## 3 Experimental section
### General information
To determine the learning rates and momenta of the BNNs and RBFNs, these networks were trained by three different sets of learning rates and momenta with BLPs. The number of bootstraps was 30 and the number of partitions was two. Table 2 gives the prediction results by the Italian olive oil data set and the training set of the PCB data set. The training parameters of the back-propagation networks did not significantly affect the comparison of the modeling methods. These sets of learning parameters were also trained by the two synthetic data sets and same results were obtained. For each data set, there was no statistical difference of the BNN and RBFN prediction results at a 95% confidence interval by two-way ANOVA with interaction. Therefore, the learning rates and momenta were fixed respectively at 0.001 and 0.5 for all further evaluations. The PLS-DA was implemented by the non-linear iterative partial least squares (NIPALS) algorithm. The number of latent variables was determined by minimizing the root mean squared prediction error in each test. As a result, the PLS-DA was a biased reference method. The numbers of latent variables in the PLS-DA models may vary between runs. All the SVMs used the Gaussian RBF as their kernel functions. Two SVM parameters: the cost \(c\) and the RBF kernel parameter \(\gamma\) must be adjusted before each prediction. The grid search of parameter pairs (\(c\), \(\gamma\)), in which \(c=2^{i}\), \(i\) = -2, -1, 0,..., 20; \(\gamma=2^{i}\), \(j\) = -10, -9, -8,..., 10, was performed to determine their value by achieving the best training accuracies. The defaults of the remaining \begin{table}
\begin{tabular}{c c c} \hline \hline
**Data set** & **Number of hidden neurons** & **Prediction accuracy** \\ \hline Olive oil & 7 & \(95.5\pm 0.3\) \\  & 14 & \(95.9\pm 0.4\) \\  & 4 & \(87.7\pm 0.2\) \\ \hline PCB & 13 & \(99.9\pm 0.1\) \\  & 26 & \(99.9\pm 0.1\) \\  & 7 & \(99.9\pm 0.1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Average prediction accuracies of the BNN models with 95% confidence intervals of Italian olive oil and the training set of the PCB data set. The BNN was trained by different number of hidden neurons with 30 BLPs. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Data set** & **Learning rate** & **Momentum** & **BNN** & **RBFN** \\ \hline Olive oil & 0.001 & 0.5 & \(95.5\pm 0.3\) & \(92.0\pm 0.7\) \\  & \(1\times 10^{-4}\) & 0.5 & \(95.4\pm 0.3\) & \(91.9\pm 0.7\) \\  & 0.001 & 0 & \(95.6\pm 0.3\) & \(91.5\pm 0.6\) \\ \hline PCB & 0.001 & 0.5 & \(99.9\pm 0.1\) & \(92.4\pm 5.5\) \\  & \(1\times 10^{-4}\) & 0.5 & \(99.5\pm 0.2\) & \(90.8\pm 6.3\) \\  & 0.001 & 0 & \(99.9\pm 0.1\) & \(94.1\pm 4.8\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average prediction accuracies of the BNN and RBFN models with 95% confidence intervals of Italian olive oil and the PCB data sets. The BNN and RBFN were trained by three different sets of learning rates and momenta with 30 BLPs. parameters were used. Because the result of the RF algorithm is not sensitive to the parameter selected, 1,000 trees with the default setting of the number of variables to split on at each node is used in all evaluations. The BLPs generates precision measures of the classification. Bootstrapping is a method that resamples the data. Latin partition is a modified cross-validation method, in which the class distributions are maintained at constant proportions among the entire data set and the randomized splits into training and prediction sets. After the data set was partitioned during each bootstrap, it was evaluated by all the modeling methods in the study. Because bootstrapping runs the evaluation repeatedly, the confidence interval of the prediction errors can also be obtained. The number of bootstraps was 30 and the number of partitions was two for evaluating all the data sets in this study. The results are reported as prediction accuracy, which is the percentage of correctly predicted objects.