# DPS Clustering: New Results
## 5 Discussion  
The density \(P\) on a finite space \(X\) is a definition in \(X\) in the language of fuzzy mathematics of the property of limiting. From a formal point of view, \(P\) is a fuzzy structure on the direct product \(2^{X}\times X\), monotonic in the first argument (1). Fixing a level \(\alpha\in[0,1]\) for \(P\), we can define a closure operator on \(2^{X}\) and, through it, the usual topology \(\tau_{\alpha}\) on \(X\). Thus, an increasing family of topologies \(\{\tau_{\alpha},\alpha\in[0,1]\}\), arises on \(X\), starting at zero with a trivially inseparable minimal topology of sticking together points and ending with a trivially separable maximal topology of all subsets. Each density on the space \(X\) gives its own view of it and its own program of its study. It is a concept that does not lie in classical mathematics. In addition to the density \(S\) Section 2.2.2, nontrivial densities always exist in a finite metric space. The description of the most important densities and their significance for data analysis are given in [5; 11]. The study of the space \(X\) with the help of the density \(P\) defined on it began with the study of perfect sets in the topology \(\tau_{\alpha}\). The explanation for this is as follows: setting \(P\) on \(X\) defines a clustering in \(X\): \(K\) is a cluster (\(P\)-cluster) in \(X\) if it consists of exactly all points that are \(P\)-limit to it. If \(\alpha\) is the limit threshold, then the formal expression of what has been said coincides with \(\alpha\)-perfection: \(K=\{x\in X:P_{K}(x)\geq\alpha\}\). The DPS-algorithm builds all such clusters, but there are too many of them for meaningful cluster analysis in \(X\): first of all, we need \"connected components\" of the maximal \(\alpha\)-perfect subset \(X(\alpha)\) in \(X\). All this exists when \(X\) is FMS and the density \(P\) is local on it. It is in this situation that the DPS-algorithm operates. It depends on the design of the density \(P\), its level \(\alpha\) and the localization radius \(r\): \(\text{DPS}=\text{DPS}(P,\alpha,r)\). DPS operates in two stages: first, it cuts out  
Figure 8: Comparison of SDPS and SDPS\({}^{2}\) algorithms: (**a**)—initial array; (**b**)—the first stage of SDPS; (**c**)—the second stage of SDPS; (**d**)—the first stage of SDPS\({}^{2}\); (**e**)—the second stage of SDPS\({}^{2}\); (**f**)—the third stage of SDPS\({}^{2}\). a subset \(X(\alpha)\) in \(X\) (first stage), and then splits it into \(r\)-connected components, which it considers to be DPS-clusters in \(X\) (second stage). Despite generally good results and efficient applications [4; 5; 11; 12; 13], the DPS-algorithm in its current version has drawbacks: at the first stage, the cutting \(X\Rightarrow X(\alpha)\) is not always thorough and of high quality; at the second stage, the partitioning of \(X(\alpha)\) into \(r\)-connectivity components is too small and detailed due to the locality of the radius \(r\). A proper partition of \(X(\alpha)\) must come from the global view of \(X(\alpha)\) induced by the entire space \(X\). In the present paper, we have made attempts to correct or mitigate these drawbacks: our answers were at the first stage--the iterative cutting scheme \(\text{DPS}^{i}(X)\), at the second--its third stage \(\text{DPS}_{f}(X)\) of connecting the components of the \(r\)-connectivity. The iterations allowed the DPS-algorithm at the first stage to achieve the completeness of its \(\text{DPS}(X)\) result, while maintaining a high level of extremality of cutting. At the moment, the result \(\text{DPS}_{\text{C}}(X)\) of the second stage of the DPS-algorithm is considered its final result. The additional connections of components from \(\text{DPS}_{\text{C}}(X)\) proposed in the paper occur in two stages. On the first of them, which has the necessary character, \(d\)-clusters are searched in \(\text{DPS}_{\text{C}}(X)\). According to the results of the article, their search is constructive.