# The Prediction of Cu(II) Adsorption Capacity of Modified Pomelo Peels Using the PSO-ANN Model

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Results Predicted from the ANN Model

^{2}values for the training and testing sets were close to 0.99977 and 0.99987, respectively, which were generally satisfactory. The RMSEs of the ANN were 0.0217 for the training set and 0.0159 for the testing set, while the MAE values were close to 0.0140 and 0.0125, respectively. In conclusion, the common ANN model has some generalization potential and can somewhat anticipate unknown data.

#### 2.2. Results Predicted from the PSO-ANN Model

^{2}values for the training and testing sets were 0.99998 and 0.99999, respectively. The RMSEs were 0.0081 for the training set and 0.0077 for the testing set. The MAE for the training set was 0.0067, whereas it was 0.0066 for the testing set.

^{2}values of the training and testing sets for the PSO-ANN model were higher than those for the standard ANN model (Table 2). In comparison to the conventional ANN model, the RMSE and MAE values of the training and testing sets for the PSO-ANN were lower by varying degrees. Therefore, the PSO algorithm introduced into the ANN model increased the accuracy of the predicted values.

#### 2.3. Discussion

^{2}, indicating its better fitting ability compared to the classical ANN approach. This is also reflected in other indicator values such as the RMSE, MSE, and MAE (Table 2). The RMSE values of the PSO-ANN (0.0081, 0.0077, and 0.0080) were significantly lower than those of the ANN (0.0217, 0.0159, and 0.0207) for the training, testing, and all data, respectively. In addition, the PSO-ANN had a lower MAE value for each phase. Therefore, the introduction of PSO into the ANN model improved its prediction accuracy, as the testing data were not included in the model training. As shown in Figure 3, the errors between the predicted and actual values based on the ANN model were larger than those based on the PSO-ANN. This indicates that the predicted values using the PSO-ANN were closer to the actual values than those using the ANN, leading to the same conclusion. Figure 4 displays the gradient change in the iterative training for the ANN and PSO-ANN. It can be seen from this figure that the gradients of the PSO-ANN and ANN both decrease sharply in the first 30 epochs. Then, these gradients decrease much slower prior to becoming stable after 30 epochs. As shown in Figure 4, the decline curve of the PSO-ANN is smoother and converges to a higher accuracy than that observed with the ANN model. In fact, some fluctuations on the gradient curve of the ANN model are observed during its training process. In addition, the training of the ANN stops at 349 epochs, even if there is still a certain distance from the ideal error value, indicating that there is a local optimal solution problem using this approach.

_{i}calculated by the CAM for the adsorption time (0.9439) and the initial pH (0.9626).

## 3. Materials and Methods

#### 3.1. Database Description and Variable Analysis

_{a}and x

_{b}are the input variables; and $\overline{{x}_{a}}$ and $\overline{{x}_{b}}$ are the mean values of the variables. Thus, r is between −1 and 1; the larger its absolute value, the stronger the linear correlation between the input variables. If 0.9 ≤ $\left|r\right|$ ≤ 1, the information redundancy between the two variables is high; only one variable is retained, and the other is deleted.

#### 3.2. Sensitivity Factor Analysis of Input Variables

_{i}is the input variable, $y$ is the output variable, and S

_{i}is the degree of influence of each input variable on the output variable. The values of S

_{i}between the input variables and the Cu(II) adsorption capacity are shown in Figure 7. The most significant influence on the Cu(II) adsorption capacity was the initial Cu(II) concentration, followed by the temperature, while the adsorption time had the lowest impact.

#### 3.3. Methods

#### 3.3.1. ANN

_{i}denotes the input variables, ${w}_{ij}$ represents the weight, and b denotes the bias.

_{i}indicates the difference between the actual value and the predicted value, while $\frac{\partial E\left(t\right)}{\partial w\left(t\right)}$ and $\frac{\partial E\left(t\right)}{\partial b\left(t\right)}$ indicate the gradients of the output errors for each weight and bias, respectively.

#### 3.3.2. PSO

_{m}) and velocity (V

_{m}) of the m-th particle are expressed as two l-dimensional vectors.

_{1}and r

_{2}are randomly selected from the range from 0 to 1 to increase the random searchability. The individual learning factor, c

_{1}, signifies the ability of a particle to search for the best option. The global learning factor, c

_{2}, represents the searchability of the group for the best solution. The default value for the learning factors c

_{1}and c

_{2}is 2. The particle is thought to have only one global learning capability if c

_{1}= 0; at this point, the particle is capable of extended search, but not local search, and converges slowly for difficult issues. The particle is thought to possess only one cognitive capacity if c

_{2}= 0; at this point, it behaves like a blind random search and is more likely to run into the local optimal solution problem [35].

_{max}∈ [0, 1] and $\omega $

_{min}∈ [0, 1] are the maximum and minimum inertia factors, while t and t

_{max}are the current and maximum number of iterations, respectively.

#### 3.3.3. PSO-ANN

- (1)
- Data Normalization. The training and testing sets were constructed after reading the sample data.
- (2)
- Setting the Model Parameters. The maximum number of iterations, neuron numbers of each layer, functions, and termination criteria for the ANN topology were set in accordance with the features of the input data.
- (3)
- Optimization of Weights and Biases. The parameters of the PSO were set. The initial weights and biases of the ANN training model were then optimized using the particle swarm algorithm.
- (4)
- Establishment of the PSO-ANN model. The optimized weights and biases obtained from the position information of the optimal particle were assigned to the ANN as the initial values for the following training. The testing dataset was applied to verify the well-trained hybrid ANN and output the prediction result.

#### 3.4. Model Performance

^{2}) were the statistical metrics utilized to assess the performances of the neural network models. R

^{2}is widely used to assess model reliability, while the RMSE and MAE reflect the degree of model deviation. These evaluation indicators can be calculated as expressed [38,39]:

## 4. Conclusions

^{2}, MSE, RMSE, and MAE were utilized as assessment indices for describing the overall model performance. According to the training, testing, and combined data results, both the ANN and PSO-ANN demonstrated good predictive abilities, but the proposed PSO-ANN exhibited a greater prediction accuracy. Moreover, the predicted values of the PSO-ANN were closer to the actual values when comparing the model errors. This meant that the PSO effectively compensated for defects in the ANN. In conclusion, this study effectively demonstrated the enhanced validity and accuracy of the PSO-ANN in comparison to the ANN. The PSO-ANN hybrid algorithm integrated and combined the benefits of both single models used. There is great potential for the PSO-ANN to be applied for predicting the adsorption capacity of heavy metals by agricultural waste biosorbents.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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**Figure 2.**Regression analysis of (

**a**,

**c**,

**e**) ANN and (

**b**,

**d**,

**f**) PSO-ANN for training data, testing data, and all data, respectively.

Parameter | Setting |
---|---|

Input layer node | 4 |

Hidden layer node | 9 |

Output layer node | 1 |

Activation function | tansig, purelin |

Training function | trainlm |

Epochs | 1000 |

Learning rate | 0.01 |

Precision | 1 × 10^{−5} |

Epochs between display | 25 |

Momentum factor | 0.01 |

Minimum performance gradient | 1.00 × 10^{−6} |

Maximum validation failure | 6 |

ANN | PSO-ANN | ||
---|---|---|---|

R^{2} | Training | 0.99977 | 0.99998 |

Testing | 0.99987 | 0.99999 | |

MSE | Training | 0.0005 | 0.0001 |

Testing | 0.0003 | 0.0001 | |

RMSE | Training | 0.0217 | 0.0081 |

Testing | 0.0159 | 0.0077 | |

MAE | Training | 0.0140 | 0.0067 |

Testing | 0.0125 | 0.0066 |

Parameter | Setting |
---|---|

Particle swarm size | 20 |

Maximum iterations | 60 |

c_{1} | 2 |

c_{1} | 2 |

ω_{max} | 0.90 |

ω_{min} | 0.40 |

Position constraint | [−5, 5] |

Velocity constraint | [−1, 1] |

Reference | Model | Removal Ions | Dataset | Performance | |
---|---|---|---|---|---|

Training | Testing | ||||

This study | PSO-ANN | Cu(II) | 100 | R^{2} = 0.99998RMSE = 0.0081 MAE = 0.0067 | R^{2} = 0.99999RMSE = 0.0077 MAE = 0.0066 |

Zheng et al. [20] | QSA-ANN | Cu(II), Pb(II), Zn(II), As(III), Cd(II), and Ni(II) | 353 | R^{2} = 0.978RMSE = 0.051 | R^{2} = 0.960RMSE = 0.074 |

Ke et al. [21] | SVM-ANN | Cu(II), Pb(II), Zn(II), As(III), Cd(II), and Ni(II) | 353 | R^{2} = 0.995RMSE = 0.036 MAE = 0.018 | R^{2} = 0.987RMSE = 0.046 MAE = 0.026 |

Bhagat et al. [22] | ANN-M5 | Cu(II) | 95 | R^{2} = 0.9983RMSE = 1.3799 MAE = 1.0338 | R^{2} = 0.9974RMSE = 0.9283 MAE = 0.6200 |

Pooladi et al. [23] | GMDH-ANN | Pb(II) | Not reported | R^{2} = 0.94RMSE = 3.5524 MAE = 2.6958 | R^{2} = 0.9315RMSE = 5.3083 MAE = 4.041 |

Zafar et al. [24] | ANFIS | Cr(VI) | 18 | R^{2} = 0.99RMSE = 0.63 | R^{2} = 0.94RMSE = 6.23 |

Max | Min | Average | Median | Standard Deviation | Skewness | |
---|---|---|---|---|---|---|

Temperature (K) | 313.15 | 288.15 | 298.80 | 298.15 | 3.98 | 1.33 |

Initial pH | 7.00 | 2.00 | 4.87 | 5.00 | 0.80 | −1.33 |

Adsorption time (min) | 60.00 | 10.00 | 53.50 | 60.00 | 13.37 | −1.99 |

Initial Cu(II) concentration (mg/L) | 28.00 | 4.00 | 19.00 | 20.00 | 4.00 | −1.95 |

Cu(II) adsorption capacity (mg/g) | 6.6994 | 0.7726 | 4.4236 | 4.6865 | 0.9820 | −1.8389 |

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**MDPI and ACS Style**

Jiao, M.; Jacquemin, J.; Zhang, R.; Zhao, N.; Liu, H.
The Prediction of Cu(II) Adsorption Capacity of Modified Pomelo Peels Using the PSO-ANN Model. *Molecules* **2023**, *28*, 6957.
https://doi.org/10.3390/molecules28196957

**AMA Style**

Jiao M, Jacquemin J, Zhang R, Zhao N, Liu H.
The Prediction of Cu(II) Adsorption Capacity of Modified Pomelo Peels Using the PSO-ANN Model. *Molecules*. 2023; 28(19):6957.
https://doi.org/10.3390/molecules28196957

**Chicago/Turabian Style**

Jiao, Mengqing, Johan Jacquemin, Ruixue Zhang, Nan Zhao, and Honglai Liu.
2023. "The Prediction of Cu(II) Adsorption Capacity of Modified Pomelo Peels Using the PSO-ANN Model" *Molecules* 28, no. 19: 6957.
https://doi.org/10.3390/molecules28196957