## 4 Simulation
### Experimental Results  
As per previous studies [9], the optimal number of bases might be related to the number of note events, as mentioned in Section 2. Therefore, we first compare the estimated number of bases of each algorithm and evaluate the performance based on the number of actual note events. To compare the overall performance, we verify the distribution of the absolute values of errors, as shown in Figure 9. The vertical lines indicate the range between the smallest and largest errors, and the boxes indicate the quartiles: The top and bottom of the box indicates the third and first quartiles, and the horizontal bar in the middle of the box indicates the median value. As expected, smaller values indicate better performance. The graph demonstrates that the ARD-NMF algorithm has the best performance amongthe conventional algorithms. The result of the proposed RESUREv2 algorithm (1 st quartile: 3.25, median: 8.5, 3 rd quartile: 13) is slightly better than that of the ARD-NMF algorithm (1 st quartile: 4.25, median: 9.0, 3 rd quartile: 16). The detailed results of the estimated NMF ranks for the database are shown in Table 1. The underlined bold number indicates the result closest to the number of musical notes for each clip, and the (highlighted) green value indicates a result with an error within \(\pm 15\%\) of the number of musical notes. The results demonstrate that ARD-NMF (with a \(\tau\) of \(2.5\times 10^{-2}\)) performs best among the conventional algorithms, and the performance of the RESUREv2 algorithm is similar to or slightly better than that of the ARD-NMF algorithm. While the ARD-NMF performance is affected by the parameter \(\tau\) (comparing the \(\tau=0.005\) and \(\tau=0.025\) cases), the proposed algorithm does not contain any parameters. This is a further advantage of the proposed algorithm. While the RESUREv2 algorithm works well for many audio clips, it greatly overestimates the estimation results in some clips such as # 11, # 23, # 27. These clips have in common that the interval between the first and second onset times is excessively short (100 ms). In those cases, the RESUREv1 shows better performance compared to RESUREv2. While we cannot generalize with only three cases, we expect that the RESUREv1 algorithm might improve the RESUREv2 algorithm with small intervals between the first and second onsets despite the poor performance of the RESUREv1 algorithm overall. Figure 9: Distribution of absolute values of the estimation errors based on the number of notes. The top and bottom of the box indicate the third and first quartiles, and the horizontal bar within the box indicates the median value. The ends of the vertical bar indicate the maximum and minimum values. While several studies have demonstrated that the actual number of notes might be related to the optimal NMF rank for the matrix-factorization-based AMT method [9], they cannot guarantee that the number of notes is the best selection for the NMF rank. Therefore, we measure the AMT performance to evaluate the rank estimation performance. The detailed settings for evaluating the AMT performance are discussed in Section 4.1. The F1-score results are shown in Figure 10. While the differences of F1-score performances between the algorithms are not as noticeable as the number of notes estimation performance, the algorithm whose estimated rank is close to the number of notes shows better F1-score performance. ARD-NMF performs best (1 st quartile: 40.89, median: 43.23, 3 rd quartile: 48.63) among the conventional algorithms, and the RESUREv2 algorithm performs slightly better (1 st quartile: 41.20, median: 45.61, 3 rd quartile: 51.25) compared to ARD-NMF. Just as in the case of the rank estimation results, the ARD-NMF F1-score results are similar to those of the RESUREv2, but the ARD-NMF performances are affected noticeably by the parameter \(\tau\). The proposed algorithm's F1-score advantages (which represent an improvement of approximately 2%-3% over the conventional algorithm), are not considerably large.