# Sequential and Comprehensive Algorithm for Fault Detection in Semiconductor Sensors
## 3 Results and Discussion
The regression analysis was applied to the respective sector, (**C**) before and (**D**) after the threshold value. Figure 3: Representation of each sensor value. (**A**) V001. (**B**) V009. threshold, sector2: after threshold). In the case between V027 and V029, the threshold value was 11,108.39 in V027 value, and the regression equation was \(y=2.796e^{9E-05x}\) (R\({}^{2}\) = 0.207) and \(y=2E-07e^{1.5E-3x}\) (R\({}^{2}\) = 0.882) in sectors 1 and 2, respectively. The R\({}^{2}\) in sector 1 was too low, which indicates that V027 might be affected from other sensors besides V029 even when V029 did not activate. Such a case should be under more complex consideration. However, once V029 activated, V027 and V029 were highly correlated, showing that this regression equation was suitable for SVR. Therefore, we detected the faults using each regression equation in sector 2. The defect-free data points were marked in green and the defective data were marked in red. When the margin was set to be 0.3, the data in the area above 70 in V029 were detected as faults. Consequently, our algorithm detected four defective data in sector 2. Even in the case of more complex sensor pairs, it can also be represented through a linear, exponential relationship, or their combinational form. When equipped with complex mathematical calculation and expertise, it is possible to apply this method to other industries with more complex relationships. For the non-correlated sensor pair, we applied a k-means clustering method with a Voronoi diagram. Actually, the choice of the similarity measures can highly affect the results of clustering. One might look at the sensor data in terms of the time-series data [36]. However, the time-series analysis is related to high-dimensional data, which involves some challenges: (i) the curse of dimensionality and (ii) difficulty in the graphical representation. A significant task to enhance the performances of the time-series is to reduce their dimensionality while preserving the main characteristics and reflect the original similarity of such data. When treating the time-series, the similarity between two sequences of the same length can be calculated by combining the ordered point-to-point distance between them. The most used distance function is the Euclidean distance, corresponding to other distance measuring methods. Thus, in our study, we chose the lower dimension data with the Euclidean point-to-point distance, while the higher dimensions allowed many-to-one point comparisons. In general, the fault detection using an algorithm suffered from the data imbalance problem, since most data should be defect-free in the process, as previously described [37]. Such an imbalance impacted the accuracy in that the algorithm could declare the defect-free to the defective data. To overcome this problem, the data processing techniques named over-sampling and under-sampling were introduced [38]. The over-sampling method could generate the artificial defect data along the line segment between other defective data. In contrast, the under-sampling method removes some defect-free data to match the numbers between the defective and defect-free data. First, we applied the k-means clustering to the data plane with the different \(k\) values (Figure 5). Then, we normalized the sensor data to convert the minimum and maximum value of each sensor into 0 to 100. This procedure prevents the misinterpretation caused due to a geometric distortion in the clustering map. The data in the minority region were generally considered to be defective data, while the other data in the majority region were defect-free data. The optimized k value was 6, because the number of the data in the minority group was the same as the number of centroids in the majority group at \(k\) = 6. There were two minority groups (blue and pink dot) allocating 12.5% and 4.2%, respectively. The closest one from the centroid was determined as a representative data point in the majority group, and all data except for the representative data point were eliminated.