Lecture8 clustering

ClusteringClustering
1
Wu-Jun Li
Department of Computer Science and Engineering
Shanghai Jiao Tong University
Lecture 8: Clustering
Mining Massive Datasets

2
Outline
 Introduction
 Hierarchical Clustering
 Point Assignment based Clustering
 Evaluation

3
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, …
Introduction

4
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
Introduction

5
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.
Introduction

6
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.
Introduction

7
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.
Introduction

8
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.
Introduction

9
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.
Introduction

10
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.
Introduction

11
Outline
 Introduction
 Evaluation

12
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?

13
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.

14
Example
(5,3)
o
(1,2)
o
o (2,1) o (4,1)
o (0,0) o
(5,0)
x (1.5,1.5)
x (4.5,0.5)
x (1,1)
x (4.7,1.3)
o : data point
x : centroid

15
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.

16
“Closest” Point?
 Possible meanings:
1. Smallest maximum distance to the other points.
2. Smallest average distance to other points.
3. Smallest sum of squares of distances to other points.
4. Etc., etc.

17
Example
1 2
3
4
5
6
intercluster
distance
clustroid
clustroid

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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.

19
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.

20
Outline
 Introduction
 Evaluation

21
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 cjsuch that dist(xi, sj) is minimal.
 For each cluster cj
Point Assignment

22
k-Means Example (k=2)
Pick seeds
Reassign clusters
Compute centroids
x
x
Reassign clusters
x
x xx Compute centroids
Reassign clusters
Converged!
Point Assignment

23
Termination conditions
 Several possibilities, e.g.,
 A fixed number of iterations.
 Point assignment unchanged.
 Centroid positions don’t change.
Point Assignment

24
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
Point Assignment

25
Example: Picking k
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.
Point Assignment

26
Example: Picking k
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.
Point Assignment

27
Example: Picking k
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.
Point Assignment

28
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.
Point Assignment

29
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.
Point Assignment

30
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.
Point Assignment

31
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.
Point Assignment

32
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.
Point Assignment

33
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.
Point Assignment

34
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.
Point Assignment

35
“Galaxies” Picture
A cluster. Its points
are in the DS.
The centroid
Compressed sets.
Their points are in
the CS.
Points in
the RS
Point Assignment

36
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.
Point Assignment

37
Processing – (2)
3. Adjust statistics of the clusters to account for the
new points.
 Add N’s, SUM’s, SUMSQ’s.
3. Consider merging compressed sets in the CS.
4. If this is the last round, merge all compressed sets
in the CS and all RS points into their nearest
cluster.
Point Assignment

38
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?
Point Assignment

39
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.
Point Assignment

40
Mahalanobis Distance
 Normalized Euclidean distance from centroid.
 For point (x1,…,xd) and centroid (c1,…,cd):
1. Normalize in each dimension: yi = (xi-ci)/σi
2. Take sum of the squares of the yi’s.
3. Take the square root.
Point Assignment

41
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.
Point Assignment

42
Picture: Equal M.D. Regions
σ
2σ
Point Assignment

43
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.
Point Assignment

44
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.
Point Assignment

45
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
Point Assignment

46
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.
Point Assignment

47
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
Point Assignment

48
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.
Point Assignment

49
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.
Point Assignment

50
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.
Point Assignment

51
Outline
 Introduction
 Evaluation

52
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
Evaluation

53
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.
Evaluation

54
External Evaluation of Cluster Quality
 Simple measure: purity, the ratio between the
dominant class in the cluster πi and the size of
cluster ωi
 Biased because having n clusters maximizes
purity
 Others are entropy of classes in clusters (or
mutual information between classes and
clusters)
Cjn
n
Purity ijj
i
i ∈= )(max
1
)(ω
Evaluation

55
• •
• •
• •
• •
• •
• •
• •
• •
•
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
Purity example
Evaluation

56
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
Evaluation

57
Rand index and Cluster F-measure
BA
A
P
+
=
DCBA
DA
RI
+++
+
=
CA
A
R
+
=
Compare with standard Precision and Recall:
People also define and use a cluster F-
measure, which is probably a better measure.
Evaluation

58
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, . . .

59
More Information
 Christopher D. Manning, Prabhakar Raghavan, and Hinrich
Schütze. Introduction to Information Retrieval. Cambridge
University Press, 2008.
 Chapter 16, 17

60
Acknowledgement
 Slides are from
 Prof. Jeffrey D. Ullman
 Dr. Anand Rajaraman
 Dr. Jure Leskovec
 Prof. Christopher D. Manning

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