The document discusses cluster analysis and outlier analysis techniques for data mining. It covers key topics such as defining clusters and the goal of cluster analysis, different types of data that can be analyzed via clustering, major categories of clustering methods like partitioning, hierarchical, density-based, and model-based approaches. Specific clustering algorithms discussed include k-means, k-medoids, hierarchical clustering, DBSCAN, and EM. The document provides examples of clustering applications and discusses evaluating clustering quality and requirements for clustering in data mining.

1. Data Mining:
Concepts and
Techniques
Lectures 17 - 20
— Cluster and Outlier Analysis —
(Adapted from)
Jiawei Han
Department of Computer Science
University of Illinois at Urbana-Champaign
www.cs.uiuc.edu/~hanj
©2006 Jiawei Han and Micheline Kamber, All rights reserved
November 10, 2013
Data Mining: Concepts and Techniques
1

2. Chapter 7. Cluster Analysis
1. What is Cluster Analysis?
2. Types of Data in Cluster Analysis
3. A Categorization of Major Clustering Methods
4. Partitioning Methods
5. Hierarchical Methods
6. Outlier Analysis
7. Summary
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Data Mining: Concepts and Techniques
2

3. What is Cluster Analysis?

Cluster: a collection of data objects



Similar to one another within the same cluster
Dissimilar to the objects in other clusters
Cluster analysis

Finding similarities between data according to the
characteristics found in the data and grouping similar
data objects into clusters

Unsupervised learning: no predefined classes

Typical applications

As a stand-alone tool to get insight into data distribution

As a preprocessing step for other algorithms
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4. Clustering: Rich Applications and
Multidisciplinary Efforts

Pattern Recognition

Spatial Data Analysis


Create thematic maps in GIS by clustering feature
spaces
Detect spatial clusters or for other spatial mining tasks

Image Processing

Economic Science (especially market research)

WWW


Document classification
Cluster Weblog data to discover groups of similar
access patterns
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Data Mining: Concepts and Techniques
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5. 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
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Data Mining: Concepts and Techniques
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6. Quality: What Is Good 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

The quality of a clustering method is also measured by its
ability to discover some or all of the hidden patterns
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Data Mining: Concepts and Techniques
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7. Measure the Quality of Clustering





Dissimilarity/Similarity metric: Similarity is expressed in
terms of a distance function, typically metric: d (i, j)
There is a separate “quality” function that measures the
“goodness” of a cluster.
The definitions of distance functions are usually very
different for interval-scaled, binary, categorical, ordinal
ratio, and vector variables.
Weights should be associated with different variables
based on applications and data semantics.
It is hard to define “similar enough” or “good enough”

the answer is typically highly subjective.
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8. Requirements of Clustering in Data Mining

Scalability

Ability to deal with different types of attributes

Ability to handle dynamic data

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
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Data Mining: Concepts and Techniques
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9. Chapter 7. Cluster Analysis
1. What is Cluster Analysis?
2. Types of Data in Cluster Analysis
3. A Categorization of Major Clustering Methods
4. Partitioning Methods
5. Hierarchical Methods
6. Outlier Analysis
7. Summary
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Data Mining: Concepts and Techniques
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10. Data Structures


Data matrix
 (two modes)
 x11

 ...
x
 i1
 ...
x
 n1

Dissimilarity matrix
 (one mode)
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...
x1f
...
...
...
...
xif
...
...
...
...
... xnf
...
...
 0
 d(2,1)
0

 d(3,1) d ( 3,2) 0

:
:
 :
d ( n,1) d ( n,2) ...

Data Mining: Concepts and Techniques
x1p 

... 
xip 

... 
xnp 








... 0

10

11. Type of data in clustering analysis

Interval-scaled variables

Binary variables

Nominal, ordinal, and ratio variables

Variables of mixed types
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Data Mining: Concepts and Techniques
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12. Interval-valued variables

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  x + x +...+ x
n 1f
2f
nf



.



Calculate the standardized measurement (z-score)
xif − m f
zif =
sf

Using mean absolute deviation is more robust than using
standard deviation
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Data Mining: Concepts and Techniques
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13. Similarity and Dissimilarity Between
Objects


Distances are normally used to measure the similarity or
dissimilarity between two data objects
Some popular ones include: Minkowski distance:
d (i, j) = q (| x − x |q + | x − x |q +...+ | x − x |q )
i1 j1
i2 j 2
ip jp
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two pdimensional data objects, and q is a positive integer

If q = 1, d is Manhattan (or city block) distance
d (i, j) =| x − x | + | x − x | + ...+ | x − x |
i1 j1 i2 j 2
ip jp
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14. Fig 7.1

15. Similarity and Dissimilarity Between
Objects (Cont.)

If q = 2, d is Euclidean distance:
d (i, j) = (| x − x | 2 + | x − x |2 +...+ | x − x | 2 )
i1
j1
i2
j2
ip
jp

Properties


d(i,i) = 0

d(i,j) = d(j,i)


d(i,j) ≥ 0
d(i,j) ≤ d(i,k) + d(k,j)
(triangle inequality)
Also, one can use weighted distance, parametric
Pearson product moment correlation, or other
disimilarity measures
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16. Binary Variables

A contingency table for binary
Object i
data
Object j
1
0
1
q
s
0
r
t
sum q + s r + t

Distance measure for
symmetric binary variables:

Distance measure for
asymmetric binary variables:

Jaccard coefficient (similarity
measure for asymmetric
d (i, j) =
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r +s
q + +s
r
simJaccard (i, j) =
Data Mining: Concepts and Techniques
p
r +s
q +r +s +t
d (i, j) =
binary variables):
sum
q+r
s +t
q
q+r +s
16

17. 1
Dissimilarity between Binary
0

sum q + s r + t
Example
Name
Jack
Mary
Jim



1
0 sum
q
r q+
Variablesr
s
t
s +t
Gender
M
F
M
Fever
Y
Y
Y
Cough
N
N
P
Test-1
P
P
N
Test-2
N
N
N
Test-3
N
P
N
p
Test-4
N
N
N
gender is a symmetric attribute
the remaining attributes are asymmetric binary
let the values Y and P be set to 1, and the value N be set to 0
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
d ( jack , mary ) =
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18. Nominal Variables


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: # of matches, p: total # of variables
d (i, j) = p −m
p

Method 2: use a large number of binary variables

creating a new binary variable for each of the M
nominal states
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19. Ordinal Variables

An ordinal variable can be discrete or continuous

Order is important, e.g., rank

Can be treated like interval-scaled



replace xif by their rank
rif ∈ 1,..., M f }
{
map the range of each variable onto [0, 1] by replacing
i-th object in the f-th variable by
r −
1
zif = if
M
−
1
f
compute the dissimilarity using methods for intervalscaled variables
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20. Ratio-Scaled Variables


Ratio-scaled variable: a positive measurement on a
nonlinear scale, approximately at exponential scale,
such as AeBt or Ae-Bt
Methods:


treat them like interval-scaled variables—not a good
choice! (why?—the scale can be distorted)
apply logarithmic transformation
yif = loge(xif)

treat them as continuous ordinal data or treat their
ranks as interval-scaled
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21. Variables of Mixed Types


A database may contain all the six types of variables
 symmetric binary, asymmetric binary, nominal,
ordinal, interval-valued and ratio-scaled
One may use a weighted formula to combine their
p
(
(
effects
Σ =1δ f ) d ij f )
d (i, j ) = f p ij ( f )
Σ =1δ
f
ij
 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 r and
if
zif = r −1
 and treat z as interval-scaled
M −1
if
if
f
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22. Vector Objects


Vector objects: keywords in documents, gene
features in micro-arrays, etc.
Broad applications: information retrieval, biological
taxonomy, etc.

Cosine measure

A variant: Tanimoto coefficient
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23. Chapter 7. Cluster Analysis
1. What is Cluster Analysis?
2. Types of Data in Cluster Analysis
3. A Categorization of Major Clustering Methods
4. Partitioning Methods
5. Hierarchical Methods
6. Outlier Analysis
7. Summary
November 10, 2013
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24. Major Clustering Approaches (I)

Partitioning approach:

Construct various partitions and then evaluate them by some
criterion, e.g., minimizing the sum of square errors


Typical methods: k-means, k-medoids, CLARANS
Hierarchical approach:

Create a hierarchical decomposition of the set of data (or objects)
using some criterion


Typical methods: Diana, Agnes, BIRCH, ROCK, CAMELEON
Density-based approach:

Based on connectivity and density functions

Typical methods: DBSACN, OPTICS, DenClue
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25. Major Clustering Approaches (II)

Grid-based approach:



based on a multiple-level granularity structure
Typical methods: STING, WaveCluster, CLIQUE
Model-based:

A model is hypothesized for each of the clusters and tries to find the best fit
of that model to each other


Typical methods: EM, SOM, COBWEB
Frequent pattern-based:



Based on the analysis of frequent patterns
Typical methods: pCluster
User-guided or constraint-based:

Clustering by considering user-specified or application-specific constraints

Typical methods: COD (obstacles), constrained clustering
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26. Chapter 7. Cluster Analysis
1. What is Cluster Analysis?
2. Types of Data in Cluster Analysis
3. A Categorization of Major Clustering Methods
4. Partitioning Methods
5. Hierarchical Methods
6. Outlier Analysis
7. Summary
November 10, 2013
Data Mining: Concepts and Techniques
26

27. Partitioning Algorithms: Basic Concept

Partitioning method: Construct a partition of a database D of n objects
into a set of k clusters, s.t., minimize sum of squared distance
Σ ik=1Σ p∈C p − mi
2
i

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
centre (or mean) 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
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28. The K-Means Clustering Method

Given k, the k-means algorithm is implemented in four
steps:




Partition objects into k non-empty subsets
Compute seed points as the centroids of the
clusters of the current partition (the centroid is the
centre, i.e., mean point, of the cluster)
Assign each object to the cluster with the nearest
seed point based on distance
Go back to Step 2, stop when no more movement
between clusters
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29. Fig 7.3

30. Comments on the K-Means Method

Strength: Relatively efficient: O(tkn), where n is # objects, k is #
clusters, and t is # iterations. Normally, k, t << n.



Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks2 + k(n-k))
Comment: 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, the number of clusters, in advance

Unable to handle noisy data and outliers

Not suitable to discover clusters with non-convex shapes
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Data Mining: Concepts and Techniques
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31. Variations of the K-Means Method

A few variants of the k-means which differ in


Dissimilarity calculations


Selection of the initial k means
Strategies to calculate cluster means
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 and numerical data: k-prototype method
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Data Mining: Concepts and Techniques
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32. What Is the Problem of the K-Means Method?

The k-means algorithm is sensitive to outliers !

Since an object with an extremely large value may substantially
distort the distribution of the data.

K-Medoids: Instead of taking the mean value of the object in a cluster
as a reference point, medoids can be used, which is the most
centrally located object in a cluster.
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Data Mining: Concepts and Techniques
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33. The K-Medoids Clustering Method

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

Focusing + spatial data structure (Ester et al., 1995)
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34. A Typical K-Medoids Algorithm (PAM)
Total Cost = 20
10
10
10
9
9
9
8
8
8
Arbitrary
choose k
object as
initial
medoids
7
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5
4
3
2
1
0
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
0
K=2
Do loop
Until no
change
1
2
3
4
5
6
7
8
9
10
Assign
each
remainin
g object
to
nearest
medoids
7
6
5
4
3
2
1
0
0
10
Compute
total cost of
swapping
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1
0
0
0
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3
10
9
If quality is
improved.
2
Randomly select a
nonmedoid object,Oramdom
Total Cost = 26
Swapping O
and Oramdom
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35. PAM (Partitioning Around Medoids)

36. Four Cases of Potential Swapping

37. What Is the Problem with PAM?


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
PAM works efficiently for small data sets but does not
scale well for large data sets.

O(k(n-k)2 ) for each iteration
where n is # of data,k is # of clusters

Sampling based method,
CLARA(Clustering LARge Applications)
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37

38. Chapter 7. Cluster Analysis
1. What is Cluster Analysis?
2. Types of Data in Cluster Analysis
3. A Categorization of Major Clustering Methods
4. Partitioning Methods
5. Hierarchical Methods
6. Outlier Analysis
7. Summary
November 10, 2013
Data Mining: Concepts and Techniques
38

39. Hierarchical Methods

Agglomerative Hierarchical Clustering



Starts by placing each object in its own cluster and
then merges these atomic clusters into larger and
larger clusters
Divisive Hierarchical Clustering



Bottom-up strategy
Top-down strategy
Starts with all objects in one cluster and then
subdivide the cluster into smaller and smaller pieces
Stopping (or termination) conditions
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40. Stopping Conditions

Agglomerative Hierarchical Clustering


Merging continues until all objects are in a single
cluster or until certain termination criteria are satisfied
Divisive Hierarchical Clustering

Subdividing continues until each object forms a cluster
on its own or until it satisfies certain termination
criteria:


A desired number of clusters is obtained
The diameter of each cluster is within a certain
threshold
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Data Mining: Concepts and Techniques
40

41. 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
a
b
Step 1
Step 2 Step 3 Step 4
ab
abcde
c
cde
d
de
e
Step 4
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agglomerative
(AGNES)
Step 3
Step 2 Step 1 Step 0
Data Mining: Concepts and Techniques
divisive
(DIANA)
41

42. AGNES (AGglomerative NESting)

Introduced in Kaufmann and Rousseeuw (1990)

Implemented in statistical analysis packages, e.g., Splus

Use the Single-Link method and the Distance matrix.

Merge nodes that have the least distance

Go on in a non-decreasing (in terms of size) fashion

Eventually all nodes belong to the same cluster
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43. AGNES (AGglomerative NESting)

Introduced in Kaufmann and Rousseeuw (1990)

Implemented in statistical analysis packages, e.g., Splus

Use the Single-Link method and the Distance matrix.

Merge nodes that have the least distance

Go on in a non-decreasing (in terms of size) fashion

Eventually all nodes belong to the same cluster
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44. Dendrogram: Shows How the Clusters are Merged
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45. 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
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46. 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
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47. Recent Hierarchical Clustering Methods

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
Integration of hierarchical with distance-based clustering



BIRCH (1996): uses CF-tree and incrementally adjusts
the quality of sub-clusters
ROCK (1999): clustering categorical data by neighbor
and link analysis
CHAMELEON (1999): hierarchical clustering using
dynamic modeling
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48. Chapter 7. Cluster Analysis
1. What is Cluster Analysis?
2. Types of Data in Cluster Analysis
3. A Categorization of Major Clustering Methods
4. Partitioning Methods
5. Hierarchical Methods
6. Outlier Analysis
7. Summary
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49. What Is Outlier Detection?




What are outliers?
 The set of objects are considerably dissimilar from the
remainder of the data
 Example: Sports: Shane Warne, Diego Maradona, ...
Problem: Define and find outliers in large data sets
Applications:
 Credit card fraud detection
 Telecom fraud detection
 Customer segmentation
 Medical analysis
However, one person’s noise could be another
person’s signal
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50. Outlier Detection:
Statistical Approaches


Assume a model underlying distribution that generates
data set (e.g. normal distribution)
Use discordancy tests depending on
 data distribution
 distribution parameter (e.g., mean, variance)
 number of expected outliers
Drawbacks
 most tests are for single attribute
 In many cases, data distribution may not be known
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51. Outlier Detection: Distance-Based
Approach



Introduced to counter the main limitations imposed by
statistical methods
 Find those objects that do not have “enough”
neighbours, where neighbours are defined
based on distance from the given object
Distance-based outlier: A DB(p, D)-outlier is an object O
in a dataset T such that at least a fraction p of the objects
in T lies at a distance greater than D from O
Algorithms for mining distance-based outliers
 Index-based algorithm
 Nested-loop algorithm
 Cell-based algorithm
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52. Density-Based Local
Outlier Detection





Distance-based outlier detection
is based on global distance
distribution
It encounters difficulties to
identify outliers if data is not
uniformly distributed
Ex. C1 contains 400 loosely
distributed points, C2 has 100
tightly condensed points, 2
outlier points o1, o2
Distance-based method cannot
identify o2 as an outlier
Need the concept of local outlier
November 10, 2013

Local outlier factor (LOF)
 Assume outlier is not
crisp
 Each point has a LOF
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53. Outlier Detection: Deviation-Based Approach



Identifies outliers by examining the main characteristics
of objects in a group
Objects that “deviate” from this description are
considered outliers
Sequential exception technique


simulates the way in which humans can distinguish
unusual objects from among a series of supposedly
like objects
OLAP data cube technique

uses data cubes to identify regions of anomalies in
large multidimensional data
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