## 3 Results and Discussion
### The Dependence on the Number of Species
In other words, we can expect \(c_{tn}(j)=nm-m\) with \(m\) being the sample size, as specified before, in case of a perfect classifier  
\[a_{tn}=\frac{1}{n}\sum_{i=1}^{n}nm-m=\frac{n(nm-m)}{n}=m(n-1). \tag{15}\]  
The results of the prediction experiments in this contribution with an obviously not perfect classifier, reveal that \(a_{tn}\) can be fitted by a linear function \(a_{tn}(n)=b_{1}n+b_{0}\), where \(b_{1}=(49.88\pm 0.82)\times 10^{-2}\), and \(b_{0}=-(59.27\pm 0.61)\). Note that the two coefficients are relatively close to the true sample size \(m=50\). The dependence of the other elements of the confusion matrix on \(n\) are more subtle with respect to the range in which these numbers vary, and the dependence can be described by quadratic functions  
\[a_{tp}(n) = \quad d_{2}\,n^{2}-d_{1}\,n+c_{0}, \tag{16}\] \[a_{fp}(n) = -d_{2}n^{2}+d_{1}n+d_{0}\quad\text{and},\] (17) \[a_{fn}(n) = -d_{2}n^{2}+d_{1}n+d_{0}, \tag{18}\]  
with \(d_{2}=(8.02\pm 1.00)\times 10^{-4}\), \(d_{1}=(22.87\pm 1.34)\times 10^{-2}\), \(d_{0}=6.84\pm 0.37\) and \(c_{0}=43.16\pm 0.38\). Note that the first two coefficients \(d_{1}\) and \(d_{2}\) of \(a_{tp}\), \(a_{fp}\) and \(a_{fn}\) have either the same values (up to the first eight digits, which are not shown here), or just differ in sign but not in value. These coefficients are shown in detail here since we will, in the following, demonstrate a connection between Equations (16)-(18) and the functions describing the dependence of the overall performance measures. Being able to describe \(a_{tp}(n)\), \(a_{fp}(n)\) and \(a_{fn}(n)\), one can now attempt to understand the dependencies of the performance measures. Assuming that each species is classified equally well by a perfect classifier, one would expect \(a_{tp}(n)=c_{tp}(j,n)\) for all \(j\) and similar for \(a_{fp}=c_{fp}(j,n)\) and \(a_{fn}=c_{fn}\). Inserting Equations (16) and (17) in Equation (4) holds  
\[p(j,n)\approx\frac{a_{tp}(n)}{a_{tp}(n)+a_{fp}(n)}\approx\frac{d_{2}n^{2}-d_{1 }n+c_{0}}{c_{0}+d_{0}}, \tag{19}\]  
since the non-constant terms in the denominator cancel each other. Inserting this in the equation for the overall precision (Equation (5)) holds  
\[P(n)\approx\frac{d_{2}n^{2}-d_{1}n+c_{0}}{c_{0}+d_{0}}\sim a_{tp}(n), \tag{20}\]  
since all terms \(p(j)\) are identical for the perfect classifier. Consequently, one should be able to predict the scaling of \(P(n)\), knowing the coefficients \(d_{2},d_{1},d_{0}\), and \(c_{0}\). Fitting the coefficients for the quadratic function describing \(P(n)\) as in Figure 4, one obtains  
Figure 4.0  
with \(g_{2}=(0.16\pm 0.02)\times 10^{-4}\), \(g_{1}=-(0.44\pm 0.03)\times 10^{-2}\), \(g_{0}=0.86\pm 0.76\times 10^{-2}\). Note that these coefficients are very close to the coefficients of \(a_{tp}\) multiplied with a factor \(\frac{1}{d_{0}+c_{0}}=\frac{1}{50}\) as indicated by Equation (20). Hence, we can confirm numerically that the dependence of the precision on the number of classes follows the dependence of \(a_{tp}\) up to a scaling factor of \(\frac{1}{d_{0}+c_{0}}=\frac{1}{50}\). Following the same assumptions and reasoning, one obtains  
Figure 4.1  
for the recall. Here, the relation between the fitting coefficients of \(a_{tp}\) and \(R\) is confirmed by the quadratic function fitted to \(R\) in Figure 4.2 Note that, for the prediction experiments in this study, the same quadratic function is able to describe the \(n\)-dependence of precision and recall. Extending the above reasoning (i.e., \(c_{tp}(j,n)\approx atp(n)\)) to explain the \(n\)-dependence of the accuracy as given by Equation (8) yields  
Figure 4.3  
This relation was numerically confirmed by comparing the coefficients for the polynomials describing \(A\) and \(a_{tp}\). The values of coefficients \(d_{2}\), \(d_{1}\), and \(c_{0}\) are given after Equation (18). Discussing the \(n\)-dependence of the multi-class AUC and the mAP analytically is not as straightforward as the previous considerations; therefore, only numerical results are presented in this contribution.