## 3 Results
### ART Sorting Algorithm
The square roots from all of the possible pairwise product terms regarding the geometry were calculated as further optional input terms of some of the models, resulting in a maximum of 28 (full geometry) or six (half geometry) optional predictor variables. Other terms were not taken into account, because this could have led to poor classification results due to too many inputs, which would have required even more training data than actually available, as a countermeasure. In addition, higher polynomial terms may have resulted in an over-fitted model that was no longer suitable for the application to the noisy measurement data. Since using the eigenfrequency interactions would have resulted in too many predictor variables (actually 153 interaction terms) for a sufficient model prediction, we did not consider them. For the distinct description of the classification models depending on the predictor variable configurations, we introduced short model identifiers. The letters \(F\), \(G\), and \(g\) indicate if the eigenfrequencies, the full geometry, or only half of the geometric information were considered for a model, respectively. Additionally, the numbers \(I\) or \(2\) show if the geometry data were only used directly (1) or if additional interactional terms occurred (2). For example, \(F\) describes that the eigenfrequencies alone were implemented, and \(F\)-\(G2\) denotes a model that is based on the eigenfrequencies, plus the full geometric data including interactions. Before applying the final classification models to the machine-made validation parts, the theoretical classification limits were analyzed with the help of new virtual parts or synthetic validation data. These synthetic data comprise 1000 defect-free parts from the basic population. In addition, many defective parts with the same geometric dimensions, but having a through-hole of 1 mm, 1.5 mm, 2 mm, 2.5 mm, or 3 mm either in one of the cylinder rings or in the bar area, were considered. Altogether, three times 1000 parts per defect size were fully simulated and not enhanced as within the training data generation to reach the most realistic defect behavior. The correction factors were applied to the simulated eigenfrequencies. This fitting to the measurement data allowed the synthetic validation data to be put directly into the calculated classification models, instead of recalculating the models. Figure 11 shows the results for the classification models F and F-G2, both derived with the help of the synthetic ART training data without additional random noise. Each diagram in Figure 11 visualizes the classification results of the 1000 virtual parts and each single data point represents one part. The title indicates the classification model used, as well as the true quality state of the parts. The \(x\)-axis specifies the defect size that is connected with the predicted defect class, whereby the parts classified as good were plotted at a defect size 0 mm. The \(y\)-axis indicates the mean geometric deviation, which is calculated by averaging the absolute percentage deviations of the measured geometric dimensions in relation to the geometrically mean part. This metric, among others, was originally selected to evaluate if the classification results depend on the magnitude of the geometric scatter. We did not find such a relationship, but kept the metric for visual reasons. The model using additional geometric input data classified all of the parts accurately. The other model made several misclassifications of the defective parts with a through-hole in any cylinder ring. Furthermore, the more complicated model always roughly classified the defective parts with regard to the correct defect size, but the simpler model led to more classification scatter. In the end, Figure 11 illustrates a significantly better classification power of model F-G2 versus model F. However, this is only a small snapshot of all of the results.