# Optimization of State of the Art Fuzzy-Based Machine Learning Techniques for Total Dissolved Solids Prediction

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## Abstract

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_{3}. To produce more compact networks along with the model’s generalization, a hybrid model which integrates a fuzzy-based intelligent system with the grasshopper optimization algorithm, NF-GMDH-GOA, is proposed for the prediction of the monthly TDS, and the prediction results are compared with five standalone and hybrid machine learning techniques. Results show that the proposed integrated NF-GMDH-GOA was able to provide an algorithmically informed simulation (NSE = 0.970 for Rig-Cheshmeh and NSE = 0.94 Soleyman Tangeh) of the dynamics of TDS records comparable to the artificial neural network, extreme learning machine, adaptive neuro fuzzy inference system, GMDH, and NF-GMDH-PSO models. According to the results of sensitivity analysis, Sodium in natural bodies of water with maximum value of error (RMSE = 56.4) had the highest influence on the TDS prediction for both stations, and Mg with RMSE = 43.251 stood second. The results of the Wilcoxon signed rank tests also indicated that the model’s prediction means were different, as the p value calculated for the models was less than the standard significance level ($\alpha =0.05$).

## 1. Introduction

- (1)
- The literature review showed that the application of GMDH integrated with Fuzzy set theory and grasshopper optimization algorithm (GOA) in WQPs modeling had not been investigated and evaluated. It is worth mentioning that GOA belongs to the category of multi-solution-based algorithms (population-based), exploring a larger portion of the search space compared to single-solution-based ones that modify and improve a single candidate solution, so the global optimum can probably be found more easily. Multi-solution-based algorithms like GOA intrinsically have higher local optima avoidance, due to their improving multiple solutions during optimization. Also, information about the search space can be exchanged between multiple solutions, which results in quick movement towards the optimum. In this regard, the feasibility of Fuzzy-GMDH-GOA in TDS prediction was explored in the present research.
- (2)
- GOA as the standard algorithm is applied to the optimized model’s parameters to validate the capability and reliability of the Fuzzy-GMDH-GOA model. In addition, some standalone AI models such as ANN, ELM, ANFIS, and GMDH have been considered as benchmarks to evaluate the feasibility of the hybrid fuzzy-based AI model in the prediction of TDS at a monthly scale.
- (3)
- For further assessment, to compare the results of expected and observation event data, an external validation was performed. Besides, a sensitivity analysis was performed to identify the most influential parameters linked to TDS variations in the Tajan river basin.

## 2. Materials and Methods

#### 2.1. Artificial Neural Network (ANN)

#### 2.2. Extreme Learning Machine (ELM)

_{,}denotes weight vector that makes links between hidden nodes and the i-th output node. Consider Vector h(t) as:

_{l}indicates the bias for the l-th hidden node. Weight vectors w

_{l}can be calculated from uniform distributions or random samples from normal distributions. Furthermore, $H=\left[h\left(1\right)h\left(2\right)\dots h\left(n\right)\right]$ is a matrix with a dimension of $q\times n$ where the t-th column represents the output vector of the hidden layer, $h\left(t\right)\u03f5{\mathbb{R}}^{q}$, $D=\left[d\left(1\right)d\left(2\right)\dots d\left(\mathrm{n}\right)\right]$ is a matrix with a dimension of $c\times n$ where the t-th column represents the target or desired vector $d\left(t\right)\u03f5{\mathbb{R}}^{c}$ associated with the input pattern $x\left(t\right),t=1,\dots ,N$, and $M=\left[m\left(1\right)m\left(2\right)\dots m\left(c\right)\right]$ is a matrix with a dimension of $q\times c$, where the i-th column represents the weight vector ${m}_{i}\u03f5{\mathbb{R}}^{q},i=1,\dots ,c$. Linear mapping is related to these matrices:

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

#### 2.4. Group Method of Data Handling (GMDH)

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

_{i}represents the position for a particle. This vector represents a possible particle or solution, whose dimension is determined by the number of existing parameters. The parameters, x

_{i}

^{0}and v

_{i}

^{0}, indicate randomly chosen numbers associated with the position and velocity at iteration 0, related to particle i, respectively. Afterward, the vectors of particles are updated according to the fitness function. According to Equations (16) and (17), the vectors are updated [52,53]:

- First, the value of velocity from the prior iteration multiplied by the inertia weight constant, ${\mathsf{\omega}\mathrm{v}}_{\mathrm{i}}^{\mathrm{k}}$,
- Second, the difference between the particle’s current position ${\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}}$ and the best global position ${\mathrm{g}}^{\mathrm{k}}$, which is also known as social learning, and
- Third, the difference between the particle’s current position ${\mathrm{x}}_{\mathrm{i}}^{\mathrm{k}}$ and the local best particle’s position up to this point, ${\mathrm{I}}_{\mathrm{i}}^{\mathrm{k}}$, which is also known as cognitive learning.

_{x}represents a real randomly selected number of a uniformly distributed function between [0,1], and c

_{x}represents a constant value for x = 1,2. The particles cover the entire search space in the first iteration. With the increase in the number of iterations, the search space decreases. Therefore, PSO analyzes plausible zones first and ultimately improves its best solution. Over the years, there have been several versions of PSO in the literature. In this study, the standard version of PSO proposed in 2011 with the subsequent parameters was chosen:

#### 2.6. Grasshopper Optimization Algorithm (GOA)

_{c}is generated by applying Equation (19).

_{c}indicates the population size and N represents the problem’s dimension. Moreover, l

_{j}and u

_{j}are the lower and upper limits for the jth variable.

_{j}) is the fitness function, which in this article is the Mean Square Error (MSE).

_{I}is the maximum value of the cycle number.

_{d}indicates the best discovered solution, and l represents the attraction length. According to Equation (22), the normalized distance between the best discovered solution and the real search space position can be determined. The better position is saved after evaluating the newfound position with Equation (21). It should be noted that Equation (22) is modified by changing the c parameter, resulting in later iterations focusing on exploration and earlier iterations focusing on exploitation. Algorithm accuracy is improved by using this tuning procedure.

#### 2.7. Development of an Adaptive Fuzzy-GMDH Using PSO/GOA

#### 2.8. Case Study Description

^{3}/s, and 85%, respectively. The difference between minimum and maximum level of the Tajan basin is approximately 3700 m; 90% of the forest surface is covered by brown soil and the remainder is covered by widespread types such as alluvial soil [6] Various agricultural, aquacultural, aquafarming, and industrial activities are implemented in this river basin. Moreover, different operations, including damming and sand mining, are done in the river, which affect the average amount of measured TDS. Due to the high rate of rainfall and the beginning of agricultural production, TDS monitoring is needed annually in the fall and winter [23]. In the basin, there are nine active hydrometric gauging stations. For TDS modeling, data from the Soleyman Tange and Rig-Cheshme stations were collected as shown in Figure 3.

_{3}), and sodium (Na), which are provided from the Meteorological Organization of Mazandaran Province (MOMP) during March 1984–August 2016 and March 1974–August 2016 at Soleyman-Tangeh (390 monthly data record) and Rig-Cheshmeh (505 monthly data record) gauging stations, respectively, were used in the TDS modeling. In this regard, about 75% of the total dataset was used for training and the rest was set aside for testing the AI’s networks.

#### 2.9. Model Performance Criteria

_{m}should satisfy:

## 3. Results and Discussion

#### 3.1. Performance Results of Standalone and Hybrid Models

#### 3.1.1. The Case Study of Rig-Cheshmeh Station

^{2}) and least-squares regression (LSR) were presented.

#### 3.1.2. The Case Study of Soleyman-Tangeh Station

#### 3.2. Further Analysis and Discussion

_{m}= 0.804 at the Rig-Cheshmeh station, and achieved the best results in selecting the most accurate model in comparison with other models. However, although the values of n, m, K, and K′ of the ANN model were in agreement with the required conditions, the criterion of the ${R}_{m}$ value (${R}_{m}$ = 0.487) was obtained as marginally less than 0.5 and subsequently the condition was not met. In terms of R

_{m}and R-values, ANFIS was able to capture TDS variations with an acceptable level of validated criteria, rather than ELM and GMDH.

_{m}as external validation criteria. Table 5 presents criteria of R

_{m}for the proposed ANN, ELM, ANFIS, and GMDH approaches, which were 0.341, 0.483, 0.384, and 0.485, respectively. In general, statistical indices have shown the high performance of the NF-GMDH-GOA model in TDS estimation.

_{3}, Ca, Mg, and Na, have been considered whose output was WQP, predicted by NF-GMDH-GOA with the highest accuracy.

## 4. Concluding Remarks

_{3}, were included in the model. Comparing the results of the hybrid and standalone models showed that the grasshopper optimization algorithm has a major effect on the performance of NF-GMDH. At the Soleyman-Tangeh station, NF-GMDH-GOA could predict TDS with more accuracy in terms of NSE (0.948), RSD (0.223) and RMSE (9.687 mg/L) in comparison to other models at the validation stage. The accuracy of GMDH and NF-GMDH-GOA revealed that the coefficient of determination was raised from 0.892 to 0.989 for the Soleyman-Tangeh gauging station. For the Rig-Cheshmeh station, the outcomes showed that NF-GMDH-GOA showed the best performance in forecasting TDS in terms of RSD (0.174) and RMSE (10.744 mg/L). Furthermore, sensitivity analysis was utilized to determine the most significant parameters on the TDS modeling and fairy justification of the relative effectiveness of independent variables. The results of the sensitivity analysis demonstrated that Na was the effective factor on TDS values at two proposed stations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Location map of the studied basin and two applied stations (Adopted from [23]).

**Figure 4.**Plots of simulated versus TDS observations at Rig-Cheshmeh stations for the validation stage.

**Figure 5.**Error bar plots and time series of estimated vs. observed TDS by proposed techniques at the Rig-Cheshmeh station.

**Figure 6.**Scatter plots of simulated versus observed TDS values at Soleyman-Tangeh stations for the validation stage.

**Figure 7.**Time series and error bar plots of estimated vs. observed TDS by proposed techniques at Soleyman-Tangeh station.

Algorithm | Parameters | Value |
---|---|---|

PSO | Acceleration constant (C1 and C2) | 2 |

Inertia Wmax | 0.9 | |

Inertia Wmin | 0.4 | |

Number of particles | 50 | |

GOA | Seeking memory pool | 5 |

Counts of dimension to change | 0.8 | |

Seeking rang of the selected dimension | 0.2 | |

Mutative ratio | 0.9 |

Variables | Indices | Rig-Cheshmeh | Soleyman-Tangeh |
---|---|---|---|

HCO_{3} (mg/L) | Min | 1.6 | 1.2 |

Mean | 3.88 | 1.2 | |

Max | 12.2 | 0.5 | |

Std | 0.89 | 0.08 | |

Variation | 0.79 | 156 | |

Ca (mg/L) | Min | 1.1 | 3.84 |

Mean | 3.16 | 3.41 | |

Max | 7.5 | 2.07 | |

Std | 0.68 | 0.87 | |

Variation | 0.46 | 408.87 | |

Mg (mg/L) | Min | 0.1 | 7.7 |

Mean | 2.17 | 6.3 | |

Max | 6 | 4.5 | |

Std | 0.69 | 2.94 | |

Variation | 0.48 | 650 | |

Na (mg/L) | Min | 0.2 | 0.91 |

Mean | 1.54 | 0.66 | |

Max | 6.5 | 0.68 | |

Std | 0.75 | 0.42 | |

Variation | 0.57 | 63.1 | |

TDS (mg/L) | Min | 271 | 0.83 |

Mean | 446.49 | 0.44 | |

Max | 1270 | 0.46 | |

Std | 78.7 | 0.18 | |

Variation | 6194.38 | 3981.8 |

**Table 3.**Statistical evaluation of proposed models at calibration and validation stages for Rig-Cheshmeh station.

ANN | ELM | ANFIS | GMDH | NF-GMDH-PSO | NF-GMDH-GOA | |
---|---|---|---|---|---|---|

Calibration | ||||||

R | 0.947 | 0.975 | 0.973 | 0.968 | 0.980 | 0.986 |

RMSE (mg/L) | 29.991 | 18.053 | 22.222 | 20.784 | 16.099 | 13.478 |

RSD | 0.369 | 0.222 | 0.273 | 0.256 | 0.198 | 0.166 |

NSE | 0.864 | 0.951 | 0.925 | 0.934 | 0.961 | 0.972 |

Validation | ||||||

R | 0.906 | 0.935 | 0.970 | 0.924 | 0.962 | 0.985 |

RMSE (mg/L) | 27.178 | 22.439 | 14.975 | 24.579 | 20.564 | 10.744 |

RSD | 0.440 | 0.363 | 0.242 | 0.398 | 0.333 | 0.174 |

NSE | 0.805 | 0.867 | 0.941 | 0.840 | 0.888 | 0.970 |

**Table 4.**Statistical evaluation of proposed models at calibration and validation stages for Soleyman-Tangeh station.

ANN | ELM | ANFIS | GMDH | NF-GMDH-PSO | NF-GMDH-GOA | |
---|---|---|---|---|---|---|

Calibration | ||||||

R | 0.891 | 0.950 | 0.932 | 0.948 | 0.948 | 0.973 |

RMSE (mg/L) | 40.295 | 19.538 | 22.807 | 19.938 | 20.107 | 14.376 |

RSD | 0.640 | 0.310 | 0.362 | 0.317 | 0.320 | 0.228 |

NSE | 0.589 | 0.903 | 0.868 | 0.899 | 0.898 | 0.948 |

Validation | ||||||

R | 0.817 | 0.905 | 0.781 | 0.892 | 0.975 | 0.989 |

RMSE (mg/L) | 27.823 | 19.378 | 35.364 | 22.254 | 10.113 | 9.687 |

RSD | 0.655 | 0.456 | 0.833 | 0.524 | 0.238 | 0.223 |

NSE | 0.567 | 0.790 | 0.300 | 0.723 | 0.942 | 0.948 |

Metrics | ANN | ELM | ANFIS | GMDH | NF-GMDH-PSO | NF-GMDG-GOA |
---|---|---|---|---|---|---|

Rig-Cheshmeh station | ||||||

R (R > 0.8) | 0.906 | 0.935 | 0.97 | 0.924 | 0.962 | 0.985 |

K (0.85 < K < 1.15) | 1.017 | 1.001 | 0.997 | 1.003 | 0.975 | 1.002 |

K′ (0.85 < K′ < 1.15) | 0.979 | 0.996 | 1.001 | 0.994 | 1.024 | 0.998 |

m (m < 0.1) | −0.202 | −0.143 | −0.062 | −0.171 | −0.047 | −0.03 |

n (n < 0.1) | −0.185 | −0.142 | −0.062 | −0.17 | −0.042 | −0.03 |

R_{m} (R_{m} > 0.5) | 0.48 | 0.565 | 0.714 | 0.527 | 0.732 | 0.804 |

Soleyman-Tangeh station | ||||||

R (R > 0.8) | 0.817 | 0.905 | 0.781 | 0.892 | 0.975 | 0.989 |

K (0.85 < K < 1.15) | 0.965 | 0.987 | 0.943 | 1.026 | 1.004 | 1.022 |

K′ (0.85 < K′ < 1.15) | 1.031 | 0.101 | 0.055 | 0.972 | 0.996 | 0.979 |

m (m < 0.1) | −0.36 | −0.206 | −0.226 | −0.191 | −0.051 | 0.013 |

n (n < 0.1) | −0.301 | −0.212 | −0.211 | −0.116 | −0.051 | 0.015 |

R_{m} (R_{m} > 0.5) | 0.341 | 0.483 | 0.384 | 0.485 | 0.741 | 0.866 |

**Table 6.**Results of Wilcoxon signed-rank test between the proposed integrative NF-GMDH-GOA and other models.

Number | Pairwise Comparison | Z | p (<0.05) | Significance |
---|---|---|---|---|

1 | NF-GMDH-GOA vs. ANN | −4.496 | 0.003 | Yes |

2 | NF-GMDH-GOA vs. ELM | −5.621 | 0.001 | Yes |

3 | NF-GMDH-GOA vs. ANFIS | −7.255 | 0.001 | Yes |

4 | NF-GMDH-GOA vs. GMDH | −7.158 | 0.002 | Yes |

5 | NF-GMDH-GOA vs. NF-GMDH-PSO | −8.157 | 0.001 | Yes |

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**MDPI and ACS Style**

Hijji, M.; Chen, T.-C.; Ayaz, M.; Abosinnee, A.S.; Muda, I.; Razoumny, Y.; Hatamiafkoueieh, J.
Optimization of State of the Art Fuzzy-Based Machine Learning Techniques for Total Dissolved Solids Prediction. *Sustainability* **2023**, *15*, 7016.
https://doi.org/10.3390/su15087016

**AMA Style**

Hijji M, Chen T-C, Ayaz M, Abosinnee AS, Muda I, Razoumny Y, Hatamiafkoueieh J.
Optimization of State of the Art Fuzzy-Based Machine Learning Techniques for Total Dissolved Solids Prediction. *Sustainability*. 2023; 15(8):7016.
https://doi.org/10.3390/su15087016

**Chicago/Turabian Style**

Hijji, Mohammad, Tzu-Chia Chen, Muhammad Ayaz, Ali S. Abosinnee, Iskandar Muda, Yury Razoumny, and Javad Hatamiafkoueieh.
2023. "Optimization of State of the Art Fuzzy-Based Machine Learning Techniques for Total Dissolved Solids Prediction" *Sustainability* 15, no. 8: 7016.
https://doi.org/10.3390/su15087016