# Fidex and FidexGlo: From Local Explanations to Global Explanations of Deep Models +
## 6 Discussion  
In this work, we focused on explainability by propositional rules. We implemented two explainability algorithms: the first is local and the second is global and uses the local algorithm (Fidex). With tabular data, FidexGlo applied to ensembles and SVMs performed well compared to other existing methods. Then, for CNNs, it turned out that the number of rules generated was high compared to the problems with tabular data, but the number of training samples and input variables was also much larger, and so the explainability task is more complex, with more samples to be covered. As an example of execution time, FidexGlo took approximately thirteen hours to extract global rules using 128 CPUs in parallel. Finally,  
Figure 14: Samples covered by a rule of class ‘Cracks’; rule antecedents are represented by colored dots. The top left shows the centroid of the 4429 covered samples in the training set. The other pictures show three of the one thousand seventy samples in the test set that activate this rule. Figure 13: Samples covered by a rule of class ‘Cracks’; rule antecedents are represented by colored dots. The top left shows the centroid of the 4549 covered samples in the training set. The other pictures show three of the one thousand one hundred twenty samples in the test set that activate this rule. once the final rules are computed, it can be time-consuming to examine thousands of rules to explain a model; however, it is at least possible to determine precisely how a model has classified data. From there, we can even determine whether a model's classification strategy is partially or completely incorrect, and whether new training examples can be created and fed to counterbalance the errors. With linear and logistic regression, it is always possible to obtain some form of explanation thanks to the coefficients that define these models. For example, linear regression makes it easy to calculate the relevance of a given variable to the output. Note, however, that these models would give very poor results on the datasets learned in this work by CNNs. Our FidexGlo global algorithm may, in some cases, have no explanation for the test samples (no rules activated), because some regions of the input space may not have been explored with the training samples. Nevertheless, it is still possible to use our local algorithm, which will always give at least one explanation in the form of a propositional rule. Since the fidelity of samples not learned during the training phase is generally different from 100%, we could adopt a strategy where the samples that do not activate any rules of the correct class would run Fidex to obtain a faithful explanation. Our explanations of the CNN classifications are low-level because they are given at the pixel level. In the experiments, we highlighted model strategies for classifying the data. Specifically, for the crack classification, we found that when there is a crack, one or more antecedents were located on the crack, so the interpretation is straightforward. Furthermore, for the classification of the joy emotion, we found that the rules indicated rule antecedents at the level of teeth or skin folds, which typically represent happy expressions. The many rules obtained highlight different elements. The examples given in the previous figures are just a small sample, but the thousands of rules obtained give us a lot of information. Determining explanations using high-level concepts is a current trend in research on the explainability of connectionist models. A disadvantage of this type of approach is that the concepts must be defined in advance before they can be learned. In our low-level approach we do not need to define any prior knowledge, and this is an advantage over a high-level approach. Finally, we highlighted the centroid images which, for a given rule, represent the average image of the samples covered. This can be seen as a prototype and, in a way, we achieve a higher-level explanation compared to the pixel representation. Grad-CAM, LIME and SHAP are local methods.