3.Unsupervised Learning.ppt presenting machine learning

1
Machine Learning
Unsupervised Learning

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Supervised learning vs.
unsupervised learning
 Supervised learning: discover patterns in the data
that relate data attributes with a target (class)
attribute.
 These patterns are then utilized to predict the values
of the target attribute in future data instances
 Unsupervised learning: The data have no target
attribute.
 We want to explore the data to find some intrinsic
structures in them.

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What is Cluster Analysis?
 Finding groups of objects in data 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

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Applications of Cluster Analysis
 Understanding
 Group related documents
for browsing, group genes
and proteins that have
similar functionality, or
group stocks with similar
price fluctuations
 Summarization
 Reduce the size of large
data sets
Discovered Clusters Industry Group
1
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
Technology1-DOWN
2
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
Technology2-DOWN
3
Fannie-Mae-DOWN,Fed-Home-Loan-DOWN,
MBNA-Corp-DOWN,Morgan-Stanley-DOWN Financial-DOWN
4
Baker-Hughes-UP,Dresser-Inds-UP,Halliburton-HLD-UP,
Louisiana-Land-UP,Phillips-Petro-UP,Unocal-UP,
Schlumberger-UP
Oil-UP
Clustering precipitation
in Australia

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Types of Clusterings
 A clustering is a set of clusters
 Important distinction between hierarchical and
partitional sets of clusters
 Partitional Clustering
 A division data objects into non-overlapping subsets (clusters)
such that each data object is in exactly one subset
 Hierarchical clustering
 A set of nested clusters organized as a hierarchical tree

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Partitional Clustering
(Bölümsel Kümeleme)
Original Points A Partitional Clustering

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Hierarchical Clustering
(Hiyerarşik Kümeleme)
p4
p1
p3
p2
p4
p1
p3
p2
p4
p1 p2 p3
p4
p1 p2 p3
Traditional Hierarchical Clustering
Non-traditional Hierarchical Clustering Non-traditional Dendrogram
Traditional Dendrogram

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Clustering Algorithms
 K-means and its variants
 Hierarchical clustering
 Density-based clustering

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K-means clustering
 K-means is a partitional clustering algorithm
 Let the set of data points (or instances) D be
{x1, x2, …, xn},
where xi = (xi1, xi2, …, xir) is a vector in a real-valued
space X  Rr, and r is the number of attributes
(dimensions) in the data.
 The k-means algorithm partitions the given data
into k clusters.
 Each cluster has a cluster center, called centroid.
 k is specified by the user

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K-means Clustering
 Basic algorithm

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Stopping/convergence criterion
1. no (or minimum) re-assignments of data points
to different clusters,
2. no (or minimum) change of centroids, or
3. minimum decrease in the sum of squared error
(SSE),
 Ci is the jth cluster, mj is the centroid of cluster Cj (the
mean vector of all the data points in Cj), and dist(x,
mj) is the distance between data point x and centroid
mj.




k
j
C j
j
dist
SSE
1
2
)
,
(
x
m
x

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K-means Clustering – Details
 Initial centroids are often chosen randomly.
 Clusters produced vary from one run to another.
 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.
 Often the stopping condition is changed to ‘Until relatively few
points change clusters’
 Complexity is O( n * K * I * d )
 n = number of points, K = number of clusters,
I = number of iterations, d = number of attributes

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Two different K-means Clusterings
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Sub-optimal Clustering
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Optimal Clustering
Original Points

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Importance of Choosing Initial Centroids
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Iteration 6

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Iteration 6

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Evaluating K-means Clusters
 Most common measure is Sum of Squared Error (SSE)
 For each point, the error is the distance to the nearest cluster
 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
 A good clustering with smaller K can have a lower SSE than a
poor clustering with higher K




k
j
C j
j
dist
SSE
1
2
)
,
(
x
m
x

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Importance of Choosing Initial
Centroids
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Iteration 1
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Iteration 5

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Iteration 5

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Problems with Selecting Initial Points
 If there are K ‘real’ clusters then the chance of selecting
one centroid from each cluster is small.
 Chance is relatively small when K is large
 If clusters are the same size, n, then
 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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10 Clusters Example
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Iteration 1
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Iteration 3
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Iteration 4
Starting with two initial centroids in one cluster of each pair of clusters

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10 Clusters Example
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Iteration 4
Starting with two initial centroids in one cluster of each pair of clusters

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10 Clusters Example
Starting with some pairs of clusters having three initial centroids, while other have only one.
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10 Clusters Example
Starting with some pairs of clusters having three initial centroids, while other have only one.
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Solutions to Initial Centroids
Problem
 Multiple runs
 Helps, but probability is not on your side
 Sample and use hierarchical clustering to
determine initial centroids
 Select more than k initial centroids and then
select among these initial centroids
 Select most widely separated
 Postprocessing
 Bisecting K-means

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Pre-processing and Post-processing
 Pre-processing
 Normalize the data
 Eliminate outliers
 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
 ISODATA

26
Limitations of K-means
 K-means has problems when clusters are of
differing
 Sizes
 Densities
 Non-globular shapes
 K-means has problems when the data contains
outliers.

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Limitations of K-means: Differing Sizes
Original Points K-means (3 Clusters)

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Limitations of K-means: Differing Density

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Limitations of K-means: Non-globular
Shapes

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Overcoming K-means Limitations
Original Points K-means Clusters
One solution is to use many clusters.
Find parts of clusters, but need to put together.

31

32

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 Produces a set of nested clusters organized as a
hierarchical tree
 Can be visualized as a dendrogram
 A tree like diagram that records the sequences of
merges or splits
1 3 2 5 4 6
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5

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Strengths of Hierarchical
Clustering
 Do not have to assume any particular number of
clusters
 Any desired number of clusters can be obtained by
‘cutting’ the dendogram at the proper level
 They may correspond to meaningful taxonomies
 Example in biological sciences (e.g., animal kingdom,
phylogeny reconstruction, …)

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 Two main types of hierarchical clustering
 Agglomerative:
 Start with the points as individual clusters
 At each step, merge the closest pair of clusters until only one
cluster (or k clusters) left
 Divisive:
 Start with one, all-inclusive cluster
 At each step, split a cluster until each cluster contains a point (or
there are k clusters)
 Traditional hierarchical algorithms use a similarity or
distance matrix
 Merge or split one cluster at a time

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Agglomerative Clustering Algorithm
 More popular hierarchical clustering technique
 Basic algorithm is straightforward
1. Compute the proximity matrix
2. Let each data point be a cluster
3. Repeat
4. Merge the two closest clusters
5. Update the proximity matrix
6. Until only a single cluster remains
 Key operation is the computation of the proximity of
two clusters
 Different approaches to defining the distance between
clusters distinguish the different algorithms

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Starting Situation
 Start with clusters of individual points and a
proximity matrix
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
. Proximity Matrix
...
p1 p2 p3 p4 p9 p10 p11 p12

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Intermediate Situation
 After some merging steps, we have some clusters
C1
C4
C2 C5
C3
C2
C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix
...
p1 p2 p3 p4 p9 p10 p11 p12

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Intermediate Situation
 We want to merge the two closest clusters (C2 and C5) and
update the proximity matrix.
C1
C4
C2 C5
C3
C2
C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix
...
p1 p2 p3 p4 p9 p10 p11 p12

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After Merging
 The question is “How do we update the proximity matrix?”
C1
C4
C2 U C5
C3
? ? ? ?
?
?
?
C2
U
C5
C1
C1
C3
C4
C2 U C5
C3 C4
Proximity Matrix
...
p1 p2 p3 p4 p9 p10 p11 p12

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How to Define Inter-Cluster Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.
Similarity?
 MIN
 MAX
 Group Average
 Distance Between Centroids
 Other methods driven by an objective
function
 Ward’s Method uses squared error
Proximity Matrix

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p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.
Proximity Matrix
 MIN
 MAX
 Group Average
function

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p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.
Proximity Matrix
 MIN
 MAX
 Group Average
function

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p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.
Proximity Matrix
 MIN
 MAX
 Group Average
function

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p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.
Proximity Matrix
 MIN
 MAX
 Group Average
function
 

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Cluster Similarity: MIN or Single
Link
 Similarity of two clusters is based on the two
most similar (closest) points in the different
clusters
 Determined by one pair of points, i.e., by one link in
the proximity graph.
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 0.70 1.00 0.40 0.30
I4 0.65 0.60 0.40 1.00 0.80
I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5

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Hierarchical Clustering: MIN
Nested Clusters Dendrogram
1
2
3
4
5
6
1
2
3
4
5
3 6 2 5 4 1
0
0.05
0.1
0.15
0.2

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Strength of MIN
Original Points Two Clusters
• Can handle non-elliptical shapes

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Limitations of MIN
• Sensitive to noise and outliers

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Cluster Similarity: MAX or Complete
Linkage
 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
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 0.70 1.00 0.40 0.30
I4 0.65 0.60 0.40 1.00 0.80
I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5

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Strength of MAX
• Less susceptible to noise and outliers

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Limitations of MAX
•Tends to break large clusters
•Biased towards globular clusters (globular -- küresel)

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Cluster Similarity: Group Average
 Proximity of two clusters is the average of pairwise proximity
between points in the two clusters.
 Need to use average connectivity for scalability since total
proximity favors large clusters
|
|Cluster
|
|Cluster
)
p
,
p
proximity(
)
Cluster
,
Cluster
proximity(
j
i
Cluster
p
Cluster
p
j
i
j
i
j
j
i
i





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 0.70 1.00 0.40 0.30
I4 0.65 0.60 0.40 1.00 0.80
I5 0.20 0.50 0.30 0.80 1.00 1 2 3 4 5

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Hierarchical Clustering: Group
Average
Nested Clusters Dendrogram
3 6 4 1 2 5
0
0.05
0.1
0.15
0.2
0.25
1
2
3
4
5
6
1
2
5
3
4

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Hierarchical Clustering: Group
Average
 Compromise between Single and Complete
Link
 Strengths
 Less susceptible to noise and outliers
 Limitations
 Biased towards globular (küresel) clusters

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Cluster Similarity: Ward’s Method
 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
 Less susceptible to noise and outliers
 Biased towards globular clusters
 Hierarchical analogue of K-means
 Can be used to initialize K-means

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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
 For cluster analysis, the analogous question is how to
evaluate the “goodness” of the resulting clusters?
 But “clusters are in the eye of the beholder”!
 Then why do we want to evaluate them?
 To avoid finding patterns in noise
 To compare clustering algorithms
 To compare two sets of clusters
 To compare two clusters

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Clusters found in Random Data
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Points
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DBSCAN
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Complete
Link

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1. Determining the clustering tendency of a set of data, i.e., distinguishing
whether non-random structure actually exists in the data.
2. Comparing the results of a cluster analysis to externally known results,
e.g., to externally given class labels.
3. Evaluating how well the results of a cluster analysis fit the data without
reference to external information.
- Use only the data
4. Comparing the results of two different sets of cluster analyses to
determine which is better.
5. Determining the ‘correct’ number of clusters.
For 2, 3, and 4, we can further distinguish whether we want to evaluate
the entire clustering or just individual clusters.
Different Aspects of Cluster Validation

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 Numerical measures that are applied to judge various aspects of
cluster validity, are classified into the following three types.
 External Index: Used to measure the extent to which cluster labels
match externally supplied class labels.
 Entropy
 Internal Index: Used to measure the goodness of a clustering
structure without respect to external information.
 Sum of Squared Error (SSE)
 Relative Index: Used to compare two different clusterings or
clusters.
 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
 However, sometimes criterion is the general strategy and index is the
numerical measure that implements the criterion.
Measures of Cluster Validity

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 Two matrices
 Proximity Matrix (Yakınlık matrisi)
 “Incidence” Matrix (Tekrar Oranı Matrisi)
 One row and one column for each data point
 An entry is 1 if the associated pair of points belong to the same cluster
 An entry is 0 if the associated pair of points belongs to different clusters
 Compute the correlation between the two matrices
 Since the matrices are symmetric, only the correlation between
n(n-1) / 2 entries needs to be calculated.
 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.
Measuring Cluster Validity Via
Correlation

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Measuring Cluster Validity Via
Correlation
 Correlation of incidence and proximity matrices
for the K-means clusterings of the following two
data sets.
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Corr = -0.9235 Corr = -0.5810

63
 Order the similarity matrix with respect to cluster
labels and inspect visually.
Using Similarity Matrix for Cluster Validation
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Points
Points
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100
Similarity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

64
Using Similarity Matrix for Cluster
Validation
 Clusters in random data are not so crisp
Points
Points
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100
Similarity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
DBSCAN
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y

65
Points
Points
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100
Similarity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Validation
K-means
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
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0.8
0.9
1
x
y

66
Validation
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Points
Points
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100
Similarity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Complete Link

67
Validation
1
2
3
5
6
4
7
DBSCAN
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
500 1000 1500 2000 2500 3000
500
1000
1500
2000
2500
3000

68
 Cluster Cohesion: Measures how closely related
are objects in a cluster
 Example: SSE
 Cluster Separation: Measure how distinct or well-
separated a cluster is from other clusters
 Example: Squared Error
 Cohesion is measured by the within cluster sum of squares (SSE)
 Separation is measured by the between cluster sum of squares
 Where |Ci| is the size of cluster i
Internal Measures: Cohesion and Separation
 



i C
x
i
i
m
x
WSS 2
)
(
 

i
i
i m
m
C
BSS 2
)
(

69
Hierarchical Clustering: Comparison
Group Average
Ward’s Method
1
2
3
4
5
6
1
2
5
3
4
MIN MAX
1
2
3
4
5
6
1
2
5
3
4
1
2
3
4
5
6
1
2 5
3
4
1
2
3
4
5
6
1
2
3
4
5

70
 Clusters in more complicated figures aren’t well separated
 Internal Index: Used to measure the goodness of a clustering
structure without respect to external information
 SSE
 SSE is good for comparing two clusterings or two clusters
(average SSE).
 Can also be used to estimate the number of clusters
Internal Measures: SSE
2 5 10 15 20 25 30
0
1
2
3
4
5
6
7
8
9
10
K
SSE
5 10 15
-6
-4
-2
0
2
4
6

71
Internal Measures: SSE
 SSE curve for a more complicated data set
1
2
3
5
6
4
7
SSE of clusters found using K-means

72
 Need a framework to interpret any measure.
 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.
 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
Framework for Cluster Validity

73
 Example
 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
Statistical Framework for SSE
0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.034
0
5
10
15
20
25
30
35
40
45
50
SSE
Count
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y

74
 Correlation of incidence and proximity matrices for the
K-means clusterings of the following two data sets.
Statistical Framework for Correlation
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Corr = -0.9235 Corr = -0.5810

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