## 3 Results and Discussion
### Modeling and Prediction by BP-ANN  
BBD experimental sets were used as input data for developing a BP-ANN model. Four fifth of experimental data (1-24) were utilized for training, while the rest (25-29) was used as the test set (Table 3). The mean absolute error between the experimental and predicted removal efficiency of MG is 0.54% with a high correlation (\(R^{2}\) = 0.9999), which suggests that the BP-ANN model possesses a good prediction strength (Figure 12). The number of neurons in the hidden layer has a direct influence on the fitting performance of the network. Nevertheless, too many neurons in the hidden layer could lead to overfitting [40]. As shown in Figure 13, the optimum number of neurons in hidden layer was selected on the basis of mean square error (MSE), and one neuron was found to be the suitable structure in the hidden layer with the minimum MSE (0.0018). Thus, the final topological structure of the BP-ANN was 4-1-1 (four neurons in the input layer, one neuron in the hidden layer and one neuron in output layer). \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Source** & **Sum of Squares** & **df** & **Mean Square** & **\(F\)-Value** & **\(p\)-Value** & \\ \hline Model & 5669.53 & 14 & 404.97 & 52.57 & \(<\)0.0001 & significant \\ A & 1306.67 & 1 & 1306.67 & 169.63 & \(<\)0.0001 & \\ B & 1283.61 & 1 & 1283.61 & 166.64 & \(<\)0.0001 & \\ C & 1937.26 & 1 & 1937.26 & 251.49 & \(<\)0.0001 & \\ D & 58.52 & 1 & 58.52 & 7.60 & 0.0155 & \\ AB & 80.82 & 1 & 80.82 & 10.49 & 0.0059 & \\ AC & 290.70 & 1 & 290.70 & 37.74 & \(<\)0.0001 & \\ AD & 154.50 & 1 & 154.50 & 20.06 & 0.0005 & \\ BC & 22.61 & 1 & 22.61 & 2.94 & 0.1087 & \\ BD & 27.56 & 1 & 27.56 & 3.58 & 0.0794 & \\ CD & 4.84 & 1 & 4.84 & 0.63 & 0.4412 & \\ A\({}^{2}\) & 249.84 & 1 & 249.84 & 32.43 & \(<\)0.0001 & \\ B\({}^{2}\) & 63.24 & 1 & 63.24 & 8.21 & 0.0125 & \\ C\({}^{2}\) & 71.50 & 1 & 71.50 & 9.28 & 0.0087 & \\ D\({}^{2}\) & 25.14 & 1 & 25.14 & 3.26 & 0.0924 & \\ Residual & 107.84 & 14 & 7.70 & - & - & \\ Lack of Fit & 107.84 & 10 & 10.78 & - & - \\ Pure Error & 0 & 4 & 0 & - & - \\ Total & 5777.37 & 28 & - & - & - \\ \hline \end{tabular}
\end{table} 
Table 2: Analysis of variance (ANOVA) for the experimental results from the Box–Behnken design (BBD). \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Runs** & **A** & **B (\({}^{\circ}\)C)** & **C (min)** & **D (mg/L)** & **Experimental Value (\%)** & **Predicted Value (\%)** & **Absolute Error (\%)** \\ \hline
1 & 6 & 25 & 60 & 700 & 72.21 & 72.45 & 0.24 \\
2 & 5 & 25 & 60 & 800 & 42.94 & 42.85 & 0.09 \\
3 & 4 & 25 & 90 & 800 & 51.56 & 51.59 & 0.03 \\
4 & 6 & 20 & 90 & 700 & 67.84 & 67.74 & 0.10 \\
5 & 5 & 25 & 90 & 700 & 63.12 & 63. The relationship between the MSE and the number of epochs showed that the MSE reached a minimum (\(10^{-5}\)) when the training was converged after 909 epochs (Figure 14). In Table 4, we present the values of weights and biases in input-hidden layer (\(w_{i}\) and \(b_{i}\)) and hidden-output layer (\(w_{j}\) and \(b_{j}\)) by the BP-ANN model used in this study. According to the results calculated with the Garson equation (Equation (10)), contact time appears to be the most influential variables followed by temperature and initial pH. Initial concentration has the least influence on the removal efficiency of MG among the independent variables. This finding is in accordance with the ANOVA results where contact time is deemed significant to the removal efficiency of MG (Table 5).