©2006 Jiawei Han and Micheline Kamber, All rights reserved
www.cs.uiuc.edu/~hanj
University of Illinois at Urbana-Champaign
Department of Computer Science
Jiawei Han
— Chapter 7 —
Concepts and Techniques
Data Mining:
1
6. Density-Based Methods
5. Hierarchical Methods
4 Partitioning Methods
4.
3. A Categorization of Major Clustering Methods
2. Types of Data in Cluster Analysis
1. What is Cluster Analysis?
Chapter 7. Cluster Analysis
2
Typical applications
As a preprocessing step for other algorithms
As a stand-alone tool to get insight into data distribution
Unsupervised learning: no predefined classes
Finding similarities between data according to the characteristics
found in the data and grouping similar data objects into clusters
Cluster analysis
distance (or similarity) measures
Dissimilar to the objects in other clusters
Similar to one another within the same cluster
Cluster: a collection of data objects
What is Cluster Analysis?
3
clustered along continent faults
Earth-quake studies: Observed earth quake epicenters should be
type, value, and geographical location
City-planning: Identifying groups of houses according to their house
a high average claim cost
Insurance: Identifying groups of motor insurance policy holders with
observation database
Land use: Identification of areas of similar land use in an earth
programs
bases, and then use this knowledge to develop targeted marketing
Marketing: Help marketers discover distinct groups in their customer
Examples of Clustering Applications
4
Example
–
clustering for ontology aligment
5
Discovery of clusters with arbitrary shape
Able to deal with noise and outliers
Insensitive to order of input records
High dimensionality
Incorporation of user-specified constraints
Interpretability and usability
Minimal requirements for domain knowledge to
determine input parameters
Ability to deal with different types of attributes
Scalability
Requirements of Clustering in Data Mining
6
6. Density-Based Methods
5. Hierarchical Methods
4 Partitioning Methods
4.
3. A Categorization of Major Clustering Methods
2. Types of Data in Cluster Analysis
1. What is Cluster Analysis?
Chapter 7. Cluster Analysis
7
Dissimilarity matrix
Distance table
(one mode)
Data matrix
n objects, p attributes
(two modes)
One row represents
one object
...
x if
...
...
...
...
...
0
:
...
:
d ( n ,2 )
...
...
...
...
...
d ( 3,2 )
0
x nf
x 1f
...
 0
 d(2,1)

 d(3,1 )

 :
 d ( n ,1)
 x 11

 ...
x
 i1
 ...
x
 n1
Data Structures






... 0 
x 1p 

... 
x ip 

... 
x np 

8
Variables of mixed types
a positive measurement on a nonlinear scale (growth)
Ratio variables
ranking is important (e.g. medals(gold,silver,bronze))
Ordinal
(color/red,yellow,blue,green)
A generalization of the binary variable in that it can take more than 2 states
Nominal variables
Variables with 2 states (on/off, yes/no)
Binary variables
Continuous measurements (weight, temperature, …)
Interval-scaled variables
Type of data in clustering analysis
9
+ ... +
.
xnf )
xif − m f
zif =
sf
Calculate the standardized measurement (z-score)
m f = 1n (x1 f + x2 f
s f = 1n (| x1 f − m f | + | x2 f − m f | +...+ | xnf − m f |)
Calculate the mean absolute deviation:
where
Sometimes we need to standardize the data
Interval-valued variables
10
d(i,j) ≥ 0
d(i,i) = 0
d(i,j) = d(j,i)
d(i,j) ≤ d(i,k) + d(k,j)
Distances are normally used to measure the
similarity or dissimilarity between two data
objects
Properties
p
Distances between objects
11
d (i, j) =| x − x | + | x − x | +...+ | x − x |
i1 j1
i2 j 2
ip
jp
Manhattan distance:
d (i, j) = (| x − x |2 + | x − x |2 +...+ | x − x |2 )
i1
j1
i2
j2
ip
jp
Euclidean distance:
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp)
are two p-dimensional data objects,
Distances between objects
12
If q = 1, d is Manhattan distance
If q = 2, d is Euclidean distance
q is a positive integer
d (i, j) = q (| x − x |q + | x − x |q +...+ | x − x |q )
i1
j1
i2
j2
ip
jp
Minkowski distance:
Distances between objects
13
test); 1 is the important state, 0 the other
important than the other (e.g. outcome of disease
asymmetric binary variables: one state is more
important; 0/1
symmetric binary variables: both states are equally
Binary Variables
14
sum
a +b
c+d
p
a: number of attributes having 1 for object i and 1 for object j
b: number of attributes having 1 for object i and 0 for object j
c: number of attributes having 0 for object i and 1 for object j
d: number of attributes having 0 for object i and 0 for object j
p = a+b+c+d
1
0
1
a
b
Object i
0
c
d
sum a + c b + d
Object j
Contingency tables for Binary Variables
15
0
1
a
b
Object i
0
c
d
sum a + c b + d
1
Object j
a +b
c+d
p
sum
d (i, j ) =
b+c
a+b+c+d
Distance measure for symmetric binary
variables
16
0
a +b
c+d
p
sum
Jaccard coefficient = 1- d(i,j) =
b+c
a+b+c
sim Jaccard (i, j ) =
d (i, j ) =
1
a
b
Object i
0
c
d
sum a + c b + d
1
Object j
a
a+b+c
Distance measure for asymmetric binary
variables
17
Y
Y
Y
N
N
P
P
P
N
N
N
N
N
P
N
d ( jack , mary
) =
0 + 1
= 0 . 33
2 + 0 + 1
1 + 1
d ( jack , jim ) =
= 0 . 67
1 + 1 + 1
1 + 2
d ( jim , mary ) =
= 0 . 75
1 + 1 + 2
gender is a symmetric attribute
the remaining attributes are asymmetric binary
distance based on these
let the values Y and P be set to 1, and the value N be set to 0
Jack M
Mary F
Jim
M
N
N
N
Example Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4
Dissimilarity between Binary Variables
18
d ( i , j ) = p −p m
m: # of matches, p: total # of variables
creating a new binary variable for each of the M
nominal states
Method 2: use a large number of binary variables
Method 1: Simple matching
Nominal or Categorical Variables
19
if
compute the dissimilarity using methods for intervalscaled variables
z
r if − 1
=
M f − 1
map the range of each variable onto [0, 1] by replacing
i-th object in the f-th variable by
r iff ∈ {1,..., M f }
Can be treated like interval-scaled
replace xif by their rank
Order is important, e.g., rank
An ordinal variable can be discrete or continuous
Ordinal Variables
20
treat them as continuous ordinal data, treat their rank
as interval-scaled
yif = log(xif)
apply logarithmic transformation
treat them like interval-scaled variables—not a good
choice! (why?—the scale can be distorted)
Methods:
Ratio-scaled variable: a positive measurement on a
nonlinear scale, approximately at exponential scale,
such as AeBt or Ae-Bt
Ratio-Scaled Variables
21
δ
δ
d
p
( f )
( f )
f = 1
ij
ij
p
( f )
f = 1
ij
delta(i,j) = 0 iff (i) x-value is missing or (ii) x-values are 0 and f
asymmetric binary attribute
Σ
f is binary or nominal:
dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise
f is interval-based: use the normalized distance
f is ordinal or ratio-scaled
compute ranks rif and
z if = r if − 1
M f −1
and treat zif as interval-scaled
d (i, j ) =
Σ
A database may contain all the six types of variables
symmetric binary, asymmetric binary, nominal, ordinal, interval
and ratio
One may use a weighted formula to combine their effects:
Variables of Mixed Types
22
Cosine measure
Broad applications: information retrieval, biologic
taxonomy, etc.
Vector objects: keywords in documents, gene
features in micro-arrays, etc.
Vector Objects
23
receptor
cloning
sim(d,q) = d . q
|d| x |q|
adrenergic
Q (1,1,1)
( , , )
Doc1 (1,1,0)
Doc2 (0,1,0)
Vector model for information retrieval (simplified)
24
6. Density-Based Methods
5. Hierarchical Methods
4 Partitioning Methods
4.
3. A Categorization of Major Clustering Methods
2. Types of Data in Cluster Analysis
1. What is Cluster Analysis?
Chapter 7. Cluster Analysis
25
Typical methods: DBSCAN, OPTICS, DenClue
Based on connectivity and density functions
Density-based approach:
Typical methods: Diana, Agnes, BIRCH, ROCK, CAMELEON
some criterion
Create a hierarchical decomposition of the set of data (or objects) using
Hierarchical approach:
Typical methods: k-means, k-medoids, CLARANS
e.g., minimizing the sum of square errors
Construct various partitions and then evaluate them by some criterion,
Partitioning approach:
Major Clustering Approaches (I)
26
Typical methods: COD (obstacles), constrained clustering
Clustering by considering user-specified or application-specific constraints
User-guided or constraint-based:
Typical methods: pCluster
Based on the analysis of frequent patterns
Frequent pattern-based:
Typical methods: EM, SOM, COBWEB
fit of that model to each other
A model is hypothesized for each of the clusters and tries to find the best
Model-based:
Typical methods: STING, WaveCluster, CLIQUE
based on a multiple-level granularity structure
Grid-based approach:
Major Clustering Approaches (II)
27
Medoid: one chosen, centrally located object in the cluster
dis(Ki, Kj) = dis(Mi, Mj)
Medoid: distance between the medoids of two clusters, i.e.,
dis(Ki, Kj) = dis(Ci, Cj)
Centroid: distance between the centroids of two clusters, i.e.,
element in the other, i.e., dis(Ki, Kj) = avg(tip, tjq)
Average: avg di
distance
t
b
between
t
an element
l
t iin one cluster
l t and
d an
and an element in the other, i.e., dis(Ki, Kj) = max(tip, tjq)
Complete link: largest distance between an element in one cluster
and an element in the other, i.e., dis(Ki, Kj) = min(tip, tjq)
Single link: smallest distance between an element in one cluster
Typical Alternatives to Calculate the Distance
between Clusters
28
Σ N (t − cm ) 2
Rm = i =1 ip
N
Σ N Σ N (t − t ) 2
Dm = i =1 i =1 ip iq
N ( N −1)
all pairs of points in the cluster
Diameter: square root of average mean squared distance between
cluster to its centroid
N
Radius: square root of average distance from any point of the
)
Cm =
ip
Centroid: the “middle” of a cluster
ΣiN= 1(t
Centroid, Radius and Diameter of a
Cluster (for numerical data sets)
29
6. Density-Based Methods
5. Hierarchical Methods
4 Partitioning Methods
4.
3. A Categorization of Major Clustering Methods
2. Types of Data in Cluster Analysis
1. What is Cluster Analysis?
Chapter 7. Cluster Analysis
30
k-medoids or PAM (Partition around medoids) (Kaufman &
Rousseeuw’87): Each cluster is represented by one of the objects
in the cluster
k-means (MacQueen’67): Each cluster is represented by the center
of the cluster
Heuristic methods: k-means and k-medoids algorithms
Global optimal: exhaustively enumerate all partitions
Given a k, find a partition of k clusters that optimizes the chosen
partitioning criterion
Σ km=1Σtmi∈Km (Cm − tmi ) 2
Partitioning method: Construct a partition of a database D of n objects
into a set of k clusters, s.t., min sum of squared distance
Partitioning Algorithms: Basic Concept
31
2. Repeat
Update cluster means (calculate mean value of the
objects for each cluster)
Until no change
Assign each object to the cluster to which the
object is most similar based on mean values of the
objects in the cluster
1. arbitrarily choose k objects as initial cluster centers
Given k, data D
The K-Means Clustering Method
32
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Example
Arbitrarily choose K
objects as initial
cluster centers
K=2
0
Assign
each
h
object
to most
similar
center
3
4
5
9
10
Update
the
cluster
means
0
0
1
2
3
4
6
7
8
0
1
2
3
4
5
0
6
10
5
2
9
7
1
8
6
0
7
reassign
6
8
5
7
4
9
3
8
2
10
1
1
2
3
4
9
0
Update
the
cluster
means
5
6
7
8
9
10
10
0
1
2
3
4
5
6
7
8
9
10
0
1
1
2
2
The K-Means Clustering Method
3
3
4
4
5
5
7
8
9
6
7
8
9
reassign
6
10
10
33
Not suitable to discover clusters with non-convex shapes
Unable to handle noisy data and outliers
Need to specify k, the number of clusters, in advance
Applicable only when mean is defined, then what about categorical
data?
Weakness
Comment: Often terminates at a local optimum. The global optimum
may be found using techniques such as: deterministic annealing and
genetic algorithms
Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks2 + k(n-k))
Strength: Relatively efficient: O(tkn), where n is # objects, k is #
clusters, and t is # iterations. Normally, k, t << n.
Comments on the K-Means Method
34
0
10
0
9
1
1
8
2
2
7
3
3
6
4
4
5
5
5
4
6
6
3
7
7
2
8
8
1
9
9
0
10
10
centrally located object in a cluster.
0
1
2
3
4
5
6
7
8
9
10
cluster as a reference point, medoids can be used, which is the most
K-Medoids: Instead of taking the mean value of the object in a
distort the distribution of the data.
Since an object with an extremely large value may substantially
The k-means algorithm is sensitive to outliers !
What Is the Problem of the K-Means Method?
35
PAM (Partitioning Around Medoids, 1987)
CLARA (Kaufmann & Rousseeuw, 1990)
CLARANS (Ng & Han, 1994): Randomized sampling
well for large data sets
PAM works effectively for small data sets, but does not scale
total distance of the resulting clustering
of the medoids by one of the non-medoids if it improves the
starts from an initial set of medoids and iteratively replaces one
Find representative objects, called medoids, in clusters
The K-Medoids Clustering Method
36
0
1
2
3
4
Until no
change
Do loop
K=2
0
1
2
3
4
5
6
7
8
9
5
6
7
8
9
10
If quality is
improved.
Swapping O
and O
Arbitrary
choose k
objects
as initial
medoids
modmar
10
2
3
4
5
6
7
8
9
10
0
0
6
1
1
5
2
4
3
5
6
7
8
2
3
0
4
2
Compute
total cost of
swapping
0
3
1
10
4
5
6
7
8
0
9
9
8
10
7
9
Total Cost = 26
1
1
2
3
4
5
10
0
1
2
3
4
5
0
7
7
6
8
8
Assign
each
remaining
object to
nearest
medoids
9
9
6
10
10
2
3
4
5
6
7
8
0
1
2
3
4
5
6
7
8
Randomly select a
nonmedoid object,O
1
9
10
Total Cost = 20
A Typical K-Medoids Algorithm (PAM) - idea
9
37
10
modmar
Algorithm
repeat steps 2-3 until there is no change
Then assign each non-selected object to the most similar
representative object
If TCih < 0, i is replaced by h
Select a pair i and h, which corresponds to the minimum
swapping cost
For each pair of non-selected object h and selected object i,
calculate the total swapping
pp g cost TCih
Select k representative objects arbitrarily
PAM (Kaufman and Rousseeuw, 1987)
PAM (Partitioning Around Medoids) (1987)
38
1
2
3
4
5
6
7
9
10
0
1
h
2
3
4
i
5
j
6
t
7
8
9
Cjih = d(j, t) - d(j, i)
0
1
2
3
4
5
6
7
8
9
10
10
Cjih = d(j, h) - d(j, i)
8
0
1
2
t
0
1
2
3
3
i
4
4
j
5
5
6
6
t
h
h
i
7
7
8
8
j
9
9
10
10
Cjih = d(j, h) - d(j, t)
0
1
2
3
4
5
6
7
8
9
10
Cjih = 0
0
0
1
0
3
4
1
h
5
6
2
i
j
7
8
2
3
4
5
6
7
t
9
9
8
10
10
PAM Clustering: Total swapping cost TCih=∑jCjih
39
where n is # of data,k is # of clusters
O(k(n-k)2 ) for each iteration
Pam works efficiently for small data sets but does not
scale well for large data sets.
Pam is more robust than k-means in the presence of
noise and outliers because a medoid is less influenced by
outliers or other extreme values than a mean
CLARA(Clustering LARge Applications)
Sampling based method,
What Is the Problem with PAM?
40
Built in statistical analysis packages, such as S+
A good clustering based on samples will not
necessarily represent a good clustering of the whole
data set if the sample is biased
Efficiency depends on the sample size
Weakness:
Strength: deals with larger data sets than PAM
It draws multiple samples of the data set, applies PAM on
each sample, and gives the best clustering as the output
CLARA (Kaufmann and Rousseeuw in 1990)
CLARA (Clustering Large Applications) (1990)
41
Calculate average dissimilarity of the clustering. If the value is
smaller than the current minimum, use this value as current
minimum and retain the k medoids as best so far.
assign each non-selected object in the entire data set to the
most similar mediod
Draw sample of s objects from the entire data set and
perform PAM to find k mediods of the sample
Repeat n times:
Algorithm (n=5, s = 40+2k)
CLARA (Clustering Large Applications) (1990)
42
…
…
Node represents k objects (medoids), a potential solution
for the clustering.
Nodes are neighbors if the sets of objects differ by one object.
Each node has k(n-k) neighbors.
Cost differential between two neighbors is TCih
(with Oi and Oh are the differing nodes in the mediod sets)
…
…
Graph abstraction
43
Searches in the original graph
Searches part of the graph
Uses the neighbors to guide the search
CLARANS
CLARA searches in smaller graphs (as it uses PAM
on samples of the entire data set)
PAM searches for node in the graph with
minimum cost
Graph Abstraction
44
It is more efficient and scalable than both PAM
and CLARA
CLARANS (A Clustering Algorithm based on
Randomized Search) (Ng and Han’94)
CLARANS (“Randomized” CLARA) (1994)
45
Compare the cost of current node with minimum cost so far. If the
cost is lower, set minumum cost to cost of the current node, and
bestnode to the current node.
Consider random neighbor S of the current node and calculate the
cost differential. If S has lower cost, then set S to current and repeat
this step. If S does not have lower cost, repeat this step (check at
most Maxneighbor neighbors)
Take arbitrary node in the graph
Repeat numlocal times: (find local minimum)
Maxneighbor: maximum number of neighbors to compare
Numlocal: number of local minima to be found
Algorithm
CLARANS (“Randomized” CLARA) (1994)
46
6. Density-Based Methods
5. Hierarchical Methods
4 Partitioning Methods
4.
3. A Categorization of Major Clustering Methods
2. Types of Data in Cluster Analysis
1. What is Cluster Analysis?
Chapter 7. Cluster Analysis
47
Step 4
e
d
c
b
a
Step 0
Step 3
ab
Step 1
Step 2 Step 1 Step 0
de
cde
abcde
Step 2 Step 3 Step 4
divisive
(DIANA)
agglomerative
gg
(AGNES)
Use distance matrix as clustering criteria. This method
does not require the number of clusters k as an input,
but needs a termination condition
Hierarchical Clustering
48
d (1, 2 ), 5 = max{d1, 5 , d 2 ,5 } = max{9,8} = 9
2
1
2
3
5
d (1, 2 ), 4 = max{d1 , 4 , d 2 , 4 } = max{10,9} = 10
(1,2)  0



3 6 0

4 10 7 0 


5  9 5 4 0
4
8
0
3
9




0

7 0 
5 4 0
(1,2) 3 4 5
d (1, 2 ), 3 = max{d1, 3 , d 2, 3 } = max{6,3} = 6
1 0
2  2
3 6

4 10
5  9
1 2 3 4 5
Complete-link Clustering Example
49
2
5
1
2
3
4
(1,2)  0


3  6 0 
(4,5) 10 7 0
d 3 ,( 4 , 5 ) = max{d3 , 4 , d 3, 5 } = max{7,5} = 7
(1,2)  0



3 6 0

4 10 7 0 


5  9 5 4 0
(1,2) 3 (4,5)
4
8
0
3
9




0

7 0 
5 4 0
(1,2) 3 4 5
d (1, 2 ), ( 4 ,5 ) = max{d (1, 2 ), 4 , d (1, 2 ), 5 } = max{10,9} = 10
1 0
2  2
3 6

4 10
5  9
1 2 3 4 5
Complete-link Clustering Example
50
8
0
3
9




0

7 0 
5 4 0
th=9
10
6
th=5
(1,2)  0



3 6 0

4 10 7 0 


5  9 5 4 0
(1,2) 3 4 5
d (1, 2 , 3 ),( 4, 5) = max{d(1, 2), ( 4, 5) , d 3, ( 4, 5)} = 10
1 0
2  2
3 6

4 10
5  9
1 2 3 4 5
4
2
1
2
3
4
5
(1,2)  0


3  6 0 
(4,5) 10 7 0
(1,2) 3 (4,5)
Complete-link Clustering Example
51
Introduced in Kaufmann and Rousseeuw (1990)
Implemented in statistical analysis packages, e.g., Splus
Use the Single-Link method and the dissimilarity matrix.
Merge nodes that have the least dissimilarity
Go on in a non-descending fashion
Eventually all nodes belong to the same cluster
AGNES (Agglomerative Nesting)
52
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
AGNES (Agglomerative Nesting)
53
A clustering of the data objects is obtained by cutting the
d d
dendrogram
att the
th desired
d i d level,
l l then
th each
h connected
t d
component forms a cluster.
Decompose data objects into a several levels of nested
partitioning (tree of clusters), called a dendrogram.
54
Dendrogram: Shows How the Clusters are Merged
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
10
Eventually each node forms a cluster on its own
10
Inverse order of AGNES
10
Implemented in statistical analysis packages, e.g., Splus
10
Introduced in Kaufmann and Rousseeuw (1990)
DIANA (Divisive Analysis)
55
can never undo what was done previously
do not scale well: time complexity of at least O(n2),
where n is the number of total objects
CHAMELEON (1999): hierarchical clustering using
dynamic modeling
ROCK (1999): clustering categorical data by neighbor
and link analysis
BIRCH (1996): uses CF-tree and incrementally adjusts
the quality of sub-clusters
Integration of hierarchical with distance-based clustering
Major weakness of agglomerative clustering methods
Recent Hierarchical Clustering Methods
56
Phase 2: use an arbitrary clustering algorithm to cluster
the leaf nodes of the CF-tree
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)
Weakness: handles only numeric data, and sensitive to the
order of the data record, not always natural clusters.
Scales linearly: finds a good clustering with a single scan
and improves the quality with a few additional scans
Incrementally construct a CF (Clustering Feature) tree, a
hierarchical data structure for multiphase clustering
Birch: Balanced Iterative Reducing and Clustering using
Hierarchies (Zhang, Ramakrishnan & Livny, SIGMOD’96)
BIRCH (1996)
57
SS: ∑ Ni=1=Xi2
LS: ∑ Ni=1=Xi
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
N: Number of data points
4
5
6
7
8
9
10
(3,4)
(2,6)
(4,5)
(4,7)
(3,8)
CF = (5, (16,30),(54,190))
Clustering Feature: CF = (N, LS, SS)
Clustering Feature Vector in BIRCH
58
threshold: max diameter of sub-clusters stored at the leaf nodes
Branching factor: specify the maximum number of children.
A CF tree has two parameters
A leaf node represents a cluster made of the subclusters represented by its
entries
A nonleaf node represents a cluster made of the subclusters represented by
its children
A nonleaf node in a tree has children and stores the sums of the CFs of
their children
A CF tree is a height-balanced tree that stores the clustering features for a
hierarchical clustering
g
registers crucial measurements for computing cluster and utilizes storage
efficiently
summary of the statistics for a given subcluster: the 0-th, 1st and 2nd
moments of the subcluster from the statistical point of view.
Clustering feature:
CF-Tree in BIRCH
59
CFk next
Leaf node
child11 child12 child13
CF1
child2 child3
child1
Non-leaf node
CF2 CF3
CF2 CF3
CF1
prev CFa CFb
T=7
B=6
Root
prev CFl CFm
child15
CF5
child6
CF6
The CF Tree Structure
CFq next
Leaf node
60
2
2
O(n + nmmma + n log n)
ROCK: RObust Clustering using linKs
S. Guha, R. Rastogi & K. Shim, ICDE’99
Major ideas
Use links to measure similarity/proximity
maximize the sum of the number of links
between points within a cluster, minimize the
sum of the number of links for points in
different clusters
Computational complexity:
ROCK: Clustering Categorical Data
61
Sim ( T 1 , T 2 ) =
{ a , b , c , d , e}
{c}
=
1
= 0 .2
5
Traditional measures for categorical data may not work well, e.g.,
Jaccard coefficient
Example: Two groups (clusters) of transactions
C1. <a, b, c, d, e>: {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d},
{a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}
C2. <a, b, f, g>: {a, b, f}, {a, b, g}, {a, f, g}, {b, f, g}
Jaccard coefficient may lead to wrong clustering result
C1: 0.2 ({a, b, c}, {b, d, e}} to 0.5 ({a, b, c}, {a, b, d})
C1 & C2: could be as high as 0.5 ({a, b, c}, {a, b, f})
Jaccard coefficient-based similarity function:
T ∩ T2
S im ( T1 , T2 ) = 1
T1 ∪ T2
Ex. Let T1 = {a, b, c}, T2 = {c, d, e}
Similarity Measure in ROCK
62
Link(pi,pj)
Link(pi
pj) is the number of common neighbors
between pi and pj
(sim and t between 0 and 1)
iff sim(p1,p2) >= t
Neighbor: p1 and p2 are neighbors
Link Measure in ROCK
63
{a, b, d}, {a, b, e}, {a, b, g}, {a, b, c}, {a, b, f}
link(T1, T3) = 5, since they have 5 common neighbors
{a, c, d}, {a, c, e}, {b, c, d}, {b, c, e}
link(T1, T2) = 4, since they have 4 common neighbors
Let T1 = {a,
{a b,
b c}
c}, T2 = {c,
{c d
d, e}
e}, T3 = {a
{a, b,
b f}
and sim the Jaccard coefficient similarity and t=0.5
C2 <a, b, f, g>: {a, b, f}, {a, b, g}, {a, f, g}, {b, f, g}
C1 <a, b, c, d, e>: {a, b, c}, {a, b, d}, {a, b, e}, {a, c, d},
{a, c, e}, {a, d, e}, {b, c, d}, {b, c, e}, {b, d, e}, {c, d, e}
Links: # of common neighbors
Link Measure in ROCK
64
G(Ci,Cj)
goodness measure for merging
g g Ci and Cjj
=g
= Link(Ci,Cj) divided by the expected number of
cross links
Link(Ci,Cj) = the number of cross links between
clusters Ci and Cj
Link Measure in ROCK
65
Algorithm: sampling-based clustering
Draw random sample
Hierarchical clustering with links using
goodness measure of merging
Label data in disk: a point is assigned to the
cluster for which it has the most neighbors
after normalization
The ROCK Algorithm
66
Measures the similarity based on a dynamic model
2. Use an agglomerative hierarchical clustering algorithm: find the
genuine clusters by repeatedly combining these sub-clusters
1. Use a graph partitioning algorithm: cluster objects into a large
number of relatively small sub-clusters
A two-phase algorithm
Two clusters are merged only if the interconnectivity and closeness
(proximity) between two clusters are high relative to the internal
interconnectivity of the clusters and closeness of items within the
clusters
CHAMELEON: by G. Karypis, E.H. Han, and V. Kumar’99
CHAMELEON: Hierarchical Clustering Using
Dynamic Modeling (1999)
67
Data Set
Sparse Graph
Construct
Final Clusters
Partition the Graph
Merge Partition
Overall Framework of CHAMELEON
68
2.
Dynamic notion of neighborhood: in regions with high density, a
neighborhood radius is small, while in sparse regions the neighborhood
radius is large
Edge weight is density of the region
Edge between two nodes if points corresponding to either of the nodes
are among the k-most similar points of the point corresponding to the
other node
Based on k-nearest neighbor graph
1. Use a graph partitioning algorithm: cluster objects into a large
number of relatively small sub-clusters
A two-phase algorithm
CHAMELEON: Hierarchical Clustering Using
Dynamic Modeling (1999)
69
Merge if both measures are above user-defined thresholds
Closeness of clusters Ci and Cj: average similarity between
points in Ci that are connected to points in Cj
Interconnectivity between clusters Ci and Cj: normalized sum of
the weights of the edges that connect nodes in Ci and Cj
2. Use an agglomerative hierarchical clustering algorithm:
find the genuine clusters by repeatedly combining these
sub-clusters
1.
A two-phase algorithm
CHAMELEON: Hierarchical Clustering Using
Dynamic Modeling (1999)
70
CHAMELEON (Clustering Complex Objects)
71
6. Density-Based Methods
5. Hierarchical Methods
4 Partitioning Methods
4.
3. A Categorization of Major Clustering Methods
2. Types of Data in Cluster Analysis
1. What is Cluster Analysis?
Chapter 7. Cluster Analysis
72
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)
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
Density-Based Clustering Methods
73
core point condition:
|NEps (q)| >= MinPts
p belongs to NEps(q)
q
p
Eps = 1 cm
MinPts = 5
Directly density-reachable: A point p is directly densityreachable from a point q w.r.t. Eps, MinPts if
{ belongs
{q
b l
tto D | dist(p,q)
di t( ) <= E
Eps}}
MinPts: Minimum number of points in an Epsneighborhood of that point
Eps: Maximum radius of the neighborhood
NEps(p):
( )
Two parameters:
Density-Based Clustering: Basic Concepts
74
A point p is density-reachable from
a point q w.r.t. 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
A point p is density-connected to a
point q w.r.t. Eps, MinPts if there
is a point o such that both, p and
q are density-reachable from o
w.r.t. Eps and MinPts
Density-connected
Density-reachable:
p
q
o
p1
p
Density-Reachable and Density-Connected
q
75
Core
Border
MinPts = 5
Eps = 1cm
Outlier
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
DBSCAN: Density Based Spatial Clustering of
Applications with Noise
76
Continue the process until all of the points have been
processed.
If p is a border point, no points are density-reachable
from p and DBSCAN visits the next point of the database.
If p is a core point, a cluster is formed.
Retrieve all points density-reachable from p w.r.t. Eps
and MinPts.
Arbitrary select a point p
DBSCAN: The Algorithm
77
DBSCAN: Sensitive to Parameters
78
CHAMELEON (Clustering Complex Objects)
79
OPTICS: Ordering Points To Identify the Clustering
Structure
Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99)
Produces a special order of the database wrt its
y
clustering
g structure
density-based
This cluster-ordering contains info equivalent to the
density-based clusterings corresponding to a broad
range of parameter settings
Good for both automatic and interactive cluster analysis,
including finding intrinsic clustering structure
Can be represented graphically or using visualization
techniques
OPTICS: A Cluster-Ordering Method (1999)
80
p2
r(p1, o) = 1.5cm. r(p2,o) = 4cm
o
p1
ε = 3 cm
MinPts = 5
Core Distance of p wrt MinPts: smallest distance eps’
between p and an object in its eps-neighborhood such that
p would be a core object for eps’ and MinPts. Otherwise,
undefined.
Reachability Distance of p wrt o:
Max (core-distance (o), d (o, p)) if o is core object.
U d fi d otherwise
Undefined
th
i
Max (core-distance (o), d (o, p))
OPTICS: Some Extension from
DBSCAN
81
Compute core distance for o
Write object o to ordered file and mark o as processed
If o is not a core object,
j , restart at ((1))
(o is a core object …)
Put neighbors of o in Seedlist and order
(2) Find neighbors (eps-neighborhood)
Take new object from Seedlist with smallest reachability and restart
at (2)
If neighbor n is not yet in SeedList then add (n, reachability from
o) else if reachability from o < current reachability, then update
reachability + order SeedList wrt reachability
(1) Select non-processed object o
OPTICS
82
εಫ
ε
undefined
Reachability
-distance
ε
Cluster-order
of the objects
83
Using statistical density functions
Major features
But needs a large number of parameters
Significant faster than existing algorithm (e.g., DBSCAN)
Allows a compact mathematical description of arbitrarily shaped
clusters in high-dimensional data sets
Good for data sets with large
g amounts of noise
Solid mathematical foundation
DENsity-based CLUstEring by Hinneburg & Keim (KDD’98)
DENCLUE: Using Statistical Density
Functions
84
2σ 2
(x) =
∑
e
i =1
N
2σ
2
−
d ( x , xi ) 2
∇f
D
Gaussian
( x, xi ) = ∑i =1 ( xi − x) ⋅ e
N
2σ 2
Clusters can be determined mathematically by identifying density
attractors. Density attractors are local maxima of the overall density
function
d ( x , xi ) 2
f
D
Gaussian
−
Overall density of the data space can be calculated as the sum of
the influence function of all data points
fGaussian (x, y) = e
Influence function: describes the impact of a data point within its
neighborhood
d ( x, y)2
−
Uses grid cells but only keeps information about grid cells that do
actually contain data points and manages these cells in a tree-based
access structure
Denclue: Technical Essence
85
Density Attractor
86
Arbitrary-shape cluster for a set of significant density attractors X
for threshold k: points that are density attracted to some density
attractor in X
Points that are attracted to a density attractor with density less
than k are called outliers
Center-defined cluster for a significant density attractor x for
threshold k: points that are density attracted by x
Set of significant density attractors X for threshold k: for each pair
of density attractors x1, x2 in X there is a path from x1 to x2 such
that each point on the path has density larger than or equal to k
Significant density attractor for threshold k: density attractor with
density larger than or equal to k
Denclue: Technical Essence
87
Center-Defined and Arbitrary
88