 DBSCAN

DBSCAN (for densitybased spatial clustering of applications with noise) is a data clustering algorithm proposed by Martin Ester, HansPeter Kriegel, Jörg Sander and Xiaowei Xu in 1996.^{[1]} It is a densitybased clustering algorithm because it finds a number of clusters starting from the estimated density distribution of corresponding nodes. DBSCAN is one of the most common clustering algorithms and also most cited in scientific literature.^{[2]} OPTICS can be seen as a generalization of DBSCAN to multiple ranges, effectively replacing the ε parameter with a maximum search radius.
Contents
Basic idea
DBSCAN's definition of a cluster is based on the notion of density reachability. Basically, a point q is directly densityreachable from a point p if it is not farther away than a given distance ε (i.e., is part of its εneighborhood), and if p is surrounded by sufficiently many points such that one may consider p and q be part of a cluster. q is called densityreachable (note: this is different from "directly densityreachable") from p if there is a sequence of points with p_{1} = p and p_{n} = q where each p_{i + 1} is directly densityreachable from p_{i}. Note that the relation of densityreachable is not symmetric (since q might lie on the edge of a cluster, having insufficiently many neighbors to count as a genuine cluster element), so the notion of densityconnected is introduced: two points p and q are densityconnected if there is a point o such that both p and q are density reachable from o.
A cluster, which is a subset of the points of the database, satisfies two properties:
 All points within the cluster are mutually densityconnected.
 If a point is densityconnected to any point of the cluster, it is part of the cluster as well.
Algorithm
DBSCAN requires two parameters: ε (eps) and the minimum number of points required to form a cluster (minPts). It starts with an arbitrary starting point that has not been visited. This point's εneighborhood is retrieved, and if it contains sufficiently many points, a cluster is started. Otherwise, the point is labeled as noise. Note that this point might later be found in a sufficiently sized εenvironment of a different point and hence be made part of a cluster.
If a point is found to be part of a cluster, its εneighborhood is also part of that cluster. Hence, all points that are found within the εneighborhood are added, as is their own εneighborhood. This process continues until the cluster is completely found. Then, a new unvisited point is retrieved and processed, leading to the discovery of a further cluster or noise.
Pseudocode
DBSCAN(D, eps, MinPts) C = 0 for each unvisited point P in dataset D mark P as visited N = regionQuery(P, eps) if sizeof(N) < MinPts mark P as NOISE else C = next cluster expandCluster(P, N, C, eps, MinPts) expandCluster(P, N, C, eps, MinPts) add P to cluster C for each point P' in N if P' is not visited mark P' as visited N' = regionQuery(P', eps) if sizeof(N') >= MinPts N = N joined with N' if P' is not yet member of any cluster add P' to cluster C
Complexity
DBSCAN visits each point of the database, possibly multiple times (e.g., as candidates to different clusters). For practical considerations, however, the time complexity is mostly governed by the number of regionQuery invocations. DBSCAN executes exactly one such query for each point, and if an indexing structure is used that executes such a neighborhood query in O(log n), an overall runtime complexity of is obtained. Without the use of an accelerating index structure, the run time complexity is O(n^{2}). Often the distance matrix of size (n^{2} − n) / 2 is materialized to avoid distance recomputations. This however also needs O(n^{2}) memory.
Advantages
 DBSCAN does not require you to know the number of clusters in the data a priori, as opposed to kmeans.
 DBSCAN can find arbitrarily shaped clusters. It can even find clusters completely surrounded by (but not connected to) a different cluster. Due to the MinPts parameter, the socalled singlelink effect (different clusters being connected by a thin line of points) is reduced.
 DBSCAN has a notion of noise.
 DBSCAN requires just two parameters and is mostly insensitive to the ordering of the points in the database. (Only points sitting on the edge of two different clusters might swap cluster membership if the ordering of the points is changed, and the cluster assignment is unique only up to isomorphism.)
Disadvantages
 DBSCAN can only result in a good clustering as good as its distance measure is in the function regionQuery(P,ε). The most common distance metric used is the euclidean distance measure. Especially for highdimensional data, this distance metric can be rendered almost useless due to the so called "Curse of dimensionality", rendering it hard to find an appropriate value for ε. This effect however is present also in any other algorithm based on the euclidean distance.
 DBSCAN cannot cluster data sets well with large differences in densities, since the minPtsε combination cannot be chosen appropriately for all clusters then.
See the section on extensions below for algorithmic modifications to handle these issues.
Parameter estimation
Every data mining task has the problem of parameters. Every parameter influences the algorithm in specific ways. For DBSCAN the parameters ε and MinPts are needed. The parameters must be specified by the user of the algorithms since other data sets and other questions require different parameters. An initial value for ε can be determined by a kdistance graph. As a rule of thumb, k can be derived from the number of dimensions in the data set D as . However, larger values are usually better for data sets with noise.
Although this parameter estimation gives a sufficient initial parameter set the resulting clustering can turn out to be not the expected partitioning. Therefore research has been performed on incrementally optimizing the parameters against a specific target value.
OPTICS can be seen as a generalization of DBSCAN that replaces the ε parameter with a maximum value that mostly effects performance. MinPts then essentially becomes the minimum cluster size to find. While the algorithm is a lot easier to parameterize then DBSCAN, the results are a bit more difficult to use, as it will usually produce a hierarchical clustering instead of the simple data partitioning that DBSCAN produces.
Generalization
Generalized DBSCAN or GDBSCAN ^{[3]}^{[4]} is a generalization by the same authors to arbitrary "neighborhood" and "dense" predicates. The ε and minpts parameters are removed from the original algorithm and moved to the predicates. For example on polygon data, the "neighborhood" could be any intersecting polygon, whereas the density predicate uses the polygon areas instead of just the object count.
Extensions
The basic DBSCAN algorithm has been used as a base for many other developments, such as parallelisation by Domenica Arlia and Massimo Coppola or an enhancement of the data set background to support uncertain data presented by Dirk Habich and Peter Benjamin Volk. The basic idea has been extended to hierarchical clustering by the OPTICS algorithm. DBSCAN is also used as part of subspace clustering algorithms like PreDeCon and SUBCLU.
Availability
An example implementation of DBSCAN is available in the ELKI framework. Notice that this implementation is not optimized for speed but for extensibility. Thus, this implementation can use various index structures for subquadratic performance and supports various distance functions and arbitrary data types, but it may be outperformed by lowlevel optimized implementations on small data sets.
External links
 DBSCAN clustering using C++ by Antonio Gulli (No index support, quadratic time and memory complexity, vector data only, some distance functions)
 R package fpc includes DBSCAN (No index support, quadratic complexity, accepts distance matrices)
References
 ^ Martin Ester, HansPeter Kriegel, Jörg Sander, Xiaowei Xu (1996). "A densitybased algorithm for discovering clusters in large spatial databases with noise". In Evangelos Simoudis, Jiawei Han, Usama M. Fayyad. Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (KDD96). AAAI Press. pp. 226–231. ISBN 1577350049. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.71.1980.
 ^ [1] Most cited data mining articles according to Microsoft academic search; DBSCAN is on rank 24, when accessed on: 4/18/2010
 ^ Sander, Jörg; Ester, Martin; Kriegel, HansPeter; Xu, Xiaowei (1998). "DensityBased Clustering in Spatial Databases: The Algorithm GDBSCAN and Its Applications". Data Mining and Knowledge Discovery (Berlin: SpringerVerlag) 2 (2): 169–194. doi:10.1023/A:1009745219419. http://www.springerlink.com/content/n22065n21n1574k6.
 ^ Sander, Jörg (1998). Generalized DensityBased Clustering for Spatial Data Mining. München: Herbert Utz Verlag. ISBN 3896754696.
Further reading
 Domenica Arlia, Massimo Coppola. "Experiments in Parallel Clustering with DBSCAN". EuroPar 2001: Parallel Processing: 7th International EuroPar Conference Manchester, UK August 28–31, 2001, Proceedings. Springer Berlin.
Categories: Data clustering algorithms
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