Download Clustering

Data Mining
Cluster Analysis: Advanced Concepts
and Algorithms
ref. Chapter 9
Introduction to Data Mining
by
Tan, Steinbach, Kumar
1
Outline
 
Prototype-based
–  Fuzzy c-means
–  Mixture Model Clustering
 
Density-based
–  Grid-based clustering
–  Subspace clustering
 
Graph-based
–  Chameleon
 
Scalable Clustering Algorithms
–  Cure and Birch
 
Characteristics of Clustering Algorithms
Data e Web Mining
2
Hard (Crisp) vs Soft (Fuzzy) Clustering
  Hard
(Crisp) vs. Soft (Fuzzy) clustering
–  Generalize K-means objective function (for all the N
k
points)
k N
2
∑ wij = 1
SSE = ∑ ∑ w ij ( x i − c j ) ,
j=1
j=1 i=1
  wij
: weight with which object xi belongs to cluster Cj
€
€
–  To minimize SSE, repeat the following steps:
  Fixed
cj and determine wij (cluster assignment)
  Fixed
wij and recompute cj
–  Hard clustering: wij ∈ {0,1}
Data e Web Mining
3
Hard (Crisp) vs Soft (Fuzzy) Clustering
c1
x
c2
1
2
5
SSE(x) is minimized when wx1 = 1, wx2 = 0
Data e Web Mining
4
Fuzzy C-means
  Objective
p: fuzzifier (p > 1)
function
k
N
2
SSE = ∑ ∑ w ( x i − c j ) ,
p
ij
j=1 i=1
  wij
k
∑w
ij
=1
j=1
: weight with which object xi belongs to cluster Cj
– € To minimize objective function, repeat the following:
  Fixed
cj and determine wij
  Fixed
wij and recompute cj
€
–  Fuzzy clustering: wij ∈[0,1]
Data e Web Mining
5
Fuzzy C-means
c1
x
c2
1
2
5
SSE(x) is minimized when wx1 = 0.9, wx2 = 0.1
Data e Web Mining
6
Fuzzy C-means
  Objective
function:
k
p: fuzzifier (p > 1)
k
N
2
SSE = ∑ ∑ w ( x i − c j ) ,
p
ij
j=1 i=1
  Initialization:
∑w
ij
=1
j=1
choose the weights wij randomly
  Repeat:
€
–  Update centroids:
€
–  Update weights:
Data e Web Mining
7
Fuzzy K-means Applied to Sample Data
Data e Web Mining
8
Hard (Crisp) vs Soft (Probabilistic) Clustering
 
Idea is to model the set of data points as arising from a
mixture of distributions
–  Typically, normal (Gaussian) distribution is used
–  But other distributions have been very profitably used.
 
Clusters are found by estimating the parameters of the
statistical distributions
–  Can use a k-means like algorithm, called the EM algorithm, to
estimate these parameters
 
Actually, k-means is a special case of this approach
–  Provides a compact representation of clusters
–  The probabilities with which point belongs to each cluster provide
a functionality similar to fuzzy clustering.
Data e Web Mining
9
Probabilistic Clustering: Example
Informal example: consider
modeling the points that
generate the following
histogram.
  Looks like a combination of two
normal distributions
  Suppose we can estimate the
mean and standard deviation of each normal distribution.
 
–  This completely describes the two clusters
–  We can compute the probabilities with which each point belongs
to each cluster
–  Can assign each point to the
cluster (distribution) in which it
is most probable.
Data e Web Mining
10
Probabilistic Clustering: EM Algorithm
Initialize the parameters
Repeat
For each point, compute its probability under each distribution
Using these probabilities, update the parameters of each distribution
Until there is not change
Very similar to of K-means
  Consists of assignment and update steps
  Can use random initialization
 
–  Problem of local minima
For normal distributions, typically use K-means to initialize
  If using normal distributions, can find elliptical as well as
spherical shapes.
 
Data e Web Mining
11
Probabilistic Clustering: EM Algorithm
  Choose
K seeds: means of a gaussian
distribution
  Estimation: calculate probability of belonging to a
cluster based on distance
  Maximization: move mean of gaussian to centroid
of data set, weighted by the contribution of each
point
  Repeat till means don’t move
Data e Web Mining
12
Probabilistic Clustering Applied to Sample Data
Data e Web Mining
13
Grid-based Clustering
  A
type of density-based clustering
Data e Web Mining
14
Grid-based Clustering
  Issues
–  how to discretize the dimensions
  equal
width vs. equal frequency discretization
–  density of cells containing the points close to the
border of a cluster can be very low
  these
cells are discarded. A possible solution is to reduce the
size of cells, but this may yield additional problems
Data e Web Mining
15
Subspace Clustering
  Until
now, we found clusters by considering all of
the attributes
  Some
clusters may involve only a subset of
attributes, i.e., subspaces of the data
–  Example:
  In
a document collection, documents can be represented as
vectors, where the dimensions correspond to terms
  When
k-means is used to find document clusters, the
resulting clusters can typically be characterized by 10 or so
terms
Data e Web Mining
16
Example
  Three
clear clusters.
–  The circle points are not a cluster in three dimensions
–  If the dimensions are discretized (equal width), these
points are included in low density cells
Data e Web Mining
17
Histograms to determine density
  Equi-width
Contiguous
intervals to
be clustered
discretized
space
  Density
Threshold =
6%
Data e Web Mining
18
Example
Data e Web Mining
19
Example
Data e Web Mining
20
Example
Data e Web Mining
21
Example : remarks
  The
circles do not form a cluster in the three
dimensions, but they may form a cluster in some
subspaces
  A cluster in the three dimensions is part of a
cluster (maybe a larger one) in the subspaces
Data e Web Mining
22
Clique Algorithm - Overview
  A
grid-based clustering algorithm that
methodically finds subspace clusters
–  Partitions the data space into rectangular units of
equal volume
–  Measures the density of each unit by the fraction of
points it contains
–  A unit is dense if the fraction of overall points it
contains is above a user specified threshold, τ
–  A cluster is a group of collections of contiguous
(touching) dense units
Data e Web Mining
23
Clique Algorithm
  It
is impractical to check each subspace to see if
it is dense, due to the exponential number of
them
–  2n subspaces, if n are the dimensions
  Monotone
property of density-based clusters:
–  If a set of points forms a density based cluster in k
dimensions, then the same set of points is also part of
a density based cluster in all possible subsets of those
dimensions
  Very
similar to the Apriori algorithm for frequent
itemset mining
  Can find overlapping clusters
Data e Web Mining
24
Clique Algorithm
Data e Web Mining
25
Limitations of Clique
  Time
complexity is exponential in number of
dimensions
–  Especially if “too many” dense units are generated at
lower stages
  May
fail if clusters are of widely differing
densities, since the threshold is fixed
–  Determining appropriate threshold and unit interval
length can be challenging
Data e Web Mining
26
Graph-Based Clustering: General Concepts
  Graph-Based
clustering uses the proximity graph
–  Start with the proximity matrix
–  Consider each point as a node in a graph
–  Each edge between two nodes has a weight which is
the proximity between the two points
–  Initially the proximity graph is fully connected
–  MIN (single-link) and MAX (complete-link) can be
viewed as starting with this graph
  In
the simplest case, clusters are connected
components in the graph.
Data e Web Mining
27
Graph-Based Clustering: Chameleon
 
Based on several key ideas
– 
Sparsification of the proximity graph
– 
Partitioning the data into clusters that are relatively
pure subclusters of the “true” clusters
– 
Merging based on preserving characteristics of
clusters
Data e Web Mining
28
Graph-Based Clustering: Sparsification
 
The amount of data that needs to be
processed is drastically reduced, thus making
the algorithm more scalable
– 
Sparsification can eliminate more than 99% of the
entries in a proximity matrix
– 
The amount of time required to cluster the data is
drastically reduced
The size of the problems that can be handled is
increased
– 
Data e Web Mining
29
Graph-Based Clustering: Sparsification …
 
 
Clustering may work better
– 
Sparsification techniques keep the connections to the most
similar (nearest) neighbors of a point while breaking the
connections to less similar points.
– 
The nearest neighbors of a point tend to belong to the same
class as the point itself.
– 
This reduces the impact of noise and outliers and sharpens
the distinction between clusters.
Sparsification facilitates the use of graph
partitioning algorithms (or algorithms based
on graph partitioning algorithms)
– 
Chameleon and Hypergraph-based Clustering
Data e Web Mining
30
Sparsification in the Clustering Process
Data e Web Mining
31
Limitations of Current Merging Schemes
  Existing
merging schemes in hierarchical
clustering algorithms are static in nature
–  MIN or CURE:
  Merge
two clusters based on their closeness (or minimum
distance)
–  GROUP-AVERAGE:
  Merge
two clusters based on their average connectivity
Data e Web Mining
32
Limitations of Current Merging Schemes
(a)
(b)
(c)
(d)
Closeness schemes
will merge (a) and (b)
Average connectivity schemes
will merge (c) and (d)
Data e Web Mining
33
Chameleon: Clustering Using Dynamic Modeling
 
 
Adapt to the characteristics of the data set to find the
natural clusters
Use a dynamic model to measure the similarity between
clusters
–  Main properties are the relative closeness and relative interconnectivity of the cluster
–  Two clusters are combined if the resulting cluster shares certain
properties with the constituent clusters
–  The merging scheme preserves self-similarity
Data e Web Mining
34
Experimental Results: CHAMELEON
Data e Web Mining
35
Experimental Results: CHAMELEON
Data e Web Mining
36
Experimental Results: CHAMELEON
Data e Web Mining
37
CURE: a Scalable Algorithm
  Agglomerative
hierarchical clustering algorithms
vary in terms of how the proximity of two clusters
are computed
  MIN
(single link)
–  susceptible to noise/outliers
  MAX (complete link)/GROUP AVERAGE/Centroid:
–  may not work well with non-globular clusters
  CURE
(Clustering Using REpresentatives)
algorithm tries to handle both problems
–  It is a graph-based algorithm
–  Starts with a proximity matrix/proximity graph
Data e Web Mining
38
CURE Algorithm
  Represents
a cluster using multiple
representative points
–  Goals: scalability, by choosing points that capture the
geometry and shape of clusters
–  Representative points are found by selecting a
constant number of points from a cluster
 
The first representative point is chosen to be the point farthest
from the center of the cluster
 
Remaining representative points are chosen so that they are
farthest from all previously chosen points
Data e Web Mining
39
CURE Algorithm
 
“Shrink” representative points toward the center of the
cluster by a factor, α
×
 
×
Shrinking representative points toward the center helps
avoid problems with noise and outliers
–  shrinking factor: α
 
Cluster similarity is the similarity of the closest pair of
representative points (MIN) from different clusters
Data e Web Mining
40
CURE Algorithm
  Uses
agglomerative hierarchical scheme to
perform clustering;
–  α = 0: similar to centroid-based
–  α = 1: somewhat similar to single-link (MIN)
  CURE
is better able to handle clusters of arbitrary
shapes and sizes
Data e Web Mining
41
Experimental Results: CURE (10 clusters)
Data e Web Mining
42
Experimental Results: CURE (9 clusters)
Data e Web Mining
43
Experimental Results: CURE
Picture from CURE, Guha, Rastogi, Shim.
Data e Web Mining
44
Experimental Results: CURE
(centroid)
(single link)
Picture from CURE, Guha, Rastogi, Shim.
Data e Web Mining
45
CURE Cannot Handle Differing Densities
Original Points
CURE
Data e Web Mining
46
BIRCH: a Scalable Algorithm
  Balanced
Iterative Reducing and Clustering using
Hierarchies
  Scales
linearly: finds a good clustering with a
single scan and improves the quality with a few
additional scans
  Weakness:
handles only numeric data (Euclidean
space), and is sensitive to the order of the data
record
Data e Web Mining
47
BIRCH
→
  Clustering
Feature (CF):
→
(N,LS,SS )
–  Number of point, Linear Sum of points, Sum of
Squares of points
–  CF incrementally updated, to be used for computing
centroids, variance (used for measuring the diameter
of the cluster) €
–  Also used for computing distances between clusters
Data e Web Mining
48
BIRCH
  CF
is a compact storage
for data on points in
a cluster
  Has enough
information to
calculate the
intra-cluster
distances
  Additivity theorem allows us to merge subclusters
–  C3 = C1  C2
–  CFC3= <nC1+ nC2 , LSC1+ LSC2, SSC1+SSC2>
Data e Web Mining
49
BIRCH
 
Basic steps of BIRCH
–  Load the data into memory by creating a CF tree that
“summarizes” the data (see the following slide)
–  Perform global clustering.
  Produces
a better clustering than the initial step.
  An agglomerative, hierarchical technique was selected.
–  Redistribute the data points using the centroids of
clusters discovered in the global clustering phase, and
thus, discover a new (and hopefully better) set of
clusters.
Data e Web Mining
50
BIRCH
 
BIRCH maintains a balanced CF-Tree
–  Branching Factor B: max entry number in a non-leaf
–  Max size leaf L: max entry number in a leaf
–  Threshold T: the diameter of a leaf < T
CF Tree
Root
B=7
CF1
CF2
CF3
CF6
L=6
child1
child2
child3
child6
Non-leaf node
CF1
CF2
CF3
CF5
child1
child2
child3
child5
Leaf node
prev
CF1
CF2
CF6
Leaf node
next
prev
CF1
CF2
Data e Web Mining
CF4
next
51
Characteristics of Data
  High
dimensionality
  Size of data set
  Sparsity of attribute values
  Noise and Outliers
  Types of attributes and type of data sets
  Differences in attribute scale
  Properties of the data space
–  Can you define a meaningful centroid
Data e Web Mining
52
Characteristics of Clusters
  Data
distribution
  Shape
  Differing sizes
  Differing densities
  Poor separation
  Relationship of clusters
  Subspace clusters
Data e Web Mining
53
Characteristics of Clustering Algorithms
  Order
dependence
  Non-determinism
  Parameter selection
  Scalability
  Underlying model
  Optimization based approach
Data e Web Mining
54