# Development of the Algorithmic Basis of the FCAZ Method for Earthquake-Prone Area Recognition
## 2 Materials and Methods: SDPS Algorithm
### Second Stage  
Figure 6b shows the result of the first stage \(\mathrm{SDPS}_{1}(X)\) for the array \(X\) shown in Figure 6a. It is clear that it needed \"reasonable\" partitioning. In the SDPS algorithm, the second and third stages are devoted to this.  
At the second stage, the set \(X(\alpha)\) is partitioned into non-intersecting \(r\)-connected components (Figure 6c) which, of course, must be included in any reasonable partition of \(X(\alpha)\). There are two reasons for this, and they are given below:  
**Definition 2**.: _Points \(x\) and \(y\) in \(X(\alpha)\) are called \(r\)-connected if in \(X(\alpha)\), there is a chain of \(r\) close-to-each-other points \(x_{0},\ldots,x_{n}\) with a start \(x=x_{0}\), end \(y=x_{n}\), and distances \(d(x_{i},x_{i+1})\leq r,i=0,\ldots,n-1\)._  
The \(r\) connectivity relation is an equivalence that splits \(X(\alpha)\) into disjunctive \(r\) connectivity components which, depending on the context, will be denoted below as \(c\) or \(c_{k}\), \(k=1,\ldots,k^{*}=k^{*}(X(\alpha),r)\).  
Figure 4: Direct dependence on the radius of localization of the first stage of SDPS when moving from \(\mathrm{SDPS}(\alpha,r,p)(X)\) (**a**) to \(\mathrm{SDPS}(\alpha,r,p)(X)\) (**b**) with \(r>r\) on the array \(X\) (Figure 3a). Finer black condensations were added to the red, yellow and green ones.  
Figure 5: The inverse nature of the dependence of the first stage of SDPS on the center parameter. For \(p>p\), the result of \(\mathrm{SDPS}(\alpha,r,\bar{p})(X)\) (**b**) was more rigorous than the result of \(\mathrm{SDPS}(\alpha,r,p)(X)\) (**a**) on array \(X\) (Figure 3a).  
The second stage of the SDPS algorithm, which we will denote as SDPS\({}_{2}\), consists of constructing the \(c\) components. Their collection, as well as the partition \(X(\alpha)\) associated with them, will be denoted identically as \(C_{r}(X(\alpha))\), and thus  
\[C_{r}(X(\alpha))=\{c_{k}|_{1}^{k^{*}}\}\text{ and }C_{r}(X(\alpha))\leftrightarrow X (\alpha)=\vee_{k=1}^{k^{*}}c_{k}. \tag{4}\]  
This is the result of the second stage of the SDPS algorithm: \(\text{SDPS}_{2}(X)=C_{r}(\text{SDPS}(X))\).  
Rationale (first reason): \(r\) is the localization radius in the SDPS algorithm, and therefore any points that are \(r\) close to each other are considered close and must necessarily be included in the same partition component \(X(\alpha)\) (the partition should not break close points).  
Rationale (second reason): the components of the \(r\) connectivity are separated from each other by more than \(r\), so the density \(S\) of each of them at any point from the other component is equal to zero. Hence, the conclusion is that each component of the \(r\) connection in \(X(\alpha)\) is itself discretely perfect, since it independently provides the necessary level \(\alpha\) of the density \(S\) at each of its points and is equal to zero at other points.  
Figure 6 shows that the result of cutting (red dots in Figure 6b) was split in the second stage into 24 \(r\)-connected components, shown in Figure 6c with different colors.