## Abstract

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.

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## References

Aggarwall CC, Reddy CK (2013) Data clustering: algorithms and applications. CRC data mining and knowledge discovery series. Chapman & Hall, London

Akinlar C, Topal C (2013) Edcircles: a real-time circle detector with a false detection control. Pattern Recognit 46:725–740

Amami R, Smiti A (2017) An incremental method combining density clustering and support vector machines for voice pathology detection. Comput Electr Eng 57:257–265

Andrade G, Ramos G, Madeira D, Sachetto R, Ferreira R, Rocha L (2013) G-DBSCAN: a GPU accelerated algorithm for density-based clustering. Procedia Comput Sci 18:369–378

Ankerst M, Breunig MM, Kriegel HP, Sander J (1999) OPTICS: ordering points to identify the clustering structure. ACM Sigmod Rec 28:49–60

Bagirov AM, Ugon J, Webb D (2011) Fast modified global \(k\)-means algorithm for incremental cluster construction. Pattern Recognit 44:866–876

Bakr AM, Ghanem NM, Ismail MA (2015) Efficient incremental density-based algorithm for clustering large datasets. Alex Eng J 54:1147–1154

Bezdek JC, Keller J, Krisnapuram R, Pal NR (2005) Fuzzy models and algorithms for pattern recognition and image processing. Springer, New York

Birant D, Kut A (2007) ST-DBSCAN: an algorithm for clustering spatial–temporal data. Data Knowl Eng 60:208–221

Cuesta-Albertos JA, Gordaliza A, Matrán C (1997) Trimmed \(k\)-means: an attempt to robustify quantizers. Ann Stat 25(2):553–576

Darong H, Peng W (2012) Grid-based DBSCAN algorithm with referential parameters. Phys Procedia 24:1166–1170

Ertöz L, Steinbach M, Kumar V (2003) Finding clusters of different sizes, shapes, and densities in noisy, high dimensional data. In: Proceedings of second SIAM international conference on data mining, San Francisco

Ester M, Krieogel H, Sander J (1996) A density-based algorithm for discovering clusters in large spatial databases with noise. In: 2nd International conference on knowledge discovery and data mining (KDD-96), Portland, pp 226–231

Frigui H (2005) Unsupervised learning of arbitrarily shaped clusters using ensembles of Gaussian models. Pattern Anal Appl 8:32–49

Fritz H, García-Escudero LA, Mayo-Iscar A (2013) A fast algorithm for robust constrained clustering. Comput Stat Data Anal 61:124–136

Grbić R, Grahovac D, Scitovski R (2016) A method for solving the multiple ellipses detection problem. Pattern Recognit 60:824–834

Grbić R, Nyarko EK, Scitovski R (2013) A modification of the DIRECT method for Lipschitz global optimization for a symmetric function. J Glob Optim 57:1193–1212

Gunawan A (2013). A Faster Algorithm for DBSCAN. Ph.D. thesis, Technische Universiteit Eindhoven

Hubert L, Arabie P (1985) Comparing partitions. J Classif 2:193–218

Jiang H, Li J, Yi S, Wang X, Hu X (2011) A new hybrid method based on partitioning-based DBSCAN and ant clustering. Expert Syst Appl 38:9373–9381

Jones DR (2001) The direct global optimization algorithm. In: Floudas CA, Pardalos PM (eds) The encyclopedia of optimization. Kluwer Academic Publishers, Dordrect, pp 431–440

Jones DR, Perttunen CD, Stuckman BE (1993) Lipschitzian optimization without the Lipschitz constant. J Optim Theory Appl 79:157–181

Karami A, Johansson R (2014) Choosing DBSCAN parameters automatically using differential evolution. Int J Comput Appl 91:1–11

Kogan J (2007) Introduction to clustering large and high-dimensional data. Cambridge University Press, New York

Kumar KM, Reddy ARM (2016) A fast DBSCAN clustering algorithm by accelerating neighbor searching using groups method. Pattern Recognit 58:39–48

Lai HP, Visani M, Boucher A, Ogier JM (2012) An experimental comparison of clustering methods for content-based indexing of large image databases. Pattern Anal Appl 15:345–366

Li Z, Zhang Y, Gong H, Liu G, Li W, Tang X (2017) An automatic and efficient coronary arteries extraction method in CT angiographies. Biomed Signal Process Control 36:221–233

Louhichi S, Gzara M, Ben-Abdallah H (2017) Unsupervised varied density based clustering algorithm using spline. Pattern Recognit Lett 93:48–57

MacQueen JB (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, pp 281–297

Marošević T, Sabo K, Taler P (2013) A mathematical model for uniform distribution voters per constituencies. Croat Oper Res Rev 4:53–64

McCallum A, Nigam K, Ungar LH (2000) Efficient clustering of high-dimensional data sets with application to reference matching. In: International conference on knowledge discovery and data mining. DBLP

Mimaroglu S, Aksehirli E (2011) Improving DBSCAN’s execution time by using a pruning technique on bit vectors. Pattern Recognit Lett 32:1572–1580

Morales-Esteban A, Martínez-Álvarez F, Scitovski S, Scitovski R (2014) A fast partitioning algorithm using adaptive Mahalanobis clustering with application to seismic zoning. Comput Geosci 73:132–141

Sabo K, Scitovski R (2015) An approach to cluster separability in a partition. Inf Sci 305:208–218

Sabo K, Scitovski R, Vazler I (2013) One-dimensional center-based \(l_1\)-clustering method. Optim Lett 7:5–22

Scitovski R (2017) A new global optimization method for a symmetric Lipschitz continuous function and application to searching for a globally optimal partition of a one-dimensional set. J Glob Optim 68:713–727

Scitovski R, Marošević T (2014) Multiple circle detection based on center-based clustering. Pattern Recognit Lett 52:9–16

Scitovski R, Sabo K (2014) Analysis of the \(k\)-means algorithm in the case of data points occurring on the border of two or more clusters. Knowl Based Syst 57:1–7

Scitovski R, Scitovski S (2013) A fast partitioning algorithm and its application to earthquake investigation. Comput Geosci 59:124–131

Scitovski R, Vidović I, Bajer D (2016) A new fast fuzzy partitioning algorithm. Expert Syst Appl 51:143–150

Späth H (1983) Cluster-formation und analyse. R. Oldenburg Verlag, München

Steinbach M, Tan PN, Potter VKC, Klooster S (2002) Data mining for the discovery of ocean climate indices, In: Mining scientific datasets workshop, 2nd Annual SIAM international conference on data mining

Teboulle M, Berkhin P, Dhilon I, Guan Y, Kogan J (2006) Clustering with entropy-like \(k\)-means algorithms. In: Kogan J, Nicholas C, Teboulle M (eds) Grouping multidimensional data. Springer, Berlin, pp 127–160

Theodoridis S, Koutroumbas K (2009) Pattern recognition, 4th edn. Academic Press, Burlington

Vendramin L, Campello RJGB, Hruschka ER (2009) On the comparison of relative clustering validity criteria, In: Proceedings of the SIAM international conference on data mining, SDM 2009, April 30–May 2, 2009. SIAM, Sparks, pp 733–744

Viswanath P, Babu VS (2009) Rough-DBSCAN: a fast hybrid density based clustering method for large data sets. Pattern Recognit Lett 30:1477–1488

Wolfram Research I (2016) Mathematica, version 11.0 edition. Wolfram Research, Inc., Champaign

Xie J, Gao H, Xie W, Liu X, Grant PW (2016) Robust clustering by detecting density peaks and assigning points based on fuzzy weighted \(K\)-nearest neighbors. Inf Sci 354:19–40

Zaki MJ, Meira W Jr (2014) Data mining and analysis: fundamental concepts and algorithms. Cambridge University Press, New York

Zhu Y, Ting KM, Carman MJ (2016) Density-ratio based clustering for discovering clusters with varying densities. Pattern Recognit 60:983–997

## Acknowledgements

The author would like to thank the referees and the journal editors for their careful reading of the paper and insightful comments that helped us improve the paper. Especially, the author would like to thank Mrs. Katarina Moržan for significantly improving the use of English in the paper. This work was supported by the Croatian Science Foundation through research Grant IP-2016-06-6545 “The optimization and statistical models and methods in recognizing properties of data sets measured with errors” and research Grant IP-2016-06-8350 “Methodological framework for efficient energy management by intelligent data analytics”.

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Scitovski, R., Sabo, K. **DBSCAN**-like clustering method for various data densities.
*Pattern Anal Applic* **23**, 541–554 (2020). https://doi.org/10.1007/s10044-019-00809-z

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DOI: https://doi.org/10.1007/s10044-019-00809-z

### Keywords

- Clustering
- DBSCAN
- Incremental algorithm
- Various data densities
- Clusters merging
- Least Squares distance-like function