## 7 Measure of Misclassification  
The definition of five classes of emission profiles may result in misclassifications for completely different profiles. A new but not implemented emission profile is assigned to one of the defined classes even if it does not fully match. A simple measure of misclassification was defined in order to overcome these inherent shortcomings [22]. The measure of misclassification, \(G_{m}\), is based on the average distance between all emission profiles in one group, compared to the distance of the new emission profile.  
The Euclidian distance \(d\left(x_{i},x_{j}\right)\) was used to calculate the dissimilarity between two emission profiles \(x_{i}\) and \(x_{j}\) within one group. The average distance between one emission profile and all others profiles of one group can be calculated as follows:  
\[\overline{d}_{i}=\overline{d}(x_{i})=\frac{1}{N}\sum_{j=1}^{N}d\left(x_{i},x_ {j}\right)\ \ \text{with}\ \ \ i,j=1,2,\ldots,N\ \ \ \text{and}\ \ \ \ i\
eq j \tag{4}\]  
Figure 10: Robustness for the classification of 360 emission profiles; 100 repetitions, with random splits into 50% training samples and 50% test samples; box whisker plots of average and per class classification rates for training samples (white) and test samples (grey).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Class** & **D0** & **D1** & **D2** & **D3** & **D4** & **Average** \\ \hline Number of & 0 & 4 & 1 & 1 & 2 & 1.6 \\ misclassifications & 100 & 94.9 & 93.3 & 93.3 & 95.8 & 95.5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Classification results for the test samples of Table 1.  
and results for \(N\) emission profiles within one group in \(\overline{d}_{i}=\overline{d}_{1},\overline{d}_{2},\ldots,\overline{d}_{N}\) average distances. The 25 th percentile \(\overline{d}_{[25]}\) and the 75 th percentile \(\overline{d}_{[25]}\) are calculated. A newly classified emission profile, \(y\), can be defined as an outlier, similarly to the definition of the box-whisker plots in [39]:  
\[\overline{d}(y)>\overline{d}_{[75]}+1.5\left(\overline{d}_{[75]}-\overline{d }_{[25]}\right) \tag{5}\]  
The measure of misclassification for newly quantified emission profiles is calculated as follows:  
\[G_{m}(y)=\frac{\overline{d}(y)-\overline{d}_{[75]}}{\overline{d}_{[75]}- \overline{d}_{[25]}} \tag{6}\]  
High values of \(G_{m}\gg 1\) are used as an indicator showing that newly classified emission profiles strongly differ from the training samples of the assigned class. The measures of misclassification are exemplarily presented in Figure 10(c) for test samples of the classes D1 and D2. Two assignments with \(G_{m}=1.5\) and one assignment with \(G_{m}=4.0\) indicate that the emission profiles differed from the training samples of the respective classes. The actual emission profiles, which are displayed in Figure 10(b),d, emphasized that the higher the measure, the more likely the misclassification of the classified emission profile.  
Nevertheless, emission profiles should be manually reviewed by the user of the classification system. Additional measurements or the definition of a new class of emission profiles may be needed, depending on the individual results.  
Figure 11: Measures of misclassification for the classes D1 and D2, and emission profiles of training samples (grey), and test samples (blue/red).