The document discusses several density-based and grid-based clustering algorithms. DBSCAN is described as a density-based method that forms clusters as maximal sets of density-connected points. OPTICS extends DBSCAN to produce a special ordering of the database with respect to density-based clustering structure. DENCLUE uses density functions to allow mathematically describing arbitrarily shaped clusters. Grid-based methods like STING, WaveCluster, and CLIQUE partition space into a grid structure to perform fast clustering.

1. Clustering
Density and Grid Based
1

2. 2
Density based methods
 Clusters – dense regions of objects
 Low density regions – Noise
 DBSCAN
 Density Based Spatial Clustering of Applications with
Noise
 OPTICS
 Ordering Points To Identify the Clustering Structure
 DENCLUE
 DENsity Based CLUstEring

3. 3
DBSCAN
 Cluster – maximal set of density connected points
 Grows regions with sufficiently high density into clusters
 ε-neighborhood
 MinPts and Core object
 Directly Density Reachable
 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
p
q
MinPts = 5

4. 4
DBSCAN
 Density Reachable
 An object p is density reachable from q, if there is
a chain of objects p1, …pn, p1=q and pn=p such that
pi+1 is directly density reachable from pi
p
q
p1

5. 5
DBSCAN
 Density Connected
An object p is density connected to object
q if there is an object o such that both p and q are
density reachable from o.
p q
o

6. 6
DBSCAN
 Arbitrarily select a point p
 Retrieve all points density-reachable from p
 If p is a core point, a cluster is formed.
 If p is a border point, no points are density-reachable
from p, then DBSCAN visits the next point of the
database.
 Continue the process until all of the points have been
processed.
 Complexity : O(n log n) / O(n2
)

7. 7
OPTICS: A Cluster-Ordering Method
 OPTICS: Ordering Points To Identify the Clustering
Structure
 Produces a special order of the database with respect
to its density-based clustering structure
 Good for both automatic and interactive cluster
analysis, including finding intrinsic clustering structure
 Can be represented graphically or using visualization
techniques

8. OPTICS
 In DBSCAN, for a constant MinPts value, density based
clusters with respect to a higher density (lower value of
ε) are completely contained in lower density sets.
 DBSCAN is extended so that Objects are processed in a
specific order.
 Selects an object that is density-reachable with respect to lowest
ε value
 Core distance of an object p : smallest ε’ value that makes {p} a
core object
 Reachability distance of an object q with respect to p = max
(core-distance of p, d(p,q))
8

9. OPTICS
 Complexity : O(n log n)
9

10. 10
ε
ε
Reachability-
distance
Cluster-order
of the objects
undefined
ε ‘
OPTICS

11. 11
DENCLUE: using density functions
 DENsity-based CLUstEring
 Major features
 Solid mathematical foundation
 Good for data sets with large amounts of noise
 Allows a compact mathematical description of arbitrarily
shaped clusters in high-dimensional data sets
 Significantly faster than existing algorithm (faster than
DBSCAN by a factor of up to 45)
 But needs a large number of parameters

12. 12
 Influence function: describes the impact of a data point within its
neighborhood.
 x, y – objects in Fd
– d-dimensional input space
 Influence of object y on x is:
 Can be determined by distance:
 Overall density of the data space can be calculated as the sum of
the influence function of all data points.
 Clusters can be determined mathematically by identifying density
attractors.
 Density attractors are local maximal of the overall density function.
DENCLUE
),()( yxfxf B
y
B =
otherwise1or),(0),( σ>= yxdifyxfsquare
f x y eGaussian
d x y
( , )
( , )
=
−
2
2
2σ

13. 13
 Density attractor – Local maxima of overall density
function
 A point x is said to be density attracted to a density
attractor x* if there exists a set of points x0, x1,..xk
such that x0 = x and xk =x* and the gradient of xi-1 is
in the direction of xi
DENCLUE

14. DENCLUE
 Center defined clusters
 For a density attractor x* - a subset of points that are
density attracted by x* and where density function x* is
no less than threshold ξ
 Others are outliers
 Arbitrary shape cluster
 Set of density attractors and set of Cs
 There should be a path from each density attractor to
another where density function value for each point is
no less that ξ
14

15. DENCLUE
15

16. 16
Grid Based Methods
 Uses a Multi-resolution grid data structure
 Quantizes space into a finite number of cells
that form a grid structure
 Fast processing time
 STING
 WaveCluster
 CLIQUE – CLustering In QUEst

17. 17
STING
 STatistical Information Grid
 Spatial area is divided into rectangular cells
 Several levels of cells – at different levels of
resolution
 High level cell is partitioned into several lower
level cells
 Statistical attributes are stored in cell
 Mean, Maximum, Minimum

18. 18
STING

19. 19
STING
 Parameters of higher level cells are computed
from those at lower levels
 To answer queries
 Identify level
 Estimate cell’s relevance to query
 Process relevant cells at lower levels
 Continue to lowest level

20. 20
STING
 Computation is query independent
 Parallel processing – supported
 Data is processed in a single pass
 Quality depends on granularity

21. 21
WaveCluster
 A multi-resolution clustering approach which applies
wavelet transform to the feature space
 A wavelet transform is a signal processing technique
that decomposes a signal into different frequency sub-
band.
 Both grid-based and density-based
 Input parameters:
 # of grid cells for each dimension
 the wavelet, and the # of applications of wavelet
transform.

22. 22
WaveCluster
 Using wavelet transform to find clusters
 Summarises the data by imposing a multidimensional
grid structure onto data space
 These multidimensional spatial data objects are
represented in a n-dimensional feature space
 Apply wavelet transform on feature space to find the
dense regions in the feature space
 Apply wavelet transform multiple times which result in
clusters at different scales from fine to coarse

23. 23
Quantization

24. 24
Transformation

25. 25
WaveCluster
 Reasons for using Wavelet transformation in clustering
 Unsupervised clustering
It uses filters to emphasize region where points cluster, but
simultaneously to suppress weaker information in their boundary
 Effective removal of outliers
 Multi-resolution
 Cost efficiency
 Major features:
 Complexity O(N)
 Detect arbitrary shaped clusters at different scales
 Not sensitive to noise, not sensitive to input order
 Only applicable to low dimensional data

26. 26
CLIQUE (Clustering In QUEst)
 Automatically identifying subspaces of a high dimensional data space
that allow better clustering than original space
 CLIQUE can be considered as both density-based and grid-based
 It partitions each dimension into the same number of equal length
interval
 It partitions an m-dimensional data space into non-overlapping
rectangular units
 A unit is dense if the fraction of total data points contained in the unit
exceeds the input model parameter
 A cluster is a maximal set of connected dense units within a
subspace