# Evaluating Three Supervised Machine Learning Algorithms (LM, BR, and SCG) for Daily Pan Evaporation Estimation in a Semi-Arid Region

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## Abstract

**:**

^{2}(greater than 84%), Nash-Sutcliff efficiency (greater than 0.8) and normalized RMSE (less than 0.1) all indicate the reliability of the estimates provided for the daily pan evaporation. In the comparison between the studied training algorithms, two algorithms, BR and SCG, in most cases, showed better performance than the powerful and common LM algorithm. The obtained results suggest that future researchers in this field consider BR and SCG training algorithms for the supervised training of MLP for the numerical estimation of pan evaporation by the MLP model.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Region and the Data

#### 2.2. Multilayer Perceptron (MLP) Neural Network

#### 2.3. Learning Algorithms for MLP Neural Network

#### 2.3.1. Levenberg–Marquardt (LM)

#### 2.3.2. Bayesian Regularization

#### 2.3.3. Scaled Conjugate Gradient

#### 2.4. Evaluating the Estimations

## 3. Results

^{2}value, which can be seen in bold in the table. Therefore, on this basis, from now on, in the entire article, the scenarios related to each arrangement of components with the symbols T (temperature-based), F (pressure-based), RH (humidity-based), T-F (temperature–pressure-based), T-RH (temperature–humidity-based), F-RH (pressure–humidity-based) and T-F-RH (temperature–pressure–humidity-based) are shown (according to the components column in Table 2). It should be noted that according to the principle of parsimony, in cases where the R

^{2}value obtained by the multiple linear regression method did not change significantly, the scenario with the least number of input variables was considered the selected scenario. For example, in base pressure scenarios (F), scenarios S7 and S9 have R

^{2}equal to 73.4% and 73.5%, respectively. In this case, the difference in R

^{2}is very small and can be ignored, therefore, considering that S7 with 2 variables and S9 with 4 variables achieved this amount of R

^{2}, the scenario with the least number of input variables (i.e., S7) was chosen as the best F scenario. Or in the T-F-RH scenarios, the S45 scenario with 9 variables achieved R

^{2}equal to 76.7%. Meanwhile, the S51 scenario with 15 variables could only improve the performance by 0.1% (R

^{2}= 76.8%); Therefore, it is obvious that based on parsimony, the S45 scenario is introduced as the best scenario of T-F-RH.

^{2}among the three models MLP-LM, MLP-BR, and MLP-SCG shows that the models have minor performance differences. However, in almost all 7 scenarios examined, this minor difference indicates the superiority of BR and SCG algorithms in MLP model training, compared with the common LM training algorithm. According to these graphs, the weakest performance is observed in the humidity-based single-component scenario (RH) where R

^{2}is equal to 67.23%, and it is the result of MLP model training by the LM algorithm. The best estimates presented always belong to the temperature–pressure–humidity-based three-component scenario (T-F-RH), which has the highest R

^{2}among all scenarios. In this scenario, the best training of the MLP model was provided by the SCG-supervised algorithm and the weakest was provided by the LM (R

^{2}equal to 84.39% and 84.01%, respectively). Of course, in the meantime, the two-component scenarios such as T-F and F-RH also had relatively good performances, in which the amount of R

^{2}was very close to the three-component scenario of T-F-RH (T-F: 83.16% < R

^{2}< 83.95%; F-RH: 82.19% < R

^{2}< 83.82%). To check the performance among the scenarios, probability plots were drawn for the error of the models (Figure 9). This diagram is drawn simultaneously for the training and testing phases.

## 4. Discussion

^{2}was around 0.650–0.692; which is actually weaker than the results of the present study. The reason for this difference can be related to the difference in the climatic and geographical conditions of the two regions. In addition, considering different combinations of meteorological variables as input of supervised algorithm can improve hydrological modeling, which is in same direction with finding of Mohammadi et al. [60] and Moazenzadeh et al. [61], and applying different supervised learning methods can have different results under various types of climates.

^{2}equal to 98.5% had a relatively better performance than MLP-LM with R

^{2}equal to 98.3%. In the current research, the MLP-SCG model (R

^{2}equal to 84.39%) was evaluated better than the MLP-LM model (R

^{2}equal to 84.01%) for pan evaporation modeling. Additionally, in the research of Tezel and Buyukyildiz [68], the modeling of the monthly pan evaporation parameter in the southwestern part of Turkey using different learning algorithms showed that the MLP-SCG model is superior to the MLP-LM model according to the performance indicators R

^{2}, RMSE, and MAE. However, the performance of MLP-SCG and MLP-LM models in simulating monthly evaporation in research of Tezel and Buyukyildiz [68] (R

^{2}equal to 90.5% for MLP-SCG and 90% for MLP-LM) was better than the current research, which can be related to the difference in climatic and geographical conditions of the two regions.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ANN | Artificial neural network |

ANFIS | Adaptive neuro-fuzzy inference system |

T | Mean air temperature |

BR | Bayesian regularization |

R^{2} | Coefficient of determination |

Tdew | Dew point temperature |

ELM | Extreme learning machine |

FG | Fuzzy genetic |

GEP | Gene expression programming |

GRNN | Generalized regression neural network |

GDX | Gradient descent with variable learning rate backpropagation |

KSOFM | Kohonen self-organizing feature maps |

LSSVM | Least square support vector machine |

LM | levenberg marquardt |

Tmax | Maximum air temperature |

Fmax | Maximum pressure |

RHmax | Maximum relative humidity |

F | Mean pressure |

Tmin | Minimum air temperature |

Fmin | Minimum pressure |

RHmin | Minimum relative humidity |

MLP | Multilayer perceptron |

MLR | Multiple linear regression |

MARS | Multivariate adaptive regression spline |

NS | Nash Sutcliff |

NNARX | Neural network autoregressive with exogenous input |

Epan | Pan evaporation |

P | Precipitation |

QRF | Quantile regression forests |

RBNN | Radial basis neural networks |

RF | Random forests |

RH | Relative humidity |

RH03 | Relative humidity at 03:00 |

RH09 | Relative humidity at 09:00 |

RH15 | Relative humidity at 15:00 |

RVM | Relevance vector machine |

RP | Resilient backpropagation |

RMSE | Root Mean Square Error |

SCG | Scaled conjugate gradient |

SOMNN | self-organizing feature map neural network |

RS | Solar radiation |

SS | Stephens and Stewart |

S | Sunshine |

SVM | Support vector machine |

VP | Vapor pressure |

Twet | Wet-bulb temperature |

WI | Willmott’s index of agreement |

WS | Wind speed |

## Appendix A

Reference | Study Region | Models | Input Variables |
---|---|---|---|

Ashrafzadeh et al. [3] | Iran | MLP, SVM, SOMNN | Tmin, Tmax, T, RH, P, WS, S |

Kişi [7] | USA | MLP, RBNN, MLR, SS | T, RS, WS, RH |

Ali Ghorbani et al. [30] | Iran | MLP | Tmin, Tmax, WS, RH, S |

Ghorbani et al. [36] | Iran | MLP, SVM | Tmin, Tmax, WS, RH, S |

Kim et al. [33] | Iran | MLP, KSOFM, GEP, MLR | T, WS, RH, S, RS |

Wang et al. [34] | China | MLP, GRNN, FG, LSSVM, MARS, ANFIS, MLR, SS | T, RS, S, RH, WS |

Ashrafzadeh et al. [21] | Iran | MLP, SVM | Tmax, RHmax, RHmin, WS, S |

Ehteram et al. [35] | Malaysia | MLP | T, WS, RH, RS |

Al-Mukhtar [6] | Iraq | RF, QRF, SVM, MLR, ANN | Tmax, Tmin, RH, WS |

Zounemat-Kermani et al. [28] | Turkey | NNARX, GEP, ANFIS | T, RS, RH, WS |

Deo et al. [29] | Australia | RVM, ELM, MARS | Tmax, Tmin, RS, VP, P |

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**Figure 4.**Results of Pearson correlation test between the meteorological variables and pan evaporation (sorted due to the correlation intensity).

**Figure 5.**Time series plots of the MLP outputs learned by Levenberg–Marquardt algorithm, and the observational pan evaporation.

**Figure 6.**Time series plots of the MLP outputs learned by Bayesian regularization algorithm, and the observational pan evaporation.

**Figure 7.**Time series plots of the MLP outputs learned by scaled conjugate gradient algorithm, and the observational pan evaporation.

**Figure 10.**Comparing the performance of supervised learning algorithms in each scenario, based on NRMSE and NS criteria.

**Table 1.**Details of the daily datasets of the Shiraz site are obtained from the Iranian Meteorological Organization.

Variable | Training Period (2006–2017) * | Validation Period (2018–2019) | Validation Period (2020–2021) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Min. | Max. | Average | STD. | Min. | Max. | Average | STD. | Min. | Max. | Average | STD. | |

Tmax (°C) | 2.0 | 42.6 | 26.7 | 9.4 | 9.0 | 42.4 | 27.1 | 9.5 | 5.2 | 42.4 | 27.6 | 9.2 |

Tmin (°C) | −8.1 | 26.6 | 10.2 | 7.9 | −6.0 | 27.8 | 10.5 | 7.9 | −5.8 | 25.0 | 10.2 | 7.8 |

T (°C) | −1.1 | 34.7 | 19.0 | 9.0 | 0.9 | 35.5 | 19.3 | 9.2 | 1.5 | 34.0 | 19.4 | 9.0 |

Tdew (°C) | −16.7 | 16.7 | 0.8 | 4.9 | −16.4 | 12.6 | −0.1 | 5.0 | −18.8 | 15.1 | −1.1 | 5.4 |

Twet (°C) | −4.1 | 21.4 | 10.1 | 5.0 | −1.5 | 18.4 | 10.0 | 4.7 | −1.8 | 20.0 | 9.8 | 4.7 |

Fmax (mbar) | 840.9 | 865.6 | 852.7 | 4.1 | 841.6 | 862.7 | 852.9 | 4.2 | 842.0 | 864.3 | 853.1 | 4.1 |

Fmin (mbar) | 837.7 | 861.1 | 849.5 | 3.9 | 838.8 | 859.2 | 849.5 | 4.1 | 838.8 | 860.4 | 849.7 | 4.0 |

F (mbar) | 839.9 | 862.8 | 851.0 | 3.9 | 840.5 | 860.6 | 851.2 | 4.1 | 840.7 | 861.7 | 851.3 | 3.9 |

VP (mbar) | 5.7 | 56.9 | 26.2 | 13.5 | 7.8 | 58.9 | 26.7 | 14.4 | 7.0 | 55.5 | 26.8 | 13.9 |

RHmax (%) | 14.0 | 100.0 | 60.1 | 20.6 | 11.0 | 100.0 | 59.1 | 24.2 | 14.0 | 100.0 | 56.5 | 24.8 |

RHmin (%) | 2.0 | 93.0 | 19.4 | 14.6 | 2.0 | 97.0 | 18.5 | 16.3 | 1.0 | 98.0 | 16.1 | 14.9 |

RH (%) | 7.3 | 98.3 | 36.8 | 18.2 | 6.6 | 98.9 | 36.4 | 20.6 | 7.3 | 99.3 | 33.8 | 20.0 |

RH03 (%) | 12.0 | 100.0 | 58.0 | 19.9 | 11.0 | 100.0 | 56.3 | 23.5 | 13.0 | 100.0 | 53.8 | 23.7 |

RH09 (%) | 2.0 | 100.0 | 23.9 | 17.4 | 2.0 | 100.0 | 23.3 | 19.0 | 2.0 | 100.0 | 20.5 | 18.3 |

RH15 (%) | 2.0 | 100.0 | 28.9 | 19.7 | 2.0 | 100.0 | 28.3 | 21.6 | 1.0 | 100.0 | 26.1 | 21.2 |

Epan (mm) | 0.0 | 18.8 | 7.0 | 3.9 | 0.1 | 17.8 | 6.7 | 4.2 | 0.1 | 18.2 | 6.8 | 4.2 |

Components | Scenario | Inputs | R^{2} |
---|---|---|---|

Temperature (T) | S1 | T | 67.1% |

S2 | T, Tmax | 67.1% | |

S3 | T, Tmax, Tmin | 67.4% | |

S4 | T, Tmax, Tmin, Twet | 72.4% | |

S5 * | T, Tmax, Tmin, Twet, Tdew | 74.4% | |

Pressure (F) | S6 | VP | 72.4% |

S7 | VP, F | 73.4% | |

S8 | VP, F, Fmax | 73.5% | |

S9 | VP, F, Fmax, Fmin | 73.5% | |

Relative humidity (RH) | S10 | RHmax | 57.7% |

S11 | RHmax, RH03 | 57.7% | |

S12 | RHmax, RH03, RH | 57.8% | |

S13 | RHmax, RH03, RH, RH15 | 57.8% | |

S14 | RHmax, RH03, RH, RH15, RH09 | 58.9% | |

S15 | RHmax, RH03, RH, RH15, RH09, RHmin | 59.1% | |

Temperature and pressure (T–F) | S16 | VP, T | 73.2% |

S17 | VP, T, Tmax | 73.2% | |

S18 | VP, T, Tmax, Tmin | 73.4% | |

S19 | VP, T, Tmax, Tmin, Twet | 75.4% | |

S20 | VP, T, Tmax, Tmin, Twet, F | 76.1% | |

S21 | VP, T, Tmax, Tmin, Twet, F, Fmax | 76.3% | |

S22 | VP, T, Tmax, Tmin, Twet, F, Fmax, Fmin | 76.3% | |

S23 | VP, T, Tmax, Tmin, Twet, F, Fmax, Fmin, Tdew | 76.5% | |

Temperature and relative humidity (T–RH) | S24 | T, Tmax, Tmin, RHmax | 70.4% |

S25 | T, Tmax, Tmin, RHmax, RH03 | 70.4% | |

S26 | T, Tmax, Tmin, RHmax, RH03, RH | 70.5% | |

S27 | T, Tmax, Tmin, RHmax, RH03, RH, Twet | 73.6% | |

S28 | T, Tmax, Tmin, RHmax, RH03, RH, Twet, RH15 | 73.6% | |

S29 | T, Tmax, Tmin, RHmax, RH03, RH, Twet, RH15, RH09 | 73.6% | |

S30 | T, Tmax, Tmin, RHmax, RH03, RH, Twet, RH15, RH09, RHmin | 73.7% | |

S31 | T, Tmax, Tmin, RHmax, RH03, RH, Twet, RH15, RH09, RHmin, Tdew | 75.1% | |

Pressure and relative humidity (F–RH) | S32 | VP, RHmax | 74.3% |

S33 | VP, RHmax, RH03 | 74.4% | |

S34 | VP, RHmax, RH03, RH | 74.4% | |

S35 | VP, RHmax, RH03, RH, F | 75.6% | |

S36 | VP, RHmax, RH03, RH, F, Fmax | 75.7% | |

S37 | VP, RHmax, RH03, RH, F, Fmax, Fmin | 75.7% | |

S38 | VP, RHmax, RH03, RH, F, Fmax, Fmin, RH15 | 75.8% | |

S39 | VP, RHmax, RH03, RH, F, Fmax, Fmin, RH15, RH09 | 75.8% | |

S40 | VP, RHmax, RH03, RH, F, Fmax, Fmin, RH15, RH09, RHmin | 75.8% | |

Temperature, pressure and relative humidity (T–F–RH) | S41 | VP, T, Tmax, Tmin, RHmax | 75.6% |

S42 | VP, T, Tmax, Tmin, RHmax, RH03 | 75.7% | |

S43 | VP, T, Tmax, Tmin, RHmax, RH03, RH | 75.8% | |

S44 | VP, T, Tmax, Tmin, RHmax, RH03, RH, Twet | 75.9% | |

S45 | VP, T, Tmax, Tmin, RHmax, RH03, RH, Twet, F | 76.7% | |

S46 | VP, T, Tmax, Tmin, RHmax, RH03, RH, Twet, F, Fmax | 76.7% | |

S47 | VP, T, Tmax, Tmin, RHmax, RH03, RH, Twet, F, Fmax, Fmin | 76.7% | |

S48 | VP, T, Tmax, Tmin, RHmax, RH03, RH, Twet, F, Fmax, Fmin, RH15 | 76.8% | |

S49 | VP, T, Tmax, Tmin, RHmax, RH03, RH, Twet, F, Fmax, Fmin, RH15, RH09 | 76.8% | |

S50 | VP, T, Tmax, Tmin, RHmax, RH03, RH, Twet, F, Fmax, Fmin, RH15, RH09, RHmin | 76.8% | |

S51 | VP, T, Tmax, Tmin, RHmax, RH03, RH, Twet, F, Fmax, Fmin, RH15, RH09, RHmin, Tdew | 76.8% |

**Table 3.**Evaluation metrics for the pan evaporation modeling of MLP, learned by Levenberg–Marquardt algorithm (MLP-LM).

Input Scenario | Transfer Function | Network Makeup * | Train | Validation | Test | |||
---|---|---|---|---|---|---|---|---|

RMSE | WI | RMSE | WI | RMSE | WI | |||

T | satlin | 12-10-1 | 1.853 | 0.932 | 1.836 | 0.945 | 1.832 | 0.944 |

F | satlin | 15-10-1 | 1.939 | 0.924 | 1.879 | 0.939 | 1.962 | 0.931 |

RH | satlin | 12-12-1 | 2.395 | 0.871 | 2.295 | 0.907 | 2.733 | 0.866 |

T-F | satlin | 15-10-1 | 1.773 | 0.939 | 1.739 | 0.951 | 1.779 | 0.947 |

T-RH | satlin | 18-12-1 | 1.844 | 0.934 | 1.838 | 0.945 | 1.861 | 0.942 |

F-RH | tansig | 18-18-1 | 1.799 | 0.935 | 1.686 | 0.953 | 1.791 | 0.945 |

T-F-RH ** | tansig | 12-12-1 | 1.797 | 0.936 | 1.652 | 0.956 | 1.747 | 0.949 |

**Table 4.**Evaluation metrics for the pan evaporation modeling of MLP, learned by Bayesian regularization algorithm (MLP-BR).

Input Scenario | Transfer Function | Network Makeup * | Train | Validation | Test | |||
---|---|---|---|---|---|---|---|---|

RMSE | WI | RMSE | WI | RMSE | WI | |||

T | tansig | 12-10-1 | 1.887 | 0.928 | 1.777 | 0.948 | 1.807 | 0.943 |

F | satlin | 12-12-1 | 1.892 | 0.928 | 1.821 | 0.943 | 1.880 | 0.937 |

RH | satlin | 10-10-1 | 2.410 | 0.872 | 2.199 | 0.916 | 2.646 | 0.874 |

T-F | satlin | 15-10-1 | 1.815 | 0.934 | 1.697 | 0.953 | 1.765 | 0.946 |

T-RH | tansig | 12-10-1 | 1.810 | 0.935 | 1.778 | 0.949 | 1.821 | 0.945 |

F-RH | tansig | 18-15-1 | 1.832 | 0.934 | 1.660 | 0.956 | 1.790 | 0.947 |

T-F-RH ** | tansig | 12-12-1 | 1.836 | 0.934 | 1.629 | 0.957 | 1.742 | 0.949 |

**Table 5.**Evaluation metrics for the pan evaporation modeling of MLP, learned by scaled conjugate gradient algorithm (MLP-SCG).

Input Scenario | Transfer Function | Network Makeup * | Train | Validation | Test | |||
---|---|---|---|---|---|---|---|---|

RMSE | WI | RMSE | WI | RMSE | WI | |||

T | tansig | 12-10-1 | 1.879 | 0.929 | 1.792 | 0.948 | 1.816 | 0.944 |

F | tansig | 12-12-1 | 1.915 | 0.926 | 1.832 | 0.941 | 1.927 | 0.933 |

RH | tansig | 10-10-1 | 2.394 | 0.867 | 2.245 | 0.909 | 2.648 | 0.869 |

T-F | satlin | 12-10-1 | 1.823 | 0.934 | 1.738 | 0.951 | 1.796 | 0.945 |

T-RH | satlin | 18-15-1 | 1.853 | 0.932 | 1.722 | 0.953 | 1.778 | 0.947 |

F-RH | satlin | 10-10-1 | 1.874 | 0.930 | 1.747 | 0.950 | 1.865 | 0.941 |

T-F-RH ** | satlin | 12-12-1 | 1.814 | 0.935 | 1.668 | 0.955 | 1.766 | 0.947 |

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**MDPI and ACS Style**

Aghelpour, P.; Bagheri-Khalili, Z.; Varshavian, V.; Mohammadi, B. Evaluating Three Supervised Machine Learning Algorithms (LM, BR, and SCG) for Daily Pan Evaporation Estimation in a Semi-Arid Region. *Water* **2022**, *14*, 3435.
https://doi.org/10.3390/w14213435

**AMA Style**

Aghelpour P, Bagheri-Khalili Z, Varshavian V, Mohammadi B. Evaluating Three Supervised Machine Learning Algorithms (LM, BR, and SCG) for Daily Pan Evaporation Estimation in a Semi-Arid Region. *Water*. 2022; 14(21):3435.
https://doi.org/10.3390/w14213435

**Chicago/Turabian Style**

Aghelpour, Pouya, Zahra Bagheri-Khalili, Vahid Varshavian, and Babak Mohammadi. 2022. "Evaluating Three Supervised Machine Learning Algorithms (LM, BR, and SCG) for Daily Pan Evaporation Estimation in a Semi-Arid Region" *Water* 14, no. 21: 3435.
https://doi.org/10.3390/w14213435