# Multivariate Analysis of Transient State Infrared Images in Production Line Quality Control Systems
## 4 Temporal Evolution of Principal Components in Exemplary Production Line Application  
The first question that must be addressed is how to set the shortest image acquisition time which ensures a good discrimination between fully functional and faulty devices. In the case of refrigerators, the transient thermal state which takes place after switching on takes a few minutes, usually not more than 10-15 min. The heating process has been illustrated in Figure 3 which shows typical time changes of evaporator inlet and outlet average temperatures. What is the shortest time after which quality control can be carried out using PCA approach? Let us conduct PCA analysis for 10 devices including 5 defective devices belonging to class B (red and empty point curves in Figures 4-6) and 5 non-defective devices belonging to class G (green and filled point curves in Figures 4-6). The time dependence of the first 2 PCs has been shown in Figures 4 and 5. The answer to the question asked in the beginning of this section can be given. In this example the minimum image acquisition time should be set to 5 min. It must be noticed that behavior of PCs is different--the discrimination between classes for PC1 remains, while for PC2 it reduces significantly after 10 min and practically declines after 12 min. It is also visible in Figure 6 which includes data clusters obtained after 17 min. In order to get the best separation between classes the acquisition time should be set to 8 min. The results presented in Figures 4-6 could suggest that the first two PCs, representing the most of the total variance of the data, should be used for quality control. Unfortunately, such theorem is not true. The following example concerning the same type of device but a different model shows that selection of PCs must be made very carefully and separately for each model. In this case, there are 21 devices under consideration including:  
* 14 non-defective devices belonging to class G with an exemplary infrared image presented in Figure 7a (these devices are represented by green points in Figures 8-12),
* 4 defective devices belonging to class E with an exemplary infrared image presented in Figure 7b (these devices are represented by blue points in Figures 8-12),
* 3 defective devices belonging to class B with an exemplary infrared image presented in Figure 7c (these devices are represented by red points in Figures 8-12). Figure 3: Time evolution of evaporator inlet and outlet average temperatures. Figure 4: Time dependence of the first principal component (PC1) of thermal images (10 devices). Figure 5: Time dependence of the second principal component (PC2) of thermal images (10 devices). Figure 6: Time evolution of infrared images illustrated in 2D feature space composed of PC1 and PC2. For a defective refrigerator of class E (Figure 7b), the evaporator outlet temperature is higher than in the case of non-defective devices (Figure 7a). It means that the evaporator is not working properly. For a defective refrigerator of class B (Figure 7c), the temperature difference between the evaporator inlet and outlet is higher than usual. It can be caused by an occlusion, an overcharge or undercharge of liquid refrigerant. Generally, the correct classification of devices that belong to class E is not an easy task but it is usually possible as it has been shown below. The analysis of Figures 8-12 leads to a conclusion that the classification using PC1 and PC2 is possible in this case but it requires much longer acquisition time (Figure 12), which in the production line environment can be unacceptable. Moreover, the comparison between PCs presented in Figure 11 results in a conclusion that for this acquisition time it is better to choose PC2 and PC5 in order to obtain higher class separation. Successive plots in Figures 8-12 have axes being successive PCs. Of course, the best PCs for classification are not always consecutive, e.g., PC2 and PC5 for the time of acquisition equal to 15 min.