## 4 Results and Discussion
### PCB data set  
The principal component scores of the PCB data are given in Figure 13. The principal components and mean were calculated only from the training set. The training set was labeled with upper case letters. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source regions** & **Total** & **BNN** & **SCRBFN** & **RBFN** & **RBFCCN** \\ \hline Calabria & 56 & \(50.9\pm 0.6\) & \(50.3\pm 0.5\) & \(52.5\pm 0.6\) & \(52.7\pm 0.2\) \\ South Apulia & 206 & \(203.8\pm 0.5\) & \(199.5\pm 0.6\) & \(203.5\pm 0.4\) & \(200.1\pm 0.3\) \\ Inland Sardin & 65 & 65 & \(58.3\pm 0.3\) & \(63.4\pm 0.4\) & \(64.2\pm 0.2\) \\ East Liguria & 50 & \(38.0\pm 1.0\) & \(37.4\pm 0.7\) & \(24.4\pm 3.6\) & \(35.4\pm 0.7\) \\ West Liguria & 50 & \(48.2\pm 0.3\) & \(43.3\pm 0.5\) & \(47.6\pm 0.8\) & \(48.5\pm 0.3\) \\ Umbria & 51 & \(50.4\pm 0.4\) & \(40.4\pm 0.4\) & \(48.7\pm 0.8\) & \(50.9\pm 0.1\) \\ \hline Prediction accuracy (\%) & & \(95.5\pm 0.3\) & \(89.8\pm 0.3\) & \(92.0\pm 0.7\) & \(94.5\pm 0.2\) \\ \hline \hline
**Source regions** & **Total** & **PLS-DA** & **SVM** & **RF** \\ \hline Calabria & 56 & \(40.9\pm 0.5\) & \(53.6\pm 0.4\) & \(53.2\pm 0.4\) \\ South Apulia & 206 & \(202.4\pm 0.4\) & \(202.5\pm 0.6\) & \(203.5\pm 0.7\) \\ Inland Sardin & 65 & \(63.7\pm 0.4\) & \(65\) & \(65\) \\ East Liguria & 50 & \(26.7\pm 1.0\) & \(47.3\pm 0.4\) & \(46.4\pm 0.4\) \\ West Liguria & 50 & \(48.4\pm 0.3\) & \(48.9\pm 0.4\) & \(49.0\pm 0.2\) \\ Umbria & 51 & \(47.2\pm 0.6\) & \(50.9\pm 0.1\) & \(50.9\pm 0.1\) \\ \hline Prediction accuracy (\%) & & \(89.8\pm 0.3\) & \(97.9\pm 0.2\) & \(97.9\pm 0.2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Average numbers of correctly predicted objects with 95% confidence intervals of Italian olive oil data set by different modeling methods with 30 BLPs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source of variation** & **Sum of squares** & **Degrees of freedom** & **Mean square** & **F** & **F\({}_{\text{crit}}\)** \\ \hline Source regions & 4.44 & 5 & 0.89 & 1919.9 & 2.22 \\ Modeling methods & 0.53 & 6 & \(8.79\times 10^{-2}\) & 189.8 & 2.11 \\ Interaction & 2.03 & 30 & \(6.78\times 10^{-2}\) & 146.5 & 1.47 \\ Within & 0.56 & 1218 & \(4.63\times 10^{-4}\) &The external validation set was projected onto the first two principal components, labeled with underlined lower case letters. The external validation scores were more dispersed than the training set. A part of the external validation scores were outside of the 95% confidence interval of their class because of their low quality. The principal component scores of PCB congeners that contain 0, 1, 9 and 10 chlorine atoms were uniquely different from the training set. The BLP internal validation for the training set alone was first performed. The prediction accuracies of internal validations are given in Table 8. The average prediction accuracies of the SVM, RF, RBFCCN, and BNN models were better than the average prediction accuracy of the SCRBFN, RBFN, and PLS-DA model. Figure 13: A principal component score plot for the PCB data set. The letters with upper case represents the training set. The underlined letters with lower case represents the external validation set. The external validation set was projected onto the first two principal components from the training set. Each axis is labeled with the percent total variance and the absolute eigenvalue from the training set. The 95% confidence intervals were calculated and given as an ellipse around each class from the training set. The PCB congeners are: (A) 2; (B) 3; (C) 4; (D) 5; (E) 6; (F) 7; (G) 8; (H) 9; (i) 10; (j) 1; (k) 0, the numbers denotes the number of chlorine atoms in the PCB congeners. After internal validation, the entire training set was trained and the external validation set was predicted repeatedly 30 times.