## 4 Results and Discussions
### Neural Network Analysis  
In the neural network analysis, data collected from machining tests were divided into three groups, in which 70% of datasets are randomly selected for neural network training, 15% for testing, and the remaining 15% for validation. The architecture of the network includes one input layer, one hidden layer, and one output layer. The ANN modeling was conducted using multi-layered perception (MLP) by Statistica Neural Networks software [40]. The prediction models of surface roughness were first created based on the cutting parameters that include spindle speed, feed rate, and axial depth of cutting as neurons in input layers. Then, the vibration level of machining was included in the input layer to show the effect of spindle vibration on the machined quality. The surface roughness and vibration amplitude can be the output neuron to be determined in the output layer. Other network parameters, such as learning rate, number of neurons in the hidden layer, iteration times, and error function, are obtained by trial and error. After many attempts, different ANN models are selected for comparison by observing the best ANN performance and the correlation coefficient. In this case, four models with better performance were selected, as listed in Table 5. The ANN models are labeled with the number of neurons in each layer, and the number of neurons in the hidden layer is optimized. For example, model MLP 3-12-1 has 1, 3, and 12 and neurons in the output, input, and hidden layers, respectively. The models with four input neurons--MLP 4-15-1 and MLP 4-22-1--were optimized with 15 and 22 neurons in the hidden \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Coefficients** & **Standard Deviations** & \(p\)**-Value** \\ \hline Intercept & \(2.254\times 10^{-1}\) & \(4.974\times 10^{-2}\) & \(9.330\times 10^{-6}\) \\ Spindle speed (\(n\)) & \(9.170\times 10^{-5}\) & \(1.600\times 10^{-5}\) & \(3.080\times 10^{-8}\) \\ Cutting depth (\(a_{p}\)) & \(2.954\times 10^{-1}\) & \(5.408\times 10^{-2}\) & \(1.210\times 10^{-7}\) \\ Depth \(\times\) speed (\(a_{p}\times n\)) & \(-3.700\times 10^{-5}\) & \(1.400\times 10^{-5}\) & \(7.929\times 10^{-3}\) \\ Depth \(\times\) feed (\(a_{p}\times f\)) & \(-2.600\times 10^{-4}\) & \(3.890\times 10^{-5}\) & \(2.190\times 10^{-10}\) \\ Feed \(\times\) speed (\(f\times n\)) & \(-3.300\times 10^{-8}\) & \(5.160\times 10^{-9}\) & \(5.790\times 10^{-10}\) \\ Depth \(\times\) speed \(\times\) feed (\(a_{p}\times n\times f\)) & \(4.750\times 10^{-8}\) & \(6.670\times 10^{-9}\) & \(1.360\times 10^{-11}\) \\ Vibration (\(V_{b}\)) & \(3.721\times 10^{-1}\) & \(1.043\) & \(7.217\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Regression parameters of the nonlinear polynomial model. layer, respectively. Further, Table 5 shows the sensitivity level of constructed ANN input parameters. The larger the value, the less influence this parameter has on the output parameter; conversely, the smaller the value, the greater is the influence. It is found that in models MLP4-15-1 and MLP 4-22-1 with four input variables (three cutting parameters and machining vibration), the spindle speed has the largest sensitivity values among the parameters. The statistical values for the ANN model, correlation coefficient (R), RMSE, and MAPE are used as the basis for determining the performance of the neural network, as summarized in Table 6. The value of R represents the agreement extent between the predicted values and measured values. The values of R for the models MLP 3-12-1 and MLP 3-13-1 within the training, validation, and testing data are in the range 0.9285-0.9693. For neural models with an additional variable (machining vibration), higher values of R are found in the range of 0.9611 to 0.9785. Figure 13 shows the comparison of measured and predicted data of the surface roughness for all data, which indicate that high agreements of the prediction results with a correlation coefficient are achieved by neural network models.