## 4 Results  
### SDPS Algorithm  
Historically, the set-theoretical SDPS was the first in a series of DPS-algorithms. It is based on the density \(S\) with the name \"Number of points\" (\"Number of space\") [12; 13] and conveying the degree of concentration of space \(X\) round each of its points \(x\) (the most natural understanding of density \(X\) in \(x\)). The density \(S_{A}(x)\) depends on the localization radius \(r=r_{q}(X)\) (5) and the non-negative parameter \(p\), considering the distance to \(x\) in the full-sphere \(D_{A}(x,r)\):  
\[S_{A}(x)=S_{A}(x|q,p)=\sum_{y\in D_{A}(x,r)}\left(1-\frac{d(x,y)}{r}\right)^{p}\]  
When \(p=0\), we have the usual number of points, explaining the name \(S\):  
\[S_{A}(x|q,0)=|D_{A}(x,r_{q})|\]  
The \(S\) density is \(r\)-local, and the SDPS algorithm is the implementation of the DPS scheme based on \(S\), described in Definition 6-Section 3.3.3: \(\text{SDPS}=\text{DPS}(S,r,\beta)\). The result of SDPS is condensations in \(X\equiv\text{sets locally containing ``many X''}\). They correspond to empirical clusters in terms of the most formal criteria. By varying the SDPS parameters, it is possible to obtain a fairly complete picture of the hierarchy of clusters in \(X\). **Example 2**. : _Figure 5b shows the result of selection by level \(\beta=-0.3\) for density \(S\) on the initial array \(X\) (Figure 5a), that is, the first iteration \(X^{1}(-0.3)\) of the SDPS algorithm. It contains isolated points that needs to be removed, and in this sense is inferior to the final result \(X(-0.3)\) of the SDPS algorithm on \(X\) (Figure 5c. )_  
**Example 3**. : _In the conditions of the Example 2, the inverse correlation of the SDPS algorithm performance with the parameter \(\beta\) is shown. By increasing it, we go inside the condensation, finding dense nuclei inside them (Figure 6a-c)._  
Figure 5: Application of the SDPS algorithm (\(q=-2\), \(\beta=-0.3\), \(p=0\)): (**a**) the original array; (**b**) the result of the first iteration \(X_{1}(-0.3)\) containing isolated points; and (**c**) the final result of applying SDPS. Figure 6: Inverse dependence of the result of the SDPS algorithm on the parameter \(\beta\): (**a**) the result of the SDPS algorithm at \(\beta=-0.35\); (**b**) the result of work at \(\beta=-0.15\); and (**c**) the result of work at \(\beta=0.05\). (In all cases \(q=-2\), \(p=0\)). **Example 4**. : _In the conditions of Example 2 the direct correlation of the SDPS algorithm performance with the parameter \(q\) is shown. By lowering it, we make the SDPS algorithm more local, focusses on finding smaller condensations (Figure 7a-c). All small condensations in Figure 7c are shown in black._  
**Example 5**. : _In the conditions of the Example 2, the inverse correlation of the SDPS algorithm performance with the parameter \(p\) is shown. By increasing it, we make the SDPS algorithm more stringent (Figure 8a-c)._  
The above examples illustrate the general property of SDPS algorithm dependence on parameters: the stronger the localization \((p,q)\) and the density level \(\beta\) is, the stricter the SDPS algorithm is, and its results are denser and finer. **Example 6**. : _Let us illustrate the clustering quality \(\tau(S,r,\alpha)\) introduced in Section 3.3.4 on the SDPS work in the array shown in Figure 9. Let us designate by \(M(\beta,r)\) and \(\overline{M}(\beta,r)\) the mean densities of the sets \(X_{S}(\beta,r)\) and \(\overline{X_{S}(\beta,r)}\) at their points, then the result of their fuzzy comparison \(\tau(\beta,r)=n\big{(}\overline{M}(\beta,r),M(\beta,r)\big{)}\) can be considered a version of the quality of the SDPS\((\beta,r)\) algorithm on the space \(X\). From left to right it is equal to \(0.858\), \(0.595\), \(0.510\) respectively.