Clusteryanam

A Brief Dealing of Clustering
Prof. Dr. N.B. Venkateswarlu
Aditya Institute of Technology and
Management
Tekkali
Venkat_ritch@yahoo.com

Cluster Analysis
• What is Cluster Analysis?
• Types of Data in Cluster Analysis
• Partitioning Methods
• Hierarchical Methods
• Density-Based Methods
• Cluster Evaluation
• Grid-Based Methods
• Model-Based Clustering Methods
• Outlier Analysis
• Summary

The Problem of Clustering
• Given a set of points, with a notion of
distance between points, group the points
into some number of clusters, so that
members of a cluster are in some sense
as close to each other as possible.

What is Cluster Analysis?
• 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

What is Cluster Analysis?
• Cluster: a collection of data objects
– Similar to one another within the same cluster
– Dissimilar to the objects in other clusters
• Cluster analysis
– Grouping a set of data objects into clusters
• Clustering is unsupervised classification: no
predefined classes
• Typical applications
– As a stand-alone tool to get insight into data
distribution
– As a preprocessing step for other algorithms

General Applications of
Clustering
• Pattern Recognition
• Spatial Data Analysis
– create thematic maps in GIS by clustering feature
spaces
– detect spatial clusters and explain them in spatial data
mining
• Image Processing
• Economic Science (especially market research)
• WWW
– Document classification
– Cluster Weblog data to discover groups of similar
access patterns

Examples of Clustering
Applications
• Marketing: Help marketers discover distinct groups in their customer
bases, and then use this knowledge to develop targeted marketing
programs
• Land use: Identification of areas of similar land use in an earth
observation database
• Insurance: Identifying groups of motor insurance policy holders with
a high average claim cost
• City-planning: Identifying groups of houses according to their house
type, value, and geographical location
• Earth-quake studies: Observed earth quake epicenters should be
clustered along continent faults

Example: SkyCat
• A catalog of 2 billion “sky objects”
represented objects by their radiation in
9 dimensions (frequency bands).
• Problem: cluster into similar objects,
e.g., galaxies, nearby stars, quasars,
etc.
• Sloan Sky Survey is a newer, better
version.

Example: Clustering CD’s
(Collaborative Filtering)
• Intuitively: music divides into categories,
and customers prefer a few categories.
– But what are categories really?
• Represent a CD by the customers who
bought it.
• Similar CD’s have similar sets of
customers, and vice-versa.

The Space of CD’s
• Think of a space with one dimension for
each customer.
– Values in a dimension may be 0 or 1 only.
• A CD’s point in this space is (x1,
x2,…, xk), where xi = 1 iff the i th
customer
bought the CD.
– Compare with the “correlated items” matrix:
rows = customers; cols. = CD’s.

Example: Clustering Documents
• Represent a document by a vector (x1,
x2,…, xk), where xi = 1 iff the i th
word (in
some order) appears in the document.
– It actually doesn’t matter if k is infinite; i.e.,
we don’t limit the set of words.
• Documents with similar sets of words may
be about the same topic.

Example: Protein
Sequences
• Objects are sequences of {C,A,T,G}.
• Distance between sequences is edit
distance, the minimum number of inserts
and deletes needed to turn one into the
other.
• Note there is a “distance,” but no
convenient space in which points “live.”

Notion of a Cluster can be
Ambiguous
How many clusters?
Four ClustersTwo Clusters
Six Clusters

Problems With Clustering
• Clustering in two dimensions looks easy.
• Clustering small amounts of data looks
easy.
• And in most cases, looks are not
deceiving.

The Curse of Dimensionality
• Many applications involve not 2, but 10 or
10,000 dimensions.
• High-dimensional spaces look different:
almost all pairs of points are at about the
same distance.
– Assuming random points within a bounding
box, e.g., values between 0 and 1 in each
dimension.

What Is Good Clustering?
• A good clustering method will produce high quality
clusters with
– high intra-class similarity
– low inter-class similarity
• The quality of a clustering result depends on both the
similarity measure used by the method and its
implementation.
• The quality of a clustering method is also measured by
its ability to discover some or all of the hidden patterns.

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

Partitional Clustering
Original Points A Partitional Clustering

Hierarchical Clustering
p4
p1
p3
p2
p4p1 p2 p3
p4p1 p2 p3
Traditional Hierarchical Clustering
Non-traditional Hierarchical Clustering Non-traditional Dendrogram
Traditional Dendrogram

Other Distinctions Between
Sets of Clusters
• Exclusive versus non-exclusive
– In non-exclusive clusterings, points may belong to multiple
clusters.
– Can represent multiple classes or ‘border’ points
• 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
• Partial versus complete
– In some cases, we only want to cluster some of the data
• Heterogeneous versus homogeneous
– Cluster of widely different sizes, shapes, and densities

Types of Clusters
• Well-separated clusters
• Center-based clusters
• Contiguous clusters
• Density-based clusters
• Property or Conceptual
• Described by an Objective Function

Types of Clusters: Well-
Separated
• Well-Separated Clusters:
– 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.
3 well-separated clusters

Types of Clusters: Center-
Based
• 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
4 center-based clusters

Types of Clusters: Contiguity-
Based
• Contiguous Cluster (Nearest neighbor or
Transitive)
– 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.
8 contiguous clusters

Types of Clusters: Density-
Based
• Density-based
– A cluster is a dense region of points, which is separated by low-
density regions, from other regions of high density.
– Used when the clusters are irregular or intertwined, and when
noise and outliers are present.
6 density-based clusters

Types of Clusters: Conceptual
Clusters
• Shared Property or Conceptual Clusters
– Finds clusters that share some common property or represent a
particular concept.
2 Overlapping Circles

Types of Clusters: Objective
Function
• 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.
• Hierarchical clustering algorithms typically have local objectives
• Partitional algorithms typically have global objectives
– A variation of the global objective function approach is to fit the
data to a parameterized model.
• Parameters for the model are determined from the data.
• Mixture models assume that the data is a ‘mixture' of a number of
statistical distributions.

Types of Clusters: Objective
Function …
• Map the clustering problem to a different domain
and solve a related problem in that domain
– Proximity matrix defines a weighted graph, where the
nodes are the points being clustered, and the
weighted edges represent the proximities between
points
– Clustering is equivalent to breaking the graph into
connected components, one for each cluster.
– Want to minimize the edge weight between clusters
and maximize the edge weight within clusters

Requirements of Clustering in
Data Mining
• Scalability
• Ability to deal with different types of attributes
• Discovery of clusters with arbitrary shape
• Minimal requirements for domain knowledge to determine input
parameters
• Able to deal with noise and outliers
• Insensitive to order of input records
• High dimensionality
• Incorporation of user-specified constraints
• Interpretability and usability

Data Structures
• Data matrix
– (two modes)
• Dissimilarity matrix
– (one mode)
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Measure the Quality of
Clustering
• Dissimilarity/Similarity metric: Similarity is expressed in
terms of a distance function, which is typically metric:
d(i, j)
• There is a separate “quality” function that measures the
“goodness” of a cluster.
• The definitions of distance functions are usually very
different for interval-scaled, boolean, categorical, ordinal
and ratio variables.
• Weights should be associated with different variables
based on applications and data semantics.
• It is hard to define “similar enough” or “good enough”
– the answer is typically highly subjective.

Type of data in clustering
analysis
• Interval-scaled variables:
• Binary variables:
• Nominal, ordinal, and ratio variables:
• Variables of mixed types:

Interval-valued variables
• Standardize data
– Calculate the mean absolute deviation:
where
– Calculate the standardized measurement (z-score)
• Using mean absolute deviation is more robust than using
standard deviation
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Similarity and Dissimilarity
Between Objects
• Distances are normally used to measure the similarity or
dissimilarity between two data objects
• Some popular ones include: Minkowski distance:
where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional
data objects, and q is a positive integer
• If q = 1, d is Manhattan distance
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Similarity and Dissimilarity
Between Objects (Cont.)
• If q = 2, d is Euclidean distance:
– Properties
• d(i,j) ≥ 0
• d(i,i) = 0
• d(i,j) = d(j,i)
• d(i,j) ≤ d(i,k) + d(k,j)
• Also, one can use weighted distance, parametric
Pearson product moment correlation, or other disimilarity
measures
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Distance Measures
• Each clustering problem is based on
some kind of “distance” between points.
• Two major classes of distance measure:
1. Euclidean
2. Non-Euclidean

Euclidean Vs. Non-
Euclidean
• A Euclidean space has some number of
real-valued dimensions and “dense”
points.
– There is a notion of “average” of two points.
– A Euclidean distance is based on the
locations of points in such a space.
• A Non-Euclidean distance is based on
properties of points, but not their “location”
in a space.

Axioms of a Distance
Measure
• d is a distance measure if it is a
function from pairs of points to reals such
that:
1. d(x,y) > 0.
2. d(x,y) = 0 iff x = y.
3. d(x,y) = d(y,x).
4. d(x,y) < d(x,z) + d(z,y) (triangle inequality ).

Some Euclidean Distances
• L2 norm : d(x,y) = square root of the sum of
the squares of the differences between x
and y in each dimension.
– The most common notion of “distance.”
• L1 norm : sum of the differences in each
dimension.
– Manhattan distance = distance if you had to
travel along coordinates only.

Examples of Euclidean Distances
x = (5,5)
y = (9,8)
L2-norm:
dist(x,y) =
√(42
+32
)
= 5
L1-norm:
dist(x,y) =
4+3 = 7
4
35

Another Euclidean Distance
• L∞ norm : d(x,y) = the maximum of the
differences between x and y in any
dimension.
• Note: the maximum is the limit as n goes
to ∞ of what you get by taking the n th
power of the differences, summing and
taking the n th
root.

Non-Euclidean Distances
• Jaccard distance for sets = 1 minus
ratio of sizes of intersection and union.
• Cosine distance = angle between
vectors from the origin to the points in
question.
• Edit distance = number of inserts and
deletes to change one string into
another.

Jaccard Distance
• Example: p1 = 10111; p2 = 10011.
– Size of intersection = 3; size of union = 4,
Jaccard measure (not distance) = 3/4.
• Need to make a distance function
satisfying triangle inequality and other
laws.
• d(x,y) = 1 – (Jaccard measure) works.

Why J.D. Is a Distance Measure
• d(x,x) = 0 because x∩x = x∪x.
• d(x,y) = d(y,x) because union and
intersection are symmetric.
• d(x,y) > 0 because |x∩y| < |x∪y|.
• d(x,y) < d(x,z) + d(z,y) trickier --- next
slide.

Triangle Inequality for J.D.
1 - |x ∩z| + 1 - |y ∩z| > 1 - |x ∩y|
|x ∪z| |y ∪z| |x ∪y|
• Remember: |a ∩b|/|a ∪b| = probability
that minhash(a) = minhash(b).
• Thus, 1 - |a ∩b|/|a ∪b| = probability that
minhash(a) ≠ minhash(b).

Triangle Inequality --- (2)
• So we need to observe that
prob[minhash(x) ≠ minhash(y)] <
prob[minhash(x) ≠ minhash(z)] +
prob[minhash(z) ≠ minhash(y)]
• Clincher: whenever minhash(x) ≠ minhash(y),
one of minhash(x) ≠ minhash(z) and
minhash(z) ≠ minhash(y) must be true.

Cosine Distance
• Think of a point as a vector from the origin
(0,0,…,0) to its location.
• Two points’ vectors make an angle, whose
cosine is the normalized dot-product of the
vectors: p1.p2/|p2||p1|.
– Example p1 = 00111; p2 = 10011.
– p1.p2 = 2; |p1| = |p2| = √3.
– cos(θ) = 2/3; θ is about 48 degrees.

Cosine-Measure Diagram
p1
p2p1.p2
θ
|p2|
dist(p1, p2) = θ = arccos(p1.p2/|p2||p1|)

Why?
p1 = (x1,y1)
p2 = (x2,0)x1
θ
Dot product is invariant under
rotation, so pick convenient
coordinate system.
x1 = p1.p2/|p2|
p1.p2 = x1*x2.
|p2| = x2.

Why C.D. Is a Distance Measure
• d(x,x) = 0 because arccos(1) = 0.
• d(x,y) = d(y,x) by symmetry.
• d(x,y) > 0 because angles are chosen
to be in the range 0 to 180 degrees.
• Triangle inequality: physical reasoning.
If I rotate an angle from x to z and then
from z to y, I can’t rotate less than from
x to y.

Edit Distance
• The edit distance of two strings is the
number of inserts and deletes of
characters needed to turn one into the
other.
• Equivalently, d(x,y) = |x| + |
y| -2|LCS(x,y)|.
– LCS = longest common subsequence =
longest string obtained both by deleting
from x and deleting from y.

Example
• x = abcde ; y = bcduve.
• Turn x into y by deleting a, then inserting
u and v after d.
– Edit-distance = 3.
• Or, LCS(x,y) = bcde.
• |x| + |y| - 2|LCS(x,y)| = 5 + 6 –2*4 = 3.

Why E.D. Is a Distance Measure
• d(x,x) = 0 because 0 edits suffice.
• d(x,y) = d(y,x) because insert/delete are
inverses of each other.
• d(x,y) > 0: no notion of negative edits.
• Triangle inequality: changing x to z and
then to y is one way to change x to y.

Variant Edit Distance
• Allow insert, delete, and mutate.
– Change one character into another.
• Minimum number of inserts, deletes, and
mutates also forms a distance measure.

Binary Variables
• A contingency table for binary data
• Simple matching coefficient (invariant, if the binary variable is
symmetric):
• Jaccard coefficient (noninvariant if the binary variable is
asymmetric):
dcba
cbjid
+++
+=),(
pdbcasum
dcdc
baba
sum
++
+
+
0
1
01
cba
cbjid
++
+=),(
Object i
Object j

Dissimilarity between Binary
Variables
• Example
– gender is a symmetric attribute
– the remaining attributes are asymmetric binary
– let the values Y and P be set to 1, and the value N be set to 0
Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4
Jack M Y N P N N N
Mary F Y N P N P N
Jim M Y P N N N N
75.0
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67.0
111
11
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10
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++
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maryjimd
jimjackd
maryjackd

Nominal Variables
• A generalization of the binary variable in that it can take
more than 2 states, e.g., red, yellow, blue, green
• Method 1: Simple matching
– m: # of matches, p: total # of variables
• Method 2: use a large number of binary variables
– creating a new binary variable for each of the M nominal states
p
mpjid −=),(

Ordinal Variables
• An ordinal variable can be discrete or continuous
• Order is important, e.g., rank
• Can be treated like interval-scaled
– replace xif by their rank
– map the range of each variable onto [0, 1] by replacing i-th
object in the f-th variable by
– compute the dissimilarity using methods for interval-scaled
variables
1
1
−
−
=
f
if
if M
r
z
},...,1{ fif
Mr ∈

Ratio-Scaled Variables
• Ratio-scaled variable: a positive measurement on a
nonlinear scale, approximately at exponential scale,
such as AeBt
or Ae-Bt
• Methods:
– treat them like interval-scaled variables—not a good choice!
(why?—the scale can be distorted)
– apply logarithmic transformation
yif = log(xif)
– treat them as continuous ordinal data treat their rank as interval-
scaled

Variables of Mixed Types
• A database may contain all the six types of variables
– symmetric binary, asymmetric binary, nominal, ordinal, interval
and ratio
• One may use a weighted formula to combine their
effects
– f is binary or nominal:
dij
(f)
= 0 if xif = xjf , or dij
(f)
= 1 o.w.
– f is interval-based: use the normalized distance
– f is ordinal or ratio-scaled
• compute ranks rif and
• and treat zif as interval-scaled
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Partitioning Algorithms: Basic
Concept
• Partitioning method: Construct a partition of a database
D of n objects into a set of k clusters
• Given a k, find a partition of k clusters that optimizes the
chosen partitioning criterion
– Global optimal: exhaustively enumerate all partitions
– Heuristic methods: k-means and k-medoids algorithms
– k-means (MacQueen’67): Each cluster is represented by the
center of the cluster
– k-medoids or PAM (Partition around medoids) (Kaufman &
Rousseeuw’87): Each cluster is represented by one of the
objects in the cluster

The K-Means Clustering
Method
• Given k, the k-means algorithm is implemented in four
steps:
– Partition objects into k nonempty subsets
– Compute seed points as the centroids of the clusters of the
current partition (the centroid is the center, i.e., mean point, of
the cluster)
– Assign each object to the cluster with the nearest seed point
– Go back to Step 2, stop when no more new assignment

The K-Means Clustering
Method
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
K=2
Arbitrarily choose K
object as initial
cluster center
Assign
each
objects
to most
similar
center
Update
the
cluster
means
Update
the
cluster
means
reassignreassign

k –Means Algorithm(s)
• Assumes Euclidean space.
• Start by picking k, the number of clusters.
• Initialize clusters by picking one point per
cluster.
– For instance, pick one point at random, then k
-1 other points, each as far away as possible
from the previous points.

Populating Clusters
1. For each point, place it in the cluster
whose current centroid it is nearest.
2. After all points are assigned, fix the
centroids of the k clusters.
3. Reassign all points to their closest
centroid.
– Sometimes moves points between clusters.

Example
1
2
3
4
5
6
7 8x
x
Clusters after first round
Reassigned
points

Getting k Right
– Try different k, looking at the change in the
average distance to centroid, as k increases.
• Average falls rapidly until right k, then
changes little.
k
Average
distance to
centroid
Best value
of k

Example
x x
x x x x
x x x x
x x x
x x
x
xx x
x x
x x x
x
x x x
x
x x
x x x x
x x x
x
x
x
Too few;
many long
distances
to centroid.

Example
x x
x x x x
x x x x
x x x
x x
x
xx x
x x
x x x
x
x x x
x
x x
x x x x
x x x
x
x
x
Just right;
distances
rather short.

Example
x x
x x x x
x x x x
x x x
x x
x
xx x
x x
x x x
x
x x x
x
x x
x x x x
x x x
x
x
x
Too many;
little improvement
in average
distance.

BFR Algorithm
• BFR (Bradley-Fayyad-Reina) is a variant
of k -means designed to handle very large
(disk-resident) data sets.
• It assumes that clusters are normally
distributed around a centroid in a
Euclidean space.
– Standard deviations in different dimensions
may vary.

BFR --- (2)
• Points are read one main-memory-full at
a time.
• Most points from previous memory loads
are summarized by simple statistics.
• To begin, from the initial load we select
the initial k centroids by some sensible
approach.

Initialization: k -Means
• Possibilities include:
1. Take a small sample and cluster optimally.
2. Take a sample; pick a random point, and
then k – 1 more points, each as far from the
previously selected points as possible.

Three Classes of Points
1. The discard set : points close enough to
a centroid to be represented statistically.
2. The compression set : groups of points
that are close together but not close to
any centroid. They are represented
statistically, but not assigned to a cluster.
3. The retained set : isolated points.

Representing Sets of Points
• For each cluster, the discard set is
represented by:
1. The number of points, N.
2. The vector SUM, whose i th
component is
the sum of the coordinates of the points in
the i th
dimension.
3. The vector SUMSQ: i th
component = sum
of squares of coordinates in i th
dimension.

Comments
• 2d + 1 values represent any number of
points.
– d = number of dimensions.
• Averages in each dimension (centroid
coordinates) can be calculated easily as
SUMi /N.
– SUMi = i th
component of SUM.

Comments --- (2)
• Variance of a cluster’s discard set in
dimension i can be computed by:
(SUMSQi /N ) – (SUMi /N )2
• And the standard deviation is the square
root of that.
• The same statistics can represent any
compression set.

“Galaxies” Picture
A cluster. Its points
are in the DS.
The centroid
Compressed sets.
Their points are in
the CS.
Points in
the RS

Processing a “Memory-
Load” of Points
1. Find those points that are “sufficiently
close” to a cluster centroid; add those
points to that cluster and the DS.
2. Use any main-memory clustering
algorithm to cluster the remaining
points and the old RS.
Clusters go to the CS; outlying points to
the RS.

Processing --- (2)
3. Adjust statistics of the clusters to
account for the new points.
4. Consider merging compressed sets in
the CS.
5. If this is the last round, merge all
compressed sets in the CS and all RS
points into their nearest cluster.

A Few Details . . .
• How do we decide if a point is “close
enough” to a cluster that we will add the
point to that cluster?
• How do we decide whether two
compressed sets deserve to be combined
into one?

How Close is Close
Enough?
• We need a way to decide whether to put
a new point into a cluster.
• BFR suggest two ways:
1. The Mahalanobis distance is less than a
threshold.
2. Low likelihood of the currently nearest
centroid changing.

Mahalanobis Distance
• Normalized Euclidean distance.
• For point (x1,…,xk) and centroid (c1,…,ck):
1. Normalize in each dimension: yi = |xi -ci|/σi
2. Take sum of the squares of the yi ’s.
3. Take the square root.

Mahalanobis Distance --- (2)
• If clusters are normally distributed in d
dimensions, then one standard deviation
corresponds to a distance √d.
– I.e., 70% of the points of the cluster will have
a Mahalanobis distance < √d.
• Accept a point for a cluster if its M.D. is <
some threshold, e.g. 4 standard
deviations.

Picture: Equal M.D. Regions
σ
2σ

Should Two CS Subclusters
Be Combined?
• Compute the variance of the combined
subcluster.
– N, SUM, and SUMSQ allow us to make that
calculation.
• Combine if the variance is below some
threshold.

Comments on the K-Means
Method
• Strength: Relatively efficient: O(tkn), where n is # objects, k is # clusters,
and t is # iterations. Normally, k, t << n.
• Comparing: PAM: O(k(n-k)2
), CLARA: O(ks2
+ k(n-k))
• Comment: Often terminates at a local optimum. The global optimum may be
found using techniques such as: deterministic annealing and genetic
algorithms
• Weakness
– Applicable only when mean is defined, then what about categorical data?
– Need to specify k, the number of clusters, in advance
– Unable to handle noisy data and outliers
– Not suitable to discover clusters with non-convex shapes

Variations of the K-Means
Method
• A few variants of the k-means which differ in
– Selection of the initial k means
– Dissimilarity calculations
– Strategies to calculate cluster means
• Handling categorical data: k-modes (Huang’98)
– Replacing means of clusters with modes
– Using new dissimilarity measures to deal with categorical objects
– Using a frequency-based method to update modes of clusters
– A mixture of categorical and numerical data: k-prototype method

What is the problem of k-
Means Method?
• The k-means algorithm is sensitive to outliers !
– Since an object with an extremely large value may substantially distort
the distribution of the data.
• K-Medoids: Instead of taking the mean value of the object in a
cluster as a reference point, medoids can be used, which is the
most centrally located object in a cluster.
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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
– Not as susceptible to initialization issues

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

Limitations of K-means: Non-
globular Shapes
Original Points K-means (2 Clusters)

The CURE Algorithm
• Problem with BFR/k -means:
– Assumes clusters are normally distributed in
each dimension.
– And axes are fixed --- ellipses at an angle are
not OK.
• CURE:
– Assumes a Euclidean distance.
– Allows clusters to assume any shape.

Example: Stanford Faculty Salaries
e e
e
e
e e
e
e e
e
e
h
h
h
h
h
h
h h
h
h
h
h h
salary
age

Starting CURE
1. Pick a random sample of points that fit
in main memory.
2. Cluster these points hierarchically ---
group nearest points/clusters.
3. For each cluster, pick a sample of
points, as dispersed as possible.
4. From the sample, pick representatives
by moving them (say) 20% toward the
centroid of the cluster.

Example: Initial Clusters
e e
e
e
e e
e
e e
e
e
h
h
h
h
h
h
h h
h
h
h
h h
salary
age

Example: Pick Dispersed Points
e e
e
e
e e
e
e e
e
e
h
h
h
h
h
h
h h
h
h
h
h h
salary
age
Pick (say) 4
remote points
for each
cluster.

Example: Pick Dispersed Points
e e
e
e
e e
e
e e
e
e
h
h
h
h
h
h
h h
h
h
h
h h
salary
age
Move points
(say) 20%
toward the
centroid.

Finishing CURE
• Now, visit each point p in the data set.
• Place it in the “closest cluster.”
– Normal definition of “closest”: that cluster with
the closest (to p ) among all the sample points
of all the clusters.

Curse of Dimensionality
• One way to look at it: in large-dimension
spaces, random vectors are
perpendicular. Why?
Argument #1: Lots of 2-dim subspaces.
There must be one where the vectors’
projections are almost perpendicular.
Argument #2: Expected value of cosine of
angle is 0.

Cosine of Angle Between
Random Vectors
• Assume vectors emanate from the origin
(0,0,…,0).
• Components are random in range [-1,1].
• (a1,a2,…,an).(b1,b2,…,bn) has expected value 0
and a standard deviation that grows as √n.
• But lengths of both vectors grow as √n.
• So dot product around √n/ (√n * √n) = 1/√n.

Random Vectors ---
Continued
• Thus, a typical pair of vectors has an
angle whose cosine is on the order of
1/√n.
• As n -> ∞, that’s 0; i.e., the angle is about
90°.

Interesting Consequence
• Suppose “random vectors are perpendicular,”
even in non-Euclidean spaces.
• Suppose we know the distance from A to B,
say d (A,B ), and we also know d (B,C ), but
we don’t know d (A,C ).
• Suppose B and C are fairly close, say in the
same cluster.
• What is d (A,C )?

Diagram of Situation
A
C
B
Approximately
perpendicular
Assuming points lie in a plane:
d (A,B )2
+ d (B,C )2
= d (A,C )2

Important Point
• Why do we assume AB is
perpendicular to AC, and not that either
of the other two angles are right-
angles?
1. AB and AC are not “random vectors”; they
each go to points that are far away from A
and close to each other.
2. If AB is longer than AC, then it is angle
ACB that is right, but both ACB and ABC
are approximately right-angles.

Dealing With a Non-
Euclidean Space
• Problem: clusters cannot be represented by
centroids.
• Why? Because the “average” of “points” might
not be a point in the space.
• Best substitute: the clustroid = point in the
cluster that minimizes the sum of the squares
of distances to the points in the cluster.

Representing Clusters in Non-
Euclidean Spaces
• Recall BFR represents a Euclidean cluster
by N, SUM, and SUMSQ.
• A non-Euclidean cluster is represented by:
– N.
– The clustroid.
– Sum of the squares of the distances from
clustroid to all points in the cluster.

Example of CoD Use
• Problem: in non-Euclidean space, we
want to decide whether to merge two
clusters.
– Each cluster represented by N, clustroid,
and “SUMSQ.”
– Also, SUMSQ for each point in the cluster,
even if it is not the clustroid.
• Merge if SUMSQ for new cluster is “low.”

Estimating SUMSQ
p
p ’s clustroid, c
other clust-
roid, b

Suppose p Were the
Clustroid of Combined
Cluster
• It’s SUMSQ would be the sum of:
1. Old SUMSQ(p) [for old cluster containing p].
2. SUMSQ(b) plus d (p,b)2
times number of
points in b ’s cluster.
• Critical point: vector p ->b assumed
perpendicular to vectors from b to all
other points in its cluster --- justifies (2).

Combining Clusters --- Continued
• We can thus estimate SUMSQ for each
point in the combined cluster. Take the
point with the least SUMSQ as the
clustroid of the new cluster --- provided
that SUMSQ is small enough.

The GRGPF Algorithm
• From Ganti et al. --- see reading list.
• Works for non-Euclidean distances.
• Works for massive (disk-resident) data.
• Hierarchical clustering.
• Clusters are grouped into a tree of disk
blocks (like a B-tree or R-tree).

Information Retained About
a Cluster
1. N, clustroid, SUMSQ.
2. The p points closest to the clustroid, and
their values of SUMSQ.
3. The p points of the cluster that are
furthest away from the clustroid, and
their SUMSQ’s.

At Interior Nodes of the Tree
• Interior nodes have samples of the
clustroids of the clusters found at
descendant leaves of this node.
• Try to keep clusters on one leaf block
close, descendants of a level-1 node
close, etc.
• Interior part of tree kept in main memory.

Picture of the Tree
cluster data cluster data
samples
main
memory
on disk

Initialization
• Take a main-memory sample of points.
• Organize them into clusters
hierarchically.
• Build the initial tree, with level-1 interior
nodes representing clusters of clusters,
and so on.
• All other points are inserted into this tree.

Inserting Points
• Start at the root.
• At each interior node, visit one or more
children that have sample clustroids near
the inserted point.
• At the leaves, insert the point into the
cluster with the nearest clustroid.

Updating Cluster Data
• Suppose we add point X to a cluster.
• Increase count N by 1.
• For each of the 2p + 1 points Y whose
SUMSQ is stored, add d (X,Y )2
.
• Estimate SUMSQ for X.

Estimating SUMSQ(X )
• If C is the clustroid, SUMSQ(X ) is, by the
CoD assumption: Nd (X,C )
2
+ SUMSQ(C )
– Based on assumption that vector from X to C
is perpendicular to vectors from C to all the
other nodes of the cluster.
• This value may allow X to replace one of
the closest or furthest nodes.

Possible Modification to
Cluster Data
• There may be a new clustroid --- one of
the p closest points --- because of the
addition of X.
• Eventually, the clustroid may migrate out
of the p closest points, and the entire
representation of the cluster needs to be
recomputed.

Splitting and Merging
Clusters
• Maintain a threshold for the radius of a
cluster = √(SUMSQ/N ).
• Split a cluster whose radius is too large.
• Adding clusters may overflow leaf
blocks, and require splits of blocks up
the tree.
– Splitting is similar to a B-tree.
– But try to keep locality of clusters.

Splitting and Merging --- (2)
• The problem case is when we have split
so much that the tree no longer fits in main
memory.
• Raise the threshold on radius and merge
clusters that are sufficiently close.

Merging Clusters
• Suppose there are nearby clusters with
clustroids C and D, and we want to
consider merging them.
• Assume that the clustroid of the combined
cluster will be one of the p furthest points
from the clustroid of one of those clusters.

Merging --- (2)
• Compute SUMSQ(X ) [from the cluster of
C ] for the combined cluster by summing:
1. SUMSQ(X ) from its own cluster.
2. SUMSQ(D ) + N [d (X,C )2
+ d (C,D )2
].
Uses the CoD to reason that the distance from X
to each point in the other cluster goes to C,
makes a right angle to D, and another right angle
to the point.

Merging --- Concluded
• Pick as the clustroid for the combined
cluster that point with the least SUMSQ.
• But if this SUMSQ is too large, do not
merge clusters.
• Hope you get enough mergers to fit the
tree in main memory.

Fastmap
• Not a clustering algorithm --- rather, a
method for applying multidimensional
scaling.
– That is, mapping the points onto a small-
dimension space, so the CoD does not apply.

Fastmap --- (2)
• Assumes non-Euclidean space.
– But like GRGFP pretends it is working in 2-
dimensional Euclidean space when it is
convenient to do so.
• Goal: map n points in much less than
O(n 2
) time.
– I.e., you cannot compute distances
between each pair of points and place
points in k-dim. space to minimize error.

Fastmap --- Key Idea
• Create a “dimension” in non-Euclidean
space by:
1. Pick a pair of points A and B that are far
apart.
Start with random A; pick most distant B.
2. Treat AB as an “axis” and project all points
onto AB, using the law of cosines.

Projecting Point C Onto AB
A B
C
x
d(A,C) d(B,C)
d(A,B)
x = [d 2
(A,C) + d 2
(A,B) – d 2
(B,C)]/2d (A,B)

Revising Distances
• Having computed the position of every
point along the pseudo-axis AB, we need
to lower the distances between points in
the “other dimensions.”

Picture
A B
C
D
dold(C,D)
x
y
dnew(C,D) =
√dold(C,D)2
– (x-y)2

But …
• We can’t afford to compute new
distances for each pseudo-dimension.
– It would take O(n 2
) time.
• Rather, for each pseudo-dimension,
store the position along the pseudo-axis
for each point, and adjust the distance
between points by square-subtract-sqrt
only when needed.
– I.e., one of the points is an axis-end.

Fastmap --- Summary
• Pick a number of dimensions k.
FOR i = 1 TO k DO BEGIN
Pick a pseudo-axis AiBi;
Compute projection of each
point onto this pseudo-axis;
END;
• Each step is O(ni ); total O(nk 2
).

Clustering a Stream (New Topic)
• Assume points enter in a stream.
• Maintain a sliding window of points.
• Queries ask for clusters of points within
some suffix of the window.
• Only important issue: where are the
cluster centroids.
– There is no notion of “all the points” in a
stream.

BDMO Approach
• BDMO = Babcock, Datar, Motwani,
O’Callaghan.
• k –means based.
• Can use less than O(N ) space for
windows of size N.
• Generalizes trick of DGIM: buckets of
increasing “weight.”

Recall DGIM
• Maintains a sequence of buckets B1, B2, …
• Buckets have timestamps (most recent
stream element in bucket).
• Sizes of buckets nondecreasing.
– In DGIM size = power of 2.
• Either 1 or 2 of each size.

Alternative Combining Rule
• Instead of “combine the 2nd
and 3rd
of any
one size” we could say:
• “Combine Bi+1 and Bi if size(Bi+1 ∪ Bi) <
size(Bi-1 ∪ Bi-2 ∪ … ∪ B1).”
– If Bi+1, Bi, and Bi-1 are the same size, inequality
must hold (almost).
– If Bi-1 is smaller, it cannot hold.

Buckets for Clustering
• In place of “size” (number of 1’s) we use
(an approximation to) the sum of the
distances from all points to the centroid of
their cluster.
• Merge consecutive buckets if the “size” of
the merged bucket is less than the sum of
the sizes of all later buckets.

Consequence of Merge
Rule
• In a stable list of buckets, any two
consecutive buckets are “bigger” than
all smaller buckets.
• Thus, “sizes” grow exponentially.
• If there is a limit on total “size,” then the
number of buckets is O(log N ).
• N = window size.
– E.g., all points are in a fixed hypercube.

Outline of Algorithm
1. What do buckets look like?
– Clusters at various levels, represented by
centroids.
2. How do we merge buckets?
– Keep # of clusters at each level small.
3. What happens when we query?
– Final clustering of all clusters of all
relevant buckets.

Organization of Buckets
• Each bucket consists of clusters at
some number of levels.
– 4 levels in our examples.
• Clusters represented by:
1. Location of centroid.
2. Weight = number of points in the cluster.
3. Cost = upper bound on sum of distances
from member points to centroid.

Processing Buckets --- (1)
• Actions determined by N (window size)
and k (desired number of clusters).
• Also uses a tuning parameter τ for
which we use 1/4 to simplify.
– 1/τ is the number of levels of clusters.

Processing Buckets --- (2)
• Initialize a new bucket with k new
points.
– Each is a cluster at level 0.
• If the timestamp of the oldest bucket is
outside the window, delete that bucket.

Level-0 Clusters
• A single point p is represented by (p,
1, 0).
• That is:
1. A point is its own centroid.
2. The cluster has one point.
3. The sum of distances to the centroid is 0.

Merging Buckets --- (1)
• Needed in two situations:
1. We have to process a query, which requires
us to (temporarily) merge some tail of the
bucket sequence.
2. We have just added a new (most recent)
bucket and we need to check the rule about
two consecutive buckets being “bigger” than
all that follow.

Merging Buckets --- (2)
• Step 1: Take the union of the clusters at
each level.
• Step 2: If the number of clusters (points)
at level 0 is now more than N 1/4
, cluster
them into k clusters.
– These become clusters at level 1.
• Steps 3,…: Repeat, going up the levels,
if needed.

Representing New Clusters
• Centroid = weighted average of
centroids of component clusters.
• Weight = sum of weights.
• Cost = sum over all component
clusters of:
1. Cost of component cluster.
2. Weight of component times distance from
its centroid to new centroid.

Example: New Centroid
+ (18,-2)
+ (3,3)
+ (12,12)
centroids
5
10
15
weights
+ (12,2)
new centroid

Example: New Costs
+ (18,-2)
+ (3,3)
+ (12,12)
5
10
15
+ (12,2)
old cost
added
true cost

Queries
• Find all the buckets within the range of the
query.
– The last bucket may be only partially within
the range.
• Cluster all clusters at all levels into k
clusters.
• Return the k centroids.

Error in Estimation
• Goal is to pick the k centroids that minimize
the true cost (sum of distances from each
point to its centroid).
• Since recorded “costs” are inexact, there
can be a factor of 2 error at each level.
• Additional error because some of last
bucket may not belong.
– But fraction of spurious points is small (why?).

Effect of Cost-Errors
1. Alter when buckets get combined.
Not really important.
2. Produce suboptimal clustering at any
stage of the algorithm.
The real measure of how bad the output is.

Speedup of Algorithm
• As given, algorithm is slow.
– Each new bucket causes O(log N ) bucket-
merger problems.
• A faster version allows the first bucket to
have not k, but N 1/2
(or in general N 2τ
)
points.
– A number of consequences, including slower
queries, more space.

The K-Medoids Clustering
Method
• Find representative objects, called medoids, in clusters
• PAM (Partitioning Around Medoids, 1987)
– starts from an initial set of medoids and iteratively replaces one of the
medoids by one of the non-medoids if it improves the total distance of
the resulting clustering
– PAM works effectively for small data sets, but does not scale well for
large data sets
• CLARA (Kaufmann & Rousseeuw, 1990)
• CLARANS (Ng & Han, 1994): Randomized sampling
• Focusing + spatial data structure (Ester et al., 1995)

Typical k-medoids algorithm
(PAM)
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
Total Cost = 20
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
K=2
Arbitrary
choose k
object as
initial
medoids
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
Assign
each
remainin
g object
to
nearest
medoids
Randomly select a
nonmedoid object,Oramdom
Compute
total cost of
swapping
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
Total Cost = 26
Swapping O
and Oramdom
If quality is
improved.
Do loop
Until no
change
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10

PAM (Partitioning Around
Medoids) (1987)
• PAM (Kaufman and Rousseeuw, 1987), built in Splus
• Use real object to represent the cluster
– Select k representative objects arbitrarily
– For each pair of non-selected object h and selected object i,
calculate the total swapping cost TCih
– For each pair of i and h,
• If TCih < 0, i is replaced by h
• Then assign each non-selected object to the most similar
representative object
– repeat steps 2-3 until there is no change

PAM Clustering: Total swapping
cost TCih=∑jCjih
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
j
i
h
t
Cjih = 0
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
t
i h
j
Cjih = d(j, h) - d(j, i)
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
h
i
t
j
Cjih = d(j, t) - d(j, i)
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
t
i
h j
Cjih = d(j, h) - d(j, t)

What is the problem with
PAM?
• Pam is more robust than k-means in the presence of
noise and outliers because a medoid is less influenced
by outliers or other extreme values than a mean
• Pam works efficiently for small data sets but does not
scale well for large data sets.
– O(k(n-k)2
) for each iteration
where n is # of data,k is # of clusters
Sampling based method,
CLARA(Clustering LARge Applications)

CLARA (Clustering Large
Applications) (1990)
• CLARA (Kaufmann and Rousseeuw in 1990)
– Built in statistical analysis packages, such as S+
• It draws multiple samples of the data set, applies PAM on each
sample, and gives the best clustering as the output
• Strength: deals with larger data sets than PAM
• Weakness:
– Efficiency depends on the sample size
– A good clustering based on samples will not necessarily represent a
good clustering of the whole data set if the sample is biased

CLARANS (“Randomized”
CLARA) (1994)
• CLARANS (A Clustering Algorithm based on Randomized Search)
(Ng and Han’94)
• CLARANS draws sample of neighbors dynamically
• The clustering process can be presented as searching a graph
where every node is a potential solution, that is, a set of k medoids
• If the local optimum is found, CLARANS starts with new randomly
selected node in search for a new local optimum
• It is more efficient and scalable than both PAM and CLARA
• Focusing techniques and spatial access structures may further
improve its performance (Ester et al.’95)

Hierarchical Clustering
• Use distance matrix as clustering criteria. This
method does not require the number of clusters
k as an input, but needs a termination condition
Step 0 Step 1 Step 2 Step 3 Step 4
b
d
c
e
a
a b
d e
c d e
a b c d e
Step 4 Step 3 Step 2 Step 1 Step 0
agglomerative
(AGNES)
divisive
(DIANA)

AGNES (Agglomerative
Nesting)
• Introduced in Kaufmann and Rousseeuw (1990)
• Implemented in statistical analysis packages, e.g., Splus
• Use the Single-Link method and the dissimilarity matrix.
• Merge nodes that have the least dissimilarity
• Go on in a non-descending fashion
• Eventually all nodes belong to the same cluster
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10

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

Starting Situation
• Start with clusters of individual points and
a proximity matrix p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
. Proximity Matrix

Intermediate Situation
• After some merging steps, we have some clusters
C1
C4
C2 C5
C3
C2C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix

Intermediate Situation
• We want to merge the two closest clusters (C2 and C5) and
update the proximity matrix.
C1
C4
C2 C5
C3
C2C1
C1
C3
C5
C4
C2
C3 C4 C5
Proximity Matrix

After Merging
• The question is “How do we update the proximity
matrix?”
C1
C4
C2 U C5
C3 ? ? ? ?
?
?
?
C2
U
C5C1
C1
C3
C4
C2 U C5
C3 C4
Proximity Matrix

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

Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.
Proximity Matrix
• MIN
• MAX
• Group Average
objective function

Similarity
p1
p3
p5
p4
p2
p1 p2 p3 p4 p5 . . .
.
.
.
Proximity Matrix
• MIN
• MAX
• Group Average
objective function
× ×

Hierarchical Clustering: Time and
Space requirements
• O(N2
) space since it uses the proximity matrix.
– N is the number of points.
• 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 for
some approaches

Hierarchical Clustering: Problems
and Limitations
• Once a decision is made to combine two
clusters, it cannot be undone
• No objective function is directly minimized
• 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

A Dendrogram Shows How the
Clusters are Merged Hierarchically
• Decompose data objects into a several levels of
nested partitioning (tree of clusters), called a
dendrogram.
• A clustering of the data objects is obtained by
cutting the dendrogram at the desired level, then
each connected component forms a cluster.

DIANA (Divisive Analysis)
• Introduced in Kaufmann and Rousseeuw (1990)
• Implemented in statistical analysis packages, e.g., Splus
• Inverse order of AGNES
• Eventually each node forms a cluster on its own
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10

More on Hierarchical
Clustering Methods
• Major weakness of agglomerative clustering methods
– do not scale well: time complexity of at least O(n2
), where n is the
number of total objects
– can never undo what was done previously
• Integration of hierarchical with distance-based clustering
– BIRCH (1996): uses CF-tree and incrementally adjusts the
quality of sub-clusters
– CURE (1998): selects well-scattered points from the cluster and
then shrinks them towards the center of the cluster by a
specified fraction
– CHAMELEON (1999): hierarchical clustering using dynamic
modeling

BIRCH (1996)
• Birch: Balanced Iterative Reducing and Clustering using
Hierarchies, by Zhang, Ramakrishnan, Livny (SIGMOD’96)
• Incrementally construct a CF (Clustering Feature) tree, a
hierarchical data structure for multiphase clustering
– Phase 1: scan DB to build an initial in-memory CF tree (a multi-level
compression of the data that tries to preserve the inherent clustering
structure of the data)
– Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes
of the CF-tree
• Scales linearly: finds a good clustering with a single scan and
improves the quality with a few additional scans
• Weakness: handles only numeric data, and sensitive to the order of
the data record.

Clustering Feature: CF = (N, LS, SS)
N: Number of data points
LS: ∑N
i=1=Xi
SS: ∑N
i=1=Xi
2
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3 4 5 6 7 8 9 10
CF = (5, (16,30),(54,190))
(3,4)
(2,6)
(4,5)
(4,7)
(3,8)
Clustering Feature Vector

CF-Tree in BIRCH
• Clustering feature:
– summary of the statistics for a given subcluster: the 0-th, 1st and
2nd moments of the subcluster from the statistical point of view.
– registers crucial measurements for computing cluster and
utilizes storage efficiently
A CF tree is a height-balanced tree that stores the
clustering features for a hierarchical clustering
– A nonleaf node in a tree has descendants or “children”
– The nonleaf nodes store sums of the CFs of their children
• A CF tree has two parameters
– Branching factor: specify the maximum number of children.
– threshold: max diameter of sub-clusters stored at the leaf nodes

CF Tree
CF1
child1
CF3
child3
CF2
child2
CF6
child6
CF1
child1
CF3
child3
CF2
child2
CF5
child5
CF1 CF2 CF6
prev next CF1 CF2 CF4
prev next
B = 7
L = 6
Root
Non-leaf node
Leaf node Leaf node

CURE (Clustering Using
REpresentatives )
• CURE: proposed by Guha, Rastogi & Shim, 1998
– Stops the creation of a cluster hierarchy if a level consists of k
clusters
– Uses multiple representative points to evaluate the distance
between clusters, adjusts well to arbitrary shaped clusters and
avoids single-link effect

Drawbacks of Distance-
Based Method
• Drawbacks of square-error based clustering method
– Consider only one point as representative of a cluster
– Good only for convex shaped, similar size and density, and if k
can be reasonably estimated

Cure: The Algorithm
• Draw random sample s.
• Partition sample to p partitions with size s/p
• Partially cluster partitions into s/pq clusters
• Eliminate outliers
– By random sampling
– If a cluster grows too slow, eliminate it.
• Cluster partial clusters.
• Label data in disk

Data Partitioning and
Clustering
– s = 50
– p = 2
– s/p = 25
x x
x
y
y y
y
x
y
x
s/pq = 5

Cure: Shrinking Representative
Points
• Shrink the multiple representative points towards the
gravity center by a fraction of α.
• Multiple representatives capture the shape of the cluster
x
y
x
y

Clustering Categorical Data:
ROCK
• ROCK: Robust Clustering using linKs,
by S. Guha, R. Rastogi, K. Shim (ICDE’99).
– Use links to measure similarity/proximity
– Not distance based
– Computational complexity:
• Basic ideas:
– Similarity function and neighbors:
Let T1 = {1,2,3}, T2={3,4,5}
O n nm m n nm a( log )2 2
+ +
Sim T T
T T
T T
( , )1 2
1 2
1 2
=
∩
∪
Sim T T( , )
{ }
{ , , , , }
.1 2
3
1 2 3 4 5
1
5
0 2= = =

CHAMELEON (Hierarchical
clustering using dynamic
modeling)
• CHAMELEON: by G. Karypis, E.H. Han, and V. Kumar’99
• Measures the similarity based on a dynamic model
– Two clusters are merged only if the interconnectivity and closeness
(proximity) between two clusters are high relative to the internal
interconnectivity of the clusters and closeness of items within the
clusters
– Cure ignores information about interconnectivity of the objects,
Rock ignores information about the closeness of two clusters
• A two-phase algorithm
1. Use a graph partitioning algorithm: cluster objects into a large number
of relatively small sub-clusters
2. Use an agglomerative hierarchical clustering algorithm: find the
genuine clusters by repeatedly combining these sub-clusters

Overall Framework of
CHAMELEON
Construct
Sparse Graph Partition the Graph
Merge Partition
Final Clusters
Data Set

Density-Based Clustering
Methods
• Clustering based on density (local cluster criterion), such
as density-connected points
• Major features:
– Discover clusters of arbitrary shape
– Handle noise
– One scan
– Need density parameters as termination condition
• Several interesting studies:
– DBSCAN: Ester, et al. (KDD’96)
– OPTICS: Ankerst, et al (SIGMOD’99).
– DENCLUE: Hinneburg & D. Keim (KDD’98)
– CLIQUE: Agrawal, et al. (SIGMOD’98)

Density Concepts
• Core object (CO)–object with at least ‘M’ objects within a
radius ‘E-neighborhood’
• Directly density reachable (DDR)–x is CO, y is in x’s ‘E-
neighborhood’
• Density reachable–there exists a chain of DDR objects
from x to y
• Density based cluster–density connected objects
maximum w.r.t. reachability

Density-Based Clustering:
Background
• Two parameters:
– Eps: Maximum radius of the neighbourhood
– MinPts: Minimum number of points in an Eps-neighbourhood of
that point
• NEps(p): {q belongs to D | dist(p,q) <= Eps}
• Directly density-reachable: A point p is directly density-
reachable from a point q wrt. Eps, MinPts if
– 1) p belongs to NEps(q)
– 2) core point condition:
|NEps (q)| >= MinPts
p
q
MinPts = 5
Eps = 1 cm

Density-Based Clustering:
Background (II)
• Density-reachable:
– A point p is density-reachable from
a point q wrt. Eps, MinPts if there is
a chain of points p1, …, pn, p1 = q, pn
= p such that pi+1 is directly density-
reachable from pi
• Density-connected
– A point p is density-connected to a
point q wrt. Eps, MinPts if there is a
point o such that both, p and q are
density-reachable from o wrt. Eps
and MinPts.
p
q
p1
p q
o

DBSCAN: Density Based Spatial
Clustering of Applications with Noise
• Relies on a density-based notion of cluster: A cluster is
defined as a maximal set of density-connected points
• Discovers clusters of arbitrary shape in spatial databases
with noise
Core
Border
Outlier
Eps = 1cm
MinPts = 5

DBSCAN
• 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.

DBSCAN: Core, Border, and
Noise Points

DBSCAN Algorithm
• Eliminate noise points
• Perform clustering on the remaining points

DBSCAN: Core, Border and
Noise Points
Original Points Point types: core,
border and noise
Eps = 10, MinPts = 4

When DBSCAN Works Well
Original Points Clusters
• Resistant to Noise
• Can handle clusters of different shapes and sizes

When DBSCAN Does NOT
Work Well
Original Points
(MinPts=4, Eps=9.75).
(MinPts=4, Eps=9.92)
• Varying densities
• High-dimensional data

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

OPTICS: A Cluster-Ordering
Method (1999)
• OPTICS: Ordering Points To Identify the
Clustering Structure
– Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99)
– Produces a special order of the database wrt its
density-based clustering structure
– This cluster-ordering contains info equiv to the
density-based clusterings corresponding to a broad
range of parameter settings
– Good for both automatic and interactive cluster
analysis, including finding intrinsic clustering structure
– Can be represented graphically or using visualization
techniques

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

Clusters found in Random Data
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
Random
Points
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
K-means
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
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 Complete
Link

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. 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.

Measures of Cluster Validity
• 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.

Measuring Cluster Validity Via
Correlation
• Two matrices
– Proximity Matrix
– “Incidence” Matrix
• 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
• 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

Using Similarity Matrix for
Cluster Validation
• Order the similarity matrix with respect to cluster
labels and inspect visually.
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

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

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
Cluster Validation
• Clusters in random data are not so crisp
K-means
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
xy

Cluster 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

Internal Measures: SSE
• 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
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

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

Framework for Cluster Validity
• 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

Statistical Framework for SSE
• 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
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

Statistical Framework for
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

Internal Measures: Cohesion and
Separation
• 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
∑ ∑
∈
−=
i Cx
i
i
mxWSS 2
)(
∑ −=
i
ii mmCBSS 2
)(

Internal Measures: Cohesion
and Separation
• Example: SSE
– BSS + WSS = constant
1 2 3 4 5
× ××
m1 m2
m
1091
9)35.4(2)5.13(2
1)5.45()5.44()5.12()5.11(
22
2222
=+=
=−×+−×=
=−+−+−+−=
Total
BSS
WSSK=2 clusters:
10010
0)33(4
10)35()34()32()31(
2
2222
=+=
=−×=
=−+−+−+−=
Total
BSS
WSSK=1 cluster:

Internal Measures: Cohesion and
Separation
• 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 separation

Internal Measures: Silhouette
Coefficient
• 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 = 1 – a/b if a < b, (or s = b/a - 1 if a ≥ b, not the usual case)
– Typically between 0 and 1.
– The closer to 1 the better.
• Can calculate the Average Silhouette width for a cluster or a
clustering
a
b

External Measures of Cluster Validity:
Entropy and Purity

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 Clustering Data, Jain and Dubes

Grid-Based Clustering
Method
• Using multi-resolution grid data structure
• Several interesting methods
– STING (a STatistical INformation Grid approach) by Wang, Yang
and Muntz (1997)
– WaveCluster by Sheikholeslami, Chatterjee, and Zhang
(VLDB’98)
• A multi-resolution clustering approach using wavelet method
– CLIQUE: Agrawal, et al. (SIGMOD’98)

STING: A Statistical
Information Grid Approach
• Wang, Yang and Muntz (VLDB’97)
• The spatial area area is divided into rectangular cells
• There are several levels of cells corresponding to
different levels of resolution

Information Grid Approach (2)
– Each cell at a high level is partitioned into a number of smaller
cells in the next lower level
– Statistical info of each cell is calculated and stored beforehand
and is used to answer queries
– Parameters of higher level cells can be easily calculated from
parameters of lower level cell
• count, mean, s, min, max
• type of distribution—normal, uniform, etc.
– Use a top-down approach to answer spatial data queries
– Start from a pre-selected layer—typically with a small number of
cells
– For each cell in the current level compute the confidence interval

Information Grid Approach (3)
– Remove the irrelevant cells from further consideration
– When finish examining the current layer, proceed to the next
lower level
– Repeat this process until the bottom layer is reached
– Advantages:
• Query-independent, easy to parallelize, incremental update
• O(K), where K is the number of grid cells at the lowest level
– Disadvantages:
• All the cluster boundaries are either horizontal or vertical,
and no diagonal boundary is detected

WaveCluster (1998)
• Sheikholeslami, Chatterjee, and Zhang (VLDB’98)
• A multi-resolution clustering approach which applies
wavelet transform to the feature space
– A wavelet transform is a signal processing technique that
decomposes a signal into different frequency sub-band.
• Both grid-based and density-based
• Input parameters:
– # of grid cells for each dimension
– the wavelet, and the # of applications of wavelet transform.

WaveCluster (1998)
• How to apply wavelet transform to find clusters
– Summaries the data by imposing a multidimensional
grid structure onto data space
– These multidimensional spatial data objects are
represented in a n-dimensional feature space
– Apply wavelet transform on feature space to find the
dense regions in the feature space
– Apply wavelet transform multiple times which result in
clusters at different scales from fine to coarse

Wavelet Transform
• Decomposes a signal into different
frequency subbands. (can be applied to
n-dimensional signals)
• Data are transformed to preserve relative
distance between objects at different
levels of resolution.
• Allows natural clusters to become more
distinguishable

WaveCluster (1998)
• Why is wavelet transformation useful for clustering
– Unsupervised clustering
It uses hat-shape filters to emphasize region where points
cluster, but simultaneously to suppress weaker information in
their boundary
– Effective removal of outliers
– Multi-resolution
– Cost efficiency
• Major features:
– Complexity O(N)
– Detect arbitrary shaped clusters at different scales
– Not sensitive to noise, not sensitive to input order
– Only applicable to low dimensional data

CLIQUE (Clustering In QUEst)
• Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98).
• Automatically identifying subspaces of a high dimensional data
space that allow better clustering than original space
• CLIQUE can be considered as both density-based and grid-based
– It partitions each dimension into the same number of equal length
interval
– It partitions an m-dimensional data space into non-overlapping
rectangular units
– A unit is dense if the fraction of total data points contained in the unit
exceeds the input model parameter
– A cluster is a maximal set of connected dense units within a subspace

CLIQUE: The Major Steps
• Partition the data space and find the number of points
that lie inside each cell of the partition.
• Identify the subspaces that contain clusters using the
Apriori principle
• Identify clusters:
– Determine dense units in all subspaces of interests
– Determine connected dense units in all subspaces of interests.
• Generate minimal description for the clusters
– Determine maximal regions that cover a cluster of connected
dense units for each cluster
– Determination of minimal cover for each cluster

Salary
(10,000)
20 30 40 50 60
age
54312670
20 30 40 50 60
age
54312670
Vacation
(week)
age
Vacation
Salary 30 50
τ = 3

Strength and Weakness of
CLIQUE
• Strength
– It automatically finds subspaces of the highest
dimensionality such that high density clusters exist in
those subspaces
– It is insensitive to the order of records in input and
does not presume some canonical data distribution
– It scales linearly with the size of input and has good
scalability as the number of dimensions in the data
increases
• Weakness
– The accuracy of the clustering result may be
degraded at the expense of simplicity of the method

Model-Based Clustering
Methods
• Attempt to optimize the fit between the data and some
mathematical model
• Statistical and AI approach
– Conceptual clustering
• A form of clustering in machine learning
• Produces a classification scheme for a set of unlabeled objects
• Finds characteristic description for each concept (class)
– COBWEB (Fisher’87)
• A popular a simple method of incremental conceptual learning
• Creates a hierarchical clustering in the form of a classification tree
• Each node refers to a concept and contains a probabilistic
description of that concept

COBWEB Clustering
Method
A classification tree

More on Statistical-Based
Clustering
• Limitations of COBWEB
– The assumption that the attributes are independent of each
other is often too strong because correlation may exist
– Not suitable for clustering large database data – skewed tree
and expensive probability distributions
• CLASSIT
– an extension of COBWEB for incremental clustering of
continuous data
– suffers similar problems as COBWEB
• AutoClass (Cheeseman and Stutz, 1996)
– Uses Bayesian statistical analysis to estimate the number of
clusters
– Popular in industry

Other Model-Based
Clustering Methods
• Neural network approaches
– Represent each cluster as an exemplar, acting as a
“prototype” of the cluster
– New objects are distributed to the cluster whose
exemplar is the most similar according to some
dostance measure
• Competitive learning
– Involves a hierarchical architecture of several units
(neurons)
– Neurons compete in a “winner-takes-all” fashion for
the object currently being presented

Model-Based Clustering
Methods

Self-organizing feature
maps (SOMs)
• Clustering is also performed by having several
units competing for the current object
• The unit whose weight vector is closest to the
current object wins
• The winner and its neighbors learn by having
their weights adjusted
• SOMs are believed to resemble processing that
can occur in the brain
• Useful for visualizing high-dimensional data in 2-
or 3-D space

What Is Outlier Discovery?
• What are outliers?
– The set of objects are considerably dissimilar from the
remainder of the data
– Example: Sports: Michael Jordon, Wayne Gretzky, ...
• Problem
– Find top n outlier points
• Applications:
– Credit card fraud detection
– Telecom fraud detection
– Customer segmentation
– Medical analysis

Outlier Discovery:
Statistical
Approaches
Assume a model underlying distribution that
generates data set (e.g. normal distribution)
• Use discordancy tests depending on
– data distribution
– distribution parameter (e.g., mean, variance)
– number of expected outliers
• Drawbacks
– most tests are for single attribute
– In many cases, data distribution may not be known

Outlier Discovery: Distance-
Based Approach
• Introduced to counter the main limitations
imposed by statistical methods
– We need multi-dimensional analysis without knowing
data distribution.
• Distance-based outlier: A DB(p, D)-outlier is an
object O in a dataset T such that at least a
fraction p of the objects in T lies at a distance
greater than D from O
• Algorithms for mining distance-based outliers
– Index-based algorithm
– Nested-loop algorithm
– Cell-based algorithm

Outlier Discovery:
Deviation-Based Approach
• Identifies outliers by examining the main characteristics
of objects in a group
• Objects that “deviate” from this description are
considered outliers
• sequential exception technique
– simulates the way in which humans can distinguish unusual
objects from among a series of supposedly like objects
• OLAP data cube technique
– uses data cubes to identify regions of anomalies in large
multidimensional data

Problems and Challenges
• Considerable progress has been made in scalable clustering
methods
– Partitioning: k-means, k-medoids, CLARANS
– Hierarchical: BIRCH, CURE
– Density-based: DBSCAN, CLIQUE, OPTICS
– Grid-based: STING, WaveCluster
– Model-based: Autoclass, Denclue, Cobweb
• Current clustering techniques do not address all the requirements
adequately
• Constraint-based clustering analysis: Constraints exist in data space
(bridges and highways) or in user queries

Constraint-Based Clustering
Analysis
• Clustering analysis: less parameters but more user-
desired constraints, e.g., an ATM allocation problem

Clustering With Obstacle
Objects
Taking obstacles into
account
Not Taking obstacles into
account

Summary
• Cluster analysis groups objects based on their similarity
and has wide applications
• Measure of similarity can be computed for various types
of data
• Clustering algorithms can be categorized into partitioning
methods, hierarchical methods, density-based methods,
grid-based methods, and model-based methods
• Outlier detection and analysis are very useful for fraud
detection, etc. and can be performed by statistical,
distance-based or deviation-based approaches
• There are still lots of research issues on cluster analysis,
such as constraint-based clustering

References (1)
• R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high
dimensional data for data mining applications. SIGMOD'98
• M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973.
• M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to identify the
clustering structure, SIGMOD’99.
• P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scietific, 1996
• M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters
in large spatial databases. KDD'96.
• M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial databases: Focusing
techniques for efficient class identification. SSD'95.
• D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine Learning,
2:139-172, 1987.
• D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach based on
dynamic systems. In Proc. VLDB’98.
• S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large databases.
SIGMOD'98.
• A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.

References (2)
• L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis.
John Wiley & Sons, 1990.
• E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets. VLDB’98.
• G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to Clustering.
John Wiley and Sons, 1988.
• P. Michaud. Clustering techniques. Future Generation Computer systems, 13, 1997.
• R. Ng and J. Han. Efficient and effective clustering method for spatial data mining. VLDB'94.
• E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large data sets.
Proc. 1996 Int. Conf. on Pattern Recognition, 101-105.
• G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution clustering
approach for very large spatial databases. VLDB’98.
• W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial Data
Mining, VLDB’97.
• T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering method for very
large databases. SIGMOD'96.

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