# Development of the Algorithmic Basis of the FCAZ Method for Earthquake-Prone Area Recognition
## 2 Materials and Methods: SDPS Algorithm
### The Third Stage  
The results of the second stage may be enough (Figures 3-5) or may not (Figure 6c). The expert \(E\) decides. He or she perceives the second stage \(\text{SDPS}_{2}(X)\) as a given that is not subject to further internal transformation, considering each component of the \(r\) connectivity \(c\in\text{SDPS}_{2}(X)\) to be a single and indivisible spot (big point).  
The spots \(c=cX(\alpha)\) are interpreted by the expert \(E\) as single exits (manifestations) of global anomalous entities in \(X\). To understand their true scale, additional connection of spots, if possible, may be needed. This is the third and final stage of the SDPS algorithm.  
The expert \(E\) considers a set of spots \(C\subset\text{SDPS}_{2}(X)\) to be one whole if any two spots in it can be connected by a chain of close (in his or her opinion) intermediate transitions.  
Figure 6: (**a**) Initial array \(X\). (**b**) Result of the first stage of the SDPS algorithm. (**c**) Result of the second stage of the SDPS algorithm with \(r\) connectivity components (uniform in color sets). (**d**) Result of the third stage of the SDPS algorithm with edges and quasi-edges (uniform in color sets).  
This gives the expert a reason to conclude that \(C\) is not random and that the set of spots (\(r\) components of connectivity) included in \(C\) is a collective but fragmentary manifestation (edge) of some global anomalous entity in \(X\).  
We denote with \(\chi_{E}\) the partition of the spot space \(\mathrm{SDPS}_{2}(X)\) into such \"non-random\" sets \(C_{i}\):  
\[\chi_{E}\leftrightarrow\mathrm{SDPS}_{2}(X)=\vee_{i=1}^{i^{+}(E)}C_{i},C_{i}= \left\{C_{ij}|_{j=1}^{i^{+}(i)}\right\}. \tag{5}\]  
This is the third stage of the SDPS algorithm (\(\mathrm{SDPS}_{3}(X)=\chi_{E}\)), and it depends on the expert \(E\)'s analysis of the initial space \(X\) and the first and second SDPS stages of it:  
\[\chi_{E}=\chi_{E}(X,\mathrm{SDPS}_{1}(X),\mathrm{SDPS}_{2}(X)). \tag{6}\]  
With such an analysis, for each set \(C_{i}\), the question of proximity within it is generally solved individually.  
In this paper, the simplest Boolean version of the third stage is presented where, based on the analysis of the space \(X\) only, the expert \(E\) develops its proximity threshold \(r_{E}\), which is weaker than \(r\) (\(r<r_{E}\)) and splits the spot space \(\mathrm{SDPS}_{2}(X)\) into \(r_{E}\) connectivity components (\(\chi_{E}=C_{r_{E}}X(\alpha)\)).  
The parameters \(r\) and \(r_{E}\) are constructed using power law averaging of non-trivial distances in \(X\):  
\[r =\left(\frac{\sum_{x\
eq y\in X}d(x,y)^{q(r)}}{|X|(|X|-1)} \right)^{1/q(r)},\] \[r_{E} =\left(\frac{\sum_{x\
eq y\in X}d(x,y)^{q(E)}}{|X|(|X|-1)} \right)^{1/q(E)}.\]  
For the parameter \(r\), numerous tests of the SDPS algorithm have established that the choice of \(q(r)\in[-3,-2]\) can be considered optimal. The studies carried out in the framework of this work show that \(q(E)\in[-2.5,-1.5]\). The intersection of the areas of parameters \(r\) and \(r_{E}\) is explained both by the fuzzy perception of proximity by the expert and by the diversity in construction of \(X\).  
Figure 6c shows the 24 \(r\) connectivity components (spots) obtained in the second stage, which combined into 15 \(r_{E}\) connectivity components (Figure 6d). Note that, in this case, \(q(r)=-3.0\), and \(q(r_{E})=-2.3\).