | 0.5479 | 0.4225 | 0.9369 | 0.9337 | 0.8178 | 0.6095 | 0.8911 | 0.8902 |

Opt inputs, α | 0.5227 | 0.4163 | 0.9423 | 0.9221 | 0.8013 | 0.6010 | 0.8960 | 0.8936 |

0.7396 | 0.5623 | 0.8704 | 0.8657 | 0.9180 | 0.6911 | 0.8449 | 0.8427 | |

ANFIS-PSO | ||||||||

Tmin | 0.7203 | 0.5499 | 0.8905 | 0.8884 | 0.8510 | 0.6482 | 0.8576 | 0.8543 |

Tmax | 0.6882 | 0.5097 | 0.9000 | 0.8986 | 0.8087 | 0.6412 | 0.8918 | 0.8893 |

Ra | 1.3282 | 1.0065 | 0.6277 | 0.6253 | 1.3177 | 0.9778 | 0.6616 | 0.6594 |

Tmin, Tmax | 0.5453 | 0.4245 | 0.9373 | 0.9352 | 0.7234 | 0.5715 | 0.9247 | 0.9225 |

Tmin, Ra | 0.7122 | 0.5466 | 0.8930 | 0.8907 | 0.8565 | 0.6624 | 0.8756 | 0.9726 |

Tmax, Ra | 0.6314 | 0.4759 | 0.9159 | 0.9124 | 0.7655 | 0.5968 | 0.9094 | 0.9071 |

Tmin, Tmax, Ra | 0.5328 | 0.4207 | 0.9401 | 0.9383 | 0.7244 | 0.5533 | 0.9275 | 0.9253 |

Opt inputs, α | 0.5146 | 0.4011 | 0.9441 | 0.9425 | 0.7135 | 0.5360 | 0.9300 | 0.9287 |

0.7091 | 0.5419 | 0.8811 | 0.8789 | 0.8451 | 0.6484 | 0.8723 | 0.8824 | |

ANFIS-HHO | ||||||||

Tmin | 0.7133 | 0.5482 | 0.8926 | 0.8901 | 0.8181 | 0.6167 | 0.8671 | 0.8643 |

Tmax | 0.6774 | 0.5040 | 0.9010 | 0.8993 | 0.8003 | 0.6042 | 0.8993 | 0.8972 |

Ra | 0.8873 | 0.6461 | 0.8338 | 0.8316 | 1.0387 | 0.7964 | 0.7984 | 0.7958 |

Tmin, Tmax | 0.5365 | 0.4244 | 0.9393 | 0.9374 | 0.7202 | 0.5452 | 0.9257 | 0.9235 |

Tmin, Ra | 0.7022 | 0.5383 | 0.8959 | 0.8932 | 0.7728 | 0.6045 | 0.8826 | 0.8804 |

Tmax, Ra | 0.6286 | 0.4746 | 0.9166 | 0.9145 | 0.7363 | 0.5749 | 0.9188 | 0.9157 |

Tmin, Tmax, Ra | 0.5108 | 0.3992 | 0.9467 | 0.9451 | 0.7093 | 0.5331 | 0.9333 | 0.9306 |

Opt inputs, α | 0.5041 | 0.3990 | 0.9464 | 0.9448 | 0.6649 | 0.5244 | 0.9396 | 0.9372 |

0.6450 | 0.4917 | 0.9090 | 0.9070 | 0.7826 | 0.5999 | 0.8956 | 0.8931 | |

ANFIS-WOA | ||||||||

Tmin | 0.6565 | 0.4804 | 0.9090 | 0.9072 | 0.8028 | 0.6025 | 0.8749 | 0.8723 |

Tmax | 0.6135 | 0.4371 | 0.9206 | 0.9183 | 0.7896 | 0.5957 | 0.9049 | 0.9027 |

Ra | 0.8862 | 0.6455 | 0.8342 | 0.8316 | 1.0365 | 0.7954 | 0.7991 | 0.7973 |

Tmin, Tmax | 0.4658 | 0.3378 | 0.9542 | 0.9524 | 0.7161 | 0.5378 | 0.9278 | 0.9252 |

Tmin, Ra | 0.6135 | 0.4524 | 0.9206 | 0.9182 | 0.7533 | 0.5795 | 0.8888 | 0.8856 |

Tmax, Ra | 0.5076 | 0.3780 | 0.9456 | 0.9427 | 0.7278 | 0.5647 | 0.9211 | 0.9202 |

Tmin, Tmax, Ra | 0.4658 | 0.3378 | 0.9542 | 0.9523 | 0.6508 | 0.5034 | 0.9433 | 0.9413 |

Opt inputs, α | 0.4636 | 0.3330 | 0.9546 | 0.9531 | 0.6370 | 0.5052 | 0.9422 | 0.9404 |

0.5841 | 0.4253 | 0.9241 | 0.9220 | 0.7642 | 0.5855 | 0.9003 | 0.8981 |

**Table 7.**Training and testing performances of the models for monthly Epan prediction—Nanxian Station using data from Jingzhou Station.

Model Inputs | Training | Test | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | NSE | RMSE | MAE | R^{2} | NSE | |

ANFIS | ||||||||

Tmin | 0.6863 | 0.5122 | 0.8467 | 0.8436 | 1.0422 | 0.9208 | 0.6846 | 0.6814 |

Tmax | 0.5509 | 0.4148 | 0.9012 | 0.8994 | 0.8229 | 0.6129 | 0.8752 | 0.8723 |

Ra | 1.0948 | 0.8537 | 0.6091 | 0.6072 | 1.2963 | 0.9326 | 0.6266 | 0.6237 |

Tmin, Tmax | 0.5229 | 0.3871 | 0.9109 | 0.9083 | 0.8157 | 0.6326 | 0.8326 | 0.8302 |

Tmin, Ra | 0.6536 | 0.4943 | 0.8608 | 0.8574 | 0.9752 | 0.8636 | 0.7236 | 0.7208 |

Tmax, Ra | 0.5340 | 0.4054 | 0.9070 | 0.9046 | 0.8122 | 0.6354 | 0.8708 | 0.8682 |

Tmin, Tmax, Ra | 0.5252 | 0.3871 | 0.9107 | 0.9081 | 0.8077 | 0.6133 | 0.8548 | 0.8517 |

Opt inputs, α | 0.5042 | 0.3775 | 0.9171 | 0.9153 | 0.8051 | 0.5966 | 0.8842 | 0.8823 |

ANFIS-PSO | ||||||||

Tmin | 0.6432 | 0.4883 | 0.8651 | 0.8623 | 0.9629 | 0.6984 | 0.8099 | 0.8073 |

Tmax | 0.5435 | 0.4107 | 0.9037 | 0.9014 | 0.8130 | 0.5933 | 0.8760 | 0.8732 |

Ra | 1.0193 | 0.7841 | 0.6612 | 0.6592 | 1.2116 | 0.8695 | 0.6841 | 0.6814 |

Tmin, Tmax | 0.5174 | 0.3839 | 0.9127 | 0.9103 | 0.8048 | 0.5793 | 0.8876 | 0.8852 |

Tmin, Ra | 0.6384 | 0.4889 | 0.8671 | 0.8652 | 0.9496 | 0.8353 | 0.8165 | 0.8137 |

Tmax, Ra | 0.5172 | 0.3938 | 0.9128 | 0.9101 | 0.8078 | 0.6172 | 0.8846 | 0.8824 |

Tmin, Tmax, Ra | 0.5032 | 0.3806 | 0.9172 | 0.9154 | 0.7661 | 0.5931 | 0.8936 | 0.8917 |

Opt inputs, α | 0.5004 | 0.3779 | 0.9183 | 0.9166 | 0.7561 | 0.5569 | 0.9049 | 0.9025 |

ANFIS-HHO | ||||||||

Tmin | 0.6377 | 0.4882 | 0.8674 | 0.8652 | 0.9155 | 0.6904 | 0.8456 | 0.8423 |

Tmax | 0.5427 | 0.4101 | 0.9040 | 0.9027 | 0.7970 | 0.5808 | 0.8956 | 0.8934 |

Ra | 0.6943 | 0.5118 | 0.8428 | 0.8403 | 0.9909 | 0.7659 | 0.8126 | 0.8102 |

Tmin, Tmax | 0.5077 | 0.3787 | 0.9159 | 0.9127 | 0.7733 | 0.5683 | 0.9124 | 0.9096 |

Tmin, Ra | 0.6318 | 0.4823 | 0.8698 | 0.8674 | 0.9181 | 0.7057 | 0.8474 | 0.8455 |

Tmax, Ra | 0.5124 | 0.3915 | 0.9144 | 0.9126 | 0.7761 | 0.5696 | 0.9102 | 0.9081 |

Tmin, Tmax, Ra | 0.4963 | 0.3692 | 0.9197 | 0.9173 | 0.7447 | 0.5520 | 0.9163 | 0.9145 |

Opt inputs, α | 0.4850 | 0.3634 | 0.9233 | 0.9204 | 0.7197 | 0.5290 | 0.9257 | 0.9223 |

ANFIS-WOA | ||||||||

Tmin | 0.5445 | 0.3947 | 0.9033 | 0.9014 | 0.9125 | 0.6886 | 0.8474 | 0.8453 |

Tmax | 0.5137 | 0.3791 | 0.9140 | 0.9126 | 0.7233 | 0.5627 | 0.9054 | 0.9024 |

Ra | 0.6925 | 0.5103 | 0.8436 | 0.8415 | 0.9854 | 0.7566 | 0.8152 | 0.8126 |

Tmin, Tmax | 0.4256 | 0.3038 | 0.9409 | 0.9382 | 0.7655 | 0.5545 | 0.9183 | 0.9157 |

Tmin, Ra | 0.5155 | 0.3794 | 0.9133 | 0.9104 | 0.9056 | 0.6803 | 0.8518 | 0.8485 |

Tmax, Ra | 0.4676 | 0.3478 | 0.9287 | 0.9257 | 0.7672 | 0.5526 | 0.9173 | 0.9152 |

Tmin, Tmax, Ra | 0.4265 | 0.3141 | 0.9407 | 0.9383 | 0.7434 | 0.5505 | 0.9216 | 0.9184 |

Opt inputs, α | 0.4157 | 0.3029 | 0.9436 | 0.9405 | 0.7085 | 0.5086 | 0.9281 | 0.9252 |

**Table 8.**Training and testing performances of the models for monthly Epan prediction—Nanxian Station using data from Yueyang Station.

Model Inputs | Training | Test | ||||||
---|---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | NSE | RMSE | MAE | R^{2} | NSE | |

ANFIS | ||||||||

Tmin | 0.6251 | 0.4813 | 0.8727 | 0.8702 | 0.9422 | 0.7294 | 0.8598 | 0.8573 |

Tmax | 0.5666 | 0.4249 | 0.8954 | 0.8923 | 0.8696 | 0.6485 | 0.8893 | 0.8871 |

Ra | 1.0969 | 0.8553 | 0.6076 | 0.6054 | 1.2981 | 0.9344 | 0.6253 | 0.6228 |

Tmin, Tmax | 0.5031 | 0.3775 | 0.9175 | 0.9156 | 0.8303 | 0.6020 | 0.9061 | 0.9043 |

Tmin, Ra | 0.5992 | 0.4639 | 0.8829 | 0.8801 | 0.9159 | 0.7081 | 0.8435 | 0.8407 |

Tmax, Ra | 0.5478 | 0.4116 | 0.9022 | 0.9004 | 0.8343 | 0.6328 | 0.8928 | 0.8902 |

Tmin, Tmax, Ra | 0.4574 | 0.3545 | 0.9318 | 0.9295 | 0.7993 | 0.6249 | 0.9231 | 0.9224 |

Opt inputs, α | 0.4845 | 0.3649 | 0.9236 | 0.9208 | 0.8245 | 0.6295 | 0.9094 | 0.9075 |

ANFIS-PSO | ||||||||

Tmin | 0.5909 | 0.4588 | 0.8861 | 0.8843 | 0.8513 | 0.6550 | 0.8617 | 0.8593 |

Tmax | 0.5560 | 0.4171 | 0.8992 | 0.8971 | 0.8379 | 0.6317 | 0.8986 | 0.8962 |

Ra | 1.0370 | 0.7993 | 0.6493 | 0.6472 | 1.2267 | 0.8791 | 0.6744 | 0.6721 |

Tmin, Tmax | 0.4762 | 0.3648 | 0.9261 | 0.9245 | 0.8131 | 0.6309 | 0.9104 | 0.9075 |

Tmin, Ra | 0.5914 | 0.4592 | 0.8859 | 0.8827 | 0.8887 | 0.6897 | 0.8540 | 0.8523 |

Tmax, Ra | 0.5301 | 0.3976 | 0.9084 | 0.9060 | 0.8163 | 0.6384 | 0.9052 | 0.9027 |

Tmin, Tmax, Ra | 0.4545 | 0.3437 | 0.9326 | 0.9304 | 0.7676 | 0.6138 | 0.9253 | 0.9228 |

Opt inputs, α | 0.4663 | 0.3575 | 0.9291 | 0.9275 | 0.8065 | 0.6248 | 0.9115 | 0.9095 |

ANFIS-HHO | ||||||||

Tmin | 0.5879 | 0.4567 | 0.8873 | 0.8852 | 0.8328 | 0.6272 | 0.8757 | 0.8724 |

Tmax | 0.5580 | 0.4201 | 0.8985 | 0.8956 | 0.8058 | 0.6170 | 0.9045 | 0.9023 |

Ra | 0.6944 | 0.5117 | 0.8427 | 0.8403 | 0.9909 | 0.7659 | 0.8126 | 0.8105 |

Tmin, Tmax | 0.4617 | 0.3594 | 0.9305 | 0.9283 | 0.7903 | 0.5989 | 0.9165 | 0.9146 |

Tmin, Ra | 0.5737 | 0.4471 | 0.8927 | 0.8904 | 0.8067 | 0.6245 | 0.8755 | 0.8721 |

Tmax, Ra | 0.5250 | 0.3953 | 0.9101 | 0.9085 | 0.7979 | 0.6116 | 0.9153 | 0.9127 |

Tmin, Tmax, Ra | 0.4399 | 0.3336 | 0.9369 | 0.9337 | 0.7665 | 0.6065 | 0.9272 | 0.9258 |

Opt inputs, α | 0.4549 | 0.3483 | 0.9328 | 0.9302 | 0.7825 | 0.6058 | 0.9167 | 0.9149 |

ANFIS-WOA | ||||||||

Tmin | 0.5377 | 0.4062 | 0.9057 | 0.9028 | 0.8079 | 0.6223 | 0.8769 | 0.8742 |

Tmax | 0.5048 | 0.3599 | 0.9169 | 0.9137 | 0.7963 | 0.6159 | 0.9097 | 0.9074 |

Ra | 0.6925 | 0.5103 | 0.8436 | 0.8405 | 0.9852 | 0.7561 | 0.8153 | 0.8125 |

Tmin, Tmax | 0.4260 | 0.3142 | 0.9408 | 0.9382 | 0.7806 | 0.5840 | 0.9225 | 0.9203 |

Tmin, Ra | 0.5009 | 0.3771 | 0.9182 | 0.9157 | 0.7995 | 0.6216 | 0.8941 | 0.8917 |

Tmax, Ra | 0.4516 | 0.3390 | 0.9335 | 0.9305 | 0.7923 | 0.6023 | 0.9199 | 0.9174 |

Tmin, Tmax, Ra | 0.4186 | 0.3113 | 0.9422 | 0.9401 | 0.7442 | 0.5771 | 0.9299 | 0.9268 |

Opt inputs, α | 0.4182 | 0.3082 | 0.9430 | 0.9416 | 0.7344 | 0.5768 | 0.9330 | 0.9283 |

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**MDPI and ACS Style**

Adnan Ikram, R.M.; Jaafari, A.; Milan, S.G.; Kisi, O.; Heddam, S.; Zounemat-Kermani, M. Hybridized Adaptive Neuro-Fuzzy Inference System with Metaheuristic Algorithms for Modeling Monthly Pan Evaporation. *Water* **2022**, *14*, 3549.
https://doi.org/10.3390/w14213549

**AMA Style**

Adnan Ikram RM, Jaafari A, Milan SG, Kisi O, Heddam S, Zounemat-Kermani M. Hybridized Adaptive Neuro-Fuzzy Inference System with Metaheuristic Algorithms for Modeling Monthly Pan Evaporation. *Water*. 2022; 14(21):3549.
https://doi.org/10.3390/w14213549

**Chicago/Turabian Style**

Adnan Ikram, Rana Muhammad, Abolfazl Jaafari, Sami Ghordoyee Milan, Ozgur Kisi, Salim Heddam, and Mohammad Zounemat-Kermani. 2022. "Hybridized Adaptive Neuro-Fuzzy Inference System with Metaheuristic Algorithms for Modeling Monthly Pan Evaporation" *Water* 14, no. 21: 3549.
https://doi.org/10.3390/w14213549