Slides for the 2016/2017 edition of the Data Mining and Text Mining Course at the Politecnico di Milano. The course is also part of the joint program with the University of Illinois at Chicago.

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

2. Prof. Pier Luca Lanzi
Readings
• Mining of Massive Datasets (Chapter 7, Section 3.5)
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3. Prof. Pier Luca Lanzi

4. Prof. Pier Luca Lanzi
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5. Prof. Pier Luca Lanzi
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.

6. Prof. Pier Luca Lanzi
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

7. Prof. Pier Luca Lanzi
Clustering searches for “natural” grouping/structure in un-labeled data
(Unsupervised Learning)

8. Prof. Pier Luca Lanzi
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
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9. Prof. Pier Luca Lanzi
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
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10. Prof. Pier Luca Lanzi
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
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11. Prof. Pier Luca Lanzi
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
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12. Prof. Pier Luca Lanzi
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
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13. Prof. Pier Luca Lanzi
Data Structures
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d(2,1) 0
d(3,1) d(3,2) 0
: : :
d(n,1) d(n,2) ... ... 0
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Outlook Temp Humidity Windy Play
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Overcast Hot High False Yes
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Data Matrix
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Dis/Similarity Matrix

14. Prof. Pier Luca Lanzi
Distance Measures

15. Prof. Pier Luca Lanzi
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)
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16. Prof. Pier Luca Lanzi
Euclidean Distances 18
There are other distance measures that have been used for Euclidean spa
For any constant r, we can define the Lr-norm to be the distance measur
defined by:
d([x1, x2, . . . , xn], [y1, y2, . . . , yn]) = (
n
i=1
|xi − yi|r
)1/r
The case r = 2 is the usual L2-norm just mentioned. Another common dista
measure is the L1-norm, or Manhattan distance. There, the distance betw
wo points is the sum of the magnitudes of the differences in each dimens
t is called “Manhattan distance” because it is the distance one would have
• Lr-norm
• Euclidean distance (r=2)
• Manhattan distance (r=1)
• L∞-norm
3.5.2 Euclidean Distances
The most familiar distance measure is the one we normally think of as “dis-
ance.” An n-dimensional Euclidean space is one where points are vectors of n
eal numbers. The conventional distance measure in this space, which we shall
efer to as the L2-norm, is defined:
d([x1, x2, . . . , xn], [y1, y2, . . . , yn]) =
n
i=1
(xi − yi)2
That is, we square the distance in each dimension, sum the squares, and take
he positive square root.
It is easy to verify the first three requirements for a distance measure are
atisfied. The Euclidean distance between two points cannot be negative, be-
cause the positive square root is intended. Since all squares of real numbers are
nonnegative, any i such that xi ̸= yi forces the distance to be strictly positive.
On the other hand, if xi = yi for all i, then the distance is clearly 0. Symmetry
ollows because (xi − yi)2
= (yi − xi)2
. The triangle inequality requires a good
deal of algebra to verify. However, it is well understood to be a property of

17. Prof. Pier Luca Lanzi
Jaccard Distance
• Jaccard distance is defined as
d(x,y) = 1 – SIM(x,y)
• SIM is the Jaccard similarity,
• Which can also be interpreted as the percentage of identical
attributes
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18. Prof. Pier Luca Lanzi
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
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19. Prof. Pier Luca Lanzi
Edit Distance
• 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
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20. Prof. Pier Luca Lanzi
Hamming Distance
• Hamming distance between two vectors is the number of
components in which they differ
• Or equivalently, given the number of variables p, and the number
m of matching components, we define
• Example: the Hamming distance between the vectors 10101 and
11110 is 3/5.
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21. Prof. Pier Luca Lanzi
Requisites for Clustering Algorithms
• 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
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22. Prof. Pier Luca Lanzi
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