*4.4. Results*

The results of the evaluation of the trend lines are presented in Figure 8. This evaluation was carried out with 20% of the total samples (1264) for initial training and the rest (5057) for online adaptation, evolution, and evaluation of the model. For the experiments, the models listed above were compared with the results obtained by eFNN-SODA with *ϑ* = 2 (the values were defined through a cross-validation procedure for *ϑ* = {2, 3, 4, 5}).

**Figure 8.** Trend lines for the Auction Fraud dataset.

The results presented in Figure 8 corroborate the efficiency of the approach in correctly classifying the situations involved in the data set that deal with fraud in auctions. The model proposed in this paper could dynamically and more assertively resolve cases of attack or nonexistence in evaluating the samples. Next to the results obtained by eFNN-SODA are the results generated by Autonomous zero-order multiple learning with pre-processing and Autonomous zero-order multiple learning. Numerically, they are similar, with a slight advantage for eFNN-SODA. A little less effective, but with good results, is the evolving fuzzy neural network with Self-Organizing Direction-Aware Data Partitioning model. The other models did not perform as effectively as the eFNN-SODA models and the Autonomous zero-order multiple learning variations.

The eFNN-SODA model solved the problem of fraud in auctions with three rules. In this case, the Self-Organizing Direction-Aware Data Partitioning fuzzification approach did not generate new neurons, as it was only necessary to change centers in the rule antecedents to solve the problem. The three fuzzy rules are presented in the following:

Rule 1. If Bidder Tendency is small with impact 0.41 or Bidding Ratio is small with impact 0.59 or Successive Outbidding is small with impact 1.00 or Last Bidding is medium with impact 0.39 or Auction Bids is small with impact 0.38 or Auction Starting Price is medium with impact 0.38 or Early Bidding is medium with impact 0.38 or Winning Ratio is small with impact 0.53 or Auction Duration is small with impact 0.38 then output is normal.

Rule 2. If Bidder Tendency is medium with impact 0.41 or Bidding Ratio is medium with impact 0.59 or Successive Outbidding is medium with impact 1.00 or Last Bidding is high with impact 0.39 or Auction Bids is high with impact 0.38 or Auction Starting Price is high with impact 0.38 or Early Bidding is high with impact 0.38 or Winning Ratio is medium with impact 0.53 or Auction Duration is medium with impact 0.38 then output is fraud.

Rule 3. If Bidder Tendency is high with impact 0.41 or Bidding Ratio is high with impact 0.59 or Successive Outbidding is high with impact 1.00 or Last Bidding is small with impact 0.39 or Auction Bids is medium with impact 0.38 or Auction Starting Price is small with impact 0.38 or Early Bidding is small with impact 0.38 or Winning Ratio is high with impact 0.53 or Auction Duration is high with impact 0.38 then output is normal.

These rules are seen as the extraction of knowledge from the evaluated data set and can serve as a conceptual analysis of the events involved in identifying or not identifying fraud in auctions. The interpretability criteria help understand and validate the generated knowledge to confirm the efficiency of the generated fuzzy rules.

The first evaluation seeks to verify whether the eFNN-SODA model was the simplest in solving auction fraud problems. This factor is confirmed because, according to Equation (46), the eFNN-SODA model had the lowest number of fuzzy rules and the highest accuracy result compared with the other models used in the experiments. eFNN-SODA assertively solved the target problem with only three fuzzy rules. As a comparison, the different models tested in the experiments generated between 10 and 28 fuzzy rules.

The distinguishability criteria (based on Equation (50)) of the formed solution are presented in Figure 9. In this figure, the evolution of the first layer of Gaussian neurons can be observed, and it is identified that the neuron was the one that suffered the most changes during the experiments. It is responsible for representing when there is no fraud in the auction (the largest class of the data set). Therefore, it is expected that with a more significant number of samples in this context, the changes in clouds will be more remarkable, consequently generating a greater impact on the comparative similarity with their previous version (before the evolution training). In this evaluation, it is possible to identify that the fuzzy rules (presented below) are distinguishable from each other, as they have different antecedents and consequents.

**Figure 9.** Similarity of Gaussian neurons.

Regarding the overlapping criteria of the generated Gaussians, it can be seen in Figure 10 (generated at the beginning of the training) that the centers of the generated fuzzy neurons are not superimposed. This confirms that there are differences between them. This criterion is also guaranteed by steps 5 and 6 of the Self-Organizing Direction-Aware Data Partitioning fuzzification approach. This figure presents the evaluation concerning the successive outbidding and bidding ratio dimensions (more relevant in the model analysis shown below).

**Figure 10.** Evaluation of overlapping of the three generated fuzzy neurons during the fuzzification process: successive outbidding versus bidding ratio.

The consistency of the generated fuzzy rules (Equation (51)) is visualized in Figure 11. In this figure, it is possible to see that the consistency relation of the fuzzy rules is violated only in a few evaluations in each of the three fuzzy rules generated. At the end of the experiment, all rules are consistent, concluding that model training corrected these inconsistencies when they appeared.

**Figure 11.** Consistency of fuzzy neurons.

Concerning the criterion of completeness of the fuzzy rules, an evaluation based on *-* − *completeness* was applied during training, according to Equation (52). The results visualized in Figure 12 confirm that the generated fuzzy ruleset meets this validation criterion throughout the experiment.

**Figure 12.** *-* − *completeness* criteria during the training.

Changes in antecedents and consequents of fuzzy rules can also be observed and visualized in eFNN-SODA. Table 1 presents feedback from the model in the last sample evaluation that generated impacts on the first and third fuzzy rules.

**Table 1.** Interpretability concerning (degree of) changes in fuzzy neurons during the evolution phase.


The evaluation of the evolving behavior of the problem features is also a relevant part of the interpretability of the results. This variation can be seen in Figure 13 and identifies how each weight generated according to the separability criterion (Equation (34)) was significant during the fraud classification process. The most relevant dimensions were successive outbidding, bidding ratio, and winning ratio (which makes sense when compared with the analysis in Figure 6). Figures 14 and 15 present a graphical evaluation based on a scatter plot of the two dimensions that best contribute to the class separability criterion.

Finally, the expansion of knowledge given by eFNN-SODA demonstrates a different way of interpreting the problem through fuzzy rules connected with the relevance of the problem's features. The Fuzzy Hoeffding Decision Tree model [66] also generated an interpretative approach, which presents a decision tree according to fuzzy techniques in constructing its leaves. The Expliclas [67] online solution (https://demos.citius.usc.es/Ex pliClas/datasets (accessed on 12 August 2022)) facilitates the acquisition of knowledge to be compared with the extraction of knowledge obtained in this paper. The decision tree formed by the Fuzzy Hoeffding Decision Tree is shown in Figure 16.

**Figure 13.** Feature separability criteria throughout the evaluations of the eFNN-SODA model.

**Figure 14.** Scatter plot of successive outbidding feature.

**Figure 15.** Scatter plot of bidding ratio feature.

**Figure 16.** Fuzzy Hoeffding Decision Tree interpretability.
