Capter10 cluster basic

Data Mining:
Concepts and
Techniques
(3 rd ed.)

— Chapter 10 —
Jiawei Han, Micheline Kamber, and Jian Pei
University of Illinois at Urbana-Champaign &
Simon Fraser University
©2013 Han, Kamber & Pei. All rights reserved.
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Chapter 10. Cluster Analysis: Basic
Concepts and Methods


Cluster Analysis: Basic Concepts



Partitioning Methods



Hierarchical Methods

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Density-Based Methods

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Grid-Based Methods



Evaluation of Clustering



Summary
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What is Cluster Analysis?
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Cluster: A collection of data objects
 similar (or related) to one another within the same group
 dissimilar (or unrelated) to the objects in other groups
Cluster analysis (or clustering, data segmentation, …)
 Finding similarities between data according to the
characteristics found in the data and grouping similar
data objects into clusters
Unsupervised learning: no predefined classes (i.e., learning
by observations vs. learning by examples: supervised)
Typical applications
 As a stand-alone tool to get insight into data distribution
 As a preprocessing step for other algorithms
4

Applications of Cluster Analysis


Data reduction


Summarization: Preprocessing for regression, PCA,
classification, and association analysis

Compression: Image processing: vector quantization
Hypothesis generation and testing
Prediction based on groups
 Cluster & find characteristics/patterns for each group


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Finding K-nearest Neighbors
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Localizing search to one or a small number of clusters

Outlier detection: Outliers are often viewed as those “far
away” from any cluster
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Clustering: Application Examples
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Biology: taxonomy of living things: kingdom, phylum, class, order,
family, genus and species
Information retrieval: document clustering
Land use: Identification of areas of similar land use in an earth
observation database
Marketing: Help marketers discover distinct groups in their customer
bases, and then use this knowledge to develop targeted marketing
programs
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
Climate: understanding earth climate, find patterns of atmospheric
and ocean
Economic Science: market resarch
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Basic Steps to Develop a Clustering Task
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Feature selection
 Select info concerning the task of interest
 Minimal information redundancy
Proximity measure
 Similarity of two feature vectors
Clustering criterion
 Expressed via a cost function or some rules
Clustering algorithms
 Choice of algorithms
Validation of the results
 Validation test (also, clustering tendency test)
Interpretation of the results
 Integration with applications
7

Quality: What Is Good Clustering?


A good clustering method will produce high quality
clusters

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high intra-class similarity: cohesive within clusters
low inter-class similarity: distinctive between clusters

The quality of a clustering method depends on


the similarity measure used by the method



its implementation, and

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Its ability to discover some or all of the hidden patterns

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Measure the Quality of Clustering


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Dissimilarity/Similarity metric

Similarity is expressed in terms of a distance function,
typically metric: d(i, j)

The definitions of distance functions are usually rather
different for interval-scaled, boolean, categorical,
ordinal ratio, and vector variables

Weights should be associated with different variables
based on applications and data semantics
Quality of clustering:

There is usually a separate “quality” function that
measures the “goodness” of a cluster.

It is hard to define “similar enough” or “good enough”

The answer is typically highly subjective
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Considerations for Cluster Analysis


Partitioning criteria
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Separation of clusters
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Exclusive (e.g., one customer belongs to only one region) vs. nonexclusive (e.g., one document may belong to more than one class)

Similarity measure
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Single level vs. hierarchical partitioning (often, multi-level
hierarchical partitioning is desirable)

Distance-based (e.g., Euclidian, road network, vector) vs.
connectivity-based (e.g., density or contiguity)

Clustering space
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Full space (often when low dimensional) vs. subspaces (often in
high-dimensional clustering)
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Requirements and Challenges
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Scalability
 Clustering all the data instead of only on samples
Ability to deal with different types of attributes
 Numerical, binary, categorical, ordinal, linked, and mixture of these
Constraint-based clustering
 User may give inputs on constraints
 Use domain knowledge to determine input parameters
Interpretability and usability
Others
 Discovery of clusters with arbitrary shape
 Ability to deal with noisy data
 Incremental clustering and insensitivity to input order
 High dimensionality
11

Major Clustering Approaches (I)
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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, CAMELEON
Density-based approach:
 Based on connectivity and density functions
 Typical methods: DBSACN, OPTICS, DenClue
Grid-based approach:
 based on a multiple-level granularity structure
 Typical methods: STING, WaveCluster, CLIQUE
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Major Clustering Approaches (II)
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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: p-Cluster
User-guided or constraint-based:
 Clustering by considering user-specified or application-specific
constraints
 Typical methods: COD (obstacles), constrained clustering
Link-based clustering:
 Objects are often linked together in various ways
 Massive links can be used to cluster objects: SimRank, LinkClus
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Grid-Based Methods

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Summary
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Partitioning Algorithms: Basic Concept


Partitioning method: Partitioning a database D of n objects into a set of
k clusters, such that the sum of squared distances is minimized (where
ci is the centroid or medoid of cluster Ci)

E = Σ ik=1Σ p∈Ci (d ( p, ci )) 2


Given k, find a partition of k clusters that optimizes the chosen
partitioning criterion
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Global optimal: exhaustively enumerate all partitions

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Heuristic methods: k-means and k-medoids algorithms

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k-means (MacQueen’67, Lloyd’57/’82): 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

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The K-Means Clustering Method


Given k, the k-means algorithm is implemented in four
steps:
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Partition objects into k nonempty subsets
Compute seed points as the centroids of the
clusters of the current partitioning (the centroid is
the center, i.e., mean point, of the cluster)
Assign each object to the cluster with the nearest
seed point
Go back to Step 2, stop when the assignment does
not change
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An Example of K-Means Clustering
K=2
Arbitrarily
partition
objects into
k groups
The initial data set


Loop if
needed

Reassign objects

Partition objects into k nonempty
subsets

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Update the
cluster
centroids

Repeat
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Compute centroid (i.e., mean
point) for each partition

Update the
cluster
centroids

Assign each object to the
cluster of its nearest centroid

Until no change
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Comments on the K-Means
Method
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Strength: Efficient: O(tkn), where n is # objects, k is # clusters, and t is
# iterations. Normally, k, t << n.
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Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks2 + k(n-k))

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Comment: Often terminates at a local optimal

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Weakness
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Applicable only to objects in a continuous n-dimensional space

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Using the k-modes method for categorical data
In comparison, k-medoids can be applied to a wide range of
data

Need to specify k, the number of clusters, in advance (there are
ways to automatically determine the best k (see Hastie et al., 2009)



Sensitive to noisy data and outliers

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Not suitable to discover clusters with non-convex shapes

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Variations of the K-Means Method
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Most of the variants of the k-means which differ in
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Dissimilarity calculations

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Selection of the initial k means

Strategies to calculate cluster means

Handling categorical data: k-modes
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Replacing means of clusters with modes

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Using new dissimilarity measures to deal with categorical objects

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Using a frequency-based method to update modes of clusters

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A mixture of categorical and numerical data: k-prototype method
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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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PAM: A Typical K-Medoids Algorithm
Total Cost = 20
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Arbitrary
choose k
object as
initial
medoids

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Assign
each
remainin
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to
nearest
medoids

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K=2

Until no
change

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If quality is
improved.

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Swapping O
and Oramdom

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Randomly select a
nonmedoid object,Oramdom

Total Cost = 26

Do loop

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Compute
total cost of
swapping

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The K-Medoid Clustering Method
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K-Medoids Clustering: Find representative objects (medoids) in clusters
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PAM (Partitioning Around Medoids, Kaufmann & Rousseeuw 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

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PAM works effectively for small data sets, but does not scale
well for large data sets (due to the computational complexity)

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Efficiency improvement on PAM
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CLARA (Kaufmann & Rousseeuw, 1990): PAM on samples

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CLARANS (Ng & Han, 1994): Randomized re-sampling
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Grid-Based Methods

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Summary
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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

Step 1

Step 2 Step 3 Step 4

ab

b

abcde

c

cde

d

de

e
Step 4

agglomerative
(AGNES)

Step 3

Step 2 Step 1 Step 0

divisive
(DIANA)
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AGNES (Agglomerative Nesting)


Introduced in Kaufmann and Rousseeuw (1990)



Implemented in statistical packages, e.g., Splus

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Use the single-link method and the dissimilarity matrix

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Merge nodes that have the least dissimilarity

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Go on in a non-descending fashion

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Eventually all nodes belong to the same cluster
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Dendrogram: Shows How Clusters are Merged
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

26

DIANA (Divisive Analysis)


Introduced in Kaufmann and Rousseeuw (1990)

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Implemented in statistical analysis packages, e.g., Splus

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Inverse order of AGNES

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Eventually each node forms a cluster on its own

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Distance between
Clusters
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X

X

Single link: smallest distance between an element in one cluster
and an element in the other, i.e., dist(Ki, Kj) = min(tip, tjq)



Complete link: largest distance between an element in one cluster
and an element in the other, i.e., dist(Ki, Kj) = max(tip, tjq)

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Average: avg distance between an element in one cluster and an
element in the other, i.e., dist(Ki, Kj) = avg(tip, tjq)

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Centroid: distance between the centroids of two clusters, i.e.,
dist(Ki, Kj) = dist(Ci, Cj)



Medoid: distance between the medoids of two clusters, i.e., dist(Ki,
Kj) = dist(Mi, Mj)


Medoid: a chosen, centrally located object in the cluster

28

Centroid, Radius and Diameter of
a Cluster (for numerical data
sets)
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Centroid: the “middle” of a cluster

Cm =

X

ΣiN= 1(t

ip

)

N

Radius: square root of average distance from any point
of the cluster to its centroid
Σ N (t − cm ) 2
Rm = i =1 ip
N
Diameter: square root of average mean squared
distance between all pairs of points in the cluster
Σ N Σ N (t − t ) 2
Dm = i =1 i =1 ip iq
N ( N −1)
29

Extensions to Hierarchical Clustering


Major weakness of agglomerative clustering methods
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Can never undo what was done previously

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Do not scale well: time complexity of at least O(n2),
where n is the number of total objects

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Integration of hierarchical & distance-based clustering
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BIRCH (1996): uses CF-tree and incrementally adjusts
the quality of sub-clusters



CHAMELEON (1999): hierarchical clustering using
dynamic modeling

30

BIRCH (Balanced Iterative Reducing
and Clustering Using Hierarchies)



Zhang, Ramakrishnan & Livny, SIGMOD’96
Incrementally construct a CF (Clustering Feature) tree, a hierarchical
data structure for multiphase clustering
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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
31

Clustering Feature Vector in BIRCH
Clustering Feature (CF): CF = (N, LS, SS)
N: Number of data points
LS: linear sum of N points:

N

∑X i

i=
1

SS: square sum of N points

CF = (5, (16,30),(54,190))
N

∑X i

i=
1

2

(3,4)
(2,6)
(4,5)
(4,7)
(3,8)

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CF-Tree in BIRCH
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Clustering feature:
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Summary of the statistics for a given subcluster: the 0-th, 1st,
and 2nd moments of the subcluster from the statistical point
of view
Registers crucial measurements for computing cluster and
utilizes storage efficiently

A CF tree is a height-balanced tree that stores the clustering
features for a hierarchical clustering

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A nonleaf node in a tree has descendants or “children”
The nonleaf nodes store sums of the CFs of their children

A CF tree has two parameters
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Branching factor: max # of children

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Threshold: max diameter of sub-clusters stored at the leaf

33

The CF Tree Structure
Root
B=7

CF1

CF2

CF3

CF6

L=6

child1

child2

child3

child6

Non-leaf node
CF1

CF2

CF3

CF5

child1

child2

child3

child5

Leaf node

prev CF1

CF2

CF6

Leaf node

next

prev CF1

CF2

CF4

next

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The Birch Algorithm


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Cluster Diameter

1
2
∑( x −x )
i
j
n( n −1)

For each point in the input
 Find closest leaf entry
 Add point to leaf entry and update CF
 If entry diameter > max_diameter, then split leaf, and possibly
parents
Algorithm is O(n)
Concerns
 Sensitive to insertion order of data points
 Since we fix the size of leaf nodes, so clusters may not be so
natural
 Clusters tend to be spherical given the radius and diameter
measures

35

CHAMELEON: Hierarchical Clustering
Using Dynamic Modeling (1999)


CHAMELEON: G. Karypis, E. H. Han, and V. Kumar, 1999



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

Graph-based, and a two-phase algorithm
1.

2.

Use a graph-partitioning algorithm: cluster objects into
a large number of relatively small sub-clusters
Use an agglomerative hierarchical clustering algorithm:
find the genuine clusters by repeatedly combining
these sub-clusters

36

KNN Graphs & Interconnectivity


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k-nearest graphs from an original data in 2D:

EC{Ci ,Cj } :The absolute inter-connectivity between Ci and Cj:
the sum of the weight of the edges that connect vertices in
Ci to vertices in Cj
Internal inter-connectivity of a cluster Ci : the size of its mincut bisector ECCi (i.e., the weighted sum of edges that
partition the graph into two roughly equal parts)
Relative Inter-connectivity (RI):
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Relative Closeness & Merge of SubClusters


Relative closeness between a pair of clusters Ci and Cj
: the absolute closeness between Ci and Cj normalized
w.r.t. the internal closeness of the two clusters Ci and Cj





and
are the average weights of the edges that belong
in the min-cut bisector of clusters Ci and Cj , respectively, and
is the average weight of the edges that connect vertices in Ci
to vertices in Cj

Merge Sub-Clusters:


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Merges only those pairs of clusters whose RI and RC are both
above some user-specified thresholds
Merge those maximizing the function that combines RI and RC
38

Overall Framework of
CHAMELEON
Construct (K-NN)
Partition the Graph

Sparse Graph

Data Set

K-NN Graph
P and q are connected if
q is among the top k
closest neighbors of p

Merge Partition

Final Clusters

Relative interconnectivity:
connectivity of c1 and c2
over internal connectivity
Relative closeness:
closeness of c1 and c2 over
internal closeness
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CHAMELEON (Clustering Complex
Objects)

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Probabilistic Hierarchical Clustering


Algorithmic hierarchical clustering

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Hard to handle missing attribute values



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Nontrivial to choose a good distance measure
Optimization goal not clear: heuristic, local search

Probabilistic hierarchical clustering

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Use probabilistic models to measure distances between clusters
Generative model: Regard the set of data objects to be clustered
as a sample of the underlying data generation mechanism to be
analyzed
Easy to understand, same efficiency as algorithmic agglomerative
clustering method, can handle partially observed data

In practice, assume the generative models adopt common distribution
functions, e.g., Gaussian distribution or Bernoulli distribution, governed
by parameters

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Generative Model


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Given a set of 1-D points X = {x1, …, xn} for clustering
analysis & assuming they are generated by a
Gaussian distribution:
The probability that a point xi ∈ X is generated by the
model
The likelihood that X is generated by the model:

The task of learning the generative model: find the
the maximum likelihood
2
parameters μ and σ such that

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Gaussian Distribution
Bean
machine:
drop ball
with pins

1-d
Gaussia
n

2-d
Gaussia
n
From wikipedia and http://home.dei.polimi.it
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A Probabilistic Hierarchical Clustering Algorithm


For a set of objects partitioned into m clusters C1, . . . ,Cm, the quality
can be measured by,



where P() is the maximum likelihood
If we merge two clusters Cj1 and Cj2 into a cluster Cj1∪Cj2, then, the
change in quality of the overall clustering is



Distance between clusters C1 and C2:



If dist(Ci, Cj) < 0, merge Ci and Cj
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Grid-Based Methods

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Summary
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Density-Based Clustering Methods


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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) (more gridbased)

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Density-Based Clustering: Basic
Concepts


Two parameters:

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Eps: Maximum radius of the neighbourhood
MinPts: Minimum number of points in an Epsneighbourhood of that point

NEps(q): {p belongs to D | dist(p,q) ≤ Eps}
Directly density-reachable: A point p is directly densityreachable from a point q w.r.t. Eps, MinPts if



p belongs to NEps(q)

p

core point condition:
|NEps (q)| ≥ MinPts

MinPts = 5
Eps = 1 cm

q
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Density-Reachable and DensityConnected


Density-reachable:




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

Density-connected


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

p
p1

q

p

q
o

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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

49

DBSCAN: The Algorithm










Arbitrary select a point p
Retrieve all points density-reachable from p w.r.t. 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
If a spatial index is used, the computational complexity of DBSCAN
is O(nlogn), where n is the number of database objects. Otherwise,
the complexity is O(n2)

50

DBSCAN: Sensitive to Parameters

DBSCAN online Demo:
http://webdocs.cs.ualberta.ca/~yaling/Cluster/Applet/Code/Cluster.html

51

OPTICS: A Cluster-Ordering Method
(1999)


OPTICS: Ordering Points To Identify the Clustering
Structure
 Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99)
 Produces a special order of the database wrt its
density-based clustering structure
 This cluster-ordering contains info equiv to the densitybased 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
52

OPTICS: Some Extension from DBSCAN




Index-based: k = # of dimensions, N: # of points
 Complexity: O(N*logN)
Core Distance of an object p: the smallest value ε such that
the ε-neighborhood of p has at least MinPts objects
Let Nε(p): ε-neighborhood of p, ε is a distance value
Core-distanceε, MinPts(p) = Undefined if card(Nε(p)) < MinPts



MinPts-distance(p), otherwise
Reachability Distance of object p from core object q is the
min radius value that makes p density-reachable from q
Reachability-distanceε, MinPts(p, q) =
Undefined if q is not a core object
max(core-distance(q), distance (q, p)), otherwise

53

Core Distance & Reachability
Distance

54

Reachabilitydistance
undefined

ε

ε

‘

ε

Cluster-order of the objects
55

Density-Based Clustering: OPTICS & Applications

demo:

http://www.dbs.informatik.uni-muenchen.de/Forschung/KDD/Clustering/OPTICS/Demo

56

DENCLUE: Using Statistical Density
Functions



DENsity-based CLUstEring by Hinneburg & Keim (KDD’98)
Using statistical density functions:

f Gaussian ( x , y ) = e


Major features

−

d ( x,y)
2σ 2

2

influence of y
on x

f
∇f

D
Gaussian

D
Gaussian

( x ) = ∑=1 e
i

d ( x , xi ) 2
−
2σ 2

( x, xi ) = ∑i =1 ( xi − x) ⋅ e



Solid mathematical foundation



Good for data sets with large amounts of noise



N

total influence
on x

N

−

d ( x , xi ) 2
2σ 2

gradient of x in
the direction of xi

Allows a compact mathematical description of arbitrarily shaped
clusters in high-dimensional data sets



Significant faster than existing algorithm (e.g., DBSCAN)



But needs a large number of parameters
57

Denclue: Technical Essence













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
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
influence function of all data points
Clusters can be determined mathematically by identifying density
attractors
Density attractors are local maximal of the overall density function
Center defined clusters: assign to each density attractor the points
density attracted to it
Arbitrary shaped cluster: merge density attractors that are connected
through paths of high density (> threshold)
58

Center-Defined and Arbitrary

60















Grid-Based Methods






Summary
61

Grid-Based Clustering Method



Using multi-resolution grid data structure
Several interesting methods
 STING (a STatistical INformation Grid
approach) by Wang, Yang and Muntz (1997)


CLIQUE: Agrawal, et al. (SIGMOD’98)




Both grid-based and subspace clustering

WaveCluster by Sheikholeslami, Chatterjee,
and Zhang (VLDB’98)


A multi-resolution clustering approach
using wavelet method
62

STING: A Statistical Information Grid
Approach




Wang, Yang and Muntz (VLDB’97)
The spatial area is divided into rectangular cells
There are several levels of cells corresponding to different
levels of resolution

63

The STING Clustering Method











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
Start from a pre-selected layer—typically with a small
number of cells
For each cell in the current level compute the confidence
interval

64

STING Algorithm and Its Analysis








Remove the irrelevant cells from further consideration
When finish examining the current layer, proceed to the
next lower level
Repeat this process until the bottom layer is reached
Advantages:
 Query-independent, easy to parallelize, incremental
update
 O(K), where K is the number of grid cells at the lowest
level
Disadvantages:
 All the cluster boundaries are either horizontal or
vertical, and no diagonal boundary is detected
65

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
66

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.

Generate minimal description for the clusters
 Determine maximal regions that cover a cluster of
connected dense units for each cluster
 Determination of minimal cover for each cluster
67

τ=3
30
40
Vacation

20
50

Salary
(10,000)
0 1 2 3 4 5 6 7

a
al
S
ry

30

Vacation
(week)
0 1 2 3 4 5 6 7

age
60
20

50
30
40
50

age
60

age

68

Strength and Weakness of CLIQUE




Strength
 automatically finds subspaces of the highest
dimensionality such that high density clusters exist in
those subspaces
 insensitive to the order of records in input and does not
presume some canonical data distribution
 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
69















Grid-Based Methods






Summary
70

Determine the Number of Clusters






Empirical method
 # of clusters: k ≈√n/2 for a dataset of n points, e.g., n = 200, k = 10
Elbow method
 Use the turning point in the curve of sum of within cluster variance
w.r.t the # of clusters
Cross validation method
 Divide a given data set into m parts
 Use m – 1 parts to obtain a clustering model
 Use the remaining part to test the quality of the clustering

E.g., For each point in the test set, find the closest centroid, and
use the sum of squared distance between all points in the test set
and the closest centroids to measure how well the model fits the
test set
 For any k > 0, repeat it m times, compare the overall quality measure
w.r.t. different k’s, and find # of clusters that fits the data the best
71

Measuring Clustering Quality


3 kinds of measures: External, internal and relative



External: supervised, employ criteria not inherent to the dataset




Internal: unsupervised, criteria derived from data itself




Compare a clustering against prior or expert-specified
knowledge (i.e., the ground truth) using certain clustering
quality measure
Evaluate the goodness of a clustering by considering how
well the clusters are separated, and how compact the
clusters are, e.g., Silhouette coefficient

Relative: directly compare different clusterings, usually those
obtained via different parameter settings for the same algorithm
72

Measuring Clustering Quality: External Methods




Clustering quality measure: Q(C, T), for a clustering C
given the ground truth T
Q is good if it satisfies the following 4 essential criteria
 Cluster homogeneity: the purer, the better
 Cluster completeness: should assign objects belong to
the same category in the ground truth to the same
cluster
 Rag bag: putting a heterogeneous object into a pure
cluster should be penalized more than putting it into a
rag bag (i.e., “miscellaneous” or “other” category)
 Small cluster preservation: splitting a small category
into pieces is more harmful than splitting a large
category into pieces
73

Some Commonly Used External Measures




Matching-based measures
 Purity, maximum matching, F-measure
Entropy-Based Measures
Ground truth partitioning T
 Conditional entropy, normalized mutual
Cluster
Cluster
C
information (NMI), variation of information C
Pair-wise measures
 Four possibilities: True positive (TP), FN, FP,
TN
 Jaccard coefficient, Rand statistic, FowlkesMallow measure
Correlation measures
 Discretized Huber static, normalized discretized
Huber static
1





1

T2

2

74

Entropy-Based Measure (I):
Conditional Entropy






Entropy of clustering C:
Entropy of partitioning T:
Entropy of T w.r.t. cluster Ci:
Conditional entropy of T
w.r.t. clustering C:
The more a cluster’s members are split into different partitions,
the higher the conditional entropy
For a perfect clustering, the conditional entropy value is 0,
where the worst possible conditional entropy value is log k


75

Entropy-Based Measure (II):

Normalized mutual information (NMI)


Mutual information: quantify the amount of shared info between
the clustering C and partitioning T:

It measures the dependency between the observed joint probability p ij of
C and T, and the expected joint probability pCi * pTj under the
independence assumption
When C and T are independent, pij = pCi * pTj, I(C, T) = 0. However, there
is no upper bound on the mutual information


Normalized mutual information (NMI)

Value range of NMI: [0,1]. Value close to 1 indicates a good clustering
76















Grid-Based Methods






Summary
77

Summary















Cluster analysis groups objects based on their similarity and has
wide applications
Measure of similarity can be computed for various types of data
Clustering algorithms can be categorized into partitioning methods,
hierarchical methods, density-based methods, grid-based methods,
and model-based methods
K-means and K-medoids algorithms are popular partitioning-based
clustering algorithms
Birch and Chameleon are interesting hierarchical clustering
algorithms, and there are also probabilistic hierarchical clustering
algorithms
DBSCAN, OPTICS, and DENCLU are interesting density-based
algorithms
STING and CLIQUE are grid-based methods, where CLIQUE is also
a subspace clustering algorithm
Quality of clustering results can be evaluated in various ways

78

CS512-Spring 2011: An Introduction


Coverage


Cluster Analysis: Chapter 11



Outlier Detection: Chapter 12



Mining Sequence Data: BK2: Chapter 8



Mining Graphs Data: BK2: Chapter 9



Social and Information Network Analysis


BK2: Chapter 9



Partial coverage: Mark Newman: “Networks: An Introduction”, Oxford U., 2010









Scattered coverage: Easley and Kleinberg, “Networks, Crowds, and Markets:
Reasoning About a Highly Connected World”, Cambridge U., 2010
Recent research papers

Mining Data Streams: BK2: Chapter 8

Requirements


One research project



One class presentation (15 minutes)



Two homeworks (no programming assignment)



Two midterm exams (no final exam)

81

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.
Beil F., Ester M., Xu X.: "Frequent Term-Based Text Clustering", KDD'02
M. M. Breunig, H.-P. Kriegel, R. Ng, J. Sander. LOF: Identifying Density-Based
Local Outliers. SIGMOD 2000.
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. VLDB’98.
V. Ganti, J. Gehrke, R. Ramakrishan. CACTUS Clustering Categorical Data
Using Summaries. KDD'99.
82

References (2)
















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.
S. Guha, R. Rastogi, and K. Shim. ROCK: A robust clustering algorithm for
categorical attributes. In ICDE'99, pp. 512-521, Sydney, Australia, March
1999.
A. Hinneburg, D.l A. Keim: An Efficient Approach to Clustering in Large
Multimedia Databases with Noise. KDD’98.
A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall,
1988.
G. Karypis, E.-H. Han, and V. Kumar. CHAMELEON: A Hierarchical
Clustering Algorithm Using Dynamic Modeling. COMPUTER, 32(8): 68-75,
1999.
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.
83

References (3)






















G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to
Clustering. John Wiley and Sons, 1988.
R. Ng and J. Han. Efficient and effective clustering method for spatial data mining.
VLDB'94.
L. Parsons, E. Haque and H. Liu, Subspace Clustering for High Dimensional Data: A
Review, SIGKDD Explorations, 6(1), June 2004
E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large
data sets. Proc. 1996 Int. Conf. on Pattern Recognition,.
G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution
clustering approach for very large spatial databases. VLDB’98.
A. K. H. Tung, J. Han, L. V. S. Lakshmanan, and R. T. Ng. Constraint-Based
Clustering in Large Databases, ICDT'01.
A. K. H. Tung, J. Hou, and J. Han. Spatial Clustering in the Presence of Obstacles,
ICDE'01
H. Wang, W. Wang, J. Yang, and P.S. Yu. Clustering by pattern similarity in large
data sets, SIGMOD’ 02.
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.
Xiaoxin Yin, Jiawei Han, and Philip Yu, “
LinkClus: Efficient Clustering via Heterogeneous Semantic Links”, in Proc. 2006 Int.
Conf. on Very Large Data Bases (VLDB'06), Seoul, Korea, Sept. 2006.
84



What Is Cluster Analysis?

What is Good Clustering? Measuring the Quality of Clustering

Major categories of clustering methods
Clustering structures

Calculating Distance between Clusters

k-Means: A Classical Partitioning Method

Alternative Methods: k-Medoids, k-Median, and its Variations

Agglomerative and Divisive Hierarchical Clustering

BIRCH: A Hierarchical, Micro-Clustering Approach

Chameleon: A Hierarchical Clustering Algorithm Using Dynamic Modeling

DBSCAN and OPTICS: Density-Based Clustering Based on Connected Regions

DENCLUE: Clustering Based on Density Distribution Functions
Link-Based Cluster Analysis

SimRank: Exploring Links in Cluster Analysis

LinkClus: Scalability in Link-Based Cluster Analysis
Grid-Based Methods

STING: STatistical INformation Grid

WaveCluster: Clustering Using Wavelet Transformation

CLIQUE: A Dimension-Growth Subspace Clustering Method
Summary
















85

85

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
6
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

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

K=2

Until no
change

10

If quality is
improved.

3

4

5

6

7

8

9

10

10

9

Swapping O
and Oramdom

2

Randomly select a
nonmedoid object,Oramdom

Total Cost = 26

Do loop

1

Compute
total cost of
swapping

8
7
6
5

9
8
7
6
5

4

4

3

3

2

2

1

1

0

0
0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

87

PAM (Partitioning Around Medoids)
(1987)


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 TC ih
For each pair of i and h ,






If TCih < 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
88

PAM Clustering: Finding the Best Cluster
Center



Case 1: p currently belongs to oj. If oj is replaced by orandom as a
representative object and p is the closest to one of the other
representative object oi, then p is reassigned to oi

89

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)
90

CLARA (Clustering Large
Applications) (1990)


CLARA (Kaufmann and Rousseeuw in 1990)



Built in statistical analysis packages, such as SPlus
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
91

CLARANS (“Randomized” CLARA)
(1994)






CLARANS (A Clustering Algorithm based on Randomized
Search) (Ng and Han’94)
 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, it starts with new randomly
selected node in search for a new local optimum
Advantages: More efficient and scalable than both PAM
and CLARA
Further improvement: Focusing techniques and spatial
access structures (Ester et al.’95)
92

ROCK: Clustering Categorical Data








ROCK: RObust Clustering using linKs
 S. Guha, R. Rastogi & K. Shim, ICDE’99
Major ideas
 Use links to measure similarity/proximity
 Not distance-based
Algorithm: sampling-based clustering
 Draw random sample
 Cluster with links
 Label data in disk
Experiments
 Congressional voting, mushroom data
93

Similarity Measure in ROCK






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 co-efficient 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 co-efficient-based similarity function: Sim( T1 , T2 ) = T1 ∩T2
T1 ∪T2


Ex. Let T1 = {a, b, c}, T2 = {c, d, e}
Sim(T 1, T 2) =

{c}

{a, b, c, d , e}

=

1
=0.2
5
94

Link Measure in ROCK




Clusters

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}
Neighbors

Two transactions are neighbors if sim(T1,T2) > threshold
Let T1 = {a, b, c}, T2 = {c, d, e}, T3 = {a, b, f}
 T connected to: {a,b,d}, {a,b,e}, {a,c,d}, {a,c,e}, {b,c,d}, {b,c,e}, {a,b,f},
1
{a,b,g}
 T connected to: {a,c,d}, {a,c,e}, {a,d,e}, {b,c,e}, {b,d,e}, {b,c,d}
2
 T connected to: {a,b,c}, {a,b,d}, {a,b,e}, {a,b,g}, {a,f,g}, {b,f,g}
3
Link Similarity

Link similarity between two transactions is the # of common neighbors






link(T1, T2) = 4, since they have 4 common neighbors




{a, c, d}, {a, c, e}, {b, c, d}, {b, c, e}

link(T1, T3) = 3, since they have 3 common neighbors


{a, b, d}, {a, b, e}, {a, b, g}

95

Measuring Clustering Quality: External Methods




Clustering quality measure: Q(C, Cg), for a clustering C
given the ground truth Cg.

Q is good if it satisfies the following 4 essential criteria
 Cluster homogeneity: the purer, the better
 Cluster completeness: should assign objects belong to
the same category in the ground truth to the same
cluster
 Rag bag: putting a heterogeneous object into a pure
cluster should be penalized more than putting it into a
rag bag (i.e., “miscellaneous” or “other” category)
 Small cluster preservation: splitting a small category
into pieces is more harmful than splitting a large
category into pieces

97

Assessing Clustering Tendency




Assess if non-random structure exists in the data by measuring the
probability that the data is generated by a uniform data distribution
Test spatial randomness by statistic test: Hopkins Static
 Given a dataset D regarded as a sample of a random variable o,
determine how far away o is from being uniformly distributed in the
data space
 Sample n points, p , …, p , uniformly from D. For each p , find its
1
n
i
nearest neighbor in D: xi = min{dist (pi, v)} where v in D






Sample n points, q1, …, qn, uniformly from D. For each qi, find its
nearest neighbor in D – {qi}: yi = min{dist (qi, v)} where v in D and v
≠ qi
Calculate the Hopkins Statistic:
If D is uniformly distributed, ∑ xi and ∑ yi will be close to each other
and H is close to 0.5. If D is clustered, H is close to 1

98

Capter10 cluster basic

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