## 5 Results
### Accuracy: SGE3 and NSGE3  
The neural network trained to predict the filter gains of the previous third-order GEQ design, called NGEQ3, produced a maximum error of \(0.29\,\mathrm{dB}\) and a mean maximum error of \(0.07\,\mathrm{dB}\) in predicting the filter gains [20]. The mean maximum error corresponded to the average of maximum errors computed from many random gain settings. In this case, it was computed using a test set of 10,000 random gain settings, each having 31 pairs of command gains and filter gains. The mean maximum error better described the typical behavior of the GEQ design than the maximum error that only occurred in one particular test setting, usually a very strange one, which may not ever be used in practice. In this case, the accuracy had to be interpreted so that the maximum error caused by the neural network never (according to our wide tests) exceeded \(0.29\,\mathrm{dB}\), and it was very likely to be about \(0.07\,\mathrm{dB}\) in practice. The maximum error of the proposed NSGE3 design was \(0.33\,\mathrm{dB}\), and the mean maximum error was 0.07 dB. In comparison to the accuracy of the NGEQ3 method above, these errors were almost the same. Surprisingly, our tests indicated that the error in the filter gain prediction did not increase the magnitude-response error as much, since the error in the gain value could actually move the magnitude response closer to the command point or the gain prediction error might occur at a frequency where the magnitude-response error was minimal to begin with. Table 5 shows the maximum and mean maximum errors obtained with different design methods: the ACGE3 method had a maximum error of 0.97 dB, and the NGEQ3 method had a max error of 1.05 dB. Thus, the increase in error was only 0.08 dB (compare the maximum errors on the first two lines of Table 5), although the neural network could increase the error in the gain of a single band by as much as 0.29 dB. Similarly, the NGE3 method was only 0.05 dB less accurate than the SGE3 method (see the last two maximum errors in Table 5), although the maximum error of individual filter gains could be as large as 0.33 dB. Table 5 also shows that the mean maximum error of the proposed NGE3 method was 0.50 dB, which was slightly better than that of the previous NGEQ3 method, which was 0.54 dB. Furthermore, the maximum errors of both of the proposed designs, SGE3 and NGE3, were smaller than that of the ACGE3 least-squares design, which was the earlier state-of-the-art method. Figures 10-13 plot the magnitude responses obtained with SGE3 (- -) and NGE3 (--) for different illustrative command gain settings (\(\O\)), along with their calculated EQ filter gain values (\(\square\) and \(\times\)). Ideally, the filter gains predicted by the neural network depicted by green crosses should lie exactly inside the black boxes representing the optimized gains, leading to identical magnitude responses. The title of each figure exposes the largest magnitude error occurring for both SGE3 and NGE3 at command point frequencies, as described above. Figure 10 shows a particular test case from the validation dataset in which SGE3 had its largest error of 0.81 dB (see Table 5, Row 3, Column 2). Here, the error occurred at 16,443 Hz, which was between the two highest command frequencies that had the same gain setting of +12 dB. The largest error of NGE3 was, in this case, slightly smaller (0.75 dB) than that of SGE3, and it occurred at the same frequency. Furthermore, the neurally-controlled filter gains (\(\times\)) matched closely those optimized using the WLS algorithm (\(\square\)). The largest error within the validation dataset for NGE3 was 0.86 (see Table 5, last row, second column). This case is shown in Figure 11. The error occurred again between adjacent command points, this time at 10,453 Hz, having the same gain value of 12 dB. In this particular case, SGE3 had an error of 0.77 dB, which occurred at the same frequency.