Download Cluster Analysis: Basic Concepts Cluster Analysis: Basic

Cluster Analysis: Basic Concepts
and Algorithms
Dr. Hui Xiong
Rutgers University
Introductionto
toData
DataMining
Mining
Introduction
08/06/20068/30/2006
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What is Cluster Analysis?
z
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
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Notion of a Cluster can be Ambiguous
How many clusters?
Six Clusters
Two Clusters
Four Clusters
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Types of Clusterings
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A clustering is a set of clusters
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Important
p
distinction between hierarchical and
partitional sets of clusters
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Partitional Clustering
– A division data objects into non-overlapping subsets (clusters)
such that each data object is in exactly one subset
z
Hi
Hierarchical
hi l clustering
l t i
– A set of nested clusters organized as a hierarchical tree
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Partitional Clustering
Original Points
A Partitional Clustering
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Hierarchical Clustering
p1
p3
p4
p2
p1 p2
Traditional Hierarchical Clustering
p3 p4
Traditional Dendrogram
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Non-traditional Hierarchical Clustering
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Non-traditional Dendrogram
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Other Distinctions Between Sets of Clusters
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Exclusive versus non-exclusive
– In non-exclusive clusterings, points may belong to multiple
clusters.
– Can
C representt multiple
lti l classes
l
or ‘b
‘border’
d ’ points
i t
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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
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Partial versus complete
– In some cases, we only want to cluster some of the data
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Heterogeneous versus homogeneous
– Clusters of widely different sizes, shapes, and densities
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Characteristics of the Input Data Are Important
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Type of proximity or density measure
– This is a derived measure, but central to clustering
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Sparseness
– Dictates type of similarity
– Adds to efficiency
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Attribute type
– Dictates type of similarity
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Type of Data
– Dictates type of similarity
– Other characteristics
characteristics, e
e.g.,
g autocorrelation
z
z
z
Dimensionality
Noise and Outliers
Type of Distribution
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Clustering Algorithms
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K-means and its variants
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Hierarchical clustering
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Density-based clustering
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K-means Clustering
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Partitional clustering approach
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Number of clusters, K, must be specified
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Each cluster is associated with a centroid ((center point)
p
)
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Each point is assigned to the cluster with the closest
centroid
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The basic algorithm is very simple
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Example of K-means Clustering
Iteration 6
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Example of K-means Clustering
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Limitations of K-means
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K-means has problems when clusters are of
differing
– Sizes
– Densities
– Non-globular shapes
z
K-means has problems when the data contains
outliers.
tli
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Limitations of K-means: Differing Sizes
Original Points
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K-means (3 Clusters)
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Limitations of K-means: Differing Density
Original Points
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K-means (3 Clusters)
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Limitations of K-means: Non-globular Shapes
Original Points
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K-means (2 Clusters)
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Hierarchical Clustering
Produces a set of nested clusters organized as a
hierarchical tree
z Can
C b
be visualized
i
li d as a d
dendrogram
d
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– A tree like diagram that records the sequences of
merges or splits
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Strengths of Hierarchical Clustering
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Do not have to assume any particular number of
clusters
– Any desired number of clusters can be obtained by
‘cutting’ the dendrogram at the proper level
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They may correspond to meaningful taxonomies
– Example in biological sciences (e.g., animal kingdom,
phylogeny reconstruction, …)
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Hierarchical Clustering
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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:
z
‹
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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How to Define Inter-Cluster Distance
p1
p2
p3
p4 p5
...
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Similarity?
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p5
MIN
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MAX
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Group Average
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Proximity Matrix
Distance Between Centroids
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Other methods driven by an objective
function
– Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
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MIN
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Group Average
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Proximity Matrix
Distance Between Centroids
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Other methods driven by an objective
function
– Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
p1
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p5
MIN
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MAX
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Group Average
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Proximity Matrix
Distance Between Centroids
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Other methods driven by an objective
function
– Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
p1
p2
p3
p4 p5
...
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p5
MIN
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Group Average
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Proximity Matrix
Distance Between Centroids
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Other methods driven by an objective
function
– Ward’s Method uses squared error
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How to Define Inter-Cluster Similarity
p1
p2
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p4 p5
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×
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MIN
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MAX
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Group Average
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Proximity Matrix
Distance Between Centroids
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Other methods driven by an objective
function
– Ward’s Method uses squared error
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Cluster Similarity: Ward’s Method
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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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Less susceptible to noise and outliers
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Biased towards g
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
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Hierarchical Clustering: Time and Space requirements
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O(N2) space since it uses the proximity matrix.
– N is the number of points.
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O(N3) time in many cases
– 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 with
some cleverness
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Hierarchical Clustering: Problems and Limitations
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Once a decision is made to combine two clusters,
it cannot be undone
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No objective function is directly minimized
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Different schemes have problems with one or
more of the following:
– Sensitivity to noise and outliers
– Difficulty handling different sized clusters and convex
shapes
– Breaking large clusters
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DBSCAN
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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
‹
These are points that are at the interior of a cluster
–
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.
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DBSCAN: Core, Border, and Noise Points
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DBSCAN Algorithm
1.
2.
3.
4.
5.
Label all points as core, border, or noise points
Eliminate noise points
Put an edge between all core points that are
within Eps of each other.
Make each group of connected core points into
a separate cluster.
Assign each border point to one of the clusters
of its associated core points.
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DBSCAN: Core, Border and Noise Points
Original Points
Point types: core,
border and noise
Eps = 10, MinPts = 4
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When DBSCAN Works Well
O i i l Points
Original
P i t
Clusters
• Resistant to Noise
• Can handle clusters of different shapes and sizes
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When DBSCAN Does NOT Work Well
(MinPts=4, Eps=9.75).
Original Points
• Varying densities
• High-dimensional data
(MinPts=4, Eps=9.92)
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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 p
points have the kth nearest neighbor
g
at farther
distance
So, plot sorted distance of every point to its kth
nearest neighbor
z
z
z
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Cluster Validity
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For supervised classification we have a variety of
measures to evaluate how good our model is
– Accuracy, precision, recall
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For cluster analysis, the analogous question is how to
evaluate the “goodness” of the resulting clusters?
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But “clusters are in the eye of the beholder”!
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Then whyy 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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Different Aspects of Cluster Validation
1.
Determining the clustering tendency of a set of data, i.e.,
distinguishing whether non-random structure actually exists in the
data.
2
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.
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Measures of Cluster Validity
z
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
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.
‹
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Often an external or internal index is used for this function, e.g.,
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.
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Measuring Cluster Validity Via Correlation
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Two matrices
–
–
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z
Ideal Similarity Matrix
‹
One row and one column for each data point
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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
–
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Proximity Matrix
Since the matrices are symmetric, only the correlation between
n(n-1) / 2 entries needs to be calculated.
High
g correlation indicates that p
points that belong
g to the
same cluster are close to each other.
Not a good measure for some density or contiguity based
clusters.
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Measuring Cluster Validity Via Correlation
Correlation of ideal similarity and proximity
matrices for the K-means clusterings of the
following two data sets
sets.
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Corr = 0.9235
Corr = 0.5810
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Using Similarity Matrix for Cluster Validation
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Order the similarity matrix with respect to cluster
labels and inspect visually.
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Using Similarity Matrix for Cluster Validation
Clusters in random data are not so crisp
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Using Similarity Matrix for Cluster Validation
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Clusters in random data are not so crisp
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Using Similarity Matrix for Cluster Validation
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Clusters in random data are not so crisp
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Using Similarity Matrix for Cluster Validation
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Internal Measures: SSE
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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
– SSE
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z
SSE is good for comparing two clusterings or two clusters
(average SSE).
Can also be used to estimate the number of clusters
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Internal Measures: SSE
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SSE curve for a more complicated data set
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SSE of clusters found using K-means
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Framework for Cluster Validity
Need a framework to interpret any measure.
z
–
For example, if our measure of evaluation has the value, 10, is that
good, fair, or poor?
Statistics provide a framework for cluster validity
z
–
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
F
i the
h results
l off two diff
different sets off cluster
l
analyses, a framework is less necessary.
z
–
However, there is the question of whether the difference between two
index values is significant
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Statistical Framework for SSE
Example
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– Compare SSE of 0.005 against three clusters in random data
– Histogram
g
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
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Statistical Framework for Correlation
Correlation of ideal similarity and proximity matrices
for the K-means clusterings of the following two data
sets.
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Corr = -0.9235
Corr = -0.5810
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Internal Measures: Cohesion and Separation
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Cluster Cohesion: Measures how closely related
are objects in a cluster
– Example: SSE
z
Cluster Separation: Measure how distinct or wellseparated a cluster is from other clusters
z
Example: Squared Error
– Cohesion is measured by the within cluster sum of squares (SSE)
WSS = ∑ ∑ ( x − mi )2
i x∈C i
– Separation is measured by the between cluster sum of squares
BSS = ∑ Ci ( m − mi ) 2
i
– Where |Ci| is the size of cluster i
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Internal Measures: Cohesion and Separation
z
Example: SSE
– BSS + WSS = constant
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
Total = 10 + 0 = 10
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
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Internal Measures: Cohesion and Separation
z
A proximity graph based approach can also be used for
cohesion and separation.
– Cluster cohesion is the sum of the weight of all links within a cluster.
– Cluster separation is the sum of the weights between nodes in the cluster
and nodes outside the cluster.
cohesion
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Internal Measures: Silhouette Coefficient
z
z
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
– Calculate b = min (average distance of i to points in another cluster)
– The silhouette coefficient for a point is then given by
s = (b – a) / max(a,b)
– Typically between 0 and 1.
b
a
– The closer to 1 the better.
better
z
Can calculate the average silhouette coefficient for a cluster or a
clustering
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External Measures of Cluster Validity: Entropy and Purity
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Final Comment on Cluster Validity
“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
f Clustering
C
Data, Jain and Dubes
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