DMTM 2015 - 06 Introduction to Clustering

Prof. Pier Luca Lanzi
Clustering: Introduction
Data Mining andText Mining (UIC 583 @ Politecnico di Milano)

Readings
•  Mining of Massive Datasets (Chapter 7, Section 3.5)
2

Clustering algorithms group a collection of data points
into “clusters” according to some distance measure
Data points in the same cluster should have
a small distance from one another
Data points in different clusters should be at
a large distance from one another.

Clustering ﬁnds “natural” grouping/structure in un-labeled data
(Unsupervised Learning)

What is Cluster Analysis?
•  A cluster is a collection of data objects
§ Similar to one another within the same cluster
§ Dissimilar to the objects in other clusters
•  Cluster analysis
§ Given a set data points try to understand their structure
§ Finds similarities between data according to the characteristics
found in the data
§ Groups similar data objects into clusters
§ It is unsupervised learning since there is no predeﬁned classes
•  Typical applications
§ Stand-alone tool to get insight into data
§ Preprocessing step for other algorithms
8

Clustering Methods
•  Hierarchical vs point assignment
•  Numeric and/or symbolic data
•  Deterministic vs. probabilistic
•  Exclusive vs. overlapping
•  Hierarchical vs. ﬂat
•  Top-down vs. bottom-up
9

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
10

What Is Good Clustering?
•  A good clustering consists of 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
•  Evaluation
§ Various measure of intra/inter cluster similarity
§ Manual inspection
§ Benchmarking on existing labels
11

Measure the Quality of Clustering
•  Dissimilarity/Similarity metric: Similarity is expressed in terms of a
distance function, 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 ratio, and vector variables
•  Weights should be associated with different variables based on
applications and data semantics
•  It is hard to deﬁne “similar enough” or “good enough” as the answer is
typically highly subjective
12

Data Structures
0
d(2,1) 0
d(3,1) d(3,2) 0
: : :
d(n,1) d(n,2) ... ... 0
!

#
#
#
#
#
#
$
%

Outlook
Temp
Humidity
Windy
Play

Sunny
Hot
High
False
No

Sunny
Hot

High

True
No

Overcast

Hot

High
False
Yes

…
…
…
…
…

x
11
... x
1f
... x
1p
... ... ... ... ...
x
i1
... x
if
... x
ip
... ... ... ... ...
x
n1
... x
nf
... x
np
!

#
#
#
#
#
#
#
#
$
%

Data Matrix
13
Dis/Similarity Matrix

Type of Data in Clustering Analysis
•  Interval-scaled variables
•  Binary variables
•  Nominal, ordinal, and ratio variables
•  Variables of mixed types
14

Distance Measures

Distance Measures
•  Given a space and a set of points on this space, a distance
measure d(x,y) maps two points x and y to a real number,
and satisﬁes three axioms
•  d(x,y) ≥
0
•  d(x,y) = 0 if and only x=y
•  d(x,y) = d(y,x)
•  d(x,y) ≤ d(x,z) + d(z,y)
16

Euclidean Distances 17
here are other distance measures that have been used for Euclidean
any constant r, we can define the Lr-norm to be the distance me
ed by:
d([x1, x2, . . . , xn], [y1, y2, . . . , yn]) = (
n
i=1
|xi − yi|r
)1/r
case r = 2 is the usual L2-norm just mentioned. Another common d
ure is the L1-norm, or Manhattan distance. There, the distance b
points is the sum of the magnitudes of the differences in each dim
called “Manhattan distance” because it is the distance one would
•  Lr-norm
•  Euclidean distance (r=2)
•  Manhattan distance (r=1)
•  L∞-norm
2 Euclidean Distances
most familiar distance measure is the one we normally think of as “dis-
e.” An n-dimensional Euclidean space is one where points are vectors of n
numbers. The conventional distance measure in this space, which we shall
to as the L2-norm, is defined:
d([x1, x2, . . . , xn], [y1, y2, . . . , yn]) =
n
i=1
(xi − yi)2
is, we square the distance in each dimension, sum the squares, and take
positive square root.
is easy to verify the first three requirements for a distance measure are
fied. The Euclidean distance between two points cannot be negative, be-
e the positive square root is intended. Since all squares of real numbers are
egative, any i such that xi ̸= yi forces the distance to be strictly positive.
he other hand, if xi = yi for all i, then the distance is clearly 0. Symmetry
ws because (xi − yi)2
= (yi − xi)2
. The triangle inequality requires a good
of algebra to verify. However, it is well understood to be a property of

Jaccard Distance
•  Jaccard distance is deﬁned as d(x,y) = 1 – SIM(x,y) where SIM is
the Jaccard similarity,
•  Which can also be interpreted as the percentage of identical
attributes
18

Cosine Distance
•  The cosine distance between x, y is the angle that the vectors to
those points make
•  This angle will be in the range 0 to 180 degrees, regardless of
how many dimensions the space has.
•  Example: given x = (1,2,-1) and y = (2,1,1) the angle between the
two vectors is 60
19

Edit Distance
•  Used when the data points are strings
•  The distance between a string x=x1x2…xn and y=y1y2…ym is the smallest
number of insertions and deletions of single characters that will transform x
into y
•  Alternatively, the edit distance d(x, y) can be compute as the longest common
subsequence (LCS) of x and y and then,

d(x,y) = |x| + |y| - 2|LCS|
•  Example: the edit distance between x=abcde and y=acfdeg is 3 (delete b,
insert f, insert g), the LCS is acde which is coherent with the previous result
20

Hamming Distance
•  Hamming distance between two vectors is the number of
components in which they differ
•  Or equivalently, given the number of variables n, and the number
m of matching components, we deﬁne
•  Example: the Hamming distance between the vectors 10101 and
11110 is 3.
21

Ordinal Variables
•  An ordinal variable can be discrete or continuous
•  Order is important, e.g., rank
•  It can be treated as an interval-scaled
§ replace xif with 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
22

Requirements of Clustering in Data Mining
•  Scalability
•  Ability to deal with different types of attributes
•  Ability to handle dynamic data
•  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
23

Curse of Dimensionality
in high dimensions, almost all pairs of points
are equally far away from one another
almost any two vectors are almost orthogonal

DMTM 2015 - 06 Introduction to Clustering

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