## 2 Materials and Methods
### Recognition of the Areas Prone to Strong Earthquakes with One Learning Class
Phase one: formation of set \(P_{\pi}(v)\) of objects that are close to \(v\) in \(W\) by characteristic \(\pi\) using the minimality threshold \(\alpha_{\pi}(v)\):  
\[P_{\pi}(v)=\{w\in W:\,\rho_{\pi}(w,v)\leq\alpha_{\pi}(v)\} \tag{3}\]  
The threshold \(\alpha_{\pi}(v)\) performs the functions of the flexible lower boundary of the set \(\{\rho_{\pi}(w,v),\ w\in W\}\) and can be obtained, for example, using Kolmogorov averaging with the value \(s<0\):  
\[\alpha_{\pi}(v)=\left(\frac{\sum_{\Xi\in W}\rho_{\pi}(\overline{w},v)^{s}}{| W|}\right)^{1/s} \tag{4}\]  
Phase two: formation on \(W\) of the value \(p_{\Pi}(w|v)\) showing the proximity of \(w\) to \(v\) based on all characteristics \(\Pi\):  
\[p_{\Pi}(w|v)=|\{\pi\in\Pi:w\in P_{\pi}(v)\}|. \tag{5}\]  
Here \(P_{\pi}(v)\) is determined by the formula (3), and the integral exponent \(p_{\Pi}(w|v)\) introduced by the formula (5) varies from 0 to \(|\Pi|\). Phase three: formation in \(W\) of a subset of \(P_{\Pi}(v)\) objects that are close to \(v\) based on all characteristics \(\Pi\) using the maximality threshold \(\beta_{\Pi}(v)\):  
\[P_{\Pi}(v)=\{w\in W:\,\,p_{\Pi}(w|v)\geq\beta_{\Pi}(v)\} \tag{6}\]  
The threshold \(\beta_{\Pi}(v)\) performs the functions of flexible upper boundary of the set of values \(\{p_{\Pi}(w|v),\ w\in W\}\). Similarly, to the formula (4), \(\beta_{\Pi}(v)\) can be constructed using Kolmogorov averaging with the value \(q>0\):  
\[\beta_{\Pi}(v)=\left(\frac{\sum_{\Xi\in W}p_{\Pi}(\overline{w}|v)^{q}}{|W|} \right)^{1/q} \tag{7}\]  
As a result, a sought set \(P_{\Pi}(B_{0})\) is obtained by the formula:  
\[P_{\Pi}(B_{0})=\cup_{v\in B_{0}}P_{\Pi}(v). \tag{8}\]  
For the quantitative assessments of the contribution of characteristics to the formation of a sought subset of high seismicity objects, the algorithm features additional computational units. In parallel with the computation of \(p_{\Pi}(w|v)\), a binary matrix is formed \(M_{\Pi}(w|v)\):  
\[M_{\Pi}(w|v)_{i,j}=\left\{ \begin{array}{cc}1,\ w_{i}\in P_{\pi_{j}}(v)\\ 0,w_{i}\
otin P_{\pi_{j}}(v)\end{array} \right.,\ i=1,\ldots,|W|,\ j=1,\ldots,| \Pi|. \tag{9}\]  
Every element of the matrix (9) determines whether or not the object \(w\in W\) belongs to the set \(P_{\pi_{j}}(v)\), \(\pi_{j}\in\Pi\) of the objects that are close to \(v\). String summation of the matrix \(M_{\Pi}(w|v)\) for all \(w\in P_{\Pi}(v)\) forms the vector \(W_{\Pi}(v)\):  
\[W_{\Pi}(v)_{j}=\sum_{k}M_{\Pi}(w|v)_{k,j},\ w_{k}\in P_{\Pi}(v),\ j=1,\ldots,| \Pi|. \tag{10}\]  
The elements of the vector (10) illustrate the contribution of characteristics \(\pi_{j}\in\Pi\) to the formation of a subset \(P_{\Pi}(v)\) of the objects that are close to \(v\in B_{0}\). The quantitative assessments of the contribution of characteristics to the formation of \(P_{\Pi}(B_{0})\) are undertaken in two phases:  
Element-by-element summation of all vectors \(W_{\Pi}(v)\), \(v\in B_{0}\) and normalization on \(|B_{0}|\) allows obtaining the average contribution of characteristics to the recognition of a sought high seismicity subset \(P_{\Pi}(B_{0})\). The sorting of \(W_{\Pi}(v)\), \(v\in B_{0}\) and the selection for each of them of three characteristics with the greatest values, followed by the summation of the number of belongings of such characteristics to the formed threes, enable assessing the contribution of such characteristics through their classification as the \"strongest.\" This class will be called Top 3 ranking. This explains the name of the algorithm Barrier-3. The set of recognition objects is represented as a disjoint union of \(W=B\coprod H\) high and low seismicity classes, \(B=P_{\Pi}(B_{0})\supseteq B_{0}\), and \(H=W\backslash B\). The Barrier-3 algorithm was employed as the EPA recognition block to recognize the areas prone to crustal earthquakes with \(M\geq 6.0\) in the Caucasus and the Altai-Sayan-Baikal region.