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Prof. Pier Luca Lanzi
Density Based Clustering
Data Mining andText Mining (UIC 583 @ Politecnico di Milano)
Prof. Pier Luca Lanzi
Prof. Pier Luca Lanzi
Prof. Pier Luca Lanzi
What is density-based clustering?
•  Clustering based on density (local cluster criterion), 
such as density-connected points
•  Major features:
§ Discover clusters of arbitrary shape
§ Handle noise
§ One scan
§ Need density parameters as termination condition
•  Several interesting studies:
§ DBSCAN: Ester, et al. (KDD’96)
§ OPTICS: Ankerst, et al (SIGMOD’99).
§ DENCLUE: Hinneburg  D. Keim (KDD’98)
§ CLIQUE: Agrawal, et al. (SIGMOD’98) (more grid-based)
4
Prof. Pier Luca Lanzi
DBSCAN: Basic Concepts
•  The neighborhood within a radius ε of a given object is called
the ε-neighborhood of the object
•  If the ε-neighborhood of an object contains at least MinPts
objects, then the object is a core object
•  An object p is directly density-reachable from object q if p is
within the ε-neighborhood of q and q is a core object
•  An object p is density-reachable from object q if there is a chain
of object p1, …, pn where p_1=p and p_n=q such that pi+1 is
directly density reachable from pi
•  An object p is density-connected to q with respect to ε and
MinPts if there is an object o such that both p and q are density
reachable from o
5
Prof. Pier Luca Lanzi
DBSCAN: Basic Concepts
•  Density = number of points within a specified radius (Eps)
•  A border point has fewer than MinPts within Eps, 
but is in the neighborhood of a core point
•  A noise point is any point that is not a core point 
or a border point
•  A density-based cluster is a set of density-connected objects that
is maximal with respect to density-reachability
6
Prof. Pier Luca Lanzi
Density-Reachable  Density-Connected
• Directly density-reachable • Density-reachable
• Density-connected
p
q
p1
p q
o
p
q
MinPts = 5
Eps = 1 cm
7
Prof. Pier Luca Lanzi
DBSCAN: Core, Border, and Noise Points 8
Prof. Pier Luca Lanzi
DBSCAN 
Density Based Spatial Clustering
•  Relies on a density-based notion of cluster: A cluster is defined
as a maximal set of density-connected points
•  Discovers clusters of arbitrary shape in spatial databases with
noise
•  The Algorithm
§ Arbitrary select a point p
§ Retrieve all points density-reachable 
from p given Eps and MinPts.
§ If p is a core point, a cluster is formed.
§ If p is a border point, no points are density-reachable from p
and DBSCAN visits the next point of the database
§ Continue the process until all of the points have been
processed
9
Prof. Pier Luca Lanzi
DBSCAN: Core, Border and Noise Points
Original Points Point types: core, border
and noise
Eps = 10, MinPts = 4
10
Prof. Pier Luca Lanzi
When DBSCAN Works Well
•  Resistant to Noise
•  Can handle clusters of different shapes and sizes
Original Points Clusters
11
Prof. Pier Luca Lanzi
When DBSCAN May Fail?
•  Varying densities
•  High-dimensional data
Original Points
(MinPts=4, Eps=9.75).
(MinPts=4, Eps=9.92)
12
Prof. Pier Luca Lanzi
Clusters found in Random Data
0 0.2 0.4 0.6 0.8 1
0
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x
y
Random
Points
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
K-means
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
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0.5
0.6
0.7
0.8
0.9
1
x
y
DBSCAN
0 0.2 0.4 0.6 0.8 1
0
0.1
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y
Complete
Link
13
Prof. Pier Luca Lanzi
Density-Based Clustering in R
library(fpc)
set.seed(665544)
n - 600
x - cbind(runif(10, 0, 10)+rnorm(n, sd=0.2), runif(10, 0,
10)+rnorm(n,sd=0.2))
par(bg=grey40)
ds - dbscan(x, 0.2, showplot=1)
14
Prof. Pier Luca Lanzi
Density-Based Clustering in R
library(fpc)
set.seed(665544)
x - seq(0,6.28,0.1)
y - sin(x)
xd - x+rnorm(630,sd=0.2)
yd - y+rnorm(630,sd=0.2)
plot(xd,yd)
par(bg=grey40)
d - cbind(xd,yd)
# this works nicely since the epsilon is
# the same size of the standard deviation (0.2)
# used to generate the data
ds - dbscan(d, 0.2, showplot=1)
# this does not work so nicely
ds - dbscan(d, 0.1, showplot=1)
15
Prof. Pier Luca Lanzi
Clustering Comparisons on Sin Data 16
hierarchical clustering kmeans clustering
Prof. Pier Luca Lanzi
Clustering Comparisons on Sin Data
(k-means with 10 clusters)
17
Prof. Pier Luca Lanzi
http://en.wikibooks.org/wiki/Data_Mining_Algorithms_In_R/Clustering/Density-Based_Clustering
Software Packages

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DMTM 2015 - 09 Density Based Clustering

  • 1. Prof. Pier Luca Lanzi Density Based Clustering Data Mining andText Mining (UIC 583 @ Politecnico di Milano)
  • 4. Prof. Pier Luca Lanzi What is density-based clustering? •  Clustering based on density (local cluster criterion), such as density-connected points •  Major features: § Discover clusters of arbitrary shape § Handle noise § One scan § Need density parameters as termination condition •  Several interesting studies: § DBSCAN: Ester, et al. (KDD’96) § OPTICS: Ankerst, et al (SIGMOD’99). § DENCLUE: Hinneburg D. Keim (KDD’98) § CLIQUE: Agrawal, et al. (SIGMOD’98) (more grid-based) 4
  • 5. Prof. Pier Luca Lanzi DBSCAN: Basic Concepts •  The neighborhood within a radius ε of a given object is called the ε-neighborhood of the object •  If the ε-neighborhood of an object contains at least MinPts objects, then the object is a core object •  An object p is directly density-reachable from object q if p is within the ε-neighborhood of q and q is a core object •  An object p is density-reachable from object q if there is a chain of object p1, …, pn where p_1=p and p_n=q such that pi+1 is directly density reachable from pi •  An object p is density-connected to q with respect to ε and MinPts if there is an object o such that both p and q are density reachable from o 5
  • 6. Prof. Pier Luca Lanzi DBSCAN: Basic Concepts •  Density = number of points within a specified radius (Eps) •  A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point •  A noise point is any point that is not a core point or a border point •  A density-based cluster is a set of density-connected objects that is maximal with respect to density-reachability 6
  • 7. Prof. Pier Luca Lanzi Density-Reachable Density-Connected • Directly density-reachable • Density-reachable • Density-connected p q p1 p q o p q MinPts = 5 Eps = 1 cm 7
  • 8. Prof. Pier Luca Lanzi DBSCAN: Core, Border, and Noise Points 8
  • 9. Prof. Pier Luca Lanzi DBSCAN Density Based Spatial Clustering •  Relies on a density-based notion of cluster: A cluster is defined as a maximal set of density-connected points •  Discovers clusters of arbitrary shape in spatial databases with noise •  The Algorithm § Arbitrary select a point p § Retrieve all points density-reachable from p given Eps and MinPts. § If p is a core point, a cluster is formed. § If p is a border point, no points are density-reachable from p and DBSCAN visits the next point of the database § Continue the process until all of the points have been processed 9
  • 10. Prof. Pier Luca Lanzi DBSCAN: Core, Border and Noise Points Original Points Point types: core, border and noise Eps = 10, MinPts = 4 10
  • 11. Prof. Pier Luca Lanzi When DBSCAN Works Well •  Resistant to Noise •  Can handle clusters of different shapes and sizes Original Points Clusters 11
  • 12. Prof. Pier Luca Lanzi When DBSCAN May Fail? •  Varying densities •  High-dimensional data Original Points (MinPts=4, Eps=9.75). (MinPts=4, Eps=9.92) 12
  • 13. Prof. Pier Luca Lanzi Clusters found in Random Data 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Random Points 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y K-means 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y DBSCAN 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Complete Link 13
  • 14. Prof. Pier Luca Lanzi Density-Based Clustering in R library(fpc) set.seed(665544) n - 600 x - cbind(runif(10, 0, 10)+rnorm(n, sd=0.2), runif(10, 0, 10)+rnorm(n,sd=0.2)) par(bg=grey40) ds - dbscan(x, 0.2, showplot=1) 14
  • 15. Prof. Pier Luca Lanzi Density-Based Clustering in R library(fpc) set.seed(665544) x - seq(0,6.28,0.1) y - sin(x) xd - x+rnorm(630,sd=0.2) yd - y+rnorm(630,sd=0.2) plot(xd,yd) par(bg=grey40) d - cbind(xd,yd) # this works nicely since the epsilon is # the same size of the standard deviation (0.2) # used to generate the data ds - dbscan(d, 0.2, showplot=1) # this does not work so nicely ds - dbscan(d, 0.1, showplot=1) 15
  • 16. Prof. Pier Luca Lanzi Clustering Comparisons on Sin Data 16 hierarchical clustering kmeans clustering
  • 17. Prof. Pier Luca Lanzi Clustering Comparisons on Sin Data (k-means with 10 clusters) 17
  • 18. Prof. Pier Luca Lanzi http://en.wikibooks.org/wiki/Data_Mining_Algorithms_In_R/Clustering/Density-Based_Clustering Software Packages