Spatial and Temporal Data
Mining
Clustering I
Vasileios Megalooikonomou
(based on notes by Jiawei Han and Micheline Kamber)
Agenda
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
Outlier Analysis
Summary
Typical Applications of Clustering
Pattern Recognition
Spatial Data Analysis
create thematic maps in GIS by clustering feature spaces
detect spatial clusters and explain them in spatial data mining
e.g., land use, city planning, earth-quake studies
Image Processing
Economic Science (especially market research) e.g.,
marketing, insurance
WWW
Document classification
Cluster Weblog data to discover groups of similar access patterns
What Is Good Clustering?
A good clustering method will produce 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.
Requirements of Clustering in Data Mining
Scalability
Ability to deal with different types of attributes
Discovery of clusters with arbitrary shape (not just spherical
clusters)
Minimal requirements for domain knowledge to determine
input parameters (such as # of clusters)
Able to deal with noise and outliers
Insensitive to order of input records
Incremental clustering
High dimensionality (especially very sparse and highly
skewed data)
Incorporation of user-specified constraints
Interpretability and usability (close to semantics)
Agenda
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
Outlier Analysis
Summary
Types of Data and Data Structures
Data matrix
x11
...
x
i1
...
x
n1
(two modes)
n objects, p variables
...
x1f
...
x1p
...
...
...
...
xif
...
...
xip
...
...
... xnf
...
...
...
xnp
d(2,1)
0
Dissimilarity matrix
d(3,1) d ( 3,2) 0
(one mode)
:
:
:
between all pairs of n objects d ( n,1) d ( n,2) ...
... 0
Measure the Quality of Clustering
Dissimilarity/Similarity metric: Similarity is expressed
in terms of a distance function, which is 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 and ratio variables.
Weights should be associated with different variables
based on applications and data semantics.
Hard to define similar enough or good enough
highly subjective.
Interval-valued variables
Continuous measurements of a roughly linear scale (e.g., weight,
height, temperature, etc)
Standardize data (to avoid dependence on the measurement units)
Calculate the mean absolute deviation of a variable f with n measurements:
s f 1n (| x1 f m f | | x2 f m f | ... | xnf m f |)
where
m f 1n (x1 f x2 f
...
xnf )
Calculate the standardized measurement (z-score)
xif m f
zif
sf
Using mean absolute deviation is more robust (to outliers) than
using standard deviation
Similarity and Dissimilarity Between Objects
Distances are normally used to measure the similarity
or dissimilarity between two data objects
Some popular ones include: Minkowski distance:
d (i, j) q (| x x |q | x x | q ... | x x |q )
i1
j1
i2
j2
ip
jp
where i = (xi1, xi2, , xip) and j = (xj1, xj2, , xjp) are two pdimensional data objects, and q is a positive integer
If q = 1, d is Manhattan distance
d (i, j) | x x | | x x | ... | x x |
i1 j1 i2 j 2
ip jp
Similarity and Dissimilarity Between Objects
If q = 2, d is Euclidean distance:
d (i, j) (| x x | 2 | x x | 2 ... | x x |2 )
i1
j1
i2
j2
ip
jp
Properties
d(i,j) 0
d(i,i) = 0
d(i,j) = d(j,i),
symmetry
d(i,j) d(i,k) + d(k,j), triangular inequality
Also one can use weighted distance, parametric
Pearson product moment correlation, or other
dissimilarity measures.
Binary Variables
A contingency table for binary data
Object j
Object i
1
0
1
a
c
0
b
d
sum a c b d
sum
a b
cd
p
Simple matching coefficient (invariant similarity, if the binary
variable is symmetric (both states same weight)): d (i, j)
Jaccard coefficient (noninvariant similarity, if the binary
bc
a bc d
variable is asymmetric (states not equally important
e.g., outcomes of a disease test)):
d (i, j)
bc
a b c
Dissimilarity between Binary Variables
Example
Name
Jack
Mary
Jim
Gender
M
F
M
Fever
Y
Y
Y
Cough
N
N
P
Test-1
P
P
N
Test-2
N
N
N
Test-3
N
P
N
Test-4
N
N
N
gender is a symmetric attribute
the remaining attributes are asymmetric binary
let the values Y and P be set to 1, and the value N be set to 0
01
0.33
2 01
11
d ( jack , jim )
0.67
111
1 2
d ( jim , mary )
0.75
11 2
d ( jack , mary )
Nominal Variables
A generalization of the binary variable in that it can take
more than 2 states, e.g., red, yellow, blue, green
Method 1: Simple matching
m: # of matches (# of variables for which i and j are in the same
state), p: total # of variables
m
d (i, j) p
p
Method 2: use a large number of binary variables
creating a new binary variable for each of the M nominal states
Ordinal Variables
An ordinal variable can be discrete or continuous
Resembles nominal var but order is important, e.g., rank
Can be treated like interval-scaled
replace xif by their rank
r {1,..., M }
if
where ordinal variable f has Mf states
and xif is fthe value of
f for the i-th object
map the range of each variable onto [0, 1] by replacing the rank of the i-th
object in the f-th variable by
rif 1
zif using
methods for interval-scaled variables
compute the dissimilarity
M f 1
Ratio-Scaled Variables
Ratio-scaled variable: a positive measurement on a nonlinear scale,
approximately at exponential scale, such as AeBt or Ae-Bt where A
and B are positive constants (e.g., decay of radioactive elements)
Methods:
treat them like interval-scaled variables not a good choice!
(why?)
apply logarithmic transformation
yif = log(xif)
and treat them as interval-valued
treat them as continuous ordinal data and treat their rank as
interval-scaled.
Variables of Mixed Types
A database may contain all the six types of variables
symmetric binary, asymmetric binary, nominal, ordinal,
interval and ratio.
One may use a weighted formula to combine their
effects.
p ( f )d ( f )
d (i, j )
f 1 ij
p
f 1
ij
(f)
ij
f is binary or nominal:
dij(f) = 0 if xif = xjf , or dij(f) = 1 otherwise.
f is interval-based: use the normalized distance
f is ordinal or ratio-scaled
compute ranks rif and z r 1
if
and treat zif as interval-scaled M 1
if
Agenda
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
Outlier Analysis
Summary
Clustering Approaches
Partitioning algorithms: Construct various partitions and then evaluate
them by some criterion (k-means, k-medoids)
Hierarchical algorithms: Create a hierarchical decomposition
(agglomerative or divisive) of the set of data (or objects) using some
criterion (CURE, Chameleon, BIRCH)
Density-based: based on connectivity and density functions grow a
cluster as long as density in the neighborhood exceeds a threshold
(DBSCAN, CLIQUE)
Grid-based: based on a multiple-level grid structure (i.e., quantized
space) (STING, CLIQUE)
Model-based: A model is hypothesized for each of the clusters and the
idea is to find the best fit of the data to the given model (EM)
Agenda
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
Outlier Analysis
Summary
Partitioning Algorithms: Basic Concept
Partitioning method: Construct a partition of a database D
of n objects into a set of k clusters
Given a k, find a partition of k clusters that optimizes the
chosen partitioning criterion
Global optimal: exhaustively enumerate all partitions
Heuristic methods: k-means and k-medoids algorithms
k-means (MacQueen67): Each cluster is represented by the
center of the cluster
k-medoids or PAM (Partition around medoids) (Kaufman &
Rousseeuw87): Each cluster is represented by one of the objects
in the cluster
The K-Means Clustering Method
Given k, the k-means algorithm is implemented in
4 steps:
1. Partition objects into k nonempty subsets
2. Compute seed points as the centroids of the clusters of
the current partition. The centroid is the center (mean
point) of the cluster.
3. Assign each object to the cluster with the nearest seed
point.
4. Go back to Step 2, stop when no more new assignment
(or fractional drop of SSE or MSE is less than a
threshold).
The K-Means Clustering Method
Example
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Comments on the K-Means Method
Strengths
Relatively efficient: O(tkn), where n is # objects, k is # clusters,
and t is # iterations. Normally, k, t << n.
Often terminates at a local optimum. The global optimum may
be found using techniques such as: deterministic annealing and
genetic algorithms
Weaknesses
Applicable only when mean is defined, then what about
categorical data?
Need to specify k, the number of clusters, in advance
Unable to handle noisy data and outliers
Not suitable to discover clusters with non-convex shapes
Variations of the K-Means Method
A few variants of the k-means which differ in
Selection of the initial k means
Dissimilarity calculations
Strategies to calculate cluster means
Handling categorical data: k-modes (Huang98)
Replacing means of clusters with modes
Using new dissimilarity measures to deal with categorical
objects
Using a frequency-based method to update modes of clusters
A mixture of categorical and numerical data: k-prototype
method
The K-Medoids Clustering Method
Find representative (i.e., the most centrally located) objects,
called medoids, in clusters
PAM (Partitioning Around Medoids, 1987)
starts from an initial set of medoids and iteratively replaces one of the
medoids by one of the non-medoids if it improves the total distance of the
resulting clustering
PAM works effectively for small data sets, but does not scale well for large
data sets
More robust than k-means
Complexity: O(k(n-k)2) for n objects and k clusters
CLARA (Kaufmann & Rousseeuw, 1990): Uses multiple samples
CLARANS (Ng & Han, 1994): Randomized sampling
Focusing + spatial data structure (Ester et al., 1995)
PAM (Partitioning Around Medoids) (1987)
PAM (Kaufman and Rousseeuw, 1987), built in Splus
Use real object to represent the cluster
1. Select k representative objects arbitrarily
2. For each pair of non-selected object h and selected object i,
calculate the total swapping cost Tcih
3. For each pair of i and h,
If TCih < 0, i is replaced by h
Then assign each non-selected object to the most similar
representative object
4. repeat steps 2-3 until there is no change
PAM Clustering: Total swapping cost TCih=jCjih
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h
i
4
3
0
0
10
Cjih = d(j, h) - d(j, i)
Cjih = 0
0
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10
7
6
5
4
3
Cjih = d(j, t) - d(j, i)
0
10
Cjih = d(j, h) - d(j, t)
0
10
CLARA (Clustering Large Applications) (1990)
CLARA (Kaufmann and Rousseeuw in 1990)(O(ks2 + k(n-k)))
Built in statistical analysis packages, such as S+
It draws multiple samples of the data set, applies PAM on each
sample, and gives the best clustering as the output
Strength: deals with larger data sets than PAM
Weakness:
efficiency depends on the sample size and
the sample: A good clustering based on samples will not
necessarily represent a good clustering of the whole data set if
the sample is biased
CLARANS (Randomized CLARA) (1994)
CLARANS (A Clustering Algorithm based on Randomized
Search) (Ng and Han94) (O(n2))
CLARANS draws a sample of neighbors dynamically
The clustering process can be presented as searching a graph
where every node is a potential solution, that is, a set of k
medoids
If the local optimum is found, CLARANS starts with new
randomly selected node in search for a new local optimum
It is more efficient and scalable than both PAM and CLARA
Focusing techniques and spatial access structures may further
improve its performance (Ester et al.95)
Agenda
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
Outlier Analysis
Summary
Hierarchical Clustering
Use distance matrix as clustering criteria. This
method does not require the number of clusters k as an
input, but needs a termination condition
Step 0
a
b
Step 1
Step 2 Step 3 Step 4
ab
abcde
cde
de
e
Step 4
agglomerative
(AGNES)
Step 3
Step 2 Step 1 Step 0
divisive
(DIANA)
AGNES (Agglomerative Nesting)
Introduced in Kaufmann and Rousseeuw (1990)
Implemented in statistical analysis packages, e.g., Splus
Use the Single-Link method and the dissimilarity matrix:
each cluster is represented by all of its objects and the similarity between clusters is
measured by the similarity of the closest pair of data points belonging to different
clusters
Merge nodes that have the least dissimilarity
Go on in a non-descending fashion
Eventually all nodes belong to the same cluster
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A Dendrogram Shows How the
Clusters are Merged Hierarchically
Decompose data objects into a several levels of nested
partitioning (tree of clusters), called a dendrogram.
A clustering of the data objects is obtained by cutting the
dendrogram at the desired level, then each connected
component forms a cluster.
DIANA (Divisive Analysis)
Introduced in Kaufmann and Rousseeuw (1990)
Implemented in statistical analysis packages, e.g., Splus
Inverse order of AGNES
Eventually each node forms a cluster on its own
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More on Hierarchical Clustering Methods
Major weakness of agglomerative clustering methods
do not scale well: time complexity of at least O(n2), where n is
the number of total objects
can never undo (backtrack) what was done previously
Integration of hierarchical with distance-based
clustering
BIRCH (1996): uses CF-tree (clustering feature tree) and
incrementally adjusts the quality of sub-clusters
CURE (1998): selects well-scattered (representative) points
from the cluster and then shrinks them towards the center of
the cluster by a specified fraction
CHAMELEON (1999): hierarchical clustering using dynamic
modeling
BIRCH (1996)
Birch: Balanced Iterative Reducing and Clustering using
Hierarchies, by Zhang, Ramakrishnan, Livny (SIGMOD96)
Incrementally construct a CF (Clustering Feature) tree, a
hierarchical data structure for multiphase clustering
Phase 1: scan DB to build an initial in-memory CF tree (a multi-level
compression of the data that tries to preserve the inherent clustering
structure of the data)
Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes of
the CF-tree
Scales linearly: finds a good clustering with a single scan and
improves the quality with a few additional scans
Weakness: handles only numeric data, and sensitive to the order
of the data record.
Clustering Feature Vector
Clustering Feature: CF = (N, LS, SS)
N: Number of data points
LS: Ni=1Xi
CF = (5, (16,30),(54,190))
SS: Ni=1Xi2
Clustering features are additive
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(3,4)
(2,6)
(4,5)
(4,7)
(3,8)
CF Tree
Root
B: Branching factor: max # children per
nonleaf node
L: Threshold: max diameter of subclusters
at leaf nodes
B=7
CF1
CF2 CF3
CF6
L=6
child1
child2 child3
child6
CF1
Non-leaf node
CF2 CF3
CF5
child1
child2 child3
child5
Leaf node
prev
CF1 CF2
CF6 next
Leaf node
prev
CF1 CF2
CF4 next
CURE (Clustering Using
REpresentatives )
CURE: proposed by Guha, Rastogi & Shim, 1998 (O(n))
Stops the creation of a cluster hierarchy if a level consists of k
clusters
Uses multiple representative points to evaluate the distance
between clusters
Adjusts well to arbitrary shaped clusters and avoids single-link
effect
Drawbacks of Distance-Based Method
Drawbacks of square-error based clustering method
Consider only one point as representative of a cluster
Good only for convex shaped, similar size and density, and
if k can be reasonably estimated
Cure: The Algorithm
Draw random sample s.
Partition sample to p partitions with size s/p
Partially cluster partitions into s/pq clusters
Eliminate outliers
By random sampling
If a cluster grows too slow, eliminate it.
Cluster partial clusters.
Label data in disk
Data Partitioning and Clustering
s = 50
p=2
s/p = 25
s/pq = 5
y
y
x
y
y
x
x
x
x
Cure: Shrinking Representative Points
y
Shrink the multiple representative points towards the
gravity center by a fraction of .
Multiple representatives capture the shape of the cluster
Clustering Categorical Data: ROCK
ROCK: Robust Clustering using linKs,
by S. Guha, R. Rastogi, K. Shim (ICDE99).
Use links to measure similarity/proximity (# points from
different clusters who have neighbors in common) ->
# of links of two points = # common neighbors
Not distance based
Hierarchical clustering
2
2
O
(
n
nm
m
n
log n)
m
a
Computational complexity:
Basic ideas:
Similarity function and neighbors:
Let T1 = {1,2,3}, T2={3,4,5}
Sim( T 1, T 2)
T1 T2
Sim( T1 , T2 )
T1 T2
{3}
1
0.2
{1,2,3,4,5}
5
CHAMELEON
CHAMELEON: hierarchical clustering using
dynamic modeling, by G. Karypis, E.H. Han and V.
Kumar99 (O(n2))
Measures the similarity based on a dynamic model
Two clusters are merged only if the interconnectivity and
closeness (proximity) between two clusters are high
relative to the internal interconnectivity of the clusters
and closeness of items within the clusters
A two phase algorithm
1. Use a graph partitioning algorithm: cluster objects into
a large number of relatively small sub-clusters
2. Use an agglomerative hierarchical clustering algorithm:
find the genuine clusters by repeatedly combining these
sub-clusters
Overall Framework of CHAMELEON
Construct
Partition the Graph
Sparse Graph
Data Set
Merge Partition
Final Clusters
It uses a k-nearest neighbor approach
to construct the sparse graph: Each vertex
represents an object, an edge exists between
two vertices if one object is among the k-most similar objects of the other
