Clustering
Mining Massive Datasets
Wu-Jun Li
Department of Computer Science and Engineering
Shanghai Jiao Tong University
Lecture 8: Clustering
1
Clustering
Outline
 Introduction
 Hierarchical Clustering
 Point Assignment based Clustering
 Evaluation
2
Clustering
Introduction
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 as close to each other as possible
 Members of different clusters are dissimilar
 Distance measure
 Euclidean, Cosine, Jaccard, edit distance, …
3
Introduction
Clustering
Example
x
x
x
x x
x x
x xx x
x x x
x x
x
x
xx x
x x
x x x
x
xx x
x
x x
x x x x
x x x
x
4
Clustering
Introduction
Application: SkyCat
 A catalog of 2 billion “sky objects” represents
objects by their radiation in 7 dimensions
(frequency bands).
 Problem: cluster into similar objects, e.g., galaxies,
nearby stars, quasars, etc.
 Sloan Sky Survey is a newer, better version.
5
Clustering
Introduction
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.
 A CD’s point in this space is (x1, x2,…, xk), where xi = 1 iff
the i th customer bought the CD.
 Similar CD’s have similar sets of customers, and
vice-versa.
6
Clustering
Introduction
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.
7
Clustering
Introduction
Example: DNA 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.
8
Clustering
Introduction
Cosine, Jaccard, and Euclidean Distances

As with CD’s, we have a choice when we
think of documents as sets of words or
shingles:
1. Sets as vectors: measure similarity by the
cosine distance.
2. Sets as sets: measure similarity by the
Jaccard distance.
3. Sets as points: measure similarity by
Euclidean distance.
9
Clustering
Introduction
Clustering Algorithms
 Hierarchical algorithms
 Agglomerative (bottom-up)
 Initially, each point in cluster by itself.
 Repeatedly combine the two “nearest”
clusters into one.
 Divisive (top-down)
 Point Assignment
 Maintain a set of clusters.
 Place points into their
“nearest” cluster.
10
Clustering
Outline
 Introduction
 Hierarchical Clustering
 Point Assignment based Clustering
 Evaluation
11
Clustering
Hierarchical Clustering
Hierarchical Clustering

Two important questions:
1. How do you represent a cluster of more than one point?
2. How do you determine the “nearness” of clusters?
12
Clustering
Hierarchical Clustering
Hierarchical Clustering – (2)
 Key problem: as you build clusters, how do you
represent the location of each cluster, to tell which
pair of clusters is closest?
 Euclidean case: each cluster has a centroid = average
of its points.
 Measure inter-cluster distances by distances of centroids.
13
Hierarchical Clustering
Clustering
Example
(5,3)
o
(1,2)
o
x (1.5,1.5)
x (1,1) o (2,1)
o (0,0)
o : data point
x : centroid
x (4.7,1.3)
o (4,1)
x (4.5,0.5)
o
(5,0)
14
Clustering
Hierarchical Clustering
And in the Non-Euclidean Case?
 The only “locations” we can talk about are the points
themselves.
 I.e., there is no “average” of two points.
 Approach 1: clustroid = point “closest” to other
points.
 Treat clustroid as if it were centroid, when computing
intercluster distances.
15
Clustering
Hierarchical Clustering
“Closest” Point?

Possible meanings:
1.
2.
3.
4.
Smallest maximum distance to the other points.
Smallest average distance to other points.
Smallest sum of squares of distances to other points.
Etc., etc.
16
Hierarchical Clustering
Clustering
Example
clustroid
1
2
6
3
4
5
clustroid
intercluster
distance
17
Clustering
Hierarchical Clustering
Other Approaches to Defining
“Nearness” of Clusters
 Approach 2: intercluster distance = minimum of
the distances between any two points, one from
each cluster.
 Approach 3: Pick a notion of “cohesion” of clusters,
e.g., maximum distance from the clustroid.
 Merge clusters whose union is most cohesive.
18
Clustering
Hierarchical Clustering
Cohesion



Approach 1: Use the diameter of the merged
cluster = maximum distance between points in the
cluster.
Approach 2: Use the average distance between
points in the cluster.
Approach 3: Use a density-based approach: take
the diameter or average distance, e.g., and divide
by the number of points in the cluster.
 Perhaps raise the number of points to a power first, e.g.,
square-root.
19
Clustering
Outline
 Introduction
 Hierarchical Clustering
 Point Assignment based Clustering
 Evaluation
20
Clustering
Point Assignment
k – Means Algorithm(s)
 Assumes Euclidean space.
 Start by picking k, the number of clusters.
 Select k points {s1, s2,… sK} as seeds.
 Example: pick one point at random, then k -1 other points,
each as far away as possible from the previous points.
 Until clustering converges (or other stopping
criterion):
 For each point xi:
 Assign xi to the cluster cj such that dist(xi, sj) is minimal.
 For each cluster cj
 sj = (cj) where (cj) is the centroid of cluster cj
21
Point Assignment
Clustering
k-Means Example (k=2)
Pick seeds
x
x
x
x
Reassign clusters
Compute centroids
Reassign clusters
Compute centroids
Reassign clusters
Converged!
22
Clustering
Point Assignment
Termination conditions
 Several possibilities, e.g.,
 A fixed number of iterations.
 Point assignment unchanged.
 Centroid positions don’t change.
23
Point Assignment
Clustering
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.
Average
distance to
centroid
Best value
of k
k
24
Point Assignment
Clustering
Example: Picking k
Too few;
many long
distances
to centroid.
x
x
x
x x
x x
x xx x
x x x
x x
x
x
xx x
x x
x x x
x
xx x
x
x x
x x x x
x x x
x
25
Point Assignment
Clustering
Example: Picking k
Just right;
distances
rather short.
x
x
x
x x
x x
x xx x
x x x
x x
x
x
xx x
x x
x x x
x
xx x
x
x x
x x x x
x x x
x
26
Point Assignment
Clustering
Example: Picking k
Too many;
little improvement
in average
x
distance.
x x
x
x
x x
x xx x
x x x
x x
x
x
xx x
x x
x x x
x
xx x
x
x x
x x x x
x x x
x
27
Clustering
Point Assignment
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.
28
Clustering
Point Assignment
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.
29
Clustering
Point Assignment
Initialization: k -Means

Possibilities include:
1. Take a small random 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.
30
Clustering
Point Assignment
Three Classes of Points

discard set (DS):


points close enough to a centroid to be summarized.
compressed set (CS):



groups of points that are close together but not close to
any centroid.
They are summarized, but not assigned to a cluster.
retained set (RS):

isolated points.
31
Clustering
Point Assignment
Summarizing Sets of Points

For each cluster, the discard set is summarized by:



The number of points, N.
The vector SUM, whose i th component is the sum of
the coordinates of the points in the i th dimension.
The vector SUMSQ, whose i th component is the sum of
squares of coordinates in i th dimension.
32
Clustering
Point Assignment
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.
33
Clustering
Point Assignment
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 compressed
set.
34
Point Assignment
Clustering
“Galaxies” Picture
Points in
the RS
Compressed sets.
Their points are in
the CS.
A cluster. Its points
are in the DS.
The centroid
35
Clustering
Point Assignment
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.
36
Clustering
Point Assignment
Processing – (2)
3. Adjust statistics of the clusters to account for the
new points.

Add N’s, SUM’s, SUMSQ’s.
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.
37
Clustering
Point Assignment
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?
38
Clustering
Point Assignment
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.
39
Clustering
Point Assignment
Mahalanobis Distance


Normalized Euclidean distance from centroid.
For point (x1,…,xd) and centroid (c1,…,cd):
1. Normalize in each dimension: yi = (xi -ci)/i
2. Take sum of the squares of the yi ’s.
3. Take the square root.
40
Clustering
Point Assignment
Mahalanobis Distance – (2)
 If clusters are normally distributed in d
dimensions, then after transformation, one
standard deviation =
.
 I.e., 70% of the points of the cluster will have a
Mahalanobis distance <
.
 Accept a point for a cluster if its M.D. is < some
threshold, e.g. 4 standard deviations.
41
Point Assignment
Clustering
Picture: Equal M.D. Regions
2

42
Clustering
Point Assignment
Should Two CS Subclusters Be Combined?
 Compute the variance of the combined subcluster.
 N, SUM, and SUMSQ allow us to make that calculation
quickly.
 Combine if the variance is below some threshold.
 Many alternatives: treat dimensions differently,
consider density.
43
Clustering
Point Assignment
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 (Clustering Using REpresentatives):
 Assumes a Euclidean distance.
 Allows clusters to assume any shape.
44
Point Assignment
Clustering
Example: Stanford Faculty Salaries
h
e
e
salary
h h h
e h
e
e
e
h
e
h
h
e
e
h
e
e
h
h
h
h
age
45
Clustering
Point Assignment
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.
46
Point Assignment
Clustering
Example: Initial Clusters
h
e
e
salary
h h h
e h
e
e
e
h
e
h
h
e
e
h
e
e
h
h
h
h
age
47
Point Assignment
Clustering
Example: Pick Dispersed Points
h
e
e
salary
h h h
e h
h
e
h
h
e
e
h
h
e
e
h
h
h
e
e
e
Pick (say) 4
remote points
for each
cluster.
age
48
Point Assignment
Clustering
Example: Pick Dispersed Points
h
e
e
salary
h h h
e h
h
e
h
h
e
e
h
h
e
e
h
h
h
e
e
e
Move points
(say) 20%
toward the
centroid.
age
49
Clustering
Point Assignment
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.
50
Clustering
Outline
 Introduction
 Hierarchical Clustering
 Point Assignment based Clustering
 Evaluation
51
Clustering
Evaluation
What Is A Good Clustering?
 Internal criterion: A good clustering will produce
high quality clusters in which:
 the intra-class (that is, intra-cluster) similarity is
high
 the inter-class similarity is low
 The measured quality of a clustering depends on
both the point representation and the similarity
measure used
52
Clustering
Evaluation
External criteria for clustering quality
 Quality measured by its ability to discover some
or all of the hidden patterns or latent classes in
gold standard data
 Assesses a clustering with respect to ground
truth … requires labeled data
 Assume documents with C gold standard classes,
while our clustering algorithms produce K
clusters, ω1, ω2, …, ωK with ni members.
53
Evaluation
Clustering
External Evaluation of Cluster Quality
 Simple measure: purity, the ratio between the
dominant class in the cluster πi and the size of
cluster ωi
1
Purity (i )  max j (nij ) j  C
ni
 Biased because having n clusters maximizes
purity
 Others are entropy of classes in clusters (or
mutual information between classes and
clusters)
54
Evaluation
Clustering
Purity example


 
 
Cluster I


 
 
Cluster II


 

Cluster III
Cluster I: Purity = 1/6 (max(5, 1, 0)) = 5/6
Cluster II: Purity = 1/6 (max(1, 4, 1)) = 4/6
Cluster III: Purity = 1/5 (max(2, 0, 3)) = 3/5
55
Evaluation
Clustering
Rand Index measures between pair
decisions. Here RI = 0.68
Number of
points
Same Cluster
in clustering
Different
Clusters in
clustering
Same class in
ground truth
A=20
C=24
Different
classes in
ground truth
B=20
D=72
56
Evaluation
Clustering
Rand index and Cluster F-measure
A D
RI 
A B C  D
Compare with standard Precision and Recall:
A
P
A B
A
R
AC
People also define and use a cluster F-measure,
which is probably a better measure.
57
Clustering
Final word and resources
 In clustering, clusters are inferred from the data without
human input (unsupervised learning)
 However, in practice, it’s a bit less clear: there are many
ways of influencing the outcome of clustering: number of
clusters, similarity measure, representation of points, . . .
58
Clustering
More Information
 Christopher D. Manning, Prabhakar Raghavan, and Hinrich
Schütze. Introduction to Information Retrieval. Cambridge
University Press, 2008.
 Chapter 16, 17
59
Clustering
Acknowledgement
 Slides are from




Prof. Jeffrey D. Ullman
Dr. Anand Rajaraman
Dr. Jure Leskovec
Prof. Christopher D. Manning
60