Mathematics unit 1

1. Cluster Analysis

2. Midterm: Monday Oct 29, 4PM
 Lecture Notes from Sept 5, 2007 until Oct 15, 2007.
Chapters from Textbook and papers discussed in class (see below detailed
list)
Specific Readings
 Textbook:
 Chapter 1
 Chapter 2: 2.1- 2.4
 Chapter 3: 3.1-3.4
 Chapter 4: 4.1.1-4.1.2, 4.2.1
 Chapter 5
 Chapter 6: 6.1-6.5, 6.9.1, 6.12, 6.13, 6.14
 Chapter 7: 7.1-7.4
 Papers:
 Apriori Paper: R. Agrawal, R. Srikant: Fast Algorithms for Mining Association
Rules. VLDB 1994
 MaxMiner Paper: R. J. Bayardo Jr: Efficiently Mining Long Patterns from
Databases. SIGMOD 1998
 SLIQ paper:M. Mehta, R. Agrawal, J. Rissanen: SLIQ: A Fast Scalable
Classifier for Data Mining. EDBT 1996

3. Cluster Analysis
 What is Cluster Analysis?
 Types of Data in Cluster Analysis
 A Categorization of Major Clustering Methods
 Partitioning Methods
 Hierarchical Methods
 Density-Based Methods
 Grid-Based Methods
 Model-Based Clustering Methods

4. Clustering High-Dimensional Data
 Clustering high-dimensional data
 Many applications: text documents, DNA micro-array data
 Major challenges:
 Many irrelevant dimensions may mask clusters
 Distance measure becomes meaningless—due to equi-distance
 Clusters may exist only in some subspaces
 Methods
 Feature transformation: only effective if most dimensions are relevant
 PCA & SVD useful only when features are highly correlated/redundant
 Feature selection: wrapper or filter approaches
 useful to find a subspace where the data have nice clusters
 Subspace-clustering: find clusters in all the possible subspaces
 CLIQUE, ProClus, and frequent pattern-based clustering

5. The Curse of Dimensionality
(graphs adapted from Parsons et al. KDD
Explorations 2004)
 Data in only one dimension is relatively
packed
 Adding a dimension “stretch” the points
across that dimension, making them
further apart
 Adding more dimensions will make the
points further apart—high dimensional
data is extremely sparse
 Distance measure becomes
meaningless—due to equi-distance

6. Why Subspace Clustering?
(adapted from Parsons et al. SIGKDD
Explorations 2004)
 Clusters may exist only in some subspaces
 Subspace-clustering: find clusters in some of
the subspaces

7. CLIQUE (Clustering In QUEst)
 Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98)
 Automatically identifying subspaces of a high dimensional data
space that allow better clustering than original space
 CLIQUE can be considered as both density-based and grid-
based
 It partitions each dimension into the same number of equal length
interval
 It partitions an m-dimensional data space into non-overlapping
rectangular units
 A unit is dense if the fraction of total data points contained in the
unit exceeds the input model parameter
 A cluster is a maximal set of connected dense units within a
subspace

8. CLIQUE: The Major Steps
 Partition the data space and find the number of
points that lie inside each cell of the partition.
 Identify the subspaces that contain clusters using
the Apriori principle
 Identify clusters
 Determine dense units in all subspaces of interests
 Determine connected dense units in all subspaces of
interests.
 Generate minimal description for the clusters
 Determine maximal regions that cover a cluster of
connected dense units for each cluster
 Determination of minimal cover for each cluster

9. Salary
(10,000)
20 30 40 50 60
age
5
4
3
1
2
6
7
0
20 30 40 50 60
age
5
4
3
1
2
6
7
0
Vacation
(week)
age
Vacation
30 50
 = 3

10. Strength and Weakness of
CLIQUE
 Strength
 automatically finds subspaces of the highest
dimensionality such that high density clusters exist in
those subspaces
 insensitive to the order of records in input and does not
presume some canonical data distribution
 scales linearly with the size of input and has good
scalability as the number of dimensions in the data
increases
 Weakness
 The accuracy of the clustering result may be degraded at
the expense of simplicity of the method

11. Frequent Pattern-Based Approach
 Clustering high-dimensional space (e.g., clustering text
documents, microarray data)
 Projected subspace-clustering: which dimensions to be
projected on?
 CLIQUE, ProClus
 Feature extraction: costly and may not be effective?
 Using frequent patterns as “features”
 “Frequent” are inherent features
 Mining freq. patterns may not be so expensive
 Typical methods
 Frequent-term-based document clustering
 Clustering by pattern similarity in micro-array data
(pClustering)

12. Clustering by Pattern Similarity (p-Clustering)
 Right: The micro-array “raw”
data shows 3 genes and their
values in a multi-dimensional
space
 Difficult to find their patterns
 Bottom: Some subsets of
dimensions form nice shift and
scaling patterns

13. Why p-Clustering?
 Microarray data analysis may need to
 Clustering on thousands of dimensions (attributes)
 Discovery of both shift and scaling patterns
 Clustering with Euclidean distance measure? — cannot find shift
patterns
 Clustering on derived attribute Aij = ai – aj? — introduces N(N-1)
dimensions
 Bi-cluster using transformed mean-squared residue score matrix (I, J)
 Where
 A submatrix is a δ-cluster if H(I, J) ≤ δ for some δ > 0
 Problems with bi-cluster
 No downward closure property,
 Due to averaging, it may contain outliers but still within δ-threshold



J
j
ij
d
J
ij
d
|
|
1



I
i
ij
d
I
Ij
d
|
|
1




J
j
I
i
ij
d
J
I
IJ
d
,
|
||
|
1

14. p-Clustering
 Given objects x, y in O and features a, b in T, pCluster is a 2 by 2
matrix
 A pair (O, T) is in δ-pCluster if for any 2 by 2 matrix X in (O, T),
pScore(X) ≤ δ for some δ > 0
 Properties of δ-pCluster
 Downward closure
 Clusters are more homogeneous than bi-cluster (thus the name:
pair-wise Cluster)
 Pattern-growth algorithm has been developed for efficient mining
 For scaling patterns, one can observe, taking logarithmic on
will lead to the pScore form
|
)
(
)
(
|
)
( yb
ya
xb
xa
yb
xb
ya
xa
d
d
d
d
d
d
d
d
pScore 











yb
xb
ya
xa
d
d
d
d
/
/

15. Cluster Analysis
 What is Cluster Analysis?
 Types of Data in Cluster Analysis
 A Categorization of Major Clustering Methods
 Partitioning Methods
 Hierarchical Methods
 Density-Based Methods
 Grid-Based Methods
 Model-Based Clustering Methods

16. Model based clustering
 Assume data generated from K probability
distributions
 Typically Gaussian distribution Soft or
probabilistic version of K-means clustering
 Need to find distribution parameters.
 EM Algorithm

17. EM Algorithm
 Initialize K cluster centers
 Iterate between two steps
 Expectation step: assign points to clusters
 Maximation step: estimate model parameters



j
j
i
j
k
i
k
k
i c
d
w
c
d
w
c
d
P )
|
Pr(
)
|
Pr(
)
(


 


m
i
k
j
i
k
i
i
k
c
d
P
c
d
P
d
m 1 )
(
)
(
1

N
c
d
w i
k
i
k
 

)
Pr(

18. Cluster Analysis
 What is Cluster Analysis?
 Types of Data in Cluster Analysis
 A Categorization of Major Clustering Methods
 Partitioning Methods
 Hierarchical Methods
 Density-Based Methods
 Grid-Based Methods
 Model-Based Clustering Methods
 Cluster Validity

19. Cluster Validity
 For supervised classification we have a variety of
measures to evaluate how good our model is
 Accuracy, precision, recall
 For cluster analysis, the analogous question is
how to evaluate the “goodness” of the resulting
clusters?
 But “clusters are in the eye of the beholder”!
 Then why do we want to evaluate them?
 To avoid finding patterns in noise
 To compare clustering algorithms
 To compare two sets of clusters
 To compare two clusters

20. Clusters found in Random Data
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Random
Points
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
K-means
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
DBSCAN
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Complete
Link

21. 1. Determining the clustering tendency of a set of data, i.e.,
distinguishing whether non-random structure actually exists
in the data.
2. Comparing the results of a cluster analysis to externally
known results, e.g., to externally given class labels.
3. Evaluating how well the results of a cluster analysis fit the
data without reference to external information.
- Use only the data
4. Comparing the results of two different sets of cluster
analyses to determine which is better.
5. Determining the ‘correct’ number of clusters.
For 2, 3, and 4, we can further distinguish whether we want
to evaluate the entire clustering or just individual clusters.
Different Aspects of Cluster Validation

22.  Numerical measures that are applied to judge various
aspects of cluster validity, are classified into the
following three types.
 External Index: Used to measure the extent to which
cluster labels match externally supplied class labels.
 Entropy
 Internal Index: Used to measure the goodness of a
clustering structure without respect to external
information.
 Sum of Squared Error (SSE)
 Relative Index: Used to compare two different clusterings
or clusters.
 Often an external or internal index is used for this function, e.g., SSE
or entropy
 Sometimes these are referred to as criteria instead of
indices
 However, sometimes criterion is the general strategy and index
is the numerical measure that implements the criterion.
Measures of Cluster Validity

23.  Two matrices
 Proximity Matrix
 “Incidence” Matrix
 One row and one column for each data point
 An entry is 1 if the associated pair of points belong to the same
cluster
 An entry is 0 if the associated pair of points belongs to different
clusters
 Compute the correlation between the two
matrices
 Since the matrices are symmetric, only the correlation
between
n(n-1) / 2 entries needs to be calculated.
 High correlation indicates that points that belong
to the same cluster are close to each other.
 Not a good measure for some density or
contiguity based clusters.
Measuring Cluster Validity Via Correlation

24. Measuring Cluster Validity Via
Correlation
 Correlation of incidence and proximity matrices
for the K-means clusterings of the following two
data sets.
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Corr = -0.9235 Corr = -0.5810

25.  Order the similarity matrix with respect to
cluster labels and inspect visually.
Using Similarity Matrix for Cluster Validation
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Points
Points
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100
Similarity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

26. Using Similarity Matrix for Cluster Validation
 Clusters in random data are not so crisp
Points
Points
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100
Similarity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
DBSCAN
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y

27. Points
Points
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100
Similarity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Using Similarity Matrix for Cluster Validation
 Clusters in random data are not so crisp
K-means
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y

28. Using Similarity Matrix for Cluster
Validation
 Clusters in random data are not so crisp
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
y
Points
Points
20 40 60 80 100
10
20
30
40
50
60
70
80
90
100
Similarity
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Complete Link

29. Using Similarity Matrix for Cluster
Validation
1
2
3
5
6
4
7
DBSCAN
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
500 1000 1500 2000 2500 3000
500
1000
1500
2000
2500
3000

30.  Clusters in more complicated figures aren’t well separated
 Internal Index: Used to measure the goodness of a
clustering structure without respect to external information
 SSE
 SSE is good for comparing two clusterings or two clusters
(average SSE).
 Can also be used to estimate the number of clusters
Internal Measures: SSE
2 5 10 15 20 25 30
0
1
2
3
4
5
6
7
8
9
10
K
SSE
5 10 15
-6
-4
-2
0
2
4
6

31. Internal Measures: SSE
 SSE curve for a more complicated data set
1
2
3
5
6
4
7
SSE of clusters found using K-means

32.  Need a framework to interpret any measure.
 For example, if our measure of evaluation has the value, 10, is
that good, fair, or poor?
 Statistics provide a framework for cluster validity
 The more “atypical” a clustering result is, the more likely it
represents valid structure in the data
 Can compare the values of an index that result from random
data or clusterings to those of a clustering result.
 If the value of the index is unlikely, then the cluster results are
valid
 These approaches are more complicated and harder to
understand.
 For comparing the results of two different sets of
cluster analyses, a framework is less necessary.
 However, there is the question of whether the difference
between two index values is significant
Framework for Cluster Validity

33.  Cluster Cohesion: Measures how closely related are
objects in a cluster
 Example: SSE
 Cluster Separation: Measure how distinct or well-
separated a cluster is from other clusters
 Example: Squared Error
 Cohesion is measured by the within cluster sum of
squares (SSE)
 Separation is measured by the between cluster sum of
squares
 Where |Ci| is the size of cluster i
Internal Measures: Cohesion and Separation
 



i C
x
i
i
m
x
WSS 2
)
(
 

i
i
i m
m
C
BSS 2
)
(

34.  A proximity graph based approach can also be used for
cohesion and separation.
 Cluster cohesion is the sum of the weight of all links within a
cluster.
 Cluster separation is the sum of the weights between nodes in
the cluster and nodes outside the cluster.
Internal Measures: Cohesion and Separation
cohesion separation

35.  Silhouette Coefficient combine ideas of both cohesion and
separation, but for individual points, as well as clusters and
clusterings
 For an individual point, i
 Calculate a = average distance of i to the points in its cluster
 Calculate b = min (average distance of i to points in another
cluster)
 The silhouette coefficient for a point is then given by
s = 1 – a/b if a < b, (or s = b/a - 1 if a  b, not the usual case)
 Typically between 0 and 1.
 The closer to 1 the better.
 Can calculate the Average Silhouette width for a cluster
or a clustering
Internal Measures: Silhouette Coefficient
a
b

36. “The validation of clustering structures is the most
difficult and frustrating part of cluster analysis.
Without a strong effort in this direction, cluster
analysis will remain a black art accessible only to
those true believers who have experience and
great courage.”
Algorithms for Clustering Data, Jain and Dubes
Final Comment on Cluster Validity