## 2 Materials and Methods
### Recognition of the Areas Prone to Strong Earthquakes with One Learning Class
In other words, the innovation we make to the statement of the problem will be building an image of the set of recognition objects \(W\) in the studied region \(S\) as in the subset of the Euclidean plane:  
\[F_{\gamma}:W\to F_{\gamma}(W)\subset S\subset R^{2}, \tag{11}\]  
where \(\gamma\) is a set of free mapping parameters \(F_{\gamma}\)[21]. A sought mapping \(F_{\gamma}\) must meet the following necessary conditions:  
1. flat set \(F_{\gamma}(B)\subset S\) is obtained for \(B\subset W\), given the fixed values of free parameters \(\gamma\),
2. flat set \(F_{\gamma}(B)\) contains high seismicity objects \(w\in B\) as points in the plane, i.e., \(\forall\ w\in B\Rightarrow w\in F_{\gamma}(B)\), and
3. the epicenters of known strong earthquakes (\(M\geq M_{0}\)) are located inside or at the borders of zones \(F_{\gamma}(B)\). That said, given possible errors in the identification of historic epicenters, they can be located near the borders \(F_{\gamma}(B)\). It is natural to view such two-dimensional sets \(F_{\gamma}(B)\) of the cardinality of the continuum as actual flat zones within which strong earthquakes can occur. The selection of values \(\gamma\) of any given variant among those that meet the conditions (a), (b), and (c) is based on the system approach using control experiments. The algorithmic system called Formalized Clustering and Zoning (FCAZ) developed by the authors [21; 22] is used as sought mapping \(F_{\gamma}\). It represents systems analysis method concerning Discrete Mathematical Analysis (DMA) [23; 24; 25; 26; 27; 28; 29; 30]. The FCAZ method enables an effective recognition of strong earthquake-prone areas based on the cluster analysis [31] of a catalog of seismic events. It represents a consistent application of Discrete Perfect Sets (DPS) algorithms [32; 33; 21; 34] and E\({}^{2}\)XT [22] (Figure 1). Unlike the EPA procedure, the FCAZ systems analysis method uses neither morphobstructural zoning nor dichotomy learning algorithms. It relies on the topological filtering of a finite set of epicenters of fairly weak earthquakes, which act as recognition objects. The fundamental difference of the FCAZ systems method from the EPA procedure is that FCAZ has a formalized block (E\({}^{2}\)XT algorithm) of passing from the classification of a finite set of point objects to sought flat high seismicity zones. The E\({}^{2}\)XT block constructs an unambiguous mapping of a set of objects identified by the DPS algorithm in the flat zones of non-zero measure. Strong earthquakes can occur within and at the boundary of such zones. Such mapping allows for the first time to switch in the problem of earthquake-prone areas recognition from the simple pattern recognition to full-fledged systems analysis. Specifically, this makes it possible to unambiguously isolate, using sharp bound, a subsystem of recognized high seismicity zones from their non-empty complement. The core of FCAZ is the DPS topological filtering algorithm (Figure 1) [21] isolating clusters as own subsets in a set \(W\). This is what makes DPS different from classical clustering algorithms. It is aimed at isolating in a finite set of Euclidean space of flat regions with a given density level \(\alpha\). It should be emphasized that DPS is effective for the considered problem exactly because it distinguishes between compact, connected groups of objects and their fuzzy, unstructured complement. In other words, DPS cuts out isolated objects, \"attracting\" the rest into dense clusters. That said, unlike classical clustering algorithms, in DPS by no means all objects end up in clusters. This is what makes DPS new and innovative as a systems analysis algorithm. The DPS algorithm has two free parameters: \(q<0\) for the calculation of localization radius \(r_{q}(W)\), which is determined as the power mean of all nontrivial pairwise distances in a set of recognition objects \(W\), and \(\beta\in[-1,1]\) is the maximality level of the necessary density of \(\alpha=\alpha(\beta,q)\) DPS clusters.