## 3 Results
### ART Sorting Algorithm  
The second step of our application scenario demonstration was the development of a suitable ART sorting algorithm for defect detection (see Figure 1). This was carried out on the synthetic ART training data, the generation of which was explained before. For the classification task in this work, the specific model type or the selection of the most powerful algorithm was not the focus. The model type itself was seen more as a required tool for the demonstration and any suitable type satisfied our needs. Common classification types were examined. The MATLAB(r) function fitcdiscr (fit discriminant analysis classifier) was found to create suitable classification models, based on quadratic discriminant analysis classification (name-value argument for model type: 'DiscrimType', 'pseudoquadratic'). We created different classification models, using synthetic ART training data either with or without additional noise, as well as with different sets of predictor variables. All of the models were designed incorporating the remarks from Section 2.5. Prior values were applied, which indicate a fifty-fifty chance for a part to be good or to be defective with a completely even defect distribution. A cost matrix, which controlled each of the training procedures through the assignment of cost values aiming at a low sum of cost, was defined as follows: 0 cost score for the correct good/defect classification; 1 cost score for confusing the good parts with defective ones or vice versa; 0.01 cost score for mixing-up the different defect classes. Although the application scenario aimed to differentiate between the good and defective parts and not between the different defect sizes or positions, each virtual part or data row within the training data was labeled according to a multiple class-system, covering one good class and many defect classes. This was chosen to ensure a solid classification functionality, after understanding the heterogeneous effects of the defect size or its position to the eigenfrequencies of a part. This led to the knowledge that the defective parts do not form a single well-defined class that unambiguously differs from the good class. To put it clearly, defining the multiple defect classes should ensure that each class contains parts that have the most homogenous defect impact characteristics. The various defect sizes of the defective trainings parts were directly used for the labeling. Then, the labels were extended with information about the part subdomain in which the defect was located. Figure 10 visualizes the defined defect subdomains while taking into account the part symmetry. Both of the cylinder rings were divided into six subdomains each, with the bar area consisting of seven subdomains. Bringing all of the combinations of the defect sizes and subdomains together, the final class system covers a total of 552 classes (1 good class + 551 defect classes, with 29 defect sizes and 19 subdomains). We assumed that more classes may allow clearer results. However, the amount of classes was limited by two factors. First, an increasing computing power requirement was recognized, as more classes were used. Second, the classification can abruptly become very bad, which is likely due to the fact that there were too few training parts in each class to properly predict a large amount of the classification model coefficients. Figure 10: Defect subdomains with regard to the position of a through-hole defect in a connecting rod taking advantage of part symmetry (red: defect domains; blue: borders of subdomains). The input or predictor variables used for the classification models comprised the available 18 eigenfrequencies of each part, as well as the geometric data as an optional input. When implementing the geometry, either the full eight geometric parameters were used, or only half of them (the inner cylinder ring diameters \(d_{\text{R1}}\) and \(d_{\text{R2}}\), the bar width \(w_{\text{B}}\), and the bar height \(h_{\text{B}}\)).