## 7 Discussion and Conclusions  
FCA is most often used to analyze the classification of text data. In our work, we consider the use of FCA as an algorithm for identifying concepts in numerical data. This can be used both when searching for compact subgroups of objects in cluster analysis (unsupervised learning) and in classification problems to refine and justify the selected classes and search for new classes of objects in the data under study. For example, FCA can be used to identify subtypes of breast cancers based on the data on gene expression considered in this work. This line of work is the goal of further research. The application of FCA for the described tasks requires an evaluation of the resulting concepts. According to the results of the work, we found that the most useful internal validation index for a concept as a compact subgroup is the silhouette index. Experiments show that a high value of the silhouette index (the specific cut-point value depends on  
Figure 4: A graphical comparison of the running time of the CloseByOne and COBWEB algorithms. the analyzed data) corresponds to the most informative subgroups (when compared with labels of previously known classes). Thus, when analyzing data with an unknown markup in advance, the silhouette index can be used as a selection criterion for concepts obtained in FCA to form compact subgroups of objects. It should be noted that these subgroups can intersect with each other, which corresponds to fuzzy clustering. The normalized intra-cluster distance can be used as an auxiliary criterion to confirm validation. It is also worth noting that the concepts in FCA have a simple logical description, which is consistent with the principles of interpretive machine learning. For comparison, the data were analyzed using the COBWEB conceptual clustering algorithm. This algorithm has significant differences from FCA both in the way of constructing a set of concepts and in the very type of concepts obtained. As a result of applying this algorithm, a hierarchy of concepts is formed, which corresponds to a clear clustering. At the same time, the concepts themselves are based on the probabilities of the values taken by the features. This makes it difficult to interpret the results and describe the concepts themselves in contrast to FCA, wherein only a part of the features can be used for a logical description. However, in general, these two algorithms give similar results in terms of the quality of the concepts they reveal. To work with data described by a large number of features, it is proposed to supplement the algorithms with a search for a support set of features. The search for a support set is the basis of test theory. In our work, the search for a support set (support set) is carried out using a greedy algorithm that allows you to quickly find an approximate solution, the accuracy of which (in terms of the objective function of minimizing the number of features) is sufficient from a practical point of view. This approach successfully complements the FCA and COBWEB algorithms and allows them to be used to analyze data described by a large number of features and interpret the result. An interesting topic for the further research is development of an algorithm for automatic selection of the set of the most informative logical concepts, which together cover the data examples. Conceptualization, I.M. and L.K. ; methodology, I.M., N.R., and G.S. ; software, N.R., S.M., and M.B. ; validation, I.M., S.M., and G.S. ; formal analysis, N.R. and G.S. ; investigation, I.M., N.R., and G.S. ; resources, I.M. ; data curation, S.M. and G.S. ; writing--original draft preparation, I.M. and M.B. ; writing--review and editing, N.R., G.S., and L.K. ; visualization, G.S. and N.R. ; supervision, I.M. and L.K. ; project administration, L.K. ; funding acquisition, L.K. All authors have read and agreed to the published version of the manuscript. This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant No. 075-15-2022-1121).