# Dynamic Line Scan Thermography Parameter Design via Gaussian Process Emulation
## 3 Results
### Model Performance  
In order to assess the accuracy of the model, we need a ground truth. We ran the simulator, as described in [2], 45,000 times. However, the movement velocity, height of the camera and the ambient temperature were kept constant at 10 mm/s, 450 mm, and 20 degC, respectively. The remaining input variables were as follows:  
1. Distance between the heat source and the camera, range 50 to 600 mm;
2. Heating power, range 50 to 800 W;
3. Start depth of the defect, range 2 to 9.8 mm;
4. Diameter of the defect, range 12 to 24 mm.  
These four tuples are the inputs of our dataset. The reason we limited the dataset to four variables is that composing a dataset of seven input variables with enough resolution to assess the accuracy of the model would take a lot more data points and thus time to simulate. Moreover, in an industrial context, one does not always have full control over the parameters we fixed in this demonstration, as they are dictated by the production process and installation itself.  
For each of those four tuples, the temperature difference between a position on the surface above a defect and a position that is not above a defect is calculated. This temperature difference is the output of our dataset.  
Via active learning, as described in Section 2, we iteratively picked data points from the dataset and moved them to the training set of the Gaussian process. The remaining data points in the dataset served as test points. After the Gaussian process was trained, two calculations on the test points were performed:  
1. The root mean square error between the posterior mean in each test point and the actual values from the simulations. This number serves as a measurement for the deviation of the model from the underlying truth.
2. The average posterior standard deviation for all remaining test points. This is a measurement for how much uncertainty there still is in the system. The point with the highest variance, i.e., the highest uncertainty, becomes the point that is moved from the test set to the training set of the Gaussian process in the next iteration.  
When both of these numbers flatline, then there is little to be gained in running more simulations. In that case, the Gaussian process is able to approximate the ground truth.  
We performed the active learning process for 500 iterations. We started with 25 training points randomly chosen from the dataset. This makes for a total of 525 data points in the Gaussian process of the last iteration. In Figure 3, the learning curves of the Gaussian process are visualized. The exact curves of the iterations depend on the initial random points that are drawn from the dataset described aboveblack. Therefore, we repeated the experiment five times.  
The hyperparameters for the covariance functions, as described in Equation (4), of the trained Gaussian processes, can be found in Table 1.  
Figure 3: Graphical visualization of the learning process of the Gaussian process for 5 runs of 500 iterations. (**a**) represents the root mean square error (RMSE) of the learned surrogate compared to the response surface created in [2]. (**b**) shows the average standard deviation of the Gaussian process posterior prediction.