## 4 Discussion
### FCAZ Recognition as the Problem of Advanced Systems Analysis
* \(W_{t_{1}}=\{w\}\) is the finite set of recognition objects (the epicenters of earthquakes with \(M\geq M_{\text{R}}\)) at the moment \(t_{1}\), \(|W_{t_{1}}|=n_{t_{1}}\);
* \(G=\{g\}\) denotes certain coverage of the considered region by square objects, on which the E\({}^{2}\)XT algorithm works;
* \(\gamma_{1}=\{\delta,C,\omega,v,q,\beta\}\) is a set of values of free parameters of FCAZ selected for an optimal recognition at the moment \(t_{1}\);
* \(B_{t_{1}}\subset G\) and \(H_{t_{1}}\subset G\) are the subsets of objects \(g\in G\) classified as high seismicity and low seismicity ones, respectively, i.e., the objects \(g\in B_{t_{1}}\) are fairly close, and the objects \(g\in H_{t_{1}}\) are fairly distant from known and potential areas prone to strong earthquakes, \(B_{t_{1}}\cup H_{t_{1}}=G\), \(B_{t_{1}}\cap H_{t_{1}}=\varnothing\). Let \(B_{0,t_{1}}\) denote the set of epicenters of strong earthquakes that had occurred by the moment \(t_{1}\). It is obvious that the higher the value of the inclusion measure of epicenters \(B_{0,t_{1}}\) in the subset of high seismicity objects \(B_{t_{1}}\), the better FCAZ recognition at the moment \(t_{1}\):  
\[\mu\big{(}B_{0,t_{1}}\subset B_{t_{1}}\big{)}=\big{|}B_{0,t_{1}}\cap B_{t_{1}} \big{|}/\big{|}B_{0,t_{1}}\big{|}. \tag{19}\]  
The quality of the FCAZ recognition problem considered above is determined by the fact that the results of future (after the appearance of new objects with time) expansions \(\text{FCAZ}_{\gamma}(W_{t}):G=B_{t}\coprod H_{t}\) tend to the limit characterized by the condition:  
\[\lim_{t\to\infty}\mu(B_{0,t}\subset B_{t})\to 1. \tag{20}\]  
Let us assume by the moment \(t_{2}=t_{1}+\Delta t\ Z_{t_{1},t_{2}}\) more strong earthquakes occurred, \(B_{0,t_{2}}=B_{0,t_{1}}\cup Z_{t_{1},t_{2}}\), i.e., bringing their total number to \(\big{|}B_{0,t_{2}}\big{|}=\big{|}B_{0,t_{1}}\cup Z_{t_{1},t_{2}}\big{|}\). That said, the total number of occurring earthquakes with \(\text{M}\geq\text{M}_{\text{R}}\) among the recognition objects increased by \(z_{t_{1},t_{2}}\) to a total of \(n_{t_{2}}=n_{t_{1}}+z_{t_{1},t_{2}}\). Let us denote this new set of objects \(W_{t_{2}}=\{w\}\). In this new situation, at the moment \(t_{1}+\Delta t\) we have important additional information in place, which was not available to us at the moment \(t_{1}\). Accordingly, it is necessary to perform FCAZ recognition this time based on \(W_{t_{2}}\) to obtain the expansion \(\text{FCAZ}_{\gamma_{2}}(W_{t_{2}}):G=B_{t_{2}}\coprod H_{t_{2}}\) (Figure 13). FCAZ recognition is determined by the selection of free parameters \(\gamma=\{\delta,C,\omega,v,q,\beta\}\). The parameters of the E\({}^{2}\)XT algorithm, as well as its result, directly depend on the recognized DPS clusters. In turn, \(\beta\) in DPS is the maximality level of density of DPS clusters, which depends on the spatial arrangement of objects. Due to the earthquakes with \(M\geq M_{\text{R}}\), which occurred over the time \(\Delta t=t_{2}-t_{1}\), the spatial distribution of objects \(w\in W_{t_{2}}\) will differ from the distribution of objects \(w\in W_{t_{1}}\). For this reason, the selection of values \(\gamma_{2}=\{\delta,C,\omega,v,q,\beta\}\) for \(\text{FCAZ}_{\gamma_{2}}(W_{t_{2}}):G=B_{t_{2}}\coprod H_{t_{2}}\) must be performed by the above-mentioned artificial intelligence blocks. These blocks ensure the selection of optimal values of input parameters accounting the spatial distribution of recognition objects at a given moment in time. It is clear that the spatial distribution of a set of objects \(W_{t_{2}}\) can be so dramatically different from the spatial distribution of \(W_{t_{1}}\) that \(B_{t_{1}}\) will not be a proper subset \(B_{t_{2}}\). In other words, the threshold (20) can fail to be achieved. To prevent this kind of situation and create a successive monotonous growing of FCAZ zones as the high seismicity areas recognized at the moment \(t_{2}\), the integration of the zones \(B_{t_{1}}\) and \(B_{t_{2}}\) should be taken, i.e., \(B_{t_{2}}=B_{t_{2}}\cup B_{t_{1}}\).