# DPS Clustering: New Results
## 3 Results: Iterative DPS
### Iterations by Extremality  
It will be remembered that the level \(\alpha(\beta)\) of density \(P\) by extremality level \(\beta\) is determined in the \(\text{DPS}(P,\beta,r)=\text{DPS}(P,\alpha(\beta),r)\) algorithm from Equation (8). Let us introduce the partition \(X=X(\beta)\vee\overline{X(\beta)}\). According to (7) and (8),  
\[\beta= n(P_{X}(X),\alpha(\beta))=\frac{\sum_{x\in X(\beta)}n(P_{X}(x), \alpha(\beta))+\sum_{x\in\overline{X(\beta)}}n(P_{X}(x),\alpha(\beta))}{|X|}=\] \[= \frac{|X(\beta)|}{|X|}\frac{\sum_{x\in X(\beta)}n(P_{X}(x),\alpha (\beta))}{|X(\beta)|}+\frac{|\overline{X(\beta)}|}{|X|}\frac{\sum_{x\in \overline{X(\beta)}}n(P_{X}(x),\alpha(\beta))}{|X(\beta)|}.\]  
From the properties of convexity, hence the inequality  
\[\max\left(\frac{\sum_{x\in X(\beta)}n(P_{X}(x),\alpha(\beta))}{|X(\beta)|}, \frac{\sum_{x\in\overline{X(\beta)}}n(P_{X}(x),\alpha(\beta))}{|X(\beta)|} \right)\geq\beta.\]  
However, the left mean \(\leq 0\) because \(\forall x\in X(\beta)\Rightarrow P_{X}(x)\geq P_{X(\beta)}(x)\geq\alpha(\beta)\). Hence, if \(\beta\geq 0\), the right mean must be \(\geq\beta\):  
\[\sum_{x\in\overline{X(\beta)}}\frac{n(P_{X}(x),\alpha(\beta))}{|\overline{X( \beta)}|}\geq\beta.\]  
Figure 3: Dependence of the DPS-algorithm result on the level of extremeness: (**a**)—initial array; (**b**)—red color shows the result \(X(\beta)\) of the DPS-algorithm, green color shows the result of the algorithm on the complement (\(\overline{X(\beta)}\)), \(\beta=0.02\), the blue cluster is the result of one more transition to the complement; \(\beta=0.02\); (**c**)—red color shows the result at \(\bar{\beta}=-0.25\). Then, from the inequalities \(\forall x\in X\Rightarrow P_{\overline{X(\beta)}}(x)\leq P_{X}(x)\) and the properties of fuzzy comparison \(n\), we have:  
\[n\Big{(}P_{\overline{X(\beta)}}\Big{(}\overline{X(\beta)}\Big{)},\alpha(\beta)\Big{)} =\frac{\sum_{x\in\overline{X(\beta)}}n(P_{\overline{X(\beta)}}(x),\alpha(\beta))}{|\overline{X(\beta)}|}\geq\] \[\geq\sum_{x\in\overline{X(\beta)}}\frac{n(P_{X}(x),\alpha(\beta) )}{|\overline{X(\beta)}|}\geq\beta.\]  
It follows that the density level \(\alpha_{1}(\beta)\), required for the operation of \(\text{DPS}(P,\beta,r)\) algorithm on the space \(\overline{X(\beta)}\):  
\[n\Big{(}P_{\overline{X(\beta)}}\Big{(}\overline{X(\beta)}\Big{)},\alpha_{1}( \beta)\Big{)}=\beta\]  
does not exceed the level \(\alpha(\beta)\), which we will consider as zero: \(\alpha(\beta)=\alpha_{0}(\beta)\), and can be strictly less than it. In addition, this, in turn, means the possible nontriviality of such operation, i.e., the set \(\overline{X(\beta)}(\beta)\). Let us denote it by \(X_{1}(\beta)\). In the situation of non-triviality \(X_{1}(\beta)\), one can define the first iteration on the extremity \(X(\beta,1)\) of the set \(X(\beta)\), which is naturally considered as its null iteration, assuming \(X(\beta,0)=X(\beta)\). Definition 5: Let \(X_{1}(\beta)\) be non-trivial, and we will call by the first iteration of the space \(X\) with respect to the extremity level \(\beta\) the disjoint union  
\[X(\beta,1)=X(\beta,0)\lor X_{1}(\beta).\]  
Note 1: Given this definition, the above can be understood as the first precondition for the existence of the first iteration on extremality when \(\beta\) is non-negative. In this connection, the zero level of \(\beta\) appears to be the most productive and interesting, at which the second precondition (non-triviality of the result) will be the weakest. A direct check shows that \(P_{X(\beta,1)}(x)\geq\alpha_{1}(\beta)\ \forall x\in X(\beta,1)\), so, by repeating the above deduction with replacement \(\alpha_{0}(\beta)\Rightarrow\alpha_{1}(\beta)\), \(X(\beta,0)\Rightarrow X(\beta,1)\), we obtain the level \(\alpha_{2}(\beta)\leq\alpha_{1}(\beta)\) of the \(\text{DPS}(P,\beta,r)\) algorithm on the complement \(\overline{X(\beta,1)}\) and the possible nontriviality of its result \(X_{2}(\beta)=\overline{X(\beta,1)}(\beta)\).