This document provides an overview of clustering techniques. It discusses what clustering is, different types of attributes that can be clustered, and major clustering approaches. The major approaches covered are partitioning algorithms, which construct partitions and evaluate them; hierarchical algorithms, which create a hierarchical decomposition; and density-based algorithms, which are based on connectivity and density. Examples of applications are also provided.
1. DBM630: Data Mining and
Data Warehousing
MS.IT. Rangsit University
Semester 2/2011
Lecture 9
Clustering
by Kritsada Sriphaew (sriphaew.k AT gmail.com)
1
2. Topics
What is Cluster Analysis?
Types of Attributes in Cluster Analysis
Major Clustering Approaches
Partitioning Algorithms
Hierarchical Algorithms
2 Data Warehousing and Data Mining by Kritsada Sriphaew
3. Classification vs. Clustering
Classification: Supervised learning
Learns a method for predicting the
instance class from pre-labeled
(classified) instances
3 Clustering Analysis
4. Clustering
Unsupervised learning:
Finds “natural” grouping of
instances given un-labeled data
4 Clustering Analysis
5. Clustering Methods
Many different method and algorithms:
For numeric and/or symbolic data
Deterministic vs. probabilistic
Exclusive vs. overlapping
Hierarchical vs. flat
Top-down vs. bottom-up
5 Clustering Analysis
6. What is Cluster Analysis ?
Cluster: a collection of data objects
High similarity of objects within a cluster
Low similarity of objects across clusters
Cluster analysis
Grouping a set of data objects into clusters
Clustering is an unsupervised classification: no predefined
classes
Typical applications
As a stand-alone tool to get insight into data distribution
As a preprocessing step for other algorithms
6 Clustering Analysis
7. General Applications of Clustering
Pattern Recognition
In biology, derive plant and animal taxonomies, categorize genes
Spatial Data Analysis
create thematic maps in GIS by clustering feature spaces
detect spatial clusters and explain them in spatial data mining
Image Processing
Economic Science (e.g. market research)
discovering distinct groups in their customer bases and characterize customer
groups based on purchasing patterns
WWW
Document classification
Cluster Weblog data to discover groups of similar access patterns
7 Clustering Analysis
8. Examples of Clustering Applications
Marketing: Help marketers discover distinct groups in their
customer bases, and then use this knowledge to develop targeted
marketing programs
Land use: Identification of areas of similar land use in an earth
observation database
Insurance: Identifying groups of motor insurance policy holders
with a high average claim cost
City-planning: Identifying groups of houses according to their
house type, value, and geographical location
Earth-quake studies: Observed earth quake epicenters should be
clustered along continent faults
8 Clustering Analysis
9. Clustering Evaluation
Manual inspection
Benchmarking on existing labels
Cluster quality measures
distance measures
high similarity within a cluster, low across clusters
9 Clustering Analysis
10. Criteria for Clustering
A good clustering method will produce high quality clusters
with
high intra-class similarity
low inter-class similarity
The quality of a clustering result depends on:
both the similarity measure used by the method and its
implementation to the new cases
ability to discover some or all of the hidden patterns
10 Clustering Analysis
11. Requirements of Clustering in Data Mining
Scalability
Ability to deal with different types of attributes
Discovery of clusters with arbitrary shape
Minimal requirements for domain knowledge to
determine input parameters
Able to deal with noise and outliers
Insensitive to order of input records
High dimensionality
Incorporation of user-specified constraints
Interpretability and usability
11 Clustering Analysis
12. The distance function
Simplest case: one numeric attribute A
Distance(X,Y) = A(X) – A(Y)
Several numeric attributes:
Distance(X,Y) = Euclidean distance between X,Y
Nominal attributes: distance is set to 1 if values are
different, 0 if they are equal
Are all attributes equally important?
Weighting the attributes might be necessary
12 Clustering Analysis
13. From Data Matrix to Similarity or Dissimilarity Matrices
Data matrix (or object-by-attribute structure)
m objects with n attributes, e.g., relational data
x11 x1 j x1n
x xij xin
i1
xm1
xmj xmn
Similarity and dissimilarity matrices
a collection of proximities for all pairs of m objects.
1 0
s 1 d 0
i1 i1
sij s ji dij d ji
13 m1
s smj 1 0 sij 1
d m1
d mj 0 0 dijAnalysis
Clustering
1
14. Distance Functions (Overview)
To transform a data matrix to similarity or dissimilarity
matrices, we need a definition of distance.
Some definitions of distance functions depend on the type of
attributes
interval-scaled attributes
Boolean attributes
nominal, ordinal and ratio attributes.
Weights should be associated with different attributes based
on applications and data semantics.
It is hard to define “similar enough” or “good enough”
the answer is typically highly subjective.
14 Clustering Analysis
15. Similarity and Dissimilarity Between Objects (I)
Distances are normally used to measure the similarity
or dissimilarity between two data objects
Some popular ones include: Minkowski distance:
dij q (| xi1 x j1 |q | xi 2 x j 2 |q ... | xip x jp |q )
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data
objects, and q is a positive integer
If q = 1, d is Manhattan distance
dij | xi1 x j1 | | xi 2 x j 2 | ... | xip x jp |
15 Clustering Analysis
16. Similarity and Dissimilarity Between Objects (II)
If q = 2, d is Euclidean distance:
dij (| xi1 x j1 |2 | xi 2 x j 2 |2 ... | xip x jp |2 )
Properties
d(i,j) >= 0
d(i,i) = 0
d(i,j) = d(j,i)
d(i,j) <= d(i,k) + d(k,j)
Also one can use weighted distance, parametric Pearson product
moment correlation, or other dissimilarity measures.
d (w | x x | w | x x | ... w | x x | )
ij 1 i1 j1
2
2 i2 j2
2
i ip jp
2
16 Clustering Analysis
17. Types of Attributes in Clustering
Interval-scaled attributes
Continuous measures of a roughly linear scale
Binary attributes
Two-state measures: 0 or 1
Nominal, ordinal, and ratio attributes
More than two states, nominal or ordinal or nonlinear scale
Mixed types
Mixture of interval-scaled, symmetric binary, asymmetric binary,
nominal, ordinal, or ratio-scaled attributes
17 Clustering Analysis: Types of Attributes
18. Interval-valued Attributes
Standardize data
Calculate the mean absolute deviation:
s f 1 (| x1 f m f | | x2 f m f | ... | xnf m f |)
n
where m f 1 (x1 f x2 f
n ... xnf )
.
Calculate the standardized measurement (mean-absolute-based z-
xif m f
score) zif sf
Using mean absolute deviation is more robust than using standard
deviation since the z-scores of outliers do not become too small.
Hence, the outliers remain detectable.
18 Clustering Analysis: Types of Attributes
19. A binary variable contains two
possible outcomes: 1
Binary Attributes (positive/present) or 0
(negative/absent).
• If there is no preference for
A contingency table for binary data which outcome should be
coded as 0 and which as 1, the
binary variable is
Object j called symmetric.
1 0 sum • If the outcomes of a binary
variable are not equally
1 a b a b important, the binary variable is
called asymmetric, such as "is
Object i 0 c d cd color-blind" for a human being.
The most important outcome
sum a c b d p is usually coded as 1 (present)
and the other is coded as 0
(absent).
Simple matching coefficient (invariant, if the binary variable is
symmetric): bc
dij
abcd
Jaccard coefficient (noninvariant if the binary variable is
asymmetric): bc
dij
19
a b cAnalysis:Types of Attributes
Clustering
20. Dissimilarity on Binary Attributes
Example
Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4
Jack M Y N P N N N
Mary F Y N P N P N
Jim M Y P N N N N
The gender is symmetric attribute and the remaining attributes are asymmetric
binary. Here, let the values Y and P be set to 1, and the value N be set to 0.
Then calculate only asymmetric binary 0 1
d ( jack , mary) 0.33
2 0 1
11
d ( jack , jim ) 0.67
111
1 2
d ( jim , mary) 0.75
11 2
20 Clustering Analysis: Types of Attributes
21. Nominal Attributes
A generalization of the binary variable in that it can take
more than 2 states, e.g., red, yellow, blue, green
Method 1: Simple matching
m is the # of matches, p is the total # of nominal attributes
pm
dij
p
Method 2: Use a large number of binary variables
creating a new binary variable for each of the M nominal
states
21 Clustering Analysis: Types of Attributes
22. Ordinal Attributes
An ordinal variable can be discrete or continuous
order is important, e.g., rank
Can be treated like interval-scaled
replacing xif by their rank rif { ,..., M f }
1
map the range of each variable onto [0, 1] by replacing
i-th object in the f-th variable by
rif 1
zif
M f 1
compute the dissimilarity using methods for interval-scaled
variables
22 Clustering Analysis: Types of Attributes
23. Ratio-Scaled Attributes
Ratio-scaled variable: a positive measurement on a nonlinear
scale, approximately at exponential scale, such as AeBt
or Ae-Bt
Methods:
(1) treat them like interval-scaled attributes
not a good choice!
(2) apply logarithmic transformation
yif = log(xif)
(3) treat them as continuous ordinal data and
treat their rank as interval-scaled.
23 Clustering Analysis: Types of Attributes
24. Mixed Types
A database may contain all the six types of attributes
symmetric binary, asymmetric binary, nominal, ordinal,
interval and ratio.
One may use a weighted formula to combine their effects.
p 1 ij f ) dij f )
( (
dij f p
f 1 ij f )
(
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
treat (normalized) zif as interval-scaled
24 Clustering Analysis: Types of Attributes
25. Major Clustering Approaches
Partitioning algorithms: Construct various partitions and then
evaluate them by some criterion
Hierarchy algorithms: Create a hierarchical decomposition of the
set of data (or objects) using some criterion
Density-based: based on connectivity and density functions
Grid-based: based on a multiple-level granularity structure
Model-based: A model is hypothesized for each of the clusters
and the idea is to find the best fit of that model to each other
25 Clustering Analysis: Clustering Approaches
26. Partitioning Approach
Construct a partition of a database D of n objects into a set of k
clusters
Given a k, find a partition of k clusters that optimizes the chosen
partitioning criterion
Global optimal: exhaustively enumerate all partitions
Heuristic methods: k-means and k-medoids algorithms
k-means (MacQueen’67): Each cluster is represented by the
center of the cluster
k-medoids or PAM (Partition around medoids) (Kaufman &
Rousseeuw’87): Each cluster is represented by one of the objects
in the cluster
26 Clustering Analysis: Partitioning Algorithms
27. The K-Means Clustering Method
(Overview)
Given k, the k-means algorithm is implemented in 4 steps:
Partition objects into k nonempty subsets
Compute seed points as the centroids of the clusters of
the current partition. The centroid is the center (mean
point) of the cluster.
Assign each object to the cluster with the nearest seed
point.
Go back to Step 2, stop when no more new assignment.
27 Clustering Analysis: Partitioning Algorithms
29. K-means example, step 1
Given k=3,
k1
Y
Pick 3 k2
initial
cluster
centers
(randomly)
k3
X
29 Clustering Analysis: Partitioning Algorithms
30. K-means example, step 2
k1
Y
k2
Assign
each point
to the closest
cluster
center k3
X
30 Clustering Analysis: Partitioning Algorithms
31. K-means example, step 3
k1 k1
Y
Move k2
each cluster
center k3
k2
to the mean
of each cluster k3
X
31 Clustering Analysis: Partitioning Algorithms
32. K-means example, step 4
Reassign k1
points Y
closest to a
different new
cluster center
k3
Q: Which k2
points are
reassigned?
X
32 Clustering Analysis: Partitioning Algorithms
33. K-means example, step 4 …
k1
Y
A: three
points with
animation k3
k2
X
33 Clustering Analysis: Partitioning Algorithms
34. K-means example, step 4b
k1
Y
re-compute
cluster
means k3
k2
X
34 Clustering Analysis: Partitioning Algorithms
35. K-means example, step 5
k1
Y
k2
move cluster
centers to k3
cluster means
X
35 Clustering Analysis: Partitioning Algorithms
36. Problems to be considered
What can be the problems with K-means clustering?
Result can vary significantly depending on initial choice of seeds
(number and position)
Can get trapped in local minimum initial cluster
Example: centers
instances
Q: What can be done?
A: To increase chance of finding global optimum: restart with
different random seeds.
What can be done about outliers?
36 Clustering Analysis: Partitioning Algorithms
37. The K-Means Clustering Method
(Strength and Weakness)
Strength
Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t
is # iterations. Normally, k, t << n
Good for finding clusters with spherical shapes
Often terminates at a local optimum. The global optimum may be
found using techniques such as: deterministic annealing and genetic
algorithms
Weakness
Applicable only when mean is defined, then what about categorical
data?
Need to specify k, no. of clusters, in advance
Unable to handle noisy data and outliers
Not suitable to discover clusters with non-convex shapes
37 Clustering Analysis: Partitioning Algorithms
38. The K-Means Clustering Method
(Variations – I)
A few variants of the k-means which differ in
Selection of the initial k means Mean of 1, 3, 5, 7, 9 is 5
Mean of 1, 3, 5, 7, 1009 is 205
Dissimilarity calculations Median of 1, 3, 5, 7, 1009 is 5
Median advantage: not affected
Strategies to calculate cluster means by extreme values
K-medoids – instead of mean, use medians of each cluster
For large databases, use sampling
Handling categorical data: k-modes (Huang’98)
Replacing means of clusters with modes
Using new dissimilarity measures to deal with categorical objects
Using a frequency-based method to update modes of clusters
A mixture of categorical/numerical data: k-prototype method
38 Clustering Analysis: Partitioning Algorithms
39. The K-Medoids Clustering Method
(Overview)
Find representative objects, called medoids, in clusters
PAM (Partitioning Around Medoids, 1987)
starts from an initial set of medoids and iteratively
replaces one of the medoids by one of the non-medoids if
it improves the total distance of the resulting clustering
PAM works effectively for small data sets, but does not
scale well for large data sets
CLARA (Kaufmann & Rousseeuw, 1990)
CLARANS (Ng & Han, 1994): Randomized sampling
39 Clustering Analysis: Partitioning Algorithms
40. The K-Medoids Clustering Method
(PAM - Partitioning Around Medoids)
PAM (Kaufman and Rousseeuw, 1987), built in Splus
Use real object to represent the cluster
Select k representative objects arbitrarily
For each pair of non-selected object h and selected object
i, calculate the total swapping cost Sih
For each pair of i and h,
If Sih < 0, i is replaced by h
Then assign each non-selected object to the most similar
representative object
repeat steps 2-3 until there is no change
40 Clustering Analysis: Partitioning Algorithms
41. PAM example
Cluster the following data set of ten objects into two
clusters i.e k = 2.
X1 2 6
X2 3 4
X3 3 8
X4 4 7
X5 6 2
X6 6 4
X7 7 3
X8 7 4
X9 8 5
X10 7 6
41 Clustering Analysis
42. PAM example, step 1
Initialise k medoids. Let assume c1 = (3,4) and c2 =
(7,4)
Calculating distance so as to associate each data
object to its nearestCost medoid. Assume that cost is Cost
calculated )using Minkowski distance metric with r =(distan
c 1
Data objects
(X
(distan 2 c
Data objects
(X ) 1.
i
ce) i
ce)
3 4 2 6 3 7 4 2 6 7
3 4 3 8 4 7 4 3 8 8
3 4 4 7 4 7 4 4 7 6
3 4 6 2 5 7 4 6 2 3
3 4 6 4 3 7 4 6 4 1
3 4 7 3 5 7 4 7 3 1
3 4 8 5 6 7 4 8 5 2
42 Clustering Analysis
3 4 7 6 6 7 4 7 6 2
44. PAM example, step 2
Selection of nonmedoid O′ randomly. Let us
assume O′ = (7,3). So now the medoids
are c1(3,4) and O′(7,3)
Calculate the cost of new medoid by using the
Cost Cost
Data objects Data objects
formula in )the step1. Total cost =
c 1
(X
(distan O′
(X )
(distan
i
ce) i
ce)
3 3+4+4+2+2+1+3+33 = 22 7
4 2 6 3 2 6 8
3 4 3 8 4 7 3 3 8 9
3 4 4 7 4 7 3 4 7 7
3 4 6 2 5 7 3 6 2 2
3 4 6 4 3 7 3 6 4 2
3 4 7 3 5 7 3 7 4 1
3 4 8 5 6 7 3 8 5 3
44 Clustering Analysis
3 4 7 6 6 7 3 7 6 3
45. PAM example, step 2b
So cost of swapping medoid from c2 to O′ is
S = current total cost – past total cost = 22-20
= 2 >0
So moving to O′ would be bad idea, so the previous
choice was good, and algorithm terminates here (i.e
there is no change in the medoids).
It may happen some data points may shift from one
cluster to another cluster depending upon their
closeness to medoid
45 Clustering Analysis
46. CLARA (Clustering Large Applications) (1990)
CLARA (Kaufmann and Rousseeuw in 1990)
Built in statistical analysis packages, such as S+
It draws multiple samples of the data set, applies PAM on each
sample, and gives the best clustering as the output
Strength: deals with larger data sets than PAM
Weakness:
Efficiency depends on the sample size
A good clustering based on samples will not necessarily represent a good
clustering of the whole data set if the sample is biased
46 Clustering Analysis: Partitioning Algorithms
47. CLARANS (“Randomized” CLARA) (1994)
CLARANS (A Clustering Algorithm based on Randomized Search) (Ng
and Han’94)
CLARANS draws sample of neighbors dynamically
The clustering process can be presented as searching a graph where
every node is a potential solution, that is, a set of k medoids
If the local optimum is found, CLARANS starts with new randomly
selected node in search for a new local optimum
It is more efficient and scalable than both PAM and CLARA
Focusing techniques and spatial access structures may further improve
its performance (Ester et al.’95)
47 Clustering Analysis: Partitioning Algorithms
48. The Partition-Based Clustering
(Discussion)
Result can vary significantly based on initial choice of seeds
Algorithm can get trapped in a local minimum
Example: four instances at the vertices of a two-
dimensional rectangle
Local minimum: two cluster centers at the midpoints
of the rectangle’s long sides
Simple way to increase chance of finding a global optimum:
restart with different random seeds
48 Clustering Analysis: Hierarchical Algorithms
49. Hierarchical Clustering
Use distance matrix as clustering criteria.
This method does not require the number of clusters k as an
input, but needs a termination condition
Step 0 Step 1 Step 2 Step 3 Step 4
Agglomerative
(AGNES)
a
ab
b abcde
c
cde
d
de
e
Divisive
Step 4 Step 3 Step 2 Step 1 Step 0 (DIANA)
49 Clustering Analysis: Hierarchical Algorithms
50. AGNES (Agglomerative Nesting)
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
10 10 10
9 9 9
8 8 8
7 7 7
6 6 6
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
0 0 0
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
50 Clustering Analysis: Hierarchical Algorithms
51. Dendrogram for Hierarchical Clustering
Decompose data objects into a several levels of nested
partitioning (tree of clusters), called a dendrogram.
A clustering of the data objects is obtained by cutting the
dendrogram at the desired level, then each connected
component forms a cluster.
51 Clustering Analysis: Hierarchical Algorithms
52. DIANA - Divisive Analysis
Introduced in Kaufmann and Rousseeuw (1990)
Implemented in statistical analysis packages, e.g., Splus
Inverse order of AGNES
Eventually each node forms a cluster on its own
10 10
10
9 9
9
8 8
8
7 7
7
6 6
6
5 5
5
4 4
4
3 3
3
2 2
2
1 1
1
0 0
0
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
52 Clustering Analysis: Hierarchical Algorithms
53. Hierarchical Clustering
Major weakness of agglomerative clustering methods
do not scale well: time complexity of at least O(n2),
where n is the number of total objects
can never undo what was done previously
53 Clustering Analysis: Hierarchical Algorithms
54. Distances - Hierarchical Clustering
(Overview)
Four measures for distance between clusters are:
Single linkage (Minimum distance):
dmin (Ci , C j ) minpCi , p'Cj p p'
Complete linkage (Maximum distance):
dmax (Ci , C j ) max pCi , p'Cj p p'
Centroid comparison (Mean distance):
d mean (Ci , C j ) mi m j
Element comparison (Average distance):
1
d avg (Ci , C j ) p p'
54
ni n j pCi p'C j Analysis: Hierarchical Algorithms
Clustering
55. Distances - Hierarchical Clustering
(Graphical Representation)
Four measures for distance between clusters are (1) single linkage, (2) complete
linkage, (3) centroid comparison and (4) element comparison
Cluster 1 (1)single Cluster 2
(4) Element comparison:
x x average distance among all
elements in two clusters
(3)centroid
(2)complete
Cluster 3
x x = centroids
55 Clustering Analysis: Hierarchical Algorithms
56. Practice
Use single and complete link agglomerative clustering
to group the data described by the following
distance matrix. Show the dendrograms.
A B C D
A 0 1 4 5
B 0 2 6
C 0 3
D 0
56 Clustering Analysis
58. Advanced Method: BIRCH (Overview)
Balanced Iterative Reducing and Clustering using Hierarchies [Tian Zhang,
Raghu Ramakrishnan, Miron Livny, 1996]
Incremental, hierarchical, one scan
Save clustering information in a tree
Each entry in the tree contains information about one cluster
New nodes inserted in closest entry in tree
Only works with "metric" attributes
Must have Euclidean coordinates
Designed for very large data sets
Time and memory constraints are explicit
Treats dense regions of data points as sub-clusters
Not all data points are important for clustering
Only one scan of data is necessary
58 Clustering Analysis
59. BIRCH (Merits)
Incremental, distance-based approach
Decisions are made without scanning all data points, or all
currently existing clusters
Does not need the whole data set in advance
Unique approach: Distance-based algorithms generally need
all the data points to work
Make best use of available memory while minimizing I/O
costs
Does not assume that the probability distributions on
attributes is independent
59 Clustering Analysis
60. BIRCH – Clustering Feature and Clustering Feature Tree
BIRTH introduces two concepts, clustering feature and clustering
feature tree (CF Tree), which are used to summarize cluster
representations.
These structures help the clustering method achieve good speed and
scalability in large databases and make it effective for incremental and
dynamic clustering of incoming object
Given n d-dimensional data objects or points in a cluster, we can
define the centroid x0, radius R and diameter D of the cluster as
follows:
60 Clustering Analysis
61. BIRCH – Centroid, Radius and Diameter
• Given a cluster of instances , we define:
• Centroid: the center of a cluster
• Radius: average distance from member points to centroid
• Diameter: average pair-wise distance within a cluster
61 Clustering Analysis
62. BIRCH – Centroid Euclidean and Manhattan distances
• The centroid Euclidean distance and centroid
Manhattan distance are defined between any two
clusters.
• Centroid Euclidean distance
• Centroid Manhattan distance
62 Clustering Analysis
63. BIRCH
(Average inter-cluster, Average intra-cluster, Variance increase)
• The average inter-cluster, the average intra-cluster, and the
variance increase distances are defined as follows
• Average inter-cluster
• Average intra-cluster
• Variance increase distances
63 Clustering Analysis
64. Clustering Feature
CF = (N,LS,SS)
N: Number of points in cluster
LS: Sum of points in the cluster
SS: Sum of squares of points in the cluster
CF Tree
Balanced search tree
Node has CF triple for each child
Leaf node represents cluster and has CF value for each
subcluster in it.
Subcluster has maximum diameter
64 Clustering Analysis
68. Properties of CF-Tree
Each non-leaf node has at most B entries
Each leaf node has at most L CF entries which each satisfy threshold
T
Node size is determined by dimensionality of data space and input
parameter P (page size)
Branching Factor
and
Thread hold
68 Clustering Analysis
69. BIRCH Algorithm (CF-Tree Insertion)
Recurse down from root, find the appropriate leaf
Follow the "closest"-CF path, w.r.t. D0 / … / D4
Modify the leaf
If the closest-CF leaf cannot absorb, make a new CF entry.
If there is no room for new leaf, split the parent node
Traverse back & up
69 Updating CFs on the path Analysis
Clustering or splitting nodes
72. Details of Each Step
Phase 1: Load data into memory
Build an initial in-memory CF-tree with the data (one scan)
Subsequent phases become fast, accurate, less order sensitive
Phase 2: Condense data
Rebuild the CF-tree with a larger T
Condensing is optional
Phase 3: Global clustering
Use existing clustering algorithm on CF entries
Helps fix problem where natural clusters span nodes
Phase 4: Cluster refining
Do additional passes over the dataset & reassign data points to the
closest centroid from phase 3
Refining is optional
72 Clustering Analysis
73. Summary of BIRCH
BIRCH works with very large data sets
Explicitly bounded by computational resources.
The computation complexity is O(n), where n is the number of
objects to be clustered.
Runs with specified amount of memory (P)
Superior to CLARANS and k-MEANS
Quality, speed, stability and scalability
73 Clustering Analysis
74. CURE (Clustering Using REpresentatives)
CURE was proposed by Guha, Rastogi & Shim, 1998
It stops the creation of a cluster hierarchy if a level consists of k
clusters
Each cluster has c representatives (instead of one)
Choose c well scattered points in the cluster
Shrink them towards the mean of the cluster by a fraction of
The representatives capture the physical shape and geometry of the
cluster
It can treat arbitrary shaped clusters and avoid single-link effect.
Merge the closest two clusters
Distance of two clusters: the distance between the two closest
representatives
74 Clustering Analysis
79. Data Partitioning and Clustering
y s = 50
p=2
s/p = 25
x s/pq = 5
y y
y y
x x
x x
79 Clustering Analysis
80. Cure: Shrinking Representative Points
Shrink the multiple representative points towards the gravity center by
a fraction of .
Multiple representatives capture the shape of the cluster
y y
x x
80 Clustering Analysis
81. Clustering Categorical Data: ROCK
ROCK: RObust Clustering using linKs,
by S. Guha, R. Rastogi, K. Shim (ICDE’99).
Use links to measure similarity/proximity
Not distance-based with categorical attributes
Computational complexity: O(n2 nmmma n2 log n)
Basic ideas (Jaccard coefficient):
Similarity function and neighbors:
T1 T2
Let T1 = {1,2,3}, T2={3,4,5} Sim(T1 , T2 )
T1 T2
{3} 1
Sim(T1, T 2) 0.2
{1,2,3,4,5} 5
81 Clustering Analysis
82. ROCK: An Example
Links: The number of common neighbors for the two points.
Using Jaccard
Use Distances to determine neighbors
(pt1,pt4) = 0, (pt1,pt2) = 0, (pt1,pt3) = 0
(pt2,pt3) = 0.6, (pt2,pt4) = 0.2
(pt3,pt4) = 0.2
Use 0.2 as threshold for neighbors
Pt2 and Pt3 have 3 common neighbors
Pt3 and Pt4 have 3 common neighbors
Pt2 and Pt4 have 3 common neighbors
Resulting clusters (1), (2,3,4) which makes more sense
82 Clustering Analysis
83. ROCK: Property & Algorithm
Links: The number of common neighbours for the
two points.
{1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,3,5}
{1,4,5}, {2,3,4}, {2,3,5}, {2,4,5}, {3,4,5}
3
{1,2,3} {1,2,4}
Algorithm
Draw random sample
Cluster with links (maybe agglomerative hierarchical)
Label data in disk
83 Clustering Analysis
84. CHAMELEON
CHAMELEON: hierarchical clustering using dynamic modeling, by G.
Karypis, E.H. Han and V. Kumar’99
Measures the similarity based on a dynamic model
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
A two phase algorithm
1. Use a graph partitioning algorithm: cluster objects into a large
number of relatively small sub-clusters
2. Use an agglomerative hierarchical clustering algorithm: find the
genuine clusters by repeatedly combining these sub-clusters
84 Clustering Analysis
85. Graph-based clustering
Sparsification techniques keep the connections to the most
similar (nearest) neighbors of a point while breaking the
connections to less similar points.
The nearest neighbors of a point tend to belong to the same
class as the point itself.
This reduces the impact of noise and outliers and sharpens
the distinction between clusters.
85 Clustering Analysis
86. Overall Framework of CHAMELEON
Construct
Sparse Graph Partition the Graph
Data Set
Merge Partition
Final Clusters
86 Clustering Analysis
87. Density-Based Clustering Methods
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)
87 Clustering Analysis
88. Density-Based Clustering (Background)
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)
2) core point condition: p MinPts = 5
|NEps (q)| >= MinPts q
Eps = 1 cm
88 Clustering Analysis
89. Density-Based Clustering (Background)
The neighborhood with in a radius Eps of a given object is call the
neighborhood of the object.
If the neighborhood of an object contains at least a minimum
number, MinPts, of objects, then the object is called a core object.
Given a set of objects, D, we say that an object p is directly density-
reachable from object q if p is within the neighborhood of q, and q
is a core object.
An object p is density-reachable from object q with respect to and
MinPts in a set of objects D, if there is a chain of object p1, …, pn,
where p1=q and pn=p such that p(i+1) is directly density-reachable from
pi with respect to and MinPts, for 1≤i≤n, from pi D.
An object p is density-connected to object q with respect to
89 Clustering Analysis
90. Density-Based Clustering p
p1
q
Density-reachable:
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
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
o
90 Clustering Analysis
91. 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
Outlier
Border
Eps = 1cm
Core MinPts = 5
91 Clustering Analysis
92. 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 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.
92 Clustering Analysis