cs4811-ch23a-K-means clustering algorithm .ppt

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Chapter 23 Probabilistic Language Processing
Clustering examples
Additional sources used in preparing the slides:
• David Grossman’s clustering slides:
http://ir.iit.edu/~dagr/IRcourse/Notes/08Clustering.pdf
• Subbarao Kambhampati’s clustering slides:
http://rakaposhi.eas.asu.edu/cse494/notes/f02-clustering.ppt
• Jeffrey Ullman’s clustering slides:
www-db.stanford.edu/~ullman/cs345-notes.html
• Ernest Davis’ clustering slides:
www.cs.nyu.edu/courses/fall02/G22.3033-008/index.htm

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Example: a cholera outbreak in London
Many years ago, during a cholera outbreak in London, a
physician plotted the location of cases on a map.
Properly visualized, the data indicated that cases
clustered around certain intersections, where there were
polluted wells, not only exposing the cause of cholera,
but indicating what to do about the problem.
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Conceptual Clustering
The clustering problem
Given
• a collection of unclassified objects, and
• a means for measuring the similarity of objects
(distance metric),
find
• classes (clusters) of objects such that some
standard of quality is met (e.g., maximize the
similarity of objects in the same class.)
Essentially, it is an approach to discover a useful
summary of the data.

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Conceptual Clustering (cont’d)
Ideally, we would like to represent clusters and
their semantic explanations. In other words, we
would like to define clusters extensionally (i.e.,
by general rules) rather than intensionally (i.e.,
by enumeration).
For instance, compare
{ X | X teaches AI at MTU CS}, and
{ John Lowther, Nilufer Onder}

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Curse of dimensionality
• While clustering looks intuitive in 2
dimensions, many applications involve 10 or
10,000 dimensions
• High-dimensional spaces look different: the
probability of random points being close drops
quickly as the dimensionality grows

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Higher dimensional examples
• Observation that customers who buy diapers are more
likely to buy beer than average allowed supermarkets to
place beer and diapers nearby, knowing many
customers would walk between them. Placing potato
chips between increased the sales of all three items.

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Sloan Digital Sky Survey
• A cool tool to “map the universe”
• Objects are represented by their radiation in 9
dimensions (each dimension represents radiation in one
band of the spectrum)
• Clustered 2 x 109
sky objects into similar objects e.g.,
stars, galaxies, quasars, etc.
• The objective was to catalog and cluster the entire
visible universe. Clustering sky objects by their
radiation levels in different bands allowed astronomers
to distinguish between galaxies, nearby stars, and many
other kinds of celestial objects.

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Clustering CDs
• Intuition: 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 CDs have similar sets of customers and
vice versa

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The space of CDs
• 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, …, xn), where xi = 1 iff the ith
customer
bought the CD
• Compare this with the correlated items matrix:
rows = customers
columns = CDs

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Clustering documents
• Query “salsa” submitted to MetaCrawler returns the following
documents among others:
 How to dance salsa
 Gourmet salsa
 Diet seen on Rachael Ray
 Michigan Salsa
• It also asks: “Are you looking for?”
 Music salsa
 Salsa recipe
 Homemade salsa recipe
 Salsa dancing
• The clusters are: dance, recipe, clubs, sauces, buy, Mexican,
bands, natural, …

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Clustering documents (cont’d)
• Documents may be thought of as points in a high-
dimensional space, where each dimension
corresponds to one possible word.
• Clusters of documents in this space often
correspond to groups of documents on the same
topic, i.e., documents with similar sets of words may
be about the same topic
• Represent a document by a vector (x1, x2, …, xn),
where xi = 1 iff the ith
word (in some order) appears in
the document
• n can be infinite

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Analyzing protein sequences
• Objects are sequences of {C, A, T, G}
• Distance between sequences is “edit
distance,” the minimum number of inserts and
deletes to turn one into the other
• Note that there is a “distance,” but no
convenient space of points

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Measuring distance
• To discuss, whether a set of points is close enough
to be considered a cluster, we need a distance
measure D(x,y) that tells how far points x and y are.
• The axioms for a distance measure D are:
1. D(x,x) = 0 A point is distance 0
from itself
2. D(x,y) = D(y,x) Distance is symmetric
3. D(x,y) ≤ D(x,z) + D(z,y) The triangle inequality
4. D(x,y) ≥ 0 Distance is positive

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K-dimensional Euclidean space
The distance between any two points, say
a = [a1, a2, … , ak] and b = [b1, b2, … , bk]
is given in some manner such as:
1. Common distance (“L2 norm”) :
i =1 (ai - bi)2
2. Manhattan distance (“L1 norm”):
i =1 |ai - bi|
3. Max of dimensions (“L norm”):
maxi =1 |ai - bi|
k
k
k
a
b
a
b
a
b

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Non-Euclidean spaces
Here are some examples where a distance measure
without a Euclidean space makes sense.
• Web pages: Roughly 108
-dimensional space where
each dimension corresponds to one word. Rather
use vectors to deal with only the words actually
present in documents a and b.
• Character strings, such as DNA sequences: Rather
use a metric based on the LCS---Lowest Common
Subsequence.
• Objects represented as sets of symbolic, rather
than numeric, features: Rather base similarity on the
proportion of features that they have in common.

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Non-Euclidean spaces (cont’d)
object1 = {small, red, rubber, ball}
object2 = {small, blue, rubber, ball}
object3 = {large, black, wooden, ball}
similarity(object1, object2) = 3 / 4
similarity(object1, object3) =
similarity(object2, object3) = 1/4
Note that it is possible to assign different
weights to features.

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Approaches to Clustering
Broadly specified, there are two classes of
clustering algorithms:
1. Centroid approaches: We guess the centroid
(central point) in each cluster, and assign points to
the cluster of their nearest centroid.
2. Hierarchical approaches: We begin assuming
that each point is a cluster by itself. We repeatedly
merge nearby clusters, using some measure of
how close two clusters are (e.g., distance between
their centroids), or how good a cluster the
resulting group would be (e.g., the average
distance of points in the cluster from the resulting
centroid.)

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The k-means algorithm
•Pick k cluster centroids.
•Assign points to clusters by picking the
closest centroid to the point in question. As
points are assigned to clusters, the centroid of
the cluster may migrate.
Example: Suppose that k = 2 and we assign
points 1, 2, 3, 4, 5, in that order. Outline circles
represent points, filled circles represent
centroids. 1 5
3
2
4

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The k-means algorithm example (cont’d)
1 5
3
2
4
1 5
3
2
4
1 5
3
2
4
1 5
3
2
4

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Issues
• How to initialize the k centroids?
Pick points sufficiently far away from any other
centroid, until there are k.
• As computation progresses, one can decide to
split one cluster and merge two, to keep the total
at k. A test for whether to do so might be to ask
whether doing so reduces the average distance
from points to their centroids.
• Having located the centroids of k clusters, we
can reassign all points, since some points that
were assigned early may actually wind up closer
to another centroid, as the centroids move about.

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Issues (cont’d)
• How to determine k?
One can try different values for k until the
smallest k such that increasing k does not
much decrease the average points of points to
their centroids.
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Determining k
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When k = 1, all the points are
in one cluster, and the average
distance to the centroid will be
high.
When k = 2, one of the clusters
will be by itself and the other
two will be forced into one
cluster. The average distance
of points to the centroid will
shrink considerably.

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Determining k (cont’d)
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When k = 3, each of the
apparent clusters should be a
cluster by itself, and the
average distance from the
points to their centroids
shrinks again.
When k = 4, then one of the
true clusters will be artificially
partitioned into two nearby
clusters. The average distance
to the centroids will drop a bit,
but not much.
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Determining k (cont’d)
This failure to drop further suggests that k = 3
is right. This conclusion can be made even if
the data is in so many dimensions that we
cannot visualize the clusters.
Average
radius
k
1 2 3 4

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The CLUSTER/2 algorithm
1. Select k seeds from the set of observed
objects. This may be done randomly or
according to some selection function.
2. For each seed, using that seed as a positive
instance and all other seeds as negative
instances, produce a maximally general
definition that covers all of the positive and
none of the negative instances (multiple
classifications of non-seed objects are
possible.)

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The CLUSTER/2 algorithm (cont’d)
3. Classify all objects in the sample according to
these descriptions. Replace each maximally
specific description that covers all objects in the
category (to decrease the likelihood that classes
overlap on unseen objects.)
4. Adjust remaining overlapping definitions.
5. Using a distance metric, select an element
closest to the center of each class.
6. Repeat steps 1-5 using the new central elements
as seeds. Stop when clusters are satisfactory.

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The CLUSTER/2 algorithm (cont’d)
7. If clusters are unsatisfactory and no
improvement occurs over several iterations,
select the new seeds closest to the edge of the
cluster.

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The steps of a CLUSTER/2 run

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Document clustering
Automatically group related documents into
clusters given some measure of similarity. For
example,
• medical documents
• legal documents
• financial documents
• web search results

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Hierarchical Agglomerative Clustering (HAC)
• Given n documents, create a n x n doc-doc
similarity matrix.
• Each document starts as a cluster of size one.
• do until there is only one cluster
 Combine the two clusters with the greatest similarity
(if X and Y are the most mergable pair of clusters,
then we create X-Y as the parent of X and Y. Hence the
name “hierarchical”.)
 Update the doc-doc matrix.

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Example
Consider A, B, C, D, E as documents with the
following similarities:
A B C D E
A - 2 7 9 4
B 2 - 9 11 14
C 7 9 - 4 8
D 9 11 4 - 2
E 4 14 8 2 -
The pair
with the
highest
similarity
is:
B-E = 14

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Example
So let’s cluster B and E. We now have the
following structure:
A C D B E
BE

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Example
Update the doc-doc matrix:
A BE C D
A - 2 7 9
BE 2 - 8 2
C 7 8 - 4
D 9 2 4 -
To compute
(A,BE):
take the
minimum of
(A,B)=2 and
(A,E)=4.
This is called
complete
linkage.

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Example
Highest link is A-D. So let’s cluster A and D. We
now have the following structure:
A D C B E
BE
AD

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Example
Update the doc-doc matrix:
AD BE C
AD - 2 4
BE 2 - 8
C 4 8 -

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Example
• Highest link is BE-C. So let’s cluster BE and C.
We now have the following structure:
A D C B E
BE
AD
BCE

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Example
• At this point, there are only two nodes that
have not been clustered. So we cluster AD and
BCE. We now have the following structure:
A D C B E
BE
AD
BCE
ABCDE
Everything
has been
clustered.

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Time complexity analysis
Hierarchical agglomerative clustering (HAC)
requires:
• O(n2
) to compute the doc-doc similarity matrix
• One node is added during each round of
clustering so there are now O(n) clustering
steps
• For each clustering step we must re-compute
the doc-doc matrix. This requires O(n) time.
• So we have: n2
+ (n)(n) = O(n2
) – so it’s
expensive!
• For 500,000 documents n2
is 250,000,000,000!!

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One pass clustering
• Choose a document and declare it to be in a
cluster of size 1.
• Now compute the distance from this cluster to
all the remaining nodes.
• Add “closest” node to the cluster. If no node
is really close (within some threshold), start a
new cluster between the two closest nodes.

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Example
• Consider the following nodes
A
E
D
B
C

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Example
• Choose node A as the first cluster
• Now compute the distance between A and the
others. B is the closest, so cluster A and B.
• Compute the centroid of the cluster just formed.
A
E
D
B
C
AB

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Example
• Compute the distance between A-B and all the
remaining clusters using the centroid of A-B.
• Let’s assume all the others are too far from AB.
Choose one of these non-clustered elements and
place it in a cluster. Let’s choose E.
A
E
D
B
C
AB

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Example
• Compute the distance from E to D and E to C.
• E to D is closer so we form a cluster of E and D.
A
E
D
B
C
AB
DE

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Example
• Compute the distance from D-E to C.
• It is within the threshold so include C in this
cluster.
A
E
D
B
C
AB
CDE
Everything
has been
clustered.

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One pass requires:
• n passes as we add node for each pass
• First pass requires n-1 comparisons
• Second pass requires n-2 comparisons
• Last pass needs 1
• So we have 1 + 2 + 3 + … + (n-1) = (n-1)(n) / 2
• (n2
- n) / 2 = O(n2
)
• The constant is lower for one pass but we are
still at n2
.

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Remember k-means clustering
• Pick k points as the seeds of k clusters
• At the onset, there are k clusters of size one.
• do until all nodes are clustered
 Pick a point and put it into the cluster whose centroid is
closest.
 Recompute the centroid of the modified cluster.

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K-means requires:
• Each node gets added to a cluster, so there
are n clustering steps
• For each addition, we need to compare to k
centroids
• We also need to recompute the centroid after
adding the new node, this takes a constant
amount of time (say c)
• The total time needed is (k + c) n = O(n)
• So it is a linear algorithm!

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But there are problems…
• K needs to be known in advance or need trials
to compute k
• Tends to go to local minima that are sensitive
to the starting centroids:
D
A B
E F
C
If the seeds are B and E, the resulting clusters
are {A,B,C} and {D,E,F}.
If the seeds are D and F, the resulting clusters
are {A,B,D,E} and {C,F}.

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Two questions for you
1. Why did the computer go to the restaurant?
2. What do you do when you have a slow
algorithm that produces quality results, and a
fast algorithm that cannot guarantee quality?
1. To get a byte.
2. Many things…
One option is to use the slow algorithm on a
portion of the problem to obtain a better
starting point for the fast algorithm.

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Buckshot clustering
• The goal is to reduce the run time by
combining HAC and k-means clustering.
• Select d documents where d is SQRT(n).
• Cluster these d documents using HAC, this
will take O(n) time.
• Use the results of HAC as initial seeds for k-
means.
• It uses HAC to bootstrap k-means.
• The overall algorithm is O(n) and avoids
problems of bad seed selection.

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Getting the k clusters
Cut where you have k clusters
A D C B E
BE
AD BCE
ABCDE

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Effect of document order
• With hierarchical clustering we get the same
clusters every time.
• With one pass clustering, we get different
clusters based on the order we process the
documents.
• With k-means clustering, we get different
clusters based on the selected seeds.

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Computing the distance (time)
• In our time complexity analysis we finessed
the time required to compute the distance
between two nodes
• Sometimes this is an expensive task
depending on the analysis required

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Computing the distance (methods)
• To compute the intra-cluster distance:
(Sum/min/max/avg) the (absolute/squared)
distance between
 All pairs of points in the cluster, or
 Between the centroid and all points in the cluster
• To compute the inter-cluster distance for HAC:
 Single-link: distance between closest neighbors
 Complete-link: distance between farthest neighbors
 Group-average: average distance between all pairs of
neighbors
 Centroid-distance: distance between centroids (most
commonly used)

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More on document clustering
• Applications
 Structuring search results
 Suggesting related pages
 Automatic directory construction / update
 Finding near identical pages
 Finding mirror pages (e.g., for propagating updates)
 Eliminate near-duplicates from results page
 Plagiarism detection
 Lost and found (find identical pages at different URLs at
different times)
• Problems
 Polysemy, e.g., “bat,” “Washington,” “Banks”
 Multiple aspects of a single topic
 Ultimately amounts to general problem of information
structuring

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Clustering vs. classification
• Clustering is when the clusters are not known
• If the system of clusters is known, and the
problem is to place a new item into the proper
cluster, this is classification

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How many possible clusterings?
If we have n points and would like to cluster
them into k clusters, then there are k clusters
the first point can go to, there are k clusters for
each of the remaining points. So the total
number of possible clusterings is kn
.
Brute force enumeration will not work. That is
why we have iterative optimization algorithms
that start with a clustering and iteratively
improve it.
Finally, note that noise (outliers) is a problem
for clustering too. One can use statistical
techniques to identify outliers.

60
Cluster structure
• Hierarchical vs flat
• Overlap
 Disjoint partitioning, e.g., partition congressmen by state
 Multiple dimensions of partitioning, each disjoint, e.g.,
partition congressmen by state; by party; by House/Senate
 Arbitrary overlap, e.g., partition bills by congressmen who
voted for them
• Exhaustive vs. non-exhaustive
• Outliers: what to do?
• How many clusters? How large?

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Measuring the quality of the clusters
A good clustering is one where
• (intra-cluster distance) the sum of distances
between objects in the same cluster are
minimized
• (inter-cluster distance) while the distances
between different clusters are maximized
The objective is to minimize: F(intra, inter)

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Related communities
• data mining (in databases, over the web)
• statistics
• clustering algorithms
• visualization
• databases

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