## 4 The Concept for the Assessment of Data Set Completeness
### Metric for Data Set Completeness
To evaluate the completeness and balance of the sections, each dimension of the feature space, meaning each variable is considered individually. The class vector \(\overset{\rightarrow}{v}_{j}\) (6) represents the number of observations \(\eta\) of the different sections that belong to one specific class \(j\) of the considered  
Figure 6: Schematic visualization of the feature space of a three-dimensional data set and illustration of selected parameters [gray box] (**a**) class 2 of variable \(X_{3}\) is highlighted (**b**) two-dimensional scatter plot of class 2 of variable \(X_{3}\) (**c**) class 3 of variable \(X_{1}\) is highlighted (**d**) class 2 of variable \(X_{2}\) is highlighted. variable \(i\) (in order to preserve greater clarity, \(\overset{\rightarrow}{v}_{j}\) and all further parameters are written without the index \(i\)). Based on this class vector, the balance of one slice of the feature space can be assessed (cf. class 2 of variable \(X_{3}\) in Figure 6a, highlighted in blue). In order to consider the distribution of observations between the classes of one variable \(i\), the variable vector \(\overset{\rightarrow}{s}\) is introduced additionally (7). The variable vector \(\overset{\rightarrow}{s}\) represents the sum of observations in each class \(j\) of variable \(i\). Based on these two vectors, each dimension is assessed individually: once across the dimension via the class vector and once along the dimension via the variable vector. Figure 6c,d visualize the class and variable vectors for dimension \(X_{1}\) and \(X_{2}\), respectively. \[\overset{\rightarrow}{v}_{j}= \begin{bmatrix}\eta_{1},\;\ldots,\;\eta_{K_{j} }\end{bmatrix} \qquad\forall i=[1\ldots n] \tag{6}\]  
\[\overset{\rightarrow}{s}= \begin{bmatrix}\sum\overset{\rightarrow}{v}_{1},\; \ldots,\sum\overset{\rightarrow}{v}_{k_{i}}\end{bmatrix} \;\forall i=[1\ldots n] \tag{7}\]  
The data set completeness \(DSC_{j}\) of each class \(j\) (8) consists of two aspects: first, the balance of the class vector \(\overset{\rightarrow}{v}_{j}\), called section balance \(SB_{j}\), and second, the relative number of observations in the considered class, called class volume \(CV_{j}\). While the section balance \(SB_{j}\) evaluates the balance of the available observations within one class, the class volume \(CV_{j}\) evaluates the relative number of observations in the considered class compared to the other classes of variable \(i\) (\(\overset{\rightarrow}{s}\)). A perfect data set completeness is characterized by a perfectly balanced distribution of observations around the sections within one class and perfectly equally distributed observations around the classes. In this case, the feature space is equally utilized. \[DSC_{j}=\alpha\cdot\ SB_{j}+(1-\alpha)\cdot\ CV_{j}\qquad\forall j=[1\ldots k_{ i}] \tag{8}\]  
The constant factor \(\alpha\) defines the importance of the two aspects. If \(\alpha<0.5\), it is more important that the class contains a higher relative number of observations rather than the sections being balanced. This could be a reasonable approach when resampling the data to balance the data set is an appropriate preprocessing step for the use case at hand. A default \(\alpha=0.5\) is proposed, given the section balance \(SB_{j}\) and the class volume \(CV_{j}\) equal importance. The metric for the section balance \(SB_{j}\) measures the balance of observations around the sections of one class. In the best case, each section contains an equal number of observations. The metric is derived by the imbalance degree of [PERSON] et al. 2017 [46]. In contrast to other imbalance measures such as imbalanced ratio (amount of data in the smallest section divided by the amount of data in the biggest section) the imbalance degree takes into account the distribution of all sections and not only the smallest and biggest. In order to do so, the distance \(d\) between the class vector \(\overset{\rightarrow}{v}_{j}\) and the equally distributed vector \(\overset{\rightarrow}{e}_{j}\) is calculated as measure for the imbalance.