# Multi-Output Regression with Generative Adversarial Networks (MOR-GANs)
## 3 Results from Synthetic Datasets
### Single-Output Regression with Random Input Values
#### 3.2.3 Confidence of Solutions from the Critic  
Sections 3.2.1 and 3.2.2 show how sampled points produced by the generator of the WGAN match the distribution seen in the test data (or sample data). During the training of the WGAN, the critic learns to determine how real an sample is. This section demonstrates how the critic can be used to determine the confidence in a sample produced by the generator, which is an indication of how reliable the method's predictions are.  
Figure 8 shows the value taken by the critic for predictions or responses made throughout domain for both the sinusoidal and annulus datasets. These are produced by finding the value of the critic for each point on a \(100\times 100\) grid that covers the same domain as the original data. As previously stated, the critic of a WGAN does not explicitly determine whether a sample is real or fake, but instead, how real a sample is. Therefore the larger the value produced by the critic, the more confidence the model has in the prediction. The critic values shown here are normalised to be between 0 and 1.  
Figure 8a shows the values of the critic produced for the sinusoidal dataset. It can be observed that the values of the critic are higher where the data of the noisy sine curve occurs (see Figure 2), which corresponds to the region mostly occupied by the training data. These values are higher where there is a higher concentration of data points, particularly  
Figure 8: Contour plots showing the values of the two critics for the sinusoidal and annulus datasets. These indicate the confidence in or reliability of the predictions and also indicate where extra training data may be required.  
Figure 7: A sine wave intersected by several lines. The test data is shown on the left, sampled points from the WGAN are shown in the middle and sampled points from the GPR posterior are shown on the right.  
around \(x=-2.0\) and \(x=2.0\). Outside of where the sine wave is located the critic value sharply decreases, therefore confidence in any prediction made here is low.  
Figure 8b shows the values of the critic produced by the annulus dataset. It can be observed that the values are higher within the annulus, which corresponds to the region occupied by the training data (see Figure 6). We can see that the critic produces lower values for coordinates predicted outside of the annulus, meaning that the confidence in predictions or responses that occur here is low.  
Figure 8a,b demonstrate how the critic can be used in conjunction with the generator to produce a confidence level in the predictions made by the generator. A lower critic value, and therefore a lower confidence, in a prediction made may indicate that extra training data is required there. A possible location requiring extra data for the sinusoidal dataset is \(x=0.5\) and for the annulus dataset is \(x=0.0\). The critic value can be used to remove solutions generated that are not realistic, thereby improving the results. The solutions shown in Figures 2 and 6 had their average \(p\)-value improved from \(<\)0.2 to \(<\)0.1 by removing the 10% of solutions generated that had the lowest value after being passed through the critic.  
Thus, the confidence level might help us to determine where to collect more experimental data or where to observe the system. It also suggests where the neural network is not predicting well, which might not be because of lack of data. Ultimately, this confidence level should be combined with the importance of the region where the confidence is being determined. This importance could be set according to how much or little influence this region may be have on the final results. If applying the GAN approach to regression to optimisation, importance could, for example, be determined from sensitivities (or adjoints) of what is important with respect to the independent variables.