Density-Based Clustering Methods Overview

 
1
 
Clustering Part2 continued
 
1.
BIRCH 
skipped
2.
Density-based Clustering --- 
DBSCAN
 and 
DENCLUE
3.
GRID-based Approaches --- STING and CLIQUE
4.
SOM 
skipped
5.
Cluster Validation 
other set of transparencies
6.
Outlier/Anomaly Detection 
other set of
transparencies
 
Slides in red will be used for Part1.
 
2
 
BIRCH (1996)
 
Birch: Balanced Iterative Reducing and Clustering using
Hierarchies,  by Zhang, Ramakrishnan, Livny (SIGMOD
96)
Incrementally construct a CF (Clustering Feature) tree, a
hierarchical data structure for multiphase clustering
Phase 1: scan DB to build an initial in-memory CF tree (a
multi-level compression of the data that tries to preserve
the inherent clustering structure of the data)
Phase 2: use an arbitrary clustering algorithm to cluster
the leaf nodes of the CF-tree
Scales linearly
: finds a good clustering with a single scan
and improves the quality with a few additional scans
Weakness:
 handles only numeric data, and sensitive to the
order of the data record.
 
3
 
Clustering Feature Vector
CF = (5, (16,30),(54,190))
 
(3,4)
(2,6)
(4,5)
(4,7)
(3,8)
 
4
 
CF Tree
 
CF
1
 
child
1
 
CF
3
 
child
3
 
CF
2
 
child
2
 
CF
5
 
child
5
 
CF
1
 
CF
2
 
CF
6
 
prev
 
next
 
CF
1
 
CF
2
 
CF
4
 
prev
 
next
 
B = 7
L = 6
 
Root
 
Non-leaf node
 
Leaf node
 
Leaf node
 
5
 
Chapter 8. 
Cluster Analysis
 
What is Cluster Analysis?
Types of Data in Cluster Analysis
A Categorization of Major Clustering Methods
Partitioning Methods
Hierarchical Methods
Density-Based Methods
Grid-Based Methods
Model-Based Clustering Methods
Outlier Analysis
Summary
 
6
 
Density-Based Clustering Methods
 
Clustering based on density (local cluster criterion),
such as density-connected points or based on an
explicitly constructed density function
Major features:
Discover clusters of arbitrary shape
Handle noise
Usually one scan
Need density estimation parameters
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)
 
7
 
DBSCAN
 
Two parameters
:
Eps
: Maximum radius of the neighbourhood
MinPts
: Minimum number of points in an Eps-
neighbourhood of that point
N
Eps
(p)
:
 
{q belongs to D | dist(p,q) <= Eps}
Directly density-reachable
: 
A point 
p
 is directly density-
reachable from a point 
q
 wrt. 
Eps
, 
MinPts
 if
1) 
p
 belongs to 
N
Eps
(q)
2) core point condition:
              
|
N
Eps
 (q)
|
 >= 
MinPts
 
8
 
Density-Based Clustering: Background (II)
 
Density-reachable:
A point 
p
 is density-reachable from
a point 
q
 wrt. 
Eps
, 
MinPts
 if there
is a chain of points 
p
1
, 
, 
p
n
, 
p
1
 =
q
, 
p
n
 = 
p
 such that 
p
i+1
 is directly
density-reachable from 
p
i
Density-connected
A point 
p
 is density-connected to a
point 
q
 wrt. 
Eps
, 
MinPts
 if there is
a point 
o 
such that both, 
p
 and 
q
are density-reachable from 
o
 wrt.
Eps
 and 
MinPts
.
 
p
 
q
 
p
1
 
9
 
DBSCAN: Density Based Spatial
Clustering of Applications with Noise
 
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
 
Density reachable
from core point
 
Not density reachable
from core point
 
10
 
DBSCAN: The Algorithm
 
Arbitrary select a point 
p
Retrieve all points density-reachable from 
p
 wrt 
Eps
and 
MinPts
.
If 
p
 is a core point, a cluster is formed.
If 
p
 ia not a core 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.
 
11
 
DENCLUE: using density functions
 
DENsity-based CLUstEring by Hinneburg & Keim  (KDD
98)
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
Significant faster than existing algorithm (faster than
DBSCAN by a factor of up to 45)
But needs a large number of parameters
 
12
 
 
Influence function: describes the impact of a data point within its
neighborhood.
Overall density of the data space can be calculated as the sum of
the influences of all data points.
Clusters can be determined mathematically by identifying density
attractors; object that are associated with the same density
attractor belong to the same cluster
Density attractors are local maximal of the overall density function.
Uses grid-cells to speed up computations; only data points in
neighboring grid-cells are used to determine the density for a
point.
 
Denclue: Technical Essence
 
13
 
DENCLUE Influence Function and its Gradient
 
Example
 
 
14
 
Example: Density Computation
 
D={x1,x2,x3,x4}
 
f
D
Gaussian
(x)= influence(x1) + influence(x2) + influence(x3) +
                   influence(x4)=0.04+0.06+0.08+0.6=0.78
 
x1
 
x2
 
x3
 
x4
 
x
 
0.6
 
0.08
 
0.06
 
0.04
 
y
 
Remark
: the density value of y would be larger than the one for x
 
Example Non-Parametric DE in R
 
Demo Rcode:
setwd("C:\\Users\\C. Eick\\Desktop")
a<-read.csv("c8.csv")
require("spatstat")
require("ppp")
d<-data.frame(a=a[,1],b=a[,2],c=a[,3])
plot(d$a,d$b)
w <- owin(poly=list (list(x=c(0,530,701,640,0),y=c(0,42,20,400,420)),
list(x=c(320,430,310), y=c(215,200,190)),list(x=c(10,70,170,20), y=c(200,220,170, 175))) )
z<-ppp(d[,1],d[,2],window=w, marks=factor(d[,3]))
plot(z)
summary(z)
q<-quadratcount(z, nx=12,ny=10)
plot(q)
den<-density(z, sigma=80)
plot(den)
den<-density(z, sigma=30)
plot(den)
den<-density(z, sigma=15)
plot(den)
den<-density(z, sigma=12)
plot(den)
den<-density(z, sigma=10)
plot(den)
den<-density(z, sigma=4)
plot(den)
documentation for function ‘density’:
http://127.0.0.1:28030/library/spatstat/html/density.ppp.html
 
15
 
16
 
Density Attractor
 
17
 
Examples of DENCLUE Clusters
 
18
 
Basic Steps DENCLUE Algorithms
 
1.
Determine density attractors
2.
Associate data objects with density
attractors using hill climbing (
 initial
clustering)
3.
Merge the initial clusters further relying
on a hierarchical clustering approach
(optional)
 
19
 
Density-based Clustering: Pros and Cons
 
+: can discover clusters of arbitrary shape
+: not sensitive to outliers and supports outlier
detection
+: can handle noise
+
: medium algorithm complexities
: finding good density estimation parameters is
frequently difficult; more difficult to use than K-means.
: usually, do not do well in clustering high-dimensional
datasets.
 
: cluster models are not well understood (yet)
 
20
 
Chapter 8. 
Cluster Analysis
 
What is Cluster Analysis?
Types of Data in Cluster Analysis
A Categorization of Major Clustering Methods
Partitioning Methods
Hierarchical Methods
Density-Based Methods
Grid-Based Methods
Model-Based Clustering Methods
Outlier Analysis
Summary
 
21
 
Steps of Grid-based Clustering Algorithms
 
Basic Grid-based Algorithm
1.
Define a set of grid-cells
2.
Assign objects to the appropriate grid cell and
compute the density of each cell.
3.
Eliminate cells, whose density is below a
certain threshold 
.
4.
Form clusters from contiguous (adjacent)
groups of dense cells (usually minimizing a
given objective function).
 
22
 
Advantages of Grid-based Clustering
Algorithms
 
fast:
No distance computations
Clustering is performed on summaries and not
individual objects; complexity is usually O(#-
populated-grid-cells) and not O(#objects)
Easy to determine which clusters are
neighboring
Shapes are limited to union of rectangular grid-
cells
 
23
 
Grid-Based Clustering Methods
 
Several interesting methods (in addition to the basic grid-
based algorithm)
STING 
(a STatistical INformation Grid approach) by
Wang, Yang and Muntz (1997)
CLIQUE
: Agrawal, et al. (SIGMOD
98)
 
24
 
STING: A Statistical Information
Grid Approach
 
Wang, Yang and Muntz (VLDB’97)
The spatial area area is divided into rectangular cells
There are several levels of cells corresponding to different
levels of resolution
 
25
 
STING: A Statistical Information
Grid Approach (2)
 
Main contribution of STING is the proposal of a data structure that
can be used for many purposes (e.g. SCMRG, BIRCH kind of uses
it)
The data structure is used to 
form clusters based on queries
Each cell at a high level is partitioned into a number of smaller
cells in the next lower level
Statistical info of each cell  is calculated and stored beforehand
and is used to answer queries
Parameters of higher level cells can be easily calculated from
parameters of lower level cell
count
, 
mean
, 
s
, 
min
, 
max
type of distribution—normal, 
uniform
, etc.
Use a top-down approach to answer spatial data queries
Clusters are formed by merging cells that match a given query
description  (
 next slide)
 
26
 
STING: Query Processing(3)
 
Used a top-down approach to answer spatial data queries
1.
Start from a pre-selected layer—typically with a small number of
cells
2.
From the pre-selected layer until you reach the bottom layer do
the following:
For each cell in the current level compute the confidence interval
indicating a cell’s relevance to a given query;
If it is relevant, include the cell in a cluster
If it irrelevant, remove cell from further consideration
otherwise, look for relevant cells at the next lower layer
3.
Combine relevant cells into relevant regions (based on grid-
neighborhood) and return the so obtained clusters as your
answers.
 
27
 
STING: A Statistical Information
Grid Approach (3)
 
Advantages:
Query-independent, easy to parallelize, incremental
update
O(K),
 where 
K
 is the number of grid cells at the
lowest level
Can be used in conjunction with a grid-based
clustering algorithm
Disadvantages:
All the cluster boundaries are either horizontal or
vertical, and no diagonal boundary is detected
 
28
 
Subspace Clustering
 
Clustering in very high-dimensional spaces is very difficult
High dimensional attribute spaces tend to be sparse
it is
hard to find any clusters
It is very difficult to create summaries from clusters in very
difficult
This creates the motivation for subspace clustering:
Find interesting subspaces (areas that are dense with
respect to the attributes belonging to the subspace)
Find clusters for each interesting
Remark: multiple, overlapping clusters might be
obtained; basically one clustering for each subspace.
 
29
 
CLIQUE (Clustering In QUEst)
 
Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98).
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
 
30
 
CLIQUE: The Major Steps
 
Partition the data space and find the number of points
that lie inside each cell of the partition.
Identify the subspaces that contain clusters using the
Apriori principle
Identify clusters
:
Determine dense units in all subspaces of interests
Determine connected dense units in all subspaces of
interests.
 
31
 
Salary
(10,000)
 
20
 
30
 
40
 
50
 
60
 
age
 
5
 
4
 
3
 
1
 
2
 
6
 
7
 
0
 
 = 3
 
32
 
Strength and Weakness of 
CLIQUE
 
Strength
It 
automatically
 finds subspaces of the
 
highest
dimensionality
 such that high density clusters exist in
those subspaces
It is 
insensitive
 to the order of records in input and does
not presume some canonical data distribution
It scales
 linearly
 with the size of input and has good
scalability as the number of dimensions in the data
increases
Weakness
The accuracy of the clustering result may be degraded at
the expense of simplicity of the method
Quite expensive
 
33
 
Self-organizing feature maps (SOMs)
 
Clustering is also performed by having several
units competing for the current object
The unit whose weight vector is closest to the
current object wins
The winner and its neighbors learn by having
their weights adjusted
SOMs are believed to resemble processing that
can occur in the brain
Useful for visualizing high-dimensional data in
2- or 3-D space
 
34
 
References (1)
 
R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of
high dimensional data for data mining applications. SIGMOD'98
M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973.
M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander.  Optics:  Ordering points to identify
the clustering structure, SIGMOD’99.
P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scietific, 1996
M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering
clusters in large spatial databases. KDD'96.
M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial databases:
Focusing techniques for efficient class identification. SSD'95.
D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine Learning,
2:139-172, 1987.
D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach based
on dynamic systems. In Proc. VLDB’98.
S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large
databases. SIGMOD'98.
A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.
 
35
 
References (2)
 
L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster
Analysis. John Wiley & Sons, 1990.
E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets.
VLDB’98.
G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to
Clustering. John Wiley and Sons, 1988.
P. Michaud. Clustering techniques. Future Generation Computer systems, 13, 1997.
R. Ng and J. Han. Efficient and effective clustering method for spatial data mining.
VLDB'94.
E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large
data sets. Proc. 1996 Int. Conf. on Pattern Recognition, 101-105.
G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution
clustering approach for very large spatial databases. VLDB’98.
W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial
Data Mining, VLDB’97.
T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering method
for very large databases. SIGMOD'96.
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Density-based clustering methods focus on clustering based on density criteria to discover clusters of arbitrary shape while handling noise efficiently. Major features include the ability to work with one scan, require density estimation parameters, and handle clusters of any shape. Notable studies in this field include DBSCAN, OPTICS, DENCLUE, and CLIQUE.

  • Density-based Clustering
  • DBSCAN
  • OPTICS
  • DENCLUE
  • CLIQUE

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  1. Clustering Part2 continued BIRCH skipped Density-based Clustering --- DBSCAN and DENCLUE GRID-based Approaches --- STING and CLIQUE SOM skipped Cluster Validation other set of transparencies Outlier/Anomaly Detection other set of transparencies 1. 2. 3. 4. 5. 6. Slides in red will be used for Part1. 1 Han/Eick: Clustering II

  2. BIRCH (1996) Birch: Balanced Iterative Reducing and Clustering using Hierarchies, by Zhang, Ramakrishnan, Livny (SIGMOD 96) Incrementally construct a CF (Clustering Feature) tree, a hierarchical data structure for multiphase clustering Phase 1: scan DB to build an initial in-memory CF tree (a multi-level compression of the data that tries to preserve the inherent clustering structure of the data) Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes of the CF-tree Scales linearly: finds a good clustering with a single scan and improves the quality with a few additional scans Weakness: handles only numeric data, and sensitive to the order of the data record. 2 Han/Eick: Clustering II

  3. Clustering Feature Vector Clustering Feature:CF = (N, LS, SS) N: Number of data points LS: Ni=1=Xi SS: Ni=1=Xi2 CF = (5, (16,30),(54,190)) (3,4) (2,6) (4,5) (4,7) (3,8) 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 3 Han/Eick: Clustering II

  4. CF Tree Root CF1 CF2 CF3 CF6 B = 7 child1 child2 child3 child6 L = 6 Non-leaf node CF1 CF2 CF3 CF5 child1 child2 child3 child5 Leaf node Leaf node prev next prev next CF1CF2 CF6 CF1CF2 CF4 4 Han/Eick: Clustering II

  5. Chapter 8. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 5 Han/Eick: Clustering II

  6. Density-Based Clustering Methods Clustering based on density (local cluster criterion), such as density-connected points or based on an explicitly constructed density function Major features: Discover clusters of arbitrary shape Handle noise Usually one scan Need density estimation parameters 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) 6 Han/Eick: Clustering II

  7. DBSCAN Two parameters: Eps: Maximum radius of the neighbourhood MinPts: Minimum number of points in an Eps- neighbourhood of that point NEps(p): {q belongs to D | dist(p,q) <= Eps} Directly density-reachable: A point p is directly density- reachable from a point q wrt. Eps, MinPts if 1) p belongs to NEps(q) p MinPts = 5 2) core point condition: |NEps (q)| >= MinPts q Eps = 1 cm 7 Han/Eick: Clustering II

  8. Density-Based Clustering: Background (II) Density-reachable: p A point p is density-reachable from a point q wrt. Eps, MinPts if there is a chain of points p1, , pn, p1 = q, pn = p such that pi+1 is directly density-reachable from pi Density-connected p1 q p q A point p is density-connected to a point q wrt. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o wrt. Eps and MinPts. o 8 Han/Eick: Clustering II

  9. DBSCAN: Density Based Spatial Clustering of Applications with Noise 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 Not density reachable from core point Outlier Density reachable from core point Border Eps = 1cm Core MinPts = 5 9 Han/Eick: Clustering II

  10. DBSCAN: The Algorithm Arbitrary select a point p Retrieve all points density-reachable from p wrt Eps and MinPts. If p is a core point, a cluster is formed. If p ia not a core 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. 10 Han/Eick: Clustering II

  11. DENCLUE: using density functions DENsity-based CLUstEring by Hinneburg & Keim (KDD 98) 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 Significant faster than existing algorithm (faster than DBSCAN by a factor of up to 45) But needs a large number of parameters 11 Han/Eick: Clustering II

  12. Denclue: Technical Essence Influence function: describes the impact of a data point within its neighborhood. Overall density of the data space can be calculated as the sum of the influences of all data points. Clusters can be determined mathematically by identifying density attractors; object that are associated with the same density attractor belong to the same cluster Density attractors are local maximal of the overall density function. Uses grid-cells to speed up computations; only data points in neighboring grid-cells are used to determine the density for a point. 12 Han/Eick: Clustering II

  13. DENCLUE Influence Function and its Gradient Example f Gaussian 2 d x y ( , ) 2 = x y ( , ) e 2 2 ( , ) d x 2 x i = N = D 2 ( ) f x e Gaussian 1 i = 2 ( , ) d x 2 x i N = D 2 ( , ) ( ) f x x x x e Gaussian i i 1 i 13 Han/Eick: Clustering II

  14. Example: Density Computation D={x1,x2,x3,x4} fDGaussian(x)= influence(x1) + influence(x2) + influence(x3) + influence(x4)=0.04+0.06+0.08+0.6=0.78 2 ( , ) d x y = 2 inf ( , ) luence x y e 2 x1 x3 0.04 0.08 y x2 x4 0.6 0.06 x Remark: the density value of y would be larger than the one for x 14 Han/Eick: Clustering II

  15. Example Non-Parametric DE in R Demo Rcode: setwd("C:\\Users\\C. Eick\\Desktop") a<-read.csv("c8.csv") require("spatstat") require("ppp") d<-data.frame(a=a[,1],b=a[,2],c=a[,3]) plot(d$a,d$b) w <- owin(poly=list (list(x=c(0,530,701,640,0),y=c(0,42,20,400,420)), list(x=c(320,430,310), y=c(215,200,190)),list(x=c(10,70,170,20), y=c(200,220,170, 175))) ) z<-ppp(d[,1],d[,2],window=w, marks=factor(d[,3])) plot(z) summary(z) q<-quadratcount(z, nx=12,ny=10) plot(q) den<-density(z, sigma=80) plot(den) den<-density(z, sigma=30) plot(den) den<-density(z, sigma=15) plot(den) den<-density(z, sigma=12) plot(den) den<-density(z, sigma=10) plot(den) den<-density(z, sigma=4) plot(den) documentation for function density : http://127.0.0.1:28030/library/spatstat/html/density.ppp.html 15 Han/Eick: Clustering II

  16. Density Attractor 16 Han/Eick: Clustering II

  17. Examples of DENCLUE Clusters 17 Han/Eick: Clustering II

  18. Basic Steps DENCLUE Algorithms 1. Determine density attractors 2. Associate data objects with density attractors using hill climbing ( initial clustering) 3. Merge the initial clusters further relying on a hierarchical clustering approach (optional) 18 Han/Eick: Clustering II

  19. Density-based Clustering: Pros and Cons +: can discover clusters of arbitrary shape +: not sensitive to outliers and supports outlier detection +: can handle noise + : medium algorithm complexities : finding good density estimation parameters is frequently difficult; more difficult to use than K-means. : usually, do not do well in clustering high-dimensional datasets. : cluster models are not well understood (yet) 19 Han/Eick: Clustering II

  20. Chapter 8. Cluster Analysis What is Cluster Analysis? Types of Data in Cluster Analysis A Categorization of Major Clustering Methods Partitioning Methods Hierarchical Methods Density-Based Methods Grid-Based Methods Model-Based Clustering Methods Outlier Analysis Summary 20 Han/Eick: Clustering II

  21. Steps of Grid-based Clustering Algorithms Basic Grid-based Algorithm Define a set of grid-cells Assign objects to the appropriate grid cell and compute the density of each cell. Eliminate cells, whose density is below a certain threshold . Form clusters from contiguous (adjacent) groups of dense cells (usually minimizing a given objective function). 1. 2. 3. 4. 21 Han/Eick: Clustering II

  22. Advantages of Grid-based Clustering Algorithms fast: No distance computations Clustering is performed on summaries and not individual objects; complexity is usually O(#- populated-grid-cells) and not O(#objects) Easy to determine which clusters are neighboring Shapes are limited to union of rectangular grid- cells 22 Han/Eick: Clustering II

  23. Grid-Based Clustering Methods Several interesting methods (in addition to the basic grid- based algorithm) STING (a STatistical INformation Grid approach) by Wang, Yang and Muntz (1997) CLIQUE: Agrawal, et al. (SIGMOD 98) 23 Han/Eick: Clustering II

  24. STING: A Statistical Information Grid Approach Wang, Yang and Muntz (VLDB 97) The spatial area area is divided into rectangular cells There are several levels of cells corresponding to different levels of resolution 24 Han/Eick: Clustering II

  25. STING: A Statistical Information Grid Approach (2) Main contribution of STING is the proposal of a data structure that can be used for many purposes (e.g. SCMRG, BIRCH kind of uses it) The data structure is used to form clusters based on queries Each cell at a high level is partitioned into a number of smaller cells in the next lower level Statistical info of each cell is calculated and stored beforehand and is used to answer queries Parameters of higher level cells can be easily calculated from parameters of lower level cell count, mean, s, min, max type of distribution normal, uniform, etc. Use a top-down approach to answer spatial data queries Clusters are formed by merging cells that match a given query description ( next slide) 25 Han/Eick: Clustering II

  26. STING: Query Processing(3) Used a top-down approach to answer spatial data queries Start from a pre-selected layer typically with a small number of cells From the pre-selected layer until you reach the bottom layer do the following: For each cell in the current level compute the confidence interval indicating a cell s relevance to a given query; If it is relevant, include the cell in a cluster If it irrelevant, remove cell from further consideration otherwise, look for relevant cells at the next lower layer Combine relevant cells into relevant regions (based on grid- neighborhood) and return the so obtained clusters as your answers. 1. 2. 3. 26 Han/Eick: Clustering II

  27. STING: A Statistical Information Grid Approach (3) Advantages: Query-independent, easy to parallelize, incremental update O(K), where K is the number of grid cells at the lowest level Can be used in conjunction with a grid-based clustering algorithm Disadvantages: All the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected 27 Han/Eick: Clustering II

  28. Subspace Clustering Clustering in very high-dimensional spaces is very difficult High dimensional attribute spaces tend to be sparse it is hard to find any clusters It is very difficult to create summaries from clusters in very difficult This creates the motivation for subspace clustering: Find interesting subspaces (areas that are dense with respect to the attributes belonging to the subspace) Find clusters for each interesting Remark: multiple, overlapping clusters might be obtained; basically one clustering for each subspace. 28 Han/Eick: Clustering II

  29. CLIQUE (Clustering In QUEst) Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD 98). 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 29 Han/Eick: Clustering II

  30. CLIQUE: The Major Steps Partition the data space and find the number of points that lie inside each cell of the partition. Identify the subspaces that contain clusters using the Apriori principle Identify clusters: Determine dense units in all subspaces of interests Determine connected dense units in all subspaces of interests. 30 Han/Eick: Clustering II

  31. Vacation (10,000) Salary (week) 6 7 6 7 5 5 4 4 3 3 1 2 1 2 age age 0 0 20 30 40 50 60 20 30 40 50 60 = 3 Vacation 30 50 age 31 Han/Eick: Clustering II

  32. Strength and Weakness of CLIQUE Strength It automatically finds subspaces of the highest dimensionality such that high density clusters exist in those subspaces It is insensitive to the order of records in input and does not presume some canonical data distribution It scales linearly with the size of input and has good scalability as the number of dimensions in the data increases Weakness The accuracy of the clustering result may be degraded at the expense of simplicity of the method Quite expensive 32 Han/Eick: Clustering II

  33. Self-organizing feature maps (SOMs) Clustering is also performed by having several units competing for the current object The unit whose weight vector is closest to the current object wins The winner and its neighbors learn by having their weights adjusted SOMs are believed to resemble processing that can occur in the brain Useful for visualizing high-dimensional data in 2- or 3-D space 33 Han/Eick: Clustering II

  34. References (1) R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. SIGMOD'98 M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973. M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to identify the clustering structure, SIGMOD 99. P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scietific, 1996 M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters in large spatial databases. KDD'96. M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial databases: Focusing techniques for efficient class identification. SSD'95. D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine Learning, 2:139-172, 1987. D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach based on dynamic systems. In Proc. VLDB 98. S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large databases. SIGMOD'98. A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988. 34 Han/Eick: Clustering II

  35. References (2) L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons, 1990. E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets. VLDB 98. G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to Clustering. John Wiley and Sons, 1988. P. Michaud. Clustering techniques. Future Generation Computer systems, 13, 1997. R. Ng and J. Han. Efficient and effective clustering method for spatial data mining. VLDB'94. E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large data sets. Proc. 1996 Int. Conf. on Pattern Recognition, 101-105. G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution clustering approach for very large spatial databases. VLDB 98. W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial Data Mining, VLDB 97. T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering method for very large databases. SIGMOD'96. 35 Han/Eick: Clustering II

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