## 3 Evaluation
### Results and Discussion  
In this section, we detail a systematic comparison of the results section obtained from the implemented methods: the IEM-FD, TVAR, and the proposed method. All of the separation techniques were implemented using the MATLAB (R2020b) programming language. The metrics employed to evaluate the implemented methods are described in Section 3.1. For evaluation, we compared the ground-truth and estimated temporal location and length of each crackle using a tolerance window of 0.6 ms. The overall results for the implemented methods are shown in Figure 6. The boxplot displays the distribution of data based on a five-number summary (\"minimum\", first quartile (Q1), median, third quartile (Q3), and \"maximum\"). It details information about the atypical values and the location of them within the entire range of possible values. At the same time, it indicates whether the data are symmetrical, how tightly the data are grouped, and how the data are skewed. Figure 6 shows these results in terms of accuracy _Acc_, sensitivity \(S_{e}\) and precision \(P_{r}\) of the IEM-FD method on the left side, TVAR method on the center and the proposed method on the right side. Additionally, Figure 7 displays the accuracy results of the compared methods as a function of the SNR value (between \(-10\) and \(10\) dB). In general terms, the values obtained from the proposed method are significantly higher in terms of accuracy \(Acc\) and precision \(P_{r}\) among the compared methods. Moreover, our method also outperforms the IEM-FD and the TVAR in terms of sensitivity (\(S_{e}\)). Regarding the boxplot accuracy values of the IEM-FD method, the median is 45.45%, the first quartile (Q1) is 32.25%, the second quartile (Q2) is 50%, the maximum is 61.54%, and the minimum is 5.66%. In fact, the performance of this method is limited by the number of false positives (i.e., normal events detected as crackles) as be seen in the precision values in Figure 6.  
In the case of the TVAR method, the following measurements are found: the median is 70%, the first quartile (Q1) is 90%, the second quartile (Q2) is 40%, the maximum is 100%, and the minimum is 0%. This variability in the results suggests that the method is too dependent on the conditions of the input signals, as can be seen in Figure 7. In fact, the  
Figure 6: Accuracy, sensitivity and precision average results evaluating all scenarios and SNRs in the dataset \(\psi\) by IEM-FD [28], TVAR [51], and the proposed method. Figure 7: Accuracy, sensitivity, and precision average results evaluating all scenarios for each SNR in the database \(\psi\) by IEM-FD [28] (red color), TVAR [51] (green color), and the proposed method (blue color), where the dashed lines represent the mean value for each metric and method. TVAR method provides the worst results among the compared methods when dealing with low SNR values. The proposed method shows the minimum standard deviation among the compared methods. In particular, the median is 100%, the first quartile (Q1) is 90.90%, the second quartile (Q2) is 100%, the maximum is 100%, and the minimum is 80%. Additionally, as can be seen in Figure 7, the results are robust independently of the input signal SNR conditions. Now, focusing on a specific type of crackles, Figure 8 presents the \(Acc\), \(S_{e}\), and \(P_{e}\) results as a function of type of crackle (coarse (right side of the figure) or fine (left side of the figure)) and SNR value. As can be seen, the performance of the proposed method is clearly superior to the other compared methods in both cases (coarse and fine crackle detection). Comparing both sides of the figure, the results reveal that the proposed method underperforms when dealing with coarse crackles under low SNR scenarios. Similar behavior is observed in the case of the TVAR method although, as previously commented, the results are limited by the number of false positives, which provokes a clear underperformance in terms of \(S_{e}\).