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Data Mining
Cluster Analysis: Basic Concepts
and Algorithms
What is Cluster Analysis?
l
Lecture Notes for Chapter 8
Finding groups of objects such that the objects in a group
will be similar (or related) to one another and different
from (or unrelated to) the objects in other groups
Inter-cluster
distances are
maximized
Intra-cluster
distances are
minimized
Introduction to Data Mining
by
Tan, Steinbach, Kumar
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
1
Applications of Cluster Analysis
l
– Group related documents
for browsing, group genes
and proteins that have
similar functionality, or
group stocks with similar
price fluctuations
l
Industry Group
Applied-Matl-DOWN,Bay-Network-Down,3-COM-DOWN,
Cabletron-Sys-DOWN,CISCO-DOWN,HP-DOWN,
DSC-Comm-DOWN,INTEL-DOWN,LSI-Logic-DOWN,
Micron-Tech-DOWN,Texas-Inst-Down,Tellabs-Inc-Down,
Natl-Semiconduct-DOWN,Oracl-DOWN,SGI-DOWN,
Sun-DOWN
Apple-Comp-DOWN,Autodesk-DOWN,DEC-DOWN,
ADV-Micro-Device-DOWN,Andrew-Corp-DOWN,
Computer-Assoc-DOWN,Circuit-City-DOWN,
Compaq-DOWN, EMC-Corp-DOWN, Gen-Inst-DOWN,
Motorola-DOWN,Microsoft-DOWN,Scientific-Atl-DOWN
1
2
Fannie-Mae-DOWN,Fed-Home-Loan-DOWN,
MBNA-Corp-DOWN,Morgan-Stanley-DOWN
3
4
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Technology2-DOWN
l
Simple segmentation
– Dividing students into different registration groups
alphabetically, by last name
Financial-DOWN
Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP,
Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP,
Schlumberger-UP
Supervised classification
– Have class label information
Technology1-DOWN
Oil-UP
l
Summarization
Results of a query
– Groupings are a result of an external specification
– Reduce the size of large
data sets
l
Introduction to Data Mining
4/18/2004
3
Notion of a Cluster can be Ambiguous
How many clusters?
Graph partitioning
– Some mutual relevance and synergy, but areas are not
identical
Clustering precipitation
in Australia
© Tan,Steinbach, Kumar
Introduction to Data Mining
What is not Cluster Analysis?
Discovered Clusters
Understanding
© Tan,Steinbach, Kumar
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
4
Types of Clusterings
Six Clusters
l
A clustering is a set of clusters
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Important distinction between hierarchical and
partitional sets of clusters
l
Partitional Clustering
– A division data objects into non-overlapping subsets (clusters)
such that each data object is in exactly one subset
l
Two Clusters
© Tan,Steinbach, Kumar
Introduction to Data Mining
Hierarchical clustering
– A set of nested clusters organized as a hierarchical tree
Four Clusters
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© Tan,Steinbach, Kumar
Introduction to Data Mining
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Partitional Clustering
Hierarchical Clustering
p1
p3
p4
p2
p1 p2
Traditional Hierarchical Clustering
p3 p4
Traditional Dendrogram
p1
p3
p4
p2
p1 p2
Original Points
Non-traditional Hierarchical Clustering
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
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Exclusive versus non-exclusive
– In non-exclusive clusterings, points may belong to multiple
clusters.
– Can represent multiple classes or ‘border’ points
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Introduction to Data Mining
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8
l
Well-separated clusters
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Center-based clusters
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Contiguous clusters
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Density-based clusters
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Property or Conceptual
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Described by an Objective Function
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Fuzzy versus non-fuzzy
– In fuzzy clustering, a point belongs to every cluster with some
weight between 0 and 1
– Weights must sum to 1
– Probabilistic clustering has similar characteristics
l
Partial versus complete
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Heterogeneous versus homogeneous
– In some cases, we only want to cluster some of the data
– Cluster of widely different sizes, shapes, and densities
© Tan,Steinbach, Kumar
Introduction to Data Mining
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Types of Clusters: Well-Separated
l
© Tan,Steinbach, Kumar
Non-traditional Dendrogram
Types of Clusters
Other Distinctions Between Sets of Clusters
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p3 p4
A Partitional Clustering
© Tan,Steinbach, Kumar
Types of Clusters: Center-Based
Well-Separated Clusters:
l
– A cluster is a set of points such that any point in a cluster is
closer (or more similar) to every other point in the cluster than
to any point not in the cluster.
Center-based
– A cluster is a set of objects such that an object in a cluster is
closer (more similar) to the “center” of a cluster, than to the
center of any other cluster
– The center of a cluster is often a centroid, the average of all
the points in the cluster, or a medoid, the most “representative”
point of a cluster
3 well-separated clusters
© Tan,Steinbach, Kumar
Introduction to Data Mining
Introduction to Data Mining
4 center-based clusters
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© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
12
Types of Clusters: Contiguity-Based
l
Types of Clusters: Density-Based
Contiguous Cluster (Nearest neighbor or
Transitive)
l
Density-based
– A cluster is a dense region of points, which is separated by
low-density regions, from other regions of high density.
– A cluster is a set of points such that a point in a cluster is
closer (or more similar) to one or more other points in the
cluster than to any point not in the cluster.
– Used when the clusters are irregular or intertwined, and when
noise and outliers are present.
8 contiguous clusters
© Tan,Steinbach, Kumar
Introduction to Data Mining
6 density-based clusters
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Introduction to Data Mining
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Types of Clusters: Objective Function
Types of Clusters: Conceptual Clusters
l
© Tan,Steinbach, Kumar
Shared Property or Conceptual Clusters
l
– Finds clusters that share some common property or represent
a particular concept.
Clusters Defined by an Objective Function
– Finds clusters that minimize or maximize an objective function.
– Enumerate all possible ways of dividing the points into clusters and
evaluate the `goodness' of each potential set of clusters by using
the given objective function. (NP Hard)
.
–
Can have global or local objectives.
u
Hierarchical clustering algorithms typically have local objectives
u
Partitional algorithms typically have global objectives
– A variation of the global objective function approach is to fit the
data to a parameterized model.
u
Parameters for the model are determined from the data.
Mixture models assume that the data is a ‘mixture' of a number of
statistical distributions.
u
2 Overlapping Circles
© Tan,Steinbach, Kumar
Introduction to Data Mining
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Types of Clusters: Objective Function …
l
© Tan,Steinbach, Kumar
– Proximity matrix defines a weighted graph, where the
nodes are the points being clustered, and the
weighted edges represent the proximities between
points
l
Type of proximity or density measure
l
Sparseness
– Dictates type of similarity
– Adds to efficiency
l
Attribute type
l
Type of Data
– Dictates type of similarity
– Dictates type of similarity
– Other characteristics, e.g., autocorrelation
l
l
– Want to minimize the edge weight between clusters
and maximize the edge weight within clusters
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– This is a derived measure, but central to clustering
– Clustering is equivalent to breaking the graph into
connected components, one for each cluster.
Introduction to Data Mining
4/18/2004
Characteristics of the Input Data Are Important
Map the clustering problem to a different domain
and solve a related problem in that domain
© Tan,Steinbach, Kumar
Introduction to Data Mining
l
17
Dimensionality
Noise and Outliers
Type of Distribution
© Tan,Steinbach, Kumar
Introduction to Data Mining
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Clustering Algorithms
Partitional clustering approach
Each cluster is associated with a centroid (center point)
Each point is assigned to the cluster with the closest
centroid
Number of clusters, K, must be specified
The basic algorithm is very simple
l
K-means and its variants
l
l
l
Hierarchical clustering
l
Density-based clustering
l
© Tan,Steinbach, Kumar
l
Introduction to Data Mining
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© Tan,Steinbach, Kumar
K-means Clustering – Details
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Often the stopping condition is changed to ‘Until relatively few
points change clusters’
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n = number of points, K = number of clusters,
I = number of iterations, d = number of attributes
© Tan,Steinbach, Kumar
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Complexity is O( n * K * I * d )
–
Original Points
2
The centroid is (typically) the mean of the points in the
cluster.
‘Closeness’ is measured by Euclidean distance, cosine
similarity, correlation, etc.
K-means will converge for common similarity measures
mentioned above.
Most of the convergence happens in the first few
iterations.
–
l
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Clusters produced vary from one run to another.
y
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Two different K-means Clusterings
Initial centroids are often chosen randomly.
–
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© Tan,Steinbach, Kumar
Importance of Choosing Initial Centroids
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Optimal Clustering
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Importance of Choosing Initial Centroids
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Evaluating K-means Clusters
l
Importance of Choosing Initial Centroids …
Most common measure is Sum of Squared Error (SSE)
Iteration 5
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– For each point, the error is the distance to the nearest cluster
– To get SSE, we square these errors and sum them.
3
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SSE = ∑ ∑ dist ( mi , x )
2
2
i =1 x∈Ci
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– x is a data point in cluster Ci and mi is the representative point for
cluster Ci
u
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can show that mi corresponds to the center (mean) of the cluster
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– Given two clusters, we can choose the one with the smallest
error
– One easy way to reduce SSE is to increase K, the number of
clusters
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A good clustering with smaller K can have a lower SSE than a poor
clustering with higher K
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© Tan,Steinbach, Kumar
Introduction to Data Mining
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Importance of Choosing Initial Centroids …
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For example, if K = 10, then probability = 10!/1010 = 0.00036
–
Sometimes the initial centroids will readjust themselves in
‘right’ way, and sometimes they don’t
–
Consider an example of five pairs of clusters
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If there are K ‘real’ clusters then the chance of selecting
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Problems with Selecting Initial Points
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10 Clusters Example
© Tan,Steinbach, Kumar
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10 Clusters Example
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Starting with two initial centroids in one cluster of each pair of clusters
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10 Clusters Example
10 Clusters Example
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Solutions to Initial Centroids Problem
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Starting with some pairs of clusters having three initial centroids, while other have only one.
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Starting with some pairs of clusters having three initial centroids, while other have only one.
© Tan,Steinbach, Kumar
Introduction to Data Mining
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Handling Empty Clusters
Multiple runs
l
Basic K-means algorithm can yield empty
clusters
l
Several strategies
– Helps, but probability is not on your side
Sample and use hierarchical clustering to
determine initial centroids
l Select more than k initial centroids and then
select among these initial centroids
l
– Choose the point that contributes most to SSE
– Choose a point from the cluster with the highest SSE
– If there are several empty clusters, the above can be
repeated several times.
– Select most widely separated
Postprocessing
l Bisecting K-means
l
– Not as susceptible to initialization issues
© Tan,Steinbach, Kumar
Introduction to Data Mining
4/18/2004
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Updating Centers Incrementally
l
l
An alternative is to update the centroids after
each assignment (incremental approach)
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Pre-processing
Post-processing
– Eliminate small clusters that may represent outliers
– Split ‘loose’ clusters, i.e., clusters with relatively high
SSE
– Merge clusters that are ‘close’ and that have relatively
low SSE
– Can use these steps during the clustering process
u
Introduction to Data Mining
4/18/2004
– Normalize the data
– Eliminate outliers
l
Each assignment updates zero or two centroids
More expensive
Introduces an order dependency
Never get an empty cluster
Can use “weights” to change the impact
© Tan,Steinbach, Kumar
Introduction to Data Mining
Pre-processing and Post-processing
In the basic K-means algorithm, centroids are
updated after all points are assigned to a centroid
–
–
–
–
–
© Tan,Steinbach, Kumar
4/18/2004
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ISODATA
© Tan,Steinbach, Kumar
Introduction to Data Mining
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Bisecting K-means
l
Bisecting K-means Example
Bisecting K-means algorithm
–
Variant of K-means that can produce a partitional or a
hierarchical clustering
© Tan,Steinbach, Kumar
Introduction to Data Mining
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Limitations of K-means
l
© Tan,Steinbach, Kumar
Introduction to Data Mining
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Limitations of K-means: Differing Sizes
K-means has problems when clusters are of
differing
– Sizes
– Densities
– Non-globular shapes
l
K-means has problems when the data contains
outliers.
K-means (3 Clusters)
Original Points
© Tan,Steinbach, Kumar
Introduction to Data Mining
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Limitations of K-means: Differing Density
© Tan,Steinbach, Kumar
Introduction to Data Mining
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Introduction to Data Mining
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Limitations of K-means: Non-globular Shapes
Original Points
K-means (3 Clusters)
Original Points
© Tan,Steinbach, Kumar
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© Tan,Steinbach, Kumar
K-means (2 Clusters)
Introduction to Data Mining
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Overcoming K-means Limitations
Original Points
Overcoming K-means Limitations
K-means Clusters
Original Points
K-means Clusters
One solution is to use many clusters.
Find parts of clusters, but need to put together.
© Tan,Steinbach, Kumar
Introduction to Data Mining
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© Tan,Steinbach, Kumar
Introduction to Data Mining
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Hierarchical Clustering
Overcoming K-means Limitations
Produces a set of nested clusters organized as a
hierarchical tree
l Can be visualized as a dendrogram
l
– A tree like diagram that records the sequences of
merges or splits
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Original Points
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Strengths of Hierarchical Clustering
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Hierarchical Clustering
Do not have to assume any particular number of
clusters
l
Two main types of hierarchical clustering
– Agglomerative:
– Any desired number of clusters can be obtained by
‘cutting’ the dendogram at the proper level
l
4
u
Start with the points as individual clusters
u At each step, merge the closest pair of clusters until only one cluster
(or k clusters) left
They may correspond to meaningful taxonomies
– Divisive:
u
– Example in biological sciences (e.g., animal kingdom,
phylogeny reconstruction, …)
Start with one, all-inclusive cluster
At each step, split a cluster until each cluster contains a point (or
there are k clusters)
u
l
Traditional hierarchical algorithms use a similarity or
distance matrix
– Merge or split one cluster at a time
© Tan,Steinbach, Kumar
Introduction to Data Mining
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© Tan,Steinbach, Kumar
Introduction to Data Mining
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Starting Situation
Agglomerative Clustering Algorithm
l
More popular hierarchical clustering technique
l
Basic algorithm is straightforward
l
Start with clusters of individual points and a
proximity matrix
p1 p2
p3
p4 p5
1.
Compute the proximity matrix
p1
2.
3.
4.
Let each data point be a cluster
Repeat
Merge the two closest clusters
p2
5.
6.
Update the proximity matrix
Until only a single cluster remains
p5
p3
p4
.
.
Proximity Matrix
.
l
Key operation is the computation of the proximity of
two clusters
–
Different approaches to defining the distance between
clusters distinguish the different algorithms
© Tan,Steinbach, Kumar
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Intermediate Situation
l
© Tan,Steinbach, Kumar
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Intermediate Situation
After some merging steps, we have some clusters
C1
C2
l
C3
C4
C5
We want to merge the two closest clusters (C2 and C5) and
update the proximity matrix.
C1 C2
C3
C4 C5
C1
C1
C2
C2
C3
C3
C3
C3
C4
C4
C4
C5
C4
C5
Proximity Matrix
Proximity Matrix
C1
C1
C5
C2
© Tan,Steinbach, Kumar
C2
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After Merging
l
...
© Tan,Steinbach, Kumar
C5
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How to Define Inter-Cluster Similarity
p1
The question is “How do we update the proximity matrix?”
C1
C1
C2 U C5
C3
C4
C2
U
C5
Similarity?
C3
C4
?
?
?
p4 p5
...
p3
p4
C3
?
l
C4
?
l
Proximity Matrix
C1
p3
p2
?
?
p2
p1
l
l
l
C2 U C5
p5
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
– Ward’s Method uses squared error
© Tan,Steinbach, Kumar
Introduction to Data Mining
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© Tan,Steinbach, Kumar
Introduction to Data Mining
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How to Define Inter-Cluster Similarity
p1
l
l
l
l
l
p2
How to Define Inter-Cluster Similarity
p3
p4 p5
p1
...
p1
p1
p2
p2
p3
p3
p4
p4
p5
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
l
l
l
l
l
– Ward’s Method uses squared error
© Tan,Steinbach, Kumar
Introduction to Data Mining
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p1
55
p2
© Tan,Steinbach, Kumar
Introduction to Data Mining
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p3
p4 p5
p1
...
×
l
l
l
l
l
×
p2
p5
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
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2
I4
0.65
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1.00
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3
– Determined by one pair of points, i.e., by one link in
the proximity graph.
I3
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Hierarchical Clustering: MIN
Similarity of two clusters is based on the two
most similar (closest) points in the different
clusters
I2
0.90
1.00
0.70
0.60
0.50
...
– Ward’s Method uses squared error
Cluster Similarity: MIN or Single Link
I1
1.00
0.90
0.10
0.65
0.20
p4 p5
p4
p5
– Ward’s Method uses squared error
l
p3
p3
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
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p2
p1
p4
l
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How to Define Inter-Cluster Similarity
p3
l
...
p5
p2
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p4 p5
MIN
.
MAX
.
Group Average
.
Proximity Matrix
Distance Between Centroids
Other methods driven by an objective
function
p1
l
p3
– Ward’s Method uses squared error
How to Define Inter-Cluster Similarity
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p2
3
0.15
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Nested Clusters
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Dendrogram
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Strength of MIN
Limitations of MIN
Original Points
Two Clusters
Original Points
• Can handle non-elliptical shapes
© Tan,Steinbach, Kumar
• Sensitive to noise and outliers
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Cluster Similarity: MAX or Complete Linkage
l
Two Clusters
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Hierarchical Clustering: MAX
Similarity of two clusters is based on the two least
similar (most distant) points in the different
clusters
– Determined by all pairs of points in the two clusters
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1
2
5
5
0.4
0.35
2
0.3
0.25
3
I1
I2
I3
I4
I5
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0.10
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0.20
I2
0.90
1.00
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I3
0.10
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1.00
0.40
0.30
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0.65
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1.00
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I5
0.20
0.50
0.30
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1.00
3
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Nested Clusters
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Dendrogram
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Strength of MAX
Original Points
6
1
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Limitations of MAX
Two Clusters
Original Points
Two Clusters
•Tends to break large clusters
• Less susceptible to noise and outliers
•Biased towards globular clusters
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© Tan,Steinbach, Kumar
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Cluster Similarity: Group Average
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Hierarchical Clustering: Group Average
Proximity of two clusters is the average of pairwise proximity
between points in the two clusters.
∑ proximity(p , p )
proximity(Clusteri , Clusterj ) =
l
i
pi∈Clusteri
p j∈Clusterj
5
j
5
Need to use average connectivity for scalability since total
proximity favors large clusters
I1
I2
I3
I4
I5
I1
1.00
0.90
0.10
0.65
0.20
I2
0.90
1.00
0.70
0.60
0.50
I3
0.10
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1.00
0.40
0.30
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I4
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1
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2
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6
1
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1.00
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3
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Nested Clusters
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4
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Dendrogram
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Hierarchical Clustering: Group Average
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4
2
|Clusteri |∗|Clusterj |
Compromise between Single and Complete
Link
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Cluster Similarity: Ward’s Method
l
Similarity of two clusters is based on the increase
in squared error when two clusters are merged
– Similar to group average if distance between points is
distance squared
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Strengths
– Less susceptible to noise and outliers
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Limitations
– Biased towards globular clusters
l
Less susceptible to noise and outliers
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Biased towards globular clusters
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Hierarchical analogue of K-means
– Can be used to initialize K-means
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Hierarchical Clustering: Comparison
1
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1
MIN
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4 1
Group Average
3
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1
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4
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O(N3) time in many cases
2
1
4
O(N2) space since it uses the proximity matrix.
– There are N steps and at each step the size, N2,
proximity matrix must be updated and searched
– Complexity can be reduced to O(N2 log(N) ) time for
some approaches
2
Ward’s Method
2
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2
MAX
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– N is the number of points.
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5
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Hierarchical Clustering: Time and Space requirements
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5
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MST: Divisive Hierarchical Clustering
Hierarchical Clustering: Problems and Limitations
l
Once a decision is made to combine two clusters,
it cannot be undone
l
No objective function is directly minimized
l
Different schemes have problems with one or
more of the following:
l
Build MST (Minimum Spanning Tree)
– Start with a tree that consists of any point
– In successive steps, look for the closest pair of points (p, q) such
that one point (p) is in the current tree but the other (q) is not
– Add q to the tree and put an edge between p and q
– Sensitivity to noise and outliers
– Difficulty handling different sized clusters and convex
shapes
– Breaking large clusters
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MST: Divisive Hierarchical Clustering
l
© Tan,Steinbach, Kumar
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DBSCAN is a density-based algorithm.
Density = number of points within a specified radius (Eps)
–
A point is a core point if it has more than a specified number
of points (MinPts) within Eps
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–
A border point has fewer than MinPts within Eps, but is in
the neighborhood of a core point
–
A noise point is any point that is not a core point or a border
point.
© Tan,Steinbach, Kumar
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Introduction to Data Mining
These are points that are at the interior of a cluster
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DBSCAN Algorithm
DBSCAN: Core, Border, and Noise Points
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–
u
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DBSCAN
Use MST for constructing hierarchy of clusters
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Eliminate noise points
Perform clustering on the remaining points
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DBSCAN: Core, Border and Noise Points
When DBSCAN Works Well
Original Points
Original Points
Point types: core,
border and noise
• Resistant to Noise
• Can handle clusters of different shapes and sizes
Eps = 10, MinPts = 4
© Tan,Steinbach, Kumar
Introduction to Data Mining
Clusters
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When DBSCAN Does NOT Work Well
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DBSCAN: Determining EPS and MinPts
Idea is that for points in a cluster, their kth nearest
neighbors are at roughly the same distance
Noise points have the kth nearest neighbor at farther
distance
So, plot sorted distance of every point to its kth
nearest neighbor
l
l
l
(MinPts=4, Eps=9.75).
Original Points
• Varying densities
• High-dimensional data
(MinPts=4, Eps=9.92)
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Cluster Validity
For supervised classification we have a variety of
measures to evaluate how good our model is
– Accuracy, precision, recall
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Introduction to Data Mining
Random
Points
For cluster analysis, the analogous question is how to
evaluate the “goodness” of the resulting clusters?
1
1
0.9
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Then why do we want to evaluate them?
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K-means
To avoid finding patterns in noise
To compare clustering algorithms
To compare two sets of clusters
To compare two clusters
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Complete
Link
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0
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1
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x
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x
1
0
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0.2
x
y
–
–
–
–
DBSCAN
0.1
0
y
But “clusters are in the eye of the beholder”!
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Clusters found in Random Data
y
l
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y
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Different Aspects of Cluster Validation
1.
Measures of Cluster Validity
Determining the clustering tendency of a set of data, i.e.,
distinguishing whether non-random structure actually exists in the
data.
Comparing the results of a cluster analysis to externally known
results, e.g., to externally given class labels.
Evaluating how well the results of a cluster analysis fit the data
without reference to external information.
2.
3.
Numerical measures that are applied to judge various aspects
of cluster validity, are classified into the following three types.
l
– External Index: Used to measure the extent to which cluster labels
match externally supplied class labels.
u
u
- Use only the data
5.
u
– However, sometimes criterion is the general strategy and index is the
numerical measure that implements the criterion.
For 2, 3, and 4, we can further distinguish whether we want to
evaluate the entire clustering or just individual clusters.
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Measuring Cluster Validity Via Correlation
© Tan,Steinbach, Kumar
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–
Proximity Matrix
–
“Incidence” Matrix
u
One row and one column for each data point
u
An entry is 1 if the associated pair of points belong to the same cluster
u
An entry is 0 if the associated pair of points belongs to different clusters
Since the matrices are symmetric, only the correlation between
n(n-1) / 2 entries needs to be calculated.
y
–
High correlation indicates that points that belong to the
same cluster are close to each other.
Not a good measure for some density or contiguity based
clusters.
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1
1
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0
1
0
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Order the similarity matrix with respect to cluster
labels and inspect visually.
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Introduction to Data Mining
Points
y
0.6
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0.1
0
0
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30
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30
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40
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40
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50
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60
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80
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0.1
0.1
50
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60
0.4
70
0.3
90
80
0.2
100
90
0.1
20
40
60
80
20
40
60
80
0
100 Similarity
Points
y
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20
x
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1
10
1
88
Clusters in random data are not so crisp
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100
1
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Points
0.7
0.8
Corr = -0.5810
© Tan,Steinbach, Kumar
1
0.8
0.6
x
10
1
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0.4
Using Similarity Matrix for Cluster Validation
Using Similarity Matrix for Cluster Validation
l
0.2
x
Corr = -0.9235
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Correlation of incidence and proximity matrices
for the K-means clusterings of the following two
data sets.
Compute the correlation between the two matrices
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Introduction to Data Mining
Measuring Cluster Validity Via Correlation
Two matrices
l
Often an external or internal index is used for this function, e.g., SSE or
entropy
Sometimes these are referred to as criteria instead of indices
l
© Tan,Steinbach, Kumar
Sum of Squared Error (SSE)
– Relative Index: Used to compare two different clusterings or
clusters.
Comparing the results of two different sets of cluster analyses to
determine which is better.
Determining the ‘correct’ number of clusters.
y
4.
Entropy
– Internal Index: Used to measure the goodness of a clustering
structure without respect to external information.
0
0
0.2
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1
x
0
100 Similarity
Points
DBSCAN
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Using Similarity Matrix for Cluster Validation
Clusters in random data are not so crisp
0.8
30
0.7
0.7
40
0.6
0.6
50
0.5
0.5
60
0.4
0.4
70
0.3
0.3
80
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90
0.1
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40
60
0
0
100 Similarity
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1
0
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20
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40
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50
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1
20
x
Points
40
60
80
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Clusters in more complicated figures aren’t well separated
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Internal Index: Used to measure the goodness of a clustering
structure without respect to external information
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SSE is good for comparing two clusterings or two clusters
(average SSE).
Can also be used to estimate the number of clusters
0.9
– SSE
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1
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Internal Measures: SSE
Using Similarity Matrix for Cluster Validation
2
0
Points
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0
0
100 Similarity
K-means
© Tan,Steinbach, Kumar
1
10
y
0.9
0.8
y
Points
0.9
20
20
Clusters in random data are not so crisp
1
1
10
100
l
Points
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Using Similarity Matrix for Cluster Validation
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1000
3
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4
1500
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2000
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5
l
10
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2500
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0
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9
7
8
3000
500
1000
1500
2000
2500
3000
7
6
DBSCAN
SSE
2
0
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4
-2
3
2
-4
1
-6
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Internal Measures: SSE
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3
For example, if our measure of evaluation has the value, 10, is that
good, fair, or poor?
Statistics provide a framework for cluster validity
–
The more “atypical” a clustering result is, the more likely it represents
valid structure in the data
–
Can compare the values of an index that result from random data or
clusterings to those of a clustering result.
6
4
u
–
5
l
7
SSE of clusters found using K-means
Introduction to Data Mining
2
Introduction to Data Mining
Need a framework to interpret any measure.
–
1
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Framework for Cluster Validity
SSE curve for a more complicated data set
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If the value of the index is unlikely, then the cluster results are valid
These approaches are more complicated and harder to understand.
For comparing the results of two different sets of cluster
analyses, a framework is less necessary.
–
However, there is the question of whether the difference between two
index values is significant
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Statistical Framework for Correlation
Statistical Framework for SSE
– Compare SSE of 0.005 against three clusters in random data
– Histogram shows SSE of three clusters in 500 sets of random data
points of size 100 distributed over the range 0.2 – 0.8 for x and y
values
50
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40
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Count
30
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25
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0
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0
1
0
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x
Corr = -0.9235
10
0.1
0.2
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0.8
1
x
0
0.016 0.018
0.02
0.022
0.024
Corr = -0.5810
0.026
0.028
0.03
0.032 0.034
SSE
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Introduction to Data Mining
Cluster Cohesion: Measures how closely related
are objects in a cluster
l
– BSS + WSS = constant
Cluster Separation: Measure how distinct or wellseparated a cluster is from other clusters
Example: Squared Error
– Cohesion is measured by the within cluster sum of squares (SSE)
1
×
m1
K=1 cluster:
m
×
2
3
×
4
m2
5
WSS= (1 − 3)2 + ( 2 − 3) 2 + (4 − 3)2 + (5 − 3)2 = 10
BSS= 4 × (3 − 3)2 = 0
2
i x∈C i
Total = 10 + 0 = 10
– Separation is measured by the between cluster sum of squares
BSS = ∑ Ci ( m − mi ) 2
K=2 clusters:
WSS= (1 − 1.5) 2 + (2 − 1.5) 2 + (4 − 4.5)2 + (5 − 4.5)2 = 1
BSS= 2 × (3 − 1.5) 2 + 2 × ( 4.5 − 3)2 = 9
Total = 1 + 9 = 10
i
– Where |Ci| is the size of cluster i
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Example: SSE
– Example: SSE
WSS = ∑ ∑ ( x − mi )
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Internal Measures: Cohesion and Separation
Internal Measures: Cohesion and Separation
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Internal Measures: Cohesion and Separation
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Correlation of incidence and proximity matrices for the
K-means clusterings of the following two data sets.
l
Example
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Internal Measures: Silhouette Coefficient
A proximity graph based approach can also be used for
cohesion and separation.
l
l
– Cluster cohesion is the sum of the weight of all links within a cluster.
Silhouette Coefficient combine ideas of both cohesion and separation,
but for individual points, as well as clusters and clusterings
For an individual point, i
– Calculate a = average distance of i to the points in its cluster
– Cluster separation is the sum of the weights between nodes in the cluster
and nodes outside the cluster.
– Calculate b = min (average distance of i to points in another cluster)
– The silhouette coefficient for a point is then given by
s = 1 – a/b if a < b,
(or s = b/a - 1
if a ≥ b, not the usual case)
– Typically between 0 and 1.
b
a
– The closer to 1 the better.
cohesion
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separation
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Can calculate the Average Silhouette width for a cluster or a
clustering
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Final Comment on Cluster Validity
External Measures of Cluster Validity: Entropy and Purity
“The validation of clustering structures is the most
difficult and frustrating part of cluster analysis.
Without a strong effort in this direction, cluster
analysis will remain a black art accessible only to
those true believers who have experience and
great courage.”
Algorithms for Clustering Data, Jain and Dubes
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