## 4 Results and Discussion
### Discussion
In each spectrum, the frequencies used for both classifiers are emphasized. Figure 6 shows ten different patterns for the concentration classification of Table 1. Each pattern is constructed with the average normalized energy density value for each of the nine frequencies at a concentration from Table 1. The similarity between some patterns is interesting. For example, the curve characterizing the 1 g/L concentration is similar to the 5 g/L concentration pattern. The same happens with the 4 g/L and 9 g/L concentration patterns. This phenomenon has been reproduced in all the experiments performed and we believe that it may be associated with the resonance of the container used. The application of the combined classifier on 143 samples from Table 2 (concentrations between 2.0 g/L and 3.0 g/L with increments of \(\pm\)0.1 g/L) provided satisfactory results. As in the previous case, a sequence of 20M random and independent iterations was carried out. As shown in Table 5, Classifiers 1 and 2 obtained an average accuracy higher than 85%, whereas their combined classifier improves the average classification accuracy to 98.98% with a standard deviation 0.0266. This result is very satisfactory, although worse than that obtained for 1 g/L concentrations. The greater the similarity among classes, the more complex the classification and the lower the precision. Figure 7 presents the patterns for the concentrations between 2.1 g/L and 2.9 g/L. These have been generated from the average and normalized energy density values for the nine frequencies of the combined classifier. Very similar values are observed in general since the variation in concentration is only \(\pm\)0.1 g/L. Extreme closeness is appreciated in some cases such as the 2.3 g/L and 2.4 g/L concentrations, also with the 2.8 g/L and 2.9 g/L concentrations. Figure 6: Spectral information patterns for fructose concentrations from 0 g/L to 9 g/L, with the average and normalized value of the sound pressure level (dB/Hz) as a function of the frequencies (kHz) used by the combined classifier. For classes with a concentration difference of \(\pm\)0.01 g/L in Table 3, the 143 samples were also analyzed by Classifiers 1, 2 and the combined system. The last columns of Table 5 show the results obtained by each classifier over 20M random and independent iterations. Classifier 1, with 4 frequencies in the 8-15 kHz range and an average classification accuracy of 98.65% was much more effective than Classifier 2, where the accuracy de-creased to 80.78%. The combination of both classifiers did not improve the accuracy of Classifier 1, so the combined system considers only the decision of the first classifier, ignoring the decision of the second one. As discussed in the beginning of the section for the concentrations of Table 1, not all frequencies contribute equally in the classification. At the high similarity concentrations of Table 3, the frequencies of Classifier 2 do not add new information in the decision process over the information of Classifier 1. As a result, all the samples were classified with an average accuracy of 98.65% and a standard deviation of 0.0272. The allocation of the four frequencies of Classifier 1 in the middle band of the spectrum, between 8 and 15 kHz, is understandable. It is the band with high mean energy density values. Figure 8 shows the position of these frequencies in the mean spectra of the nine concentrations between 2.01 g/L and 2.09 g/L. Figure 9 shows the patterns generated by Classifier 1 of the average and normalized energy density for the four frequencies. As in the previous cases there are very similar patterns, for example in the 2.01 g/L and 2.02 g/L patterns. There is also close similarity between the 2.03 g/L, 2.05 g/L, and 2.09 g/L patterns. Figure 7: Spectral information patterns for fructose concentrations from 2.1 g/L to 2.9, with the average and normalized value of the sound pressure level (dB/Hz) as a function of the frequencies (kHz) used by the combined classifier.