# On the Prediction of Evaporation in Arid Climate Using Machine Learning Model

^{*}

## Abstract

**:**

## 1. Introduction

_{m}) at two stations in India. Artificial neural network (MM-ANN) and multi-gene genetic programming (MGGP) posed the best results.

- To evaluate the performance of all four models, using the climate information of Arizona, United States, and compare the results by using statistical analysis.
- To explore the ability of the ANFIS model to improve the accuracy of daily evaporation estimation for the data set.
- To obtain the best model, in terms of accuracy and efficiency, for the arid environments in the United States.

## 2. Methodology

#### 2.1. Adaptive Neuro Fuzzy Inference System (ANFIS)

- (a)
- Can identify the relation between input and output without direct physical consideration.
- (b)
- It can work even when the training sets carry noise and/or measurement errors.
- (c)
- It can adapt situations in changing environments. Therefore, an adaptive neuro-fuzzy inference system (ANFIS) is preferred to maximize the benefit from the combination of both FIS and ANN model in one structure. ANFIS can be well-understood by the following diagram, as shown in Figure 1.

#### 2.2. Firefly Algorithm (FFA)

- Each firefly can engage another firefly.
- The attractiveness between two fireflies is calculated by the light intensity of each firefly.
- The brightness is correspondingly related to the light released by fireflies [14].

#### 2.3. Genetic Algorithm (ANFIS–GA)

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

## 3. Results and Discussion

#### 3.1. Data Description

#### 3.2. Model Accuracy Indicator

#### 3.3. Simulation Results

#### 3.4. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- ${Y}_{i\left(actual\right)}$: the output observational parameter;
- ${Y}_{i\left(model\right)}$: the y parameter predicted by the models;
- ${Y}_{i\left(model\right)}$: the mean predicted y parameter;
- M: the number of parameters;
- n: number of samples;
- ${E}_{NSC}$: the Nash–Sutcliffe test statistic;
- ${T}_{i.Actual}$: the ith value of actual data;
- ${T}_{i.Predicted}$: the ith value of predicted data.

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**Figure 2.**Location of the study area under consideration in this manuscript. (

**a**) zoom-out view; (

**b**) zoom-in view. Source: Internet.

**Figure 4.**Comparison of the target (predicted) and obtained output sample index of training data set for (

**a**) ANFIS, (

**b**) ANFIS–FFA, (

**c**) ANFIS–GA, and (

**d**) ANFIS–PSO, respectively, using the first combination of the data set.

**Figure 5.**Comparison of the target (predicted) and obtained output sample index of training data set for (

**a**) ANFIS, (

**b**) ANFIS–FFA, (

**c**) ANFIS–GA, and (

**d**) ANFIS–PSO, respectively, using the second combination of the data set.

**Figure 6.**Comparison of the target and obtained output sample index of the test data for (

**a**) ANFIS, (

**b**) ANFIS–FFA, (

**c**) ANFIS–GA, and (

**d**) ANFIS–PSO, respectively (first combination of the data set).

**Figure 7.**Comparison of the target and obtained output sample index of the test data for (

**a**) ANFIS, (

**b**) ANFIS–FFA, (

**c**) ANFIS–GA, and (

**d**) ANFIS–PSO, respectively (second combination of the data set).

Statistics | N | Min | 1st Q | X50 | 3rd Q | Max | Avg | SD | CV (%) | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|---|---|---|

All | 86 | 44 | 82.50 | 158 | 254.50 | 331 | 172.30 | 89.48 | 51.93 | 0.066 | −1.45 |

Train | 59 | 44 | 83 | 183 | 273 | 331 | 178.28 | 91.79 | 51.48 | 0.015 | −1.48 |

Test | 27 | 49 | 74.75 | 154 | 247.25 | 298 | 158.73 | 82.40 | 51.91 | 0.117 | −1.50 |

**Table 2.**Summary of model accuracy indicator test for the training data set (for the first combination data set), which was calculated in Excel.

R^{2} | VAF | RMSE | SI | MAE | MARE | RMSRE | MRE | BIAS | NASH | |
---|---|---|---|---|---|---|---|---|---|---|

ANFIS | 0.99 | 99.04 | 8.93 | 0.050 | −0.0008 | 0.044 | 0.001 | 0.001 | 0.001 | 0.99 |

FFA | 0.97 | 94.08 | 24.38 | 0.140 | 8.976 | 0.110 | 0.079 | 0.018 | 8.98 | 0.92 |

GA | 0.98 | 97.50 | 14.38 | 0.084 | 4.569 | 0.095 | 0.024 | 0.027 | 4.57 | 0.97 |

PSO | 0.99 | 98.85 | 9.73 | 0.054 | −0.167 | 0.040 | 0.001 | −0.001 | −1.69 | 0.98 |

**Table 3.**Summary of model accuracy indicator test for the testing data set (for the first combination data set), which was calculated in Excel.

R^{2} | VAF | RMSE | SI | MAE | MARE | RMSRE | MRE | BIAS | NASH | |
---|---|---|---|---|---|---|---|---|---|---|

ANFIS | 0.98 | 97.04 | 15.55 | 0.094 | −4.56 | 0.087 | 0.018 | −0.027 | −4.56 | 0.97 |

FFA | 0.97 | 93.11 | 24.39 | 0.148 | −8.98 | 0.118 | 0.154 | −0.400 | −8.98 | 0.93 |

GA | 0.98 | 97.51 | 14.38 | 0.087 | −4.57 | 0.101 | 0.033 | −0.421 | −4.57 | 0.97 |

PSO | 0.98 | 97.18 | 14.60 | 0.088 | 1.68 | 0.101 | 0.014 | 0.003 | 1.68 | 0.97 |

**Table 4.**Summary of model accuracy indicator test for the training data set (for the second combination data set), which was calculated in Excel.

R^{2} | VAF | RMSE | SI | MAE | MARE | RMSRE | MRE | BIAS | NASH | |
---|---|---|---|---|---|---|---|---|---|---|

ANFIS | 0.99 | 98.99 | 8.99 | 0.05 | 0.0002 | 0.046 | 0.0003 | 0.019 | 2.733 | 0.99 |

FFA | 0.98 | 97.93 | 12.80 | 0.07 | 0.001 | 0.082 | 0.0066 | −0.015 | 0.001 | 0.98 |

GA | 0.99 | 98.32 | 11.66 | 0.07 | 0.403 | 0.072 | 0.0073 | −0.011 | 0.403 | 0.98 |

PSO | 0.99 | 99.11 | 8.44 | 0.05 | −0.040 | 0.042 | 0.0016 | −0.001 | −0.040 | 0.99 |

**Table 5.**Summary of model accuracy indicator test for test data set (for the second combination data set), which was calculated in Excel.

R^{2} | VAF | RMSE | SI | MAE | MARE | RMSRE | MRE | BIAS | NASH | |
---|---|---|---|---|---|---|---|---|---|---|

ANFIS | 0.99 | 98.42 | 11.94 | 0.07 | 3.73 | 0.062 | 0.007 | 0.025 | 3.73 | 0.98 |

FFA | 0.98 | 97.45 | 15.03 | 0.09 | 4.25 | 0.076 | 0.018 | 0.014 | 4.25 | 0.97 |

GA | 0.98 | 97.52 | 14.63 | 0.08 | 3.50 | 0.073 | 0.024 | 0.010 | 3.50 | 0.97 |

PSO | 0.98 | 97.50 | 15.08 | 0.09 | 4.61 | 0.081 | 0.0004 | 0.024 | 4.61 | 0.97 |

Type of Model | Training Data | Test Data | ||||||
---|---|---|---|---|---|---|---|---|

MSE | RMSE | MEAN | STD | MSE | RMSE | MEAN | STD | |

GA | 146.92 | 12.12 | −2.82 | 11.89 | 206.79 | 14.38 | −4.57 | 13.89 |

ANFIS | 58.23 | 7.63 | −8.16 | 7.69 | 241.72 | 15.54 | −4.56 | 15.14 |

PSO | 58.75 | 7.66 | 0.11 | 7.73 | 213.05 | 14.59 | 1.68 | 14.77 |

FFA | 507.20 | 22.52 | −4.87 | 22.17 | 594.80 | 24.38 | −8.97 | 23.10 |

Model Name | Run Time |
---|---|

ANFIS | 5 to 10 min |

ANFIS–FFA | 30 min to 3 h |

ANFIS–PSO | 10 to 30 min |

ANFIS–GA | 10 to 30 min |

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**MDPI and ACS Style**

Jasmine, M.; Mohammadian, A.; Bonakdari, H.
On the Prediction of Evaporation in Arid Climate Using Machine Learning Model. *Math. Comput. Appl.* **2022**, *27*, 32.
https://doi.org/10.3390/mca27020032

**AMA Style**

Jasmine M, Mohammadian A, Bonakdari H.
On the Prediction of Evaporation in Arid Climate Using Machine Learning Model. *Mathematical and Computational Applications*. 2022; 27(2):32.
https://doi.org/10.3390/mca27020032

**Chicago/Turabian Style**

Jasmine, Mansura, Abdolmajid Mohammadian, and Hossein Bonakdari.
2022. "On the Prediction of Evaporation in Arid Climate Using Machine Learning Model" *Mathematical and Computational Applications* 27, no. 2: 32.
https://doi.org/10.3390/mca27020032