clustering using different methods in .pdf

2
Road map
• Basic concepts
• K-means algorithm
• Representation of clusters
• Hierarchical clustering
• Distance functions
• Data standardization
• Handling mixed attributes
• Which clustering algorithm to use?
• Cluster evaluation
• Discovering holes and data regions
• Summary

3
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.

4
Clustering
• Clustering is a technique for finding similarity groups in
data, called clusters. I.e.,
– it groups data instances that are similar to (near) each other in
one cluster and data instances that are very different (far away)
from each other into different clusters.
• Clustering is often called an unsupervised learning task as
no class values denoting an a priori grouping of the data
instances are given, which is the case in supervised
learning.
• Due to historical reasons, clustering is often considered
synonymous with unsupervised learning.
– In fact, association rule mining is also unsupervised

5
An illustration
• The data set has three natural groups of data points, i.e.,
3 natural clusters.

6
What is clustering for?
• Let us see some real-life examples
• Example 1: groups people of similar sizes
together to make “small”, “medium” and
“large” T-Shirts.
– Tailor-made for each person: too expensive
– One-size-fits-all: does not fit all.
• Example 2: In marketing, segment customers
according to their similarities
– To do targeted marketing.

7
What is clustering for? (cont…)
• Example 3: Given a collection of text
documents, we want to organize them
according to their content similarities,
– To produce a topic hierarchy
• In fact, clustering is one of the most utilized
data mining techniques.
– It has a long history, and used in almost every field,
e.g., medicine, psychology, botany, sociology,
biology, archeology, marketing, insurance, libraries,
etc.
– In recent years, due to the rapid increase of online
documents, text clustering becomes important.

8
Aspects of clustering
• A clustering algorithm
– Partitional clustering
– Hierarchical clustering
– …
• A distance (similarity, or dissimilarity) function
• Clustering quality
– Inter-clusters distance  maximized
– Intra-clusters distance  minimized
• The quality of a clustering result depends on
the algorithm, the distance function, and the
application.

Road map
9
• Basic concepts
• Summary

10
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

11
K-means algorithm
• Given k, the k-means algorithm works as
follows:
1)Randomly choose k data points (seeds) to be the
initial centroids, cluster centers
2)Assign each data point to the closest centroid
3)Re-compute the centroids using the current
cluster memberships.
4)If a convergence criterion is not met, go to 2).

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K-means algorithm – (cont …)

13
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
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x
m
x (1)

16
An example distance function

17
A disk version of k-means
• K-means can be implemented with data on
disk
– In each iteration, it scans the data once.
– as the centroids can be computed incrementally
• It can be used to cluster large datasets that do
not fit in main memory
• We need to control the number of iterations
– In practice, a limited is set (< 50).
• Not the best method. There are other scale-up
algorithms, e.g., BIRCH.

18
A disk version of k-means (cont …)

19
Strengths of k-means
• Strengths:
– Simple: easy to understand and to implement
– Efficient: Time complexity: O(tkn),
where n is the number of data points,
k is the number of clusters, and
t is the number of iterations.
– Since both k and t are small. k-means is considered a linear
algorithm.
• K-means is the most popular clustering algorithm.
• Note that: it terminates at a local optimum if SSE is used.
The global optimum is hard to find due to complexity.

20
Weaknesses of k-means
• The algorithm is only applicable if the mean is
defined.
– For categorical data, k-mode - the centroid is
represented by most frequent values.
• The user needs to specify k.
• The algorithm is sensitive to outliers
– Outliers are data points that are very far away
from other data points.
– Outliers could be errors in the data recording or
some special data points with very different values.

21
Weaknesses of k-means: Problems with
outliers

22
Weaknesses of k-means: To deal with
outliers
• One method is to remove some data points in the
clustering process that are much further away from the
centroids than other data points.
– To be safe, we may want to monitor these possible outliers over
a few iterations and then decide to remove them.
• Another method is to perform random sampling. Since in
sampling we only choose a small subset of the data
points, the chance of selecting an outlier is very small.
– Assign the rest of the data points to the clusters by distance or
similarity comparison, or classification

23
Weaknesses of k-means (cont …)
• The algorithm is sensitive to initial seeds.

24
• If we use different seeds: good results
There are some
methods to help
choose good
seeds

25
• The k-means algorithm is not suitable for discovering
clusters that are not hyper-ellipsoids (or hyper-spheres).
+

26
K-means summary
• Despite weaknesses, k-means is still the most
popular algorithm due to its simplicity,
efficiency and
– other clustering algorithms have their own lists of
weaknesses.
• No clear evidence that any other clustering
algorithm performs better in general
– although they may be more suitable for some
specific types of data or applications.
• Comparing different clustering algorithms is a
difficult task. No one knows the correct
clusters!

27
Road map
• Basic concepts
• Summary

28
Common ways to represent
clusters
• Use the centroid of each cluster to represent
the cluster.
– compute the radius and
– standard deviation of the cluster to determine its
spread in each dimension
– The centroid representation alone works well if
the clusters are of the hyper-spherical shape.
– If clusters are elongated or are of other shapes,
centroids are not sufficient

29
Using classification model
• All the data points in a
cluster are regarded to
have the same class label,
e.g., the cluster ID.
– run a supervised learning
algorithm on the data to
find a classification model.

30
Use frequent values to represent
cluster
• This method is mainly for clustering of
categorical data (e.g., k-modes clustering).
• Main method used in text clustering, where a
small set of frequent words in each cluster is
selected to represent the cluster.

31
Clusters of arbitrary shapes
• Hyper-elliptical and hyper-spherical
clusters are usually easy to
represent, using their centroid
together with spreads.
• Irregular shape clusters are hard to
represent. They may not be useful in
some applications.
– Using centroids are not suitable (upper
figure) in general
– K-means clusters may be more useful
(lower figure), e.g., for making 2 size T-
shirts.

32
Road map
• Basic concepts
• Summary

33
Hierarchical Clustering
• Produce a nested sequence of clusters, a tree, also
called Dendrogram.

34
Types of hierarchical clustering
• Agglomerative (bottom up) clustering: It builds the
dendrogram (tree) from the bottom level, and
– merges the most similar (or nearest) pair of clusters
– stops when all the data points are merged into a single cluster
(i.e., the root cluster).
• Divisive (top down) clustering: It starts with all data
points in one cluster, the root.
– Splits the root into a set of child clusters. Each child cluster is
recursively divided further
– stops when only singleton clusters of individual data points
remain, i.e., each cluster with only a single point

35
Agglomerative clustering
It is more popular then divisive methods.
• At the beginning, each data point forms a
cluster (also called a node).
• Merge nodes/clusters that have the least
distance.
• Go on merging
• Eventually all nodes belong to one cluster

36
Agglomerative clustering algorithm

37
An example: working of the
algorithm

38
Measuring the distance of two
clusters
• A few ways to measure distances of two
clusters.
• Results in different variations of the algorithm.
– Single link
– Complete link
– Average link
– Centroids
– …

39
Single link method
• The distance between two
clusters is the distance
between two closest data
points in the two clusters,
one data point from each
cluster.
• It can find arbitrarily
shaped clusters, but
– It may cause the
undesirable “chain effect”
by noisy points
Two natural clusters are
split into two

40
Complete link method
• The distance between two clusters is the distance of
two furthest data points in the two clusters.
• It is sensitive to outliers because they are far away

41
Average link and centroid methods
• Average link: A compromise between
– the sensitivity of complete-link clustering to
outliers and
– the tendency of single-link clustering to form long
chains that do not correspond to the intuitive
notion of clusters as compact, spherical objects.
– In this method, the distance between two clusters
is the average distance of all pair-wise distances
between the data points in two clusters.
• Centroid method: In this method, the distance
between two clusters is the distance between
their centroids

42
The complexity
• All the algorithms are at least O(n2). n is the
number of data points.
• Single link can be done in O(n2).
• Complete and average links can be done in
O(n2logn).
• Due the complexity, hard to use for large data
sets.
– Sampling
– Scale-up methods (e.g., BIRCH).

43
Road map
• Basic concepts
• Summary

44
Distance functions
• Key to clustering. “similarity” and
“dissimilarity” can also commonly used terms.
• There are numerous distance functions for
– Different types of data
• Numeric data
• Nominal data
– Different specific applications

45
Distance functions for numeric attributes
• Most commonly used functions are
– Euclidean distance and
– Manhattan (city block) distance
• We denote distance with: dist(xi, xj), where xi
and xj are data points (vectors)
• They are special cases of Minkowski distance.
h is positive integer.
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46
Euclidean distance and Manhattan
distance
• If h = 2, it is the Euclidean distance
• If h = 1, it is the Manhattan distance
• Weighted Euclidean distance
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47
Squared distance and Chebychev
distance
• Squared Euclidean distance: to place
progressively greater weight on data points
that are further apart.
• Chebychev distance: one wants to define two
data points as "different" if they are different
on any one of the attributes.
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48
Distance functions for binary and
nominal attributes
• Binary attribute: has two values or states but
no ordering relationships, e.g.,
– Gender: male and female.
• We use a confusion matrix to introduce the
distance functions/measures.
• Let the ith and jth data points be xi and xj
(vectors)

50
Symmetric binary attributes
• A binary attribute is symmetric if both of its
states (0 and 1) have equal importance, and
carry the same weights, e.g., male and female
of the attribute Gender
• Distance function: Simple Matching
Coefficient, proportion of mismatches of their
values
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51
Symmetric binary attributes:
example

52
Asymmetric binary attributes
• Asymmetric: if one of the states is more
important or more valuable than the other.
– By convention, state 1 represents the more
important state, which is typically the rare or
infrequent state.
– Jaccard coefficient is a popular measure
– We can have some variations, adding weights
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53
Nominal attributes
• Nominal attributes: with more than two states
or values.
– the commonly used distance measure is also
based on the simple matching method.
– Given two data points xi and xj, let the number of
attributes be r, and the number of values that
match in xi and xj be q.
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54
Distance function for text documents
• A text document consists of a sequence of sentences and
each sentence consists of a sequence of words.
• To simplify: a document is usually considered a “bag” of
words in document clustering.
– Sequence and position of words are ignored.
• A document is represented with a vector just like a
normal data point.
• It is common to use similarity to compare two
documents rather than distance.
– The most commonly used similarity function is the cosine
similarity. We will study this later.

55
Road map
• Basic concepts
• Summary

56
Data standardization
• In the Euclidean space, standardization of attributes is
recommended so that all attributes can have equal
impact on the computation of distances.
• Consider the following pair of data points
– xi: (0.1, 20) and xj: (0.9, 720).
• The distance is almost completely dominated by (720-20)
= 700.
• Standardize attributes: to force the attributes to have a
common value range
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57
Interval-scaled attributes
• Their values are real numbers following a
linear scale.
– The difference in Age between 10 and 20 is the
same as that between 40 and 50.
– The key idea is that intervals keep the same
importance through out the scale
• Two main approaches to standardize interval
scaled attributes, range and z-score. f is an
attribute ,
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58
Interval-scaled attributes (cont …)
• Z-score: transforms the attribute values so that they have
a mean of zero and a mean absolute deviation of 1. The
mean absolute deviation of attribute f, denoted by sf, is
computed as follows
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59
Ratio-scaled attributes
• Numeric attributes, but unlike interval-scaled
attributes, their scales are exponential,
• For example, the total amount of micro
organisms that evolve in a time t is
approximately given by
AeBt,
– where A and B are some positive constants.
• Do log transform:
– Then treat it as an interval-scaled attribuete
)
log( if
x

60
Nominal attributes
• Sometime, we need to transform nominal
attributes to numeric attributes.
• Transform nominal attributes to binary
attributes.
– The number of values of a nominal attribute is v.
– Create v binary attributes to represent them.
– If a data instance for the nominal attribute takes a
particular value, the value of its binary attribute is
set to 1, otherwise it is set to 0.
• The resulting binary attributes can be used as
numeric attributes, with two values, 0 and 1.

61
Nominal attributes: an example
• Nominal attribute fruit: has three values,
– Apple, Orange, and Pear
• We create three binary attributes called,
Apple, Orange, and Pear in the new data.
• If a particular data instance in the original data
has Apple as the value for fruit,
– then in the transformed data, we set the value of
the attribute Apple to 1, and
– the values of attributes Orange and Pear to 0

62
Ordinal attributes
• Ordinal attribute: an ordinal attribute is like a
nominal attribute, but its values have a
numerical ordering. E.g.,
– Age attribute with values: Young, MiddleAge and
Old. They are ordered.
– Common approach to standardization: treat is as
an interval-scaled attribute.

63
Road map
• Basic concepts
• Summary

64
Mixed attributes
• Our distance functions given are for data with
all numeric attributes, or all nominal
attributes, etc.
• Practical data has different types:
– Any subset of the 6 types of attributes,
• interval-scaled,
• symmetric binary,
• asymmetric binary,
• ratio-scaled,
• ordinal and
• nominal

65
Convert to a single type
• One common way of dealing with mixed
attributes is to
– Decide the dominant attribute type, and
– Convert the other types to this type.
• E.g, if most attributes in a data set are
interval-scaled,
– we convert ordinal attributes and ratio-scaled
attributes to interval-scaled attributes.
– It is also appropriate to treat symmetric binary
attributes as interval-scaled attributes.

66
Convert to a single type (cont …)
• It does not make much sense to convert a
nominal attribute or an asymmetric binary
attribute to an interval-scaled attribute,
– but it is still frequently done in practice by
assigning some numbers to them according to
some hidden ordering, e.g., prices of the fruits
• Alternatively, a nominal attribute can be
converted to a set of (symmetric) binary
attributes, which are then treated as numeric
attributes.

67
Combining individual distances
• This approach computes
individual attribute distances
and then combine them.
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68
Road map
• Basic concepts
• Summary

69
How to choose a clustering
algorithm
• Clustering research has a long history. A vast collection of
algorithms are available.
– We only introduced several main algorithms.
• Choosing the “best” algorithm is a challenge.
– Every algorithm has limitations and works well with certain data
distributions.
– It is very hard, if not impossible, to know what distribution the
application data follow. The data may not fully follow any “ideal”
structure or distribution required by the algorithms.
– One also needs to decide how to standardize the data, to
choose a suitable distance function and to select other
parameter values.

70
Choose a clustering algorithm (cont
…)
• Due to these complexities, the common practice is to
– run several algorithms using different distance functions and
parameter settings, and
– then carefully analyze and compare the results.
• The interpretation of the results must be based on
insight into the meaning of the original data together
with knowledge of the algorithms used.
• Clustering is highly application dependent and to certain
extent subjective (personal preferences).

71
Road map
• Basic concepts
• Summary

72
Cluster Evaluation: hard problem
• The quality of a clustering is very hard to
evaluate because
– We do not know the correct clusters
• Some methods are used:
– User inspection
• Study centroids, and spreads
• Rules from a decision tree.
• For text documents, one can read some documents in
clusters.

73
Cluster evaluation: ground truth
• We use some labeled data (for classification)
• Assumption: Each class is a cluster.
• After clustering, a confusion matrix is
constructed. From the matrix, we compute
various measurements, entropy, purity,
precision, recall and F-score.
– Let the classes in the data D be C = (c1, c2, …, ck).
The clustering method produces k clusters, which
divides D into k disjoint subsets, D1, D2, …, Dk.

74
Evaluation measures: Entropy

75
Evaluation measures: purity

77
A remark about ground truth
evaluation
• Commonly used to compare different clustering
algorithms.
• A real-life data set for clustering has no class labels.
– Thus although an algorithm may perform very well on some
labeled data sets, no guarantee that it will perform well on the
actual application data at hand.
• The fact that it performs well on some label data sets
does give us some confidence of the quality of the
algorithm.
• This evaluation method is said to be based on external
data or information.

78
Evaluation based on internal information
• Intra-cluster cohesion (compactness):
– Cohesion measures how near the data points in a
cluster are to the cluster centroid.
– Sum of squared error (SSE) is a commonly used
measure.
• Inter-cluster separation (isolation):
– Separation means that different cluster centroids
should be far away from one another.
• In most applications, expert judgments are
still the key.

79
Indirect evaluation
• In some applications, clustering is not the primary task,
but used to help perform another task.
• We can use the performance on the primary task to
compare clustering methods.
• For instance, in an application, the primary task is to
provide recommendations on book purchasing to online
shoppers.
– If we can cluster books according to their features, we might be
able to provide better recommendations.
– We can evaluate different clustering algorithms based on how
well they help with the recommendation task.
– Here, we assume that the recommendation can be reliably
evaluated.

80
Road map
• Basic concepts
• Summary

81
Holes in data space
• All the clustering algorithms only group data.
• Clusters only represent one aspect of the
knowledge in the data.
• Another aspect that we have not studied is
the holes.
– A hole is a region in the data space that contains
no or few data points. Reasons:
• insufficient data in certain areas, and/or
• certain attribute-value combinations are not possible or
seldom occur.

82
Holes are useful too
• Although clusters are important, holes in the
space can be quite useful too.
• For example, in a disease database
– we may find that certain symptoms and/or test
values do not occur together, or
– when a certain medicine is used, some test values
never go beyond certain ranges.
• Discovery of such information can be
important in medical domains because
– it could mean the discovery of a cure to a disease
or some biological laws.

83
Data regions and empty regions
• Given a data space, separate
– data regions (clusters) and
– empty regions (holes, with few or no data points).
• Use a supervised learning technique, i.e.,
decision tree induction, to separate the two
types of regions.
• Due to the use of a supervised learning
method for an unsupervised learning task,
– an interesting connection is made between the
two types of learning paradigms.

84
Supervised learning for unsupervised
learning
• Decision tree algorithm is not directly applicable.
– it needs at least two classes of data.
– A clustering data set has no class label for each data point.
• The problem can be dealt with by a simple idea.
– Regard each point in the data set to have a class label Y.
– Assume that the data space is uniformly distributed with
another type of points, called non-existing points. We give
them the class, N.
• With the N points added, the problem of partitioning the
data space into data and empty regions becomes a
supervised classification problem.

85
An example
A decision tree method is used for partitioning in (B).

86
Can it done without adding N
points?
• Yes.
• Physically adding N points increases the size of
the data and thus the running time.
• More importantly: it is unlikely that we can
have points truly uniformly distributed in a
high dimensional space as we would need an
exponential number of points.
• Fortunately, no need to physically add any N
points.
– We can compute them when needed

87
Characteristics of the approach
• It provides representations of the resulting data and
empty regions in terms of hyper-rectangles, or rules.
• It detects outliers automatically. Outliers are data points in
an empty region.
• It may not use all attributes in the data just as in a normal
decision tree for supervised learning.
– It can automatically determine what attributes are useful.
Subspace clustering …
• Drawback: data regions of irregular shapes are hard to
handle since decision tree learning only generates hyper-
rectangles (formed by axis-parallel hyper-planes), which
are rules.

88
Building the Tree
• The main computation in decision tree building is to
evaluate entropy (for information gain):
• Can it be evaluated without adding N points? Yes.
• Pr(cj) is the probability of class cj in data set D, and |C| is
the number of classes, Y and N (2 classes).
– To compute Pr(cj), we only need the number of Y (data) points
and the number of N (non-existing) points.
– We already have Y (or data) points, and we can compute the
number of N points on the fly. Simple: as we assume that the N
points are uniformly distributed in the space.
)
Pr(
log
)
Pr(
)
(
|
|
1
2 j
C
j
j c
c
D
entropy 




89
An example
• The space has 25 data (Y) points and 25 N points.
Assume the system is evaluating a possible cut S.
– # N points on the left of S is 25 * 4/10 = 10. The number of Y
points is 3.
– Likewise, # N points on the right of S is 15 (= 25 - 10).The
number of Y points is 22.
• With these numbers, entropy can be computed.

90
How many N points to add?
• We add a different number of N points at each different
node.
– The number of N points for the current node E is determined
by the following rule (note that at the root node, the number
of inherited N points is 0):

92
How many N points to add?
(cont…)
• Basically, for a Y node (which has more data
points), we increase N points so that
#Y = #N
• The number of N points is not reduced if the
current node is an N node (an N node has
more N points than Y points).
– A reduction may cause outlier Y points to form Y
nodes (a Y node has an equal number of Y points
as N points or more).
– Then data regions and empty regions may not be
separated well.

93
Building the decision tree
• Using the above ideas, a decision tree can be
built to separate data regions and empty
regions.
• The actual method is more sophisticated as a
few other tricky issues need to be handled in
– tree building and
– tree pruning.

94
Road map
• Basic concepts
• Summary

95
Summary
• Clustering is has along history and still active
– There are a huge number of clustering algorithms
– More are still coming every year.
• We only introduced several main algorithms. There are
many others, e.g.,
– density based algorithm, sub-space clustering, scale-up methods,
neural networks based methods, fuzzy clustering, co-clustering,
etc.
• Clustering is hard to evaluate, but very useful in practice.
This partially explains why there are still a large number of
clustering algorithms being devised every year.
• Clustering is highly application dependent and to some
extent subjective.

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