*2.4. Auction Fraud*

Online markets present themselves as a very popular way to trade online. E-commerce's main feature is to unite the real and virtual market for the online shopping model, being able to be divided into several sales schemes [30]. The most common business models are Business to Business, which indicates a company that does business with other companies, Business to Consumer, which is the trade carried out directly between the producer, seller, or service provider and the final customer, and Consumer to Consumer, which is based on the direct relationship between consumer and consumer.

According to the Global Payments 2022 Report (https://worldpay.globalpaymentsrep ort.com/en (accessed on 12 August 2022)) published by FIS Worldpay that examines current and future paymen<sup>t</sup> trends in 40 countries across five regions, the global e-commerce market will grow 55% by 2025.

Among the many user benefits, C2C offers minimal costs while maintaining higher margins for sellers and lower prices for buyers. There is also the convenience factor, and customers can list their products online and wait for buyers to come to them. A grea<sup>t</sup> example would be an auction.

Large companies promote online auctions, being the only intermediaries to correspond to consumers [30]. The most prominent example is eBay, a successful site since its launch in 1995. Anyone can sign up and start selling or buying. However, fraudsters use it to take advantage by finding alternatives to take advantage of. The leading illicit practices in online auctions are [31]:


According to the FBI's 2019 Internet Crime Report (FBI's Internet Crime Complaint Center (IC3) in its 2019 Internet Crime Report (available in: https://pdf.ic3.gov/2019\_IC3 Report.pdf (accessed on 15 August 2022))), there were 467,361 complaints in 2019—and more than USD 3.5 billion in losses for individual and commercial victims.

The practice of Shill Bidding, the focus of this paper, is when a seller uses a separate account, be it themselves, a friend, or a family member, or someone else, and asks them to bid on their auction to raise the auction price artificially. The item's price is higher than if a legitimate buyer placed a bid and purchased the item. Machine learning techniques have been working on solving fraud identification in auctions. Recent works proposed in [32–35] stand out.

#### *2.5. Evolving Fuzzy Systems and Interpretability*

The evolving fuzzy systems are models capable of working with advanced problem solving with a certain degree of interpretability. This problem-interpreting capacity comes from its structure, which can transform data into representations people can read and interpret. A range of models extracts knowledge through fuzzy rules, where each applies techniques and has different architectures for this purpose. The most famous examples of existing evolving fuzzy systems are evolving models based on Takagi–Sugeno, fuzzy classifiers, and fuzzy neural networks.

They differ in the type of fuzzy rules and possible interpretations of them. Several works in the evolving fuzzy system literature work on pattern classification, linear regression, and time-series forecasting (see examples in [36]). In particular, this work deals with interpretable aspects generated by an evolving fuzzy neural network.

Regarding aspects of evolution over time, several proposals were made to facilitate the understanding of the behavior of models as they analyze data. Lughofer [37] proposed criteria that ensure that the fuzzy rules generated can bring certainty to users regarding their actions during the execution of their activities. One should look for more straightforward and distinguishable models, that is, models with a smaller number of fuzzy rules solving the target problems in a coherent and interpretable way, avoiding ambiguities, and that each structure that composes the model avoids redundancy (overlapping of Gaussians may be an example of redundancy present in these models). Other criteria deal with overlapping by evaluating consequents and rule antecedents. The relationship between these two fuzzy rule components determines whether there are inconsistent rules (which can confuse evaluators). Moreover, the evaluation of completeness (a criterion that evaluates the contribution of rules with a significant distance to the sample) and coverage is noteworthy, which verifies whether all samples evaluated by the model are covered by the space of characteristics generated in the fuzzification processes. However, there are also criteria related to assessing the dimensions of the problems. These criteria for interpretability facilitate the identification of dimensions relevant to the issue, which can facilitate understanding a new problem.

On the other hand, this evaluation can also identify less irrelevant features that do not need to be in constructing fuzzy rules as they do not significantly contribute to the resolution of the situation itself. Finally, the criteria for identifying the importance of rules (how much a rule contributes or not to the identification of a target class of the problem), the interpretation of rule consequents (allowing to assess locally and globally the impacts of a fuzzy rule response for the problem solving), and knowledge expansion (identifying when the model identified new patterns and expanded its knowledge base) are fundamental to affirm that a fuzzy neural network model is interpretable. By guaranteeing these criteria, analyzing the generated fuzzy rules with a more outstanding guarantee that they represent solid knowledge about the analyzed problem is possible.

#### **3. Evolving Fuzzy Neural Network Based on Self-Organizing Direction-Aware Data Partitioning (SODA) Approach and Or-Neurons-eFNN-SODA**

The development of evolving fuzzy neural networks, models with unprecedented adaptability and freedom, allows acquiring knowledge through the information presented in a data set. This approach assists the precise demonstration of how to build a model able to identify some patterns in a problem analyzed. In this paper, the leading layer of the model proposes a calculation with data density to foster consistent neurons with Gaussian membership function through the idea of or-neurons.

These neurons are responsible for extracting knowledge based on fuzzy rules. The third layer of the model is represented by an artificial neural network model that can process the defuzzification approach. Its training is based on the concept of weight definition through an incremental approach based on recursive least squares. The model's engineering is visible in Figure 3, which is introduced in the following section.

**Figure 3.** Architecture of the fuzzy neural network proposed in this paper.
