## 2 Background and Related Work  
The development and use of evolving algorithms have been the subject of interest and study in recent years. These algorithms can develop and update unfamiliar environments and detect deviations of input data over time. Thus, it is possible to solve problems in which changes in the data stream occur, e.g., detecting changes in the user profile on social networks [7; 8; 9; 10; 17]. A particularity of evolving systems is the concept evolution; they simultaneously learn the structure and parameters through the data stream. Here, the data stream is a continuous sequence of samples, where the order of that sequence matters. Moreover, it has an infinite size; that is, the sample can be represented by \(x(t)\), where time \(t\) varies from 0 to infinity [23]. Due to their ability to adapt, evolving systems can be considered as intelligent adaptive systems, which can have structural components: artificial neurons, production rules, fuzzy rules, data clusters, or subtrees [8; 9]. It is significant to highlight that clustering data stream algorithms with online learning must deal adequately with both the shift and concept drift problems from data with non-stationary statistics. Concept drift means an unpredictable change over time in the statistical properties of the input or target variables [24]. These changes can be in the distribution of the input data (input space drift), in the relationship between the inputs and the model (joint drift), or the distribution of the respective target class (target concept drift) [8; 9; 25]. Figure 1 shows some types of concept drift, where the circles represent samples obtained over time. The changes can occur incrementally with many phases in the migration of the concept; intensely (sudden or abrupt drift, shift drift, concept evolution); or gradual (local drift), activating the system's automatic evolution mechanism, known as passive drift handling [9; 25; 26]. There is also the case where earlier concepts may appear again after some time, as shown in Figure 1d. Still another form of concept deviation is cyclic drift, where changes in data deviations can occur cyclically [9]. These changes in the dynamics of the data stream can occur due to seasonal effects, wear, contamination, change of catalyst in the chemical industry, or operational changes, among other phenomena [8; 9; 26]. Another feature of evolving algorithms and methods is the operation in the single-data-pass mode. The performed calculations are made recursively; this consists of discarding the sample after processing the algorithm, considering that only the last data is available. The algorithm performs the clustering from the current sample and statistical aggregations or model parameters, such as centroids and prototypes obtained based on previous samples [8]. Some examples of usability of evolving systems in real applications are the use of TEDA (Typicality and Eccentricity Data Analytics) for the detection of failures in industrial processes [27], and spam detection in e-mail [28]. Another example is the optimization of routing in the MANET (Mobile Ad-hoc Network) and, consequently, reducing energy consumption and increasing the network's useful life and stability using ICA (Imperialist Competitive Algorithm) [29]. Finally, in [30], a semi-supervised algorithm was used to classify data from eleven real-world datasets, some of which are electricity, chemical sensors, e-mails, and TCP type limit records. Although previous approaches are recent, evolving data stream clustering algorithms had appeared in the literature for some time.