## 3 Proposed Methodology
### Polymorphic Generative Index
Furthermore, IS uses the Inception v3 model, a pre-trained model based on the ILSVRC 2012 dataset, which implies that it is best suited to the evaluation of the 1000 object types used in the ILSVRC 2012. This means that IS is not suitable for the evaluation of all aspects of interest to us. IS evaluates performance using only the distribution of the generated images, while the FID compares the distribution of generated images with the distribution of a set of real images of the target domain. The FID uses Inception v3 as a feature extractor to extract the activation map between the real image and the generated image. Then, the FID is obtained by measuring \(d^{2}\), which is the two-Wasserstein distance between the probability distribution of the real image \(r\) and the probability distribution of the generated image \(g\). The FID is calculated as follows:  
\[\text{FID}_{(r,g)}=\text{d}^{2}=\|\mu_{r}-\mu_{g}\|_{2}^{2}-Tr\big{(}\Sigma_{r }+\Sigma_{g}-2\Sigma_{r}\Sigma_{g}\big{)} \tag{3}\]  
where \(\mu_{r}\), \(\mu_{g}\), and \(\Sigma_{r}\), \(\Sigma_{g}\) represent the mean and covariance matrices of feature vectors' output by the real images of the target domain and the generated images via the Inception v3 model, respectively. \(Tr\) is the sum of the diagonal component of the matrix. The FID obtained by measuring the distance between the activation distributions of the two images has a lower value, as similar images are obtained. The FID entails that the distance between sets of images of the same class is closer than that of other images of the class, and the distance between completely different sets of images is further than that of other images of the class. Therefore, a generated image with a low FID can be considered similar in probability distribution to the real image, which can be evaluated as generating a high-quality image given that the FID of the augmentation model is measured lower. Furthermore, the FID is able to evaluate the quality of the generated image more robustly than IS because the distance value is higher if the model generates only one image per class. For example, the generated image is more sensitive than IS when it contains noises such as blurring and mosaic compared to the original image. However, the FID has the following disadvantages. (A) ILSVRC 2012-based pre-trained models such as Inception v3 are required. As noted in the description of IS, ILSVRC 2012-based pre-trained models may not be suitable for specific domains such as WBM classification; (B) it is recommended that FID is calculated using a minimum sample size of 10,000, otherwise, the true FID of the augmentation model may be underestimated [42]. As such, it is difficult to obtain the 10,000 sample data entries recommended for FID evaluation when the deviation between normal and abnormal data is large and collection is difficult, as it is for industrial data. In this paper, we propose PGI as a new index to numerically verify the effectiveness of an augmentation model. The PGI is obtained by comparing the number of original data used in augmentation model learning to the number of original data for each class correctly inferred through the preceding classifier; the corresponding equation is as follows:  
\[\text{PGI}=\frac{N_{Correct}}{N_{Augment}} \tag{4}\]  
where \(N_{Correct}\) is the number of original wafers correctly classified through the classifier, \(N_{Augment}\) is the number of original wafers used to augment the data, and \(0\leq PGI\leq N_{Correct}\). The PGI is obtained for each defect pattern. The obtained PGI can be expressed as \(PGI_{Center}\), \(PGI_{Donnt}\), \(PGI_{Loc}\) for each defect pattern. In the present work, the PGI was used to numerically verify how many various defect patterns can be formed with respect to the original defect pattern per WBM used for data augmentation. We estimated the quantitative effect of the augmentation model by calculating the PGI.