## 3 Results and Discussion
### The Dependence on the Number of Species  
All performance measures decrease with an increased number of species as is visible in Figures 3 and 4. An intuitive explanation for this could be that species are more difficult to distinguish when more species are added to the classification task. However, looking at the definitions  
Figure 4: The \(n\)-dependence of (**a**) Precision, (**b**) AUC, (**c**) mAP, (**d**) Recall and (**e**) Accuracy can be described by fitting quadratic functions (line). The error bars (whiskers) represent the ranges from the best result to worst result obtained for 20 different randomly drawn subsets for each value of \(n\). The red marking in each box indicates the median, and the boxes show the range of the middle 50% of the results. of the performance measures (Equations (4)-(13)) we investigated whether it is possible to understand the numerical results by analytic reasoning. Consider, e.g., the precision \(P(n)\), which is defined in Equation (5). If each precision per species \(p(j)\) contributing to the average is constant and not depending on \(n\) (i.e., \(p(j)\sim c\)), one should expect \(P(n)\sim c\). This is clearly not what is observed in Figure 4. Therefore, one must assume that \(p(j)\) is dependent on \(n\), although this is not explicitly visible in Equation (4). To investigate this implicit dependence on \(n\), we visualized the average numbers of true positives  
\[a_{tp}=\frac{1}{n}\sum_{j=1}^{n}c_{tp}(j), \tag{14}\]  
for each \(n\) and, in a similar way, the averaged numbers of false positives, true negatives, and false negatives in Figure 5a-c. These elements of a confusion matrix enter (in an non-averaged form) into the computed performance measures, and thus their dependence on \(n\) influences the performance measures. The averaging in Figure 5 was done, since the amount of sound samples in each trial clearly depends linearly on \(n\). Therefore, this trivial dependence was removed, and we can monitor a non-trivial implicit dependence on \(n\). As one can see, the dependence of the averaged numbers of true positive, false positives, and false negatives can be described relatively well by a quadratic function, whereas the averaged number of true negatives increases linearly with increasing \(n\). The fact that true negatives, as can be seen in Figure 5d, behave differently than the other elements of the confusion matrix can be understood by considering the way true negatives are computed in a multi-class classification problem, using a one-vs-all configuration. Each time a sound sample was correctly not classified as the particular species \(j\) under consideration, the count of true negatives was increased by one. Therefore,  
Figure 5: The elements of a confusion matrix (**a**) averaged numbers of true positives, (**b**) averaged numbers of false positives, (**c**) averaged numbers of false negatives, (**d**) averaged numbers of true negatives) display a linear (true negatives) and quadratic (all other elements) dependence on \(n\). The error bars (whiskers) represent the ranges from the best result to worst result obtained for 20 different randomly drawn subsets for each value of \(n\). The green marking in each box indicates the median, and the boxes show the range of the middle 50% of the results. e.g., in a subset of \(m\cdot n=500\) sound samples recorded from \(n=10\) different species and each species being represented by \(m=50\) sound samples, a perfect algorithm would classify 50 samples correctly as belonging to species \(j\). Consequently, the count of true positives would be \(c_{tp}(j)=50\), and the count of true negatives \(c_{tn}(j)=450\) for a perfect classifier.