## 4 Results and Discussion
### Discussion  
A total of 20 M random and independent iterations was run for the two independent classifiers and the combined classifier on random test sets. The results are reported in Table 5. For each set of concentrations (\(\pm\)1 g/L, \(\pm\)0.1 g/L, and \(\pm\)0.01 g/L) the average value and standard deviation of the classification accuracy are given. Overall, it is observed that the combined classifier is valid for all concentrations in Tables 1-3 (with a minimum average accuracy of 98.65% over the 20 M iterations). As the average classification accuracy decreases, the difference between sample concentrations b becomes smaller: 99.82% at \(\pm\)1 g/L (Table 1), 98.98% at \(\pm\)0.1 g/L (Table 2), and 98.65% at \(\pm\)0.01 g/L (Table 3). The standard deviation also increases in this direction: 0.0123 with \(\pm\)1 g/L (Table 1), 0.0266 with \(\pm\)0.1 g/L (Table 2), and 0.0272 with \(\pm\)0.01 g/L (Table 3). This pattern meets the expected results: the difficulty of discrimination rises with higher class similarity. In the following lines, we elaborate on the results for each set of classes (fructose concentration), analyzing Tables 1-3 separately. The classification of the samples of Table 1 (0-9 g/L) has very satisfactory results with the three classifiers: Classifier 1 and 2 of Table 4 and the combined classifier of them. With the three classifiers an average accuracy of more than 97% over 20 M random iterations is obtained. It is very remarkable that Classifier 1 can characterize, with only four spectral \begin{table}
\begin{tabular}{c c c} \hline \hline
**Frequencies (kHz)** & **Classifier 1** & **Classifier 2** \\ \hline f1 & 8.4 & 3.1 \\ f2 & 11.7 & 11.2 \\ f3 & 13.8 & 12.8 \\ f4 & 14.7 & 13.0 \\ f5 & - & 14.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Characteristics of the two classifiers provided by GGA+ELM to discriminate the fructose concentrations of Tables 1–3. The classifiers make a decision according to the value of aver-age energy density on specific frequencies in the acoustic response spectrum. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**0–9 g/L (\(\pm\)1 g/L)**} & \multicolumn{2}{c}{**2.0–3.0 g/L (\(\pm\)0.1 g/L)**} & \multicolumn{2}{c}{**2.00–2.10 g/L (\(\pm\)0.01 g/L)**} \\ \cline{2-6}
**Classifier** & **Average Accuracy** & **Standard Deviation** & **Average Accuracy** & **Standard Deviation** \\ \hline
1 & 99.71 & 0.0126 & 90.32 & 0.0704 & 98.65 & 0.0272 \\ \hline
2 & 97.60 & 0.0415 & 85.89 & 0.0727 & 80.78 & 0.0824 \\ \hline Combined classifier & 99.82 & 0.0123 & 98.98 & 0.0266 & 98.65 & 0.0272 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Performance of the two classifiers provided by GGA+ELM and the voting system classifier to discriminate the fructose concentrations referred in Tables 1–3. The average values and standard deviation of the accuracy were estimated from 20 M independent and random iterations. lines, up to ten concentrations with an average accuracy of 99.71%. Combining the two decision-makers in a single classifier achieves an average accuracy of better than 99.8%. The nine frequencies of the combined classifier are located in the range (3-15) kHz. These frequencies have been highlighted in Figure 5 on the spectral information of the vibrational absorption bands for each concentration of Table 1. Note that the combination of these spectral lines allows the ten classes to be differentiated. Not all frequencies are equally important in the classification operation, some frequencies are more decisive in the classification among several classes. There may be other sets of frequencies that classify the samples with similar accuracy. The optimization algorithm offers solutions close to the optimal solution, without ensuring that there is a unique solution. Figure 5: Spectral information of vibrational absorption bands of 10 concentrations from 0 g/L to 9 g/L. The curves represent the average and normalized values of the sound pressure level (dB/Hz) over the audible frequencies range.