## 4 The Concept for the Assessment of Data Set Completeness
### Metric for Data Set Completeness  
For data set completeness, the presence and balance of the value combinations for the variables \(\overset{\rightarrow}{X_{i}}\) in \(\overset{\rightarrow}{I}\) are evaluated. High data set completeness is given when observations of the data set equally utilize the feature space. In a three-dimensional case, the utilization of the feature space is easily assessed by a graphical representation. However, such representations are limited in the number of dimensions. Assessing the data set completeness numerically requires a metric to quantify the presence and balance of the value combinations. To reduce computational complexity, the feature space is partitioned into different sections (cf. Figure 5(a)). A section is an n-dimensional bin in the feature space. These sections are observed for data set completeness rather than the single values with arbitrary precision. All sections are defined by a specific value interval of each variable. In the example of Figure 5(a), observations with a stroke between 75% and 80%, a pressure between 17.5 and 18.5 bar and a pressure difference between 0.0 and 3.0 bar belong to one section (marked as Boxexample in Figure 5(a)). To identify the value intervals of the variables--so-called classes--each variable is partitioned individually into \(k_{i}\) classes. The classes do not overlap but border to each other, representing one specific value interval. A superscript, e.g., \(\overset{j}{j}\) and \(\overset{j}{u}\) for class \(j\), denotes the lower and upper bound of each class, respectively. The lower bound of the first class represents the minimum value, whereas the upper bound of the last class represents the maximum value of the variable. Different approaches can be chosen to partition the values range of the variables. First, the partitioning of each variable can be performed based on simple rules derived from histograms. The optimal width of the classes can be calculated by quotients, based on the number of observations, e.g., Sturges-rule [43], or in combination with the spread of the data, e.g., Freedman-Diaconis rule [44]. In this case, the classes are of equal width. This might not get along with the characteristics of the data. Two different operational points (e.g., Figure 0(a)) should be in two different classes to be examined for their combinations with the other variables individually. On the other side, one operating point should not be split into two classes unnecessarily. This is not ensured when using Sturges or Freedman-Diaconis-rule. A second approach, the one-dimensional clustering such as Jenks Natural Breaks [45], partitions the variable into classes with different widths based on the data's characteristics. The clustering takes into account the potential operating points. While the clustering splits the operating points to separate classes, the Sturges rule and Freedman-Diaconis rule might split one operating point or mix several points together due to the fixed width of the classes. A third approach, avoiding wrong breaks of the value range, performs the partitioning based on expert knowledge. The expert can take specific requirements set by the use case into account. In the following, it is assumed that the experts define the classes of the variables. For future work, the implementation of a one-dimensional clustering is considered, automating the process. Splitting the n-dimensional feature space into its sections based on the classes results in the total number of \(K\) (Equation (5)). An example for this parameter and the following is given in the grey box of Figure 6 to increase clarity. \[K=\prod_{i=1}^{n}k_{i} \tag{5}\]  
All observations \(m\) of the data set \(I\) are assigned to their respective section according to the values of the variables \(\overset{\rightarrow}{X_{i}}\). The number of observations \(\eta\) within one section is considered as a feature for the calculation of the data set completeness.