DBSCAN-like clustering method for various data densities

In this paper, we propose a modification of the well-known DBSCAN algorithm, which recognizes clusters with various data densities in a given set of data points \({\mathcal {A}}=\{a^i\in {\mathbb {R}}^n:i=1,\dots ,m\}\). First, we define the parameter \(MinPts=\lfloor \ln |{\mathcal {A}}|\rfloor\) and after that, by using a standard procedure from DBSCAN algorithm, for each \(a\in {\mathcal {A}}\) we determine radius \(\epsilon _a\) of the circle containing MinPts elements from the set \({\mathcal {A}}\). We group the set of all these radii into the most appropriate number (t) of clusters by using Least Squares distance-like function applying SymDIRECT or SepDIRECT algorithm. In that way, we obtain parameters \(\epsilon _1>\dots >\epsilon _t\). Furthermore, for parameters \(\{MinPts,\epsilon _1\}\) we construct a partition starting with one cluster and then add new clusters for as long as the isolated groups of at least MinPts data points in some circle with radius \(\epsilon _1\) exist. We follow a similar procedure for other parameters \(\epsilon _2,\dots ,\epsilon _t\). After the implementation of the algorithm, a larger number of clusters appear than can be expected in the optimal partition. Along with defined criteria, some of them are merged by applying a merging process for which a detailed algorithm has been written. Compared to the standard DBSCAN algorithm, we show an obvious advantage for the case of data with various densities.