## 2 Methods
### Measuring Classification Success  
This section introduces the indices used to measure the classification performance of bird vocalizations [34; 35]. In each numerical experiment, we considered \(n\) species and a dataset of a total of \(4m\) sound samples. In more detail, \(3m\) samples were used for training, whereas the remaining \(m\) samples were employed to evaluate the quality of classifications. Within the dataset for \(n\) randomly selected species, for each species \(j=1,2,...,n\), a dataset of \(4m\) samples was processed. Here, \(3m\) samples were used for training, whereas the remaining \(m\) samples were employed to evaluate the quality of the classifications. These evaluations were then done by computing a confusion matrix for each species, containing:  
\(c_{tp}(j)\): the number of classifications that are true positives for species \(j\),  
\(c_{tn}(j)\): the number of classifications that are true negatives for species \(j\),  
\(c_{fp}(j)\): the number of classifications that are false positives for species \(j\), and  
\(c_{fn}(j)\): the number of classifications that are false negatives for species \(j\). The resulting elements of the confusion matrix enter into computation of more advanced metrics for measuring classification success, such as the precision, recall, accuracy, mean-average-precision, and receiver-operator characteristics. To understand how the entries of each species' confusion matrix influence the outcomes of these summarizing measures, more detailed descriptions of their computations are introduced in the following:  
* _Precision:_ This metric gives the measure of reliability of our predictions. The formula to compute the precision for a bird species \(j\) is \[p(j):=\frac{c_{tp}(j)}{c_{tp}(j)+c_{fp}(j)}.\] (4)  
Therefore, the precision for a species \(j\) indicates how many true positives the model predicted out of all positives. Therefore, the higher the precision, the more confident a model is about its predictions. In order to compute the precision for the entire test dataset, we average over all species  
Figure 2: The architecture of the feed forward neural network used for bird classification. The input \(x\in\mathbb{R}^{45}\) maps through three intermediate layers with \(d_{1}=256\), \(d_{2}=128\), and \(d_{3}=64\) and ReLU transfer functions. The output layer maps to \(n\) independent classes with a softmax transfer function where \(n\) is the number of bird species. \[P=\frac{1}{n}\sum_{j=1}^{n}p(j). \tag{5}\]
* _Recall:_ This metric gives the measure of predictive power of a model. The formula to compute recall for each bird species \(j\) is \[r(j):=\frac{c_{tp}(j)}{c_{tp}(j)+c_{fn}(j)}.\] (6) Thus, to recall, for a class species \(j\), would indicate, of all actual positives in the test dataset, how many did the model predict as positive. Therefore, the higher the recall, the more positive samples the model correctly classified as positive. In order to compute the recall for the entire test dataset, we averaged over all species \[R=\frac{1}{n}\sum_{j=1}^{n}r(j).\] (7)
* _Accuracy:_ While precision and recall are computed for each class separately in a multi-class classification problem, the accuracy \(A\) is computed for the entire test dataset using \[A:=\frac{\sum_{j=1}^{n}c_{tp}(j)}{m\cdot n}.\] (8) and thus, out of all test samples, how many were correctly classified. * _Area Under ROC Curve (AUC):_ An ROC curve shows the performance of a classification model at different classification thresholds. The curve is computed by plotting the true positive rate (\(r_{tp}\)) against the false positive rate (\(r_{fp}\)) at these thresholds. The true positive rate for a bird species \(j\) is defined as: \[r_{tp}(j,\rho):=\frac{c_{tp}(j,\rho)}{c_{tp}(j,\rho)+c_{fn}(j,\rho)},\] (9) and the false positive rate for a bird species \(j\) is defined as \[r_{fp}(j,\rho):=\frac{c_{fp}(j,\rho)}{c_{fp}(j,\rho)+c_{tn}(j,\rho)},\] (10) with \(\rho\) denoting a probability threshold that is varied from 0 to 1 in order to obtain the ROC curve.