## 4 Simulation
### Experimental Results
However, the proposed algorithm neither specifies a particular NMF method, nor requires sensitive design parameters; therefore, it is meaningful that its performance is improved, rather than degraded. To analyze the rank estimation and F1-score results, we plot the precision and recall of each algorithm (RESUREv2, ARD-NMF, GaP-NMF) versus the differences between the estimated ranks of the algorithms and the numbers of musical notes in audio clips, as shown in Figure 11. The x-axis of each graph indicates the estimated rank minus the number of notes. The graph demonstrates that the small estimated ranks tend to have high precision and low recall and vice versa. However, certain extremely small ranks (indicated by the dashed red circle in Figure 11a) cause low precision and recall performances. In our view, the proposed algorithm avoided severely underestimating the rank that causes both low precision and recall, resulting in better performance. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c} \hline \hline
**Clip No.** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** & **12** & **13** & **14** & **15** \\ \hline Number of Notes & 28 & 30 & 34 & 27 & 33 & 42 & 27 & 33 & 33 & 26 & 32 & 20 & 43 & 25 & 59 \\ \hline VAR-NMF & 21 & 15 & 23 & 9 & 18 & 9 & 5 & 24 & 23 & 15 & 59 & 11 & 33 & 9 & **57** \\ ARD-NMF\((\tau=5\times 10^{-3})\) & 23 & 17 & 24 & 14 & 20 & 20 & 17 & 12 & 13 & 14 & 22 & 32 & 14 & **23** & 14 \\ ARD-NMF\((\tau=2.5\times 10^{-2})\) & 35 & **27** & **38** & 12 & 28 & 15 & **30** & 19 & 20 & 18 & 12 & 27 & 26 & **24** & 29 \\ GaP-NMF & 22 & 20 & 23 & **20** & 22 & 22 & 19 & **20** & **23** & 18 & **17** & **21** & 23 & **22** & 14 \\ RESUREv1 & **25** & 22 & 15 & 9 & 13 & 19 & **25** & 58 & 58 & 12 & 47 & 13 & 14 & 15 & 30 \\ RESUREv2 & **31** & 39 & **30** & 14 & **35** & **34** & 40 & 18 & 17 & **23** & 62 & 18 & **40** & 35 & **51** \\ \hline
**Clip No.** & **16** & **17** & **18** & **19** & **20** & **21** & **22** & **23** & **24** & **25** & **26** & **27** & **28** & **29** & **30** \\ \hline Number of Notes & 52 & 23 & 49 & 19 & 39 & 46 & 21 & 23 & 27 & 28 & 29 & 45 & 36 & 43 & 18 \\ \hline VAR-NMF & 5 & 11 & 30 & 11 & 16 & 15 & 17 & **21** & 15 & 16 & 13 & 22 & 16 & 6 & 6 \\ ARD-NMF\((\tau=5\times 10^{-3})\) & 17 & 16 & 12 & 41 & 19 & 15 & 20 & 35 & 19 & **24** & 22 & 14 & 15 & 15 & 53 \\ ARD-NMF\((\tau=2.5\times 10^{-2})\) & 23 & 15 & 34 & 15 & 55 & 23 & **22** & 39 & 40 & 34 & **31** & 14 & **26** & **35** & 22 \\ GaP-NMF & 15 & **17** & 18 & 24 & 16 & 19 & **19** & **21** & 20 & 22 & 20 & 17 & 21 & 17 & **21** \\ RESUREv1 & 12 & 35 & **56** & **20** & 19 & 20 & 17 & 32 & 19 & 15 & 14 & **36** & 18 & 12 & **12** \\ RESUREv2 & **46** & 4 & 37 & 26 & **44** & **53** & **21** & 39 & **24** & **29** & 18 & 57 & 18 & 62 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Estimated NMF ranks for 30 test signals. 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. To compare the performance of the proposed algorithm with that of the conventional SURE-based rank estimation method, we measured the estimation error of the number of notes and the F1-score performance of the conventional SURE method [16]. The input data of the conventional SURE method were the same as that of the proposed method. The conventional SURE method does not require parameter settings, similar to the proposed method. The Figure 12a shows the absolute values of the estimation errors (same as Figure 9) and Figure 12b shows the F1-score results (same as Figure 10). The graphs show that the conventional SURE method fails to determine the proper number of notes, and the F1-score results are also degraded. Figure 11: Scatter plots for (**a**) precision and (**b**) recall versus the difference between the estimated rank and number of notes (estimated rank number of notes).