# Fidex and FidexGlo: From Local Explanations to Global Explanations of Deep Models +
## 5 Experiments with CNNs
### Facial Expression Recognition  
The facial expression recognition (FER) dataset that we retrieved consists of \(48\times 48\) grayscale images. Seven categories of emotions have been defined, but in this work, we only focused on two classes: 'Happy', and 'Unhappy' (represented by angry, disgust, fear, sad, surprise and neutral). The number of training samples is 28,709 and an independent test set of 7178 samples was used for testing. Moreover, the number of 'happy' class samples is 7215 for training and 1774 for testing. Again, to avoid over-training, we used 10% of randomly selected training samples as the tuning set for early stopping. FidexGlo was run with the following parameter values:  
* \(\mu=2\);
* \(\rho=20\);
* \(\phi=1\);
* \(p=0.9\);
* \(q=0.9\). The results of the fine-tuning of a VGG architecture, averaged over ten trials, are shown in Table 7. Similarly to the previous classification problem, we defined two fully connected layers on top of the VGG network, with two hundred fifty-six and two neurons respectively. The average number of rules is similar to that obtained in the MNIST problem and the average fidelity is 93.06%. Again, the average predictive accuracy of the rules, when the rules and the network agree, is higher than that obtained by the model. Figure 9: Samples covered by a rule of class ‘4’; rule antecedents are represented by colored dots. The top left shows the centroid of the 565 covered samples in the training set. The other pictures show three of the one hundred three samples in the test set that activate this rule. The following Figures illustrate several examples of rules for the 'Happy' class. Figure 10 shows four pictures with the centroid of the represented rule at the top left, which covers 27 samples of the training set. The other three pictures represent three covered samples from the testing set. In the picture representing the centroid and the three samples, the antecedents of the rules in red delineate the ends of the mouth in a smiling posture. In addition, a green antecedent is positioned over the teeth and three others in the lower part of the face. Generally, an expression of joy can be strongly marked by a smile. In Figure 11, the centroid was calculated using 31 samples from the training set. Again, it shows us that the teeth are targeted, as well as the corner of the mouth on the right. Note that the left cheek fold is targeted by a red spot. We can also see a rule antecedent on the eyebrow and a little higher up on the forehead. Finally, an antecedent is used for the chin and, at the bottom left, an antecedent is used to select images with a bright area. The pictures on the right are relatively consistent with the features highlighted by the centroid, while the picture on the bottom left has a slightly different orientation and shows the antecedents of the rules in slightly different places; in particular, the spot on the teeth is on the lower lip and the spot on the eyebrows is in the darker part of the nose. A final example of the 'Happy' class is shown in Figure 12. The centroid has the following rule antecedents: on the left eye; on the teeth; above the upper lip (left and right); below the lower lip (left); and on the bottom right. The lower-left picture has a different orientation, but the features pointed to by the rule antecedents are similar. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Acc.** & **Fid.** & **Acc. R. (a)** & **Acc. R. (b)** & **Nb. R.** & **Nb. Ant.** \\ \hline CNN-VGG & 90.81 (0.69) & 93.06 (0.67) & 86.21 (0.63) & 91.38 (0.93) & 6765.1 (72.4) & 4.4 (0.1) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Average results provided of a CNN-VGG architecture on the FER dataset (ten trials). The results are related to a two-class problem: ‘Happy’ versus ‘Unhappy’. Figure 10: Samples covered by a rule of class ‘Happy’; rule antecedents are represented by colored dots.