## 5 Application and Discussion of the Proposed Metric and Visualization
The pressure \(p\) shows a higher number of observations for higher values. The boxplot marks lower values of \(p\) (\(0<p<0.25\)) as outliers, since these values are rare. A high data set completeness requires equally distributed data around the whole feature space. E.g., high pressure \(p\) should occur with high as well as low stroke \(s\) and the other way around. Furthermore, a high-pressure difference \(\Delta p\) should be observed for high upstream pressure \(p\) as well as for low upstream pressure \(p\) and the other way around. To assess the data set completeness of this training data, each variable is partitioned into four classes, representing the operating points. The variable stroke \(s\) could also be partitioned into more classes, since no clear operating points are identifiable. The data set completeness (cf. Figure 10) shows high values for stroke and pressure difference, while the data set completeness decreases for lower values of pressure. Two aspects cause the low data set completeness for lower pressure. First, the absolute number of observations is low for a pressure between 0 and 0.25 (cf. Figure 9). Less than 1000 observations are observed for a single bar of the histogram, while a pressure between 0.75 and 1 shows frequencies of more than 5000 observations. This affects the class volume \(CV_{j}\), decreasing the data set completeness. Second, low pressure does not allow a high-pressure difference. For that reason, the sections of low-pressure and high-pressure difference of the feature space do not contain observations, which decrease the section balance \(SB_{j}\) and data set completeness in these classes, respectively. Figure 9: Descriptive statistics of the original training data set (normalized) with ca. 27,000 observations (**left**) and histograms and boxplots of the training data set (**right**). However, such physical constraints do not harm the quality of the data. Only the unbalanced distribution of the data for lower and higher pressure (class volume \(CV_{j}\)) is unfavorable. This is why the expert is still required. The results need to be interpreted appropriately. Even though the data set completeness is minimal for low pressure, the data set still is useful. On average, stroke and pressure difference achieve a data set completeness of 0.79. In contrast, pressure achieves a mean data set completeness of 0.51. If only stroke and pressure difference are considered, the data set completeness reveals a very high utilization of the feature space (cf. Figure 11a). For this two-dimensional consideration, the metric results for data set completeness can be verified based on a simple scatter plot (cf. Figure 11b) visualizing the observations between the two variables. Each section of the feature space contains several observations, given a reasonable basis to train a valid model to predict the valve's flow. Figure 11: (a) Screenshot of the visualization of data set completeness for stroke \(s\) and pressure difference \(\Delta p\) and (b) two-dimensional scatter plot of stroke \(s\) and pressure difference \(\Delta p\). Figure 10: Annotated screenshot of the visualization of data set completeness in the training data set. To model the dependencies between flow \(q\), stroke \(s\), pressure \(p\) and pressure difference \(\Delta p\), a random forest regression was trained using the flow as the dependent and the stroke, pressure and pressure difference as the independent variable. The resulting model represents the benchmark for evaluating the proposed metric and visualization for data set completeness. The model was trained in Python 3.7 with the random forest regressor of the scikit-learn package (version 0.23.1). A random search was performed, changing the number of generated decision trees, the minimum number of samples in one leaf and the tree's maximum depth. The evaluation was performed based on the coefficient of determination, RMSE and the computation time required to use the model.