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Lecture 21 of 42
Partitioning-Based Clustering
and Expectation Maximization (EM)
Monday, 10 March 2008
William H. Hsu
Department of Computing and Information Sciences, KSU
http://www.cis.ksu.edu/Courses/Spring-2008/CIS732
Readings:
Section 7.1 – 7.3, Han & Kamber 2e
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
EM Algorithm:
Example [3]
•
Expectation Step
– Suppose we observed m actual experiments, each n coin flips long
• Each experiment corresponds to one choice of coin (~)
• Let h denote the number of heads in experiment xi (a single data point)
– Q: How did we simulate the “fictional” data points, E[ log P(x | αˆ , pˆ , qˆ )]?
– A: By estimating (for 1  i  m, i.e., the real data points)


P x i | Coin  1 P Coin  1
P Coin  1 | x i  

P x i 
•
Maximization Step
n-h
αˆ  pˆ h 1- pˆ 

n- h
n- h
αˆ  pˆ h 1- pˆ   1- αˆ  qˆ h 1- qˆ 
 
– Q: What are we updating? What objective function are we maximizing?

E E E
m

– A: We are updating αˆ , pˆ , qˆ to maximize
where
,
,
E  E log P x i | αˆ , pˆ , qˆ 
αˆ pˆ qˆ
 i 1



hi
hi





P
Coin

1
|
x
1P
Coin

1
|
x
n
n
i
i 
P Coin  1 | x i 

ˆ
α
, pˆ 
, qˆ 




m
P
Coin

1
|
x

 1- P Coin  1| x i 
i

CIS 732: Machine Learning and Pattern Recognition

Kansas State University
Department of Computing and Information Sciences
EM for Unsupervised Learning
•
Unsupervised Learning Problem
– Objective: estimate a probability distribution with unobserved variables
– Use EM to estimate mixture policy (more on this later; see 6.12, Mitchell)
•
Pattern Recognition Examples
– Human-computer intelligent interaction (HCII)
• Detecting facial features in emotion recognition
• Gesture recognition in virtual environments
– Computational medicine [Frey, 1998]
• Determining morphology (shapes) of bacteria, viruses in microscopy
• Identifying cell structures (e.g., nucleus) and shapes in microscopy
– Other image processing
– Many other examples (audio, speech, signal processing; motor control; etc.)
•
Inference Examples
– Plan recognition: mapping from (observed) actions to agent’s (hidden) plans
– Hidden changes in context: e.g., aviation; computer security; MUDs
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning:
AutoClass [1]
•
Bayesian Unsupervised Learning
– Given: D = {(x1, x2, …, xn)} (vectors of indistingushed attribute values)
– Return: set of class labels that has maximum a posteriori (MAP) probability
•
Intuitive Idea
– Bayesian learning: hMAP  arg max P h | D   arg max P D | h P h 
hH
hH
– MDL/BIC (Occam’s Razor): priors P(h) express “cost of coding” each model h
– AutoClass
• Define mutually exclusive, exhaustive clusters (class labels) y1, y2, …, yJ
• Suppose: each yj (1  j  J) contributes to x
• Suppose also: yj’s contribution ~ known pdf, e.g., Mixture of Gaussians (MoG)
• Conjugate priors: priors on y of same form as priors on x
•
When to Use for Clustering
– Believe (or can assume): clusters generated by known pdf
– Believe (or can assume): clusters combined using finite mixture (later)
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning:
AutoClass [2]
•
AutoClass Algorithm [Cheeseman et al, 1988]
– Based on maximizing P(x | j, yj, J)
• j: class (cluster) parameters (e.g., mean and variance)
• yj : synthetic classes (can estimate marginal P(yj) any time)
– Apply Bayes’s Theorem, use numerical BOC estimation techniques (cf. Gibbs)
– Search objectives
• Find best J (ideally: integrate out j, yj; really: start with big J, decrease)
• Find j, yj: use MAP estimation, then “integrate in the neighborhood” of yMAP
•
EM: Find MAP Estimate for P(x | j, yj, J) by Iterative Refinement
•
Advantages over Symbolic (Non-Numerical) Methods
– Returns probability distribution over class membership
• More robust than “best” yj
• Compare: fuzzy set membership (similar but probabilistically motivated)
– Can deal with continuous as well as discrete data
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning:
AutoClass [3]
•
AutoClass Resources
– Beginning tutorial (AutoClass II): Cheeseman et al, 4.2.2 Buchanan and Wilkins
– Project page: http://ic-www.arc.nasa.gov/ic/projects/bayes-group/autoclass/
•
Applications
– Knowledge discovery in databases (KDD) and data mining
•
Infrared astronomical satellite (IRAS): spectral atlas (sky survey)
• Molecular biology: pre-clustering DNA acceptor, donor sites (mouse, human)
• LandSat data from Kansas (30 km2 region, 1024 x 1024 pixels, 7 channels)
• Positive findings: see book chapter by Cheeseman and Stutz, online
– Other typical applications: see KD Nuggets (http://www.kdnuggets.com)
•
Implementations
– Obtaining source code from project page
• AutoClass III: Lisp implementation [Cheeseman, Stutz, Taylor, 1992]
• AutoClass C: C implementation [Cheeseman, Stutz, Taylor, 1998]
– These and others at: http://www.recursive-partitioning.com/cluster.html
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning:
Competitive Learning for Feature Discovery
•
Intuitive Idea: Competitive Mechanisms for Unsupervised Learning
– Global organization from local, competitive weight update
• Basic principle expressed by Von der Malsburg
• Guiding examples from (neuro)biology: lateral inhibition
– Previous work: Hebb, 1949; Rosenblatt, 1959; Von der Malsburg, 1973; Fukushima,
1975; Grossberg, 1976; Kohonen, 1982
•
A Procedural Framework for Unsupervised Connectionist Learning
– Start with identical (“neural”) processing units, with random initial parameters
– Set limit on “activation strength” of each unit
– Allow units to compete for right to respond to a set of inputs
•
Feature Discovery
– Identifying (or constructing) new features relevant to supervised learning
– Examples: finding distinguishable letter characteristics in handwriten character
recognition (HCR), optical character recognition (OCR)
– Competitive learning: transform X into X’; train units in X’ closest to x
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning:
Kohonen’s Self-Organizing Map (SOM) [1]
•
Another Clustering Algorithm
– aka Self-Organizing Feature Map (SOFM)
– Given: vectors of attribute values (x1, x2, …, xn)
– Returns: vectors of attribute values (x1’, x2’, …, xk’)
• Typically, n >> k (n is high, k = 1, 2, or 3; hence “dimensionality reducing”)
• Output: vectors x’, the projections of input points x; also get P(xj’ | xi)
• Mapping from x to x’ is topology preserving
•
Topology Preserving Networks
– Intuitive idea: similar input vectors will map to similar clusters
– Recall: informal definition of cluster (isolated set of mutually similar entities)
– Restatement: “clusters of X (high-D) will still be clusters of X’ (low-D)”
•
Representation of Node Clusters
– Group of neighboring artificial neural network units (neighborhood of nodes)
– SOMs: combine ideas of topology-preserving networks, unsupervised learning
•
Implementation: http://www.cis.hut.fi/nnrc/ and MATLAB NN Toolkit
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning:
Kohonen’s Self-Organizing Map (SOM) [2]
•
Kohonen Network (SOM) for Clustering
– Training algorithm: unnormalized competitive learning
– Map is organized as a grid (shown here in 2D)
• Each node (grid element) has a weight vector wj
• Dimension of wj is n (same as input vector)
x’ : vector
in 2-space
x : vector
in n-space
• Number of trainable parameters (weights): m · m · n for an m-by-m SOM
• 1999 state-of-the-art: typical small SOMs 5-20, “industrial strength” > 20
– Output found by selecting j* whose wj has minimum Euclidean distance from x
• Only one active node, aka Winner-Take-All (WTA): winning node j*
• i.e., j* = arg minj || wj - x ||2
•
Update Rule
• Same as competitive learning algorithm, with one modification
• Neighborhood function associated with j* spreads the wj around

 





w
t

r
t
h
x
 w j t  if j  Neighborhood  j * 


j, j *
 j
w j t  1   
otherwise

w j t 
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning:
Kohonen’s Self-Organizing Map (SOM) [3]
•
Traditional Competitive Learning
• Only train j*
j*
• Corresponds to neighborhood of 0
•
Neighborhood of 1
Neighborhood Function hj, j*
– For 2D Kohonen SOMs, h is typically a square or hexagonal region
• j*, the winner, is at the center of Neighborhood (j*)
• hj*, j*  1
– Nodes in Neighborhood (j) updated whenever j wins, i.e., j* = j
– Strength of information fed back to wj is inversely proportional to its distance
from the j* for each x
– Often use exponential or Gaussian (normal) distribution on neighborhood to
decay weight delta as distance from j* increases
•
Annealing of Training Parameters
– Neighborhood must shrink to 0 to achieve convergence
– r (learning rate) must also decrease monotonically
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning:
SOM and Other Projections for Clustering
DimensionalityReducing
Projection (x’)
Clusters of
Similar Records
Delaunay
Triangulation
Voronoi
(Nearest Neighbor)
Diagram (y)
Cluster Formation and Segmentation Algorithm (Sketch)
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning:
Other Algorithms (PCA, Factor Analysis)
•
Intuitive Idea
– Q: Why are dimensionality-reducing transforms good for supervised learning?
– A: There may be many attributes with undesirable properties, e.g.,
• Irrelevance: xi has little discriminatory power over c(x) = yi
• Sparseness of information: “feature of interest” spread out over many xi’s
(e.g., text document categorization, where xi is a word position)
• We want to increase the “information density” by “squeezing X down”
•
Principal Components Analysis (PCA)
– Combining redundant variables into a single variable (aka component, or factor)
– Example: ratings (e.g., Nielsen) and polls (e.g., Gallup); responses to certain
questions may be correlated (e.g., “like fishing?” “time spent boating”)
•
Factor Analysis (FA)
– General term for a class of algorithms that includes PCA
– Tutorial: http://www.statsoft.com/textbook/stfacan.html
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Clustering Methods:
Design Choices
•
Intuition
– Functional (declarative) definition: easy (“We recognize a cluster when we see it”)
– Operational (procedural, constructive) definition: much harder to give
– Possible reason: clustering of objects into groups has taxonomic semantics (e.g.,
shape, size, time, resolution, etc.)
•
Possible Assumptions
– Data generated by a particular probabilistic model
– No statistical assumptions
•
Design Choices
– Distance (similarity) measure: standard metrics, transformation-invariant metrics
• L1 (Manhattan):  |xi - yi|, L2 (Euclidean):
 x
 y i  , L (Sup): max |xi - yi|
2
i
• Symmetry: Mahalanobis distance
• Shift, scale invariance: covariance matrix
– Transformations (e.g., covariance diagonalization: rotate axes to get rotational
invariance, cf. PCA, FA)
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Clustering: Applications
Data from T. Mitchell’s web site:
http://www.cs.cmu.edu/~tom/faces.html
NCSA D2K 1.0 - http://www.ncsa.uiuc.edu/STI/ALG/
http://www.cnl.salk.edu/~wiskott/Bibliographies/
FaceFeatureFinding.html
Transactional Database Mining
6500 news stories
from the WWW
in 1997
Facial Feature Extraction
Confidential and proprietary to Caterpillar; may only
be used with prior written consent from Caterpillar.
Information Retrieval:
Text Document
Categorization
ThemeScapes - http://www.cartia.com
NCSA D2K 2.0 - http://www.ncsa.uiuc.edu/STI/ALG/
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Unsupervised Learning and
Constructive Induction
•
Unsupervised Learning in Support of Supervised Learning
– Given: D  labeled vectors (x, y)
Constructive
Induction
– Return: D’  transformed training examples (x’, y’)
(x, y)
– Solution approach: constructive induction
Feature (Attribute)
Construction and
Partitioning
• Feature “construction”: generic term
• Cluster definition
•
x’ / (x1’, …, xp’)
Feature Construction: Front End
– Synthesizing new attributes
Cluster
Definition
• Logical: x1   x2, arithmetic: x1 + x5 / x2
• Other synthetic attributes: f(x1, x2, …, xn), etc.
– Dimensionality-reducing projection, feature extraction
(x’, y’) or ((x1’, y1’), …, (xp’, yp’))
– Subset selection: finding relevant attributes for a given target y
– Partitioning: finding relevant attributes for given targets y1, y2, …, yp
•
Cluster Definition: Back End
– Form, segment, and label clusters to get intermediate targets y’
– Change of representation: find an (x’, y’) that is good for learning target y
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Clustering:
Relation to Constructive Induction
•
Clustering versus Cluster Definition
– Clustering: 3-step process
– Cluster definition: “back end” for feature construction
•
Clustering: 3-Step Process
– Form
• (x1’, …, xk’) in terms of (x1, …, xn)
• NB: typically part of construction step, sometimes integrates both
– Segment
• (y1’, …, yJ’) in terms of (x1’, …, xk’)
• NB: number of clusters J not necessarily same as number of dimensions k
– Label
• Assign names (discrete/symbolic labels (v1’, …, vJ’)) to (y1’, …, yJ’)
• Important in document categorization (e.g., clustering text for info retrieval)
•
Hierarchical Clustering: Applying Clustering Recursively
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
CLUTO
•
Clustering Algorithms
– High-performance & High-quality
partitional clustering
– High-quality agglomerative clustering
– High-quality graph-partitioning-based
clustering
– Hybrid partitional & agglomerative
algorithms for building trees for very large
datasets.
•
Cluster Analysis Tools
– Cluster signature identification
– Cluster organization identification
•
Visualization Tools
– Hierarchical Trees
– High-dimensional datasets
– Cluster relations
•
Interfaces
– Stand-alone programs
– Library with a fully published API
•
Available on Windows, Sun, and Linux
http://www.cs.umn.edu/~cluto
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Today
•
Clustering
–
–
–
–
Distance Measures
Graph-based Techniques
K-Means Clustering
Tools and Software for Clustering
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Prediction, Clustering, Classification
•
What is Prediction?
– The goal of prediction is to forecast or deduce the value of an attribute
based on values of other attributes
– A model is first created based on the data distribution
– The model is then used to predict future or unknown values
•
Supervised vs. Unsupervised Classification
– Supervised Classification = Classification
• We know the class labels and the number of classes
– Unsupervised Classification = Clustering
• We do not know the class labels and may not know the number
of classes
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
What is Clustering in Data Mining?
Clustering is a process of partitioning a set of data (or
objects) in a set of meaningful sub-classes, called clusters
Helps users understand the natural grouping or structure in a data set
•
Cluster:
– a collection of data objects that are
“similar” to one another and thus can
be treated collectively as one group
– but as a collection, they are
sufficiently different from other
groups
•
Clustering
– unsupervised classification
– no predefined classes
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Requirements of Clustering Methods
•
•
•
•
•
•
•
•
Scalability
Dealing with different types of attributes
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
The curse of dimensionality
Interpretability and usability
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Applications of Clustering
•
Clustering has wide applications in Pattern Recognition
•
Spatial Data Analysis:
– create thematic maps in GIS by clustering feature spaces
– detect spatial clusters and explain them in spatial data mining
•
Image Processing
•
Market Research
•
Information Retrieval
– Document or term categorization
– Information visualization and IR interfaces
•
Web Mining
– Cluster Web usage data to discover groups of similar access patterns
– Web Personalization
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Clustering Methodologies
•
Two general methodologies
– Partitioning Based Algorithms
– Hierarchical Algorithms
•
Partitioning Based
– divide a set of N items into K clusters (top-down)
•
Hierarchical
– agglomerative: pairs of items or clusters are successively linked to produce larger
clusters
– divisive: start with the whole set as a cluster and successively divide sets into
smaller partitions
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Distance or Similarity Measures
•
Measuring Distance
– In order to group similar items, we need a way to measure the distance
between objects (e.g., records)
– Note: distance = inverse of similarity
– Often based on the representation of objects as “feature vectors”
An Employee DB
ID
1
2
3
4
5
Gender
F
M
M
F
M
Age
27
51
52
33
45
Salary
19,000
64,000
100,000
55,000
45,000
Term Frequencies for Documents
Doc1
Doc2
Doc3
Doc4
Doc5
T1
0
3
3
0
2
T2
4
1
0
1
2
T3
0
4
0
0
2
T4
0
3
0
3
3
T5
0
1
3
0
1
T6
2
2
0
0
4
Which objects are more similar?
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Distance or Similarity Measures
•
Properties of Distance Measures:
– for all objects A and B, dist(A, B)  0, and dist(A, B) = dist(B, A)
– for any object A, dist(A, A) = 0
– dist(A, C)  dist(A, B) + dist (B, C)
•
Common Distance Measures:
X  x1 , x2 ,
– Manhattan distance:
dist ( X , Y )  x  y1  x2  y2 
1
– Euclidean distance:
–
dist ( Xsimilarity:
, Y )   x1  y1  
Cosine
2
dist ( X , Y )  1  sim( X , Y )
, xn
, yn
 xn  yn
  xn  yn 
Can be normalized
to make values fall
between 0 and 1.
2
sim( X , Y ) 
 ( xi  yi )
i
 xi   yi
2
i
CIS 732: Machine Learning and Pattern Recognition
Y  y1 , y2 ,
2
i
Kansas State University
Department of Computing and Information Sciences
Distance or Similarity Measures
•
Weighting Attributes
– in some cases we want some attributes to count more than others
– associate a weight with each of the attributes in calculating distance, e.g.,
dist ( X , Y )  w1  x1  y1  
2
•
 wn  xn  yn 
2
Nominal (categorical) Attributes
– can use simple matching: distance=1 if values match, 0 otherwise
– or convert each nominal attribute to a set of binary attribute, then use the
usual distance measure
– if all attributes are nominal, we can normalize by dividing the number of
matches by the total number of attributes
•
Normalization:
– want values to fall between 0 an 1:
– other variations possible
x 'i 
CIS 732: Machine Learning and Pattern Recognition
xi  min xi
max xi  min xi
Kansas State University
Department of Computing and Information Sciences
Distance or Similarity Measures
•
Example
– max distance for age: 100000-19000 = 79000
– max distance for age: 52-27 = 25
ID
1
2
3
4
5
Gender
F
M
M
F
M
Age
27
51
52
33
45
Salary
19,000
64,000
100,000
55,000
45,000
ID
1
2
3
4
5
x 'i 
Gender
1
0
0
1
0
xi  min xi
max xi  min xi
Age
0.00
0.96
1.00
0.24
0.72
Salary
0.00
0.56
1.00
0.44
0.32
– dist(ID2, ID3) = SQRT( 0 + (0.04)2 + (0.44)2 ) = 0.44
– dist(ID2, ID4) = SQRT( 1 + (0.72)2 + (0.12)2 ) = 1.24
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Domain Specific Distance Functions
•
For some data sets, we may need to use specialized functions
– we may want a single or a selected group of attributes to be used in the
computation of distance - same problem as “feature selection”
– may want to use special properties of one or more attribute in the data
Example: Zip Codes
distzip(A, B) = 0, if zip codes are identical
distzip(A, B) = 0.1, if first 3 digits are identical
distzip(A, B) = 0.5, if first digits are identical
distzip(A, B)
= 1,exist
if firstindigits
are different
– natural distance functions
may
the data
Example: Customer Solicitation
distsolicit(A, B) = 0, if both A and B responded
distsolicit(A, B) = 0.1, both A and B were chosen but did not respond
distsolicit(A, B) = 0.5, both A and B were chosen, but only one responded
distsolicit(A, B) = 1, one was chosen, but the other was not
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Distance (Similarity) Matrix
•
Similarity (Distance) Matrix
– based on the distance or similarity measure we can construct a symmetric matrix of
distance (or similarity values)
– (i, j) entry in the matrix is the distance (similarity) between items i and j
I1
I2
I1

d 21
I2
d12

In
d1n
d2n

In
d n1 d n 2

Note that dij = dji (i.e., the matrix is
symmetric. So, we only need the lower
triangle part of the matrix.
The diagonal is all 1’s (similarity) or all
0’s (distance)
dij  similarity (or distance) of Di to D j
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Example: Term Similarities in Documents
Doc1
Doc2
Doc3
Doc4
Doc5
T1
0
3
3
0
2
T2
4
1
0
1
2
T3
0
4
0
0
2
T4
0
3
0
3
3
T5
0
1
3
0
1
T6
2
2
0
0
4
T7
1
0
3
2
0
T8
3
1
0
0
2
N
sim(Ti , Tj )   ( wik  w jk )
k 1
Term-Term
Similarity Matrix
T2
T3
T4
T5
T6
T7
T8
T1
7
16
15
14
14
9
7
T2
T3
T4
T5
T6
T7
8
12
3
18
6
17
18
6
16
0
8
6
18
6
9
6
9
3
2
16
3
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Similarity (Distance) Thresholds
– A similarity (distance) threshold may be used to mark pairs that are
“sufficiently” similar
T2
T3
T4
T5
T6
T7
T2
T3
T4
T5
T6
T7
T8
T1
7
16
15
14
14
9
7
8
12
3
18
6
17
18
6
16
0
8
6
18
6
9
6
9
3
2
16
3
T2
T3
T4
T5
T6
T7
T2
T3
T4
T5
T6
T7
T8
T1
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
0
0
0
1
0
0
0
0
0
0
1
0
CIS 732: Machine Learning and Pattern Recognition
Using a threshold
value of 10 in the
previous example
Kansas State University
Department of Computing and Information Sciences
Graph Representation
•
The similarity matrix can be visualized as an undirected graph
– each item is represented by a node, and edges represent the fact that two
items are similar (a one in the similarity threshold matrix)
T2
T3
T4
T5
T6
T7
T8
T1 T2 T3 T4 T5 T6 T7
0
1 0
1 1 1
1 0 0 0
1 1 1 1 0
0 0 0 0 0 0
0 1 0 0 0 1 0
T1
T3
T5
T4
If no threshold is used, then
matrix can be represented as
a weighted graph
CIS 732: Machine Learning and Pattern Recognition
T2
T7
T6
T8
Kansas State University
Department of Computing and Information Sciences
Simple Clustering Algorithms
•
•
•
•
If we are interested only in threshold (and not the degree of similarity or
distance), we can use the graph directly for clustering
Clique Method (complete link)
– all items within a cluster must be within the similarity threshold of all
other items in that cluster
– clusters may overlap
– generally produces small but very tight clusters
Single Link Method
– any item in a cluster must be within the similarity threshold of at least one
other item in that cluster
– produces larger but weaker clusters
Other methods
– star method - start with an item and place all related items in that cluster
– string method - start with an item; place one related item in that cluster;
then place anther item related to the last item entered, and so on
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Simple Clustering Algorithms
•
Clique Method
– a clique is a completely connected subgraph of a graph
– in the clique method, each maximal clique in the graph becomes a cluster
T1
T3
Maximal cliques (and therefore the
clusters) in the previous example are:
{T1, T3, T4, T6}
{T2, T4, T6}
{T2, T6, T8}
{T1, T5}
{T7}
T5
T4
T2
Note that, for example, {T1, T3, T4}
is also a clique, but is not maximal.
T7
T6
T8
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Simple Clustering Algorithms
•
Single Link Method
– selected an item not in a cluster and place it in a new cluster
– place all other similar item in that cluster
– repeat step 2 for each item in the cluster until nothing more can be
added
– repeat steps 1-3 for each item that remains unclustered
T1
T3
In this case the single link method
produces only two clusters:
T5
T4
T2
{T1, T3, T4, T5, T6, T2, T8}
{T7}
Note that the single link method does
not allow overlapping clusters, thus
partitioning the set of items.
T7
T6
T8
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Clustering with Existing Clusters
•
•
•
The notion of comparing item similarities can be extended to clusters themselves,
by focusing on a representative vector for each cluster
– cluster representatives can be actual items in the cluster or other “virtual”
representatives such as the centroid
– this methodology reduces the number of similarity computations in clustering
– clusters are revised successively until a stopping condition is satisfied, or
until no more changes to clusters can be made
Partitioning Methods
– reallocation method - start with an initial assignment of items to clusters and
then move items from cluster to cluster to obtain an improved partitioning
– Single pass method - simple and efficient, but produces large clusters, and
depends on order in which items are processed
Hierarchical Agglomerative Methods
– starts with individual items and combines into clusters
– then successively combine smaller clusters to form larger ones
– grouping of individual items can be based on any of the methods discussed
earlier
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
K-Means Algorithm
•
The basic algorithm (based on reallocation method):
1. select K data points as the initial representatives
2. for i = 1 to N, assign item xi to the most similar centroid (this gives K clusters)
3. for j = 1 to K, recalculate the cluster centroid Cj
4. repeat steps 2 and 3 until these is (little or) no change in clusters
•
Example: Clustering Terms
Initial (arbitrary) assignment:
C1 = {T1,T2}, C2 = {T3,T4}, C3 = {T5,T6}
Doc1
Doc2
Doc3
Doc4
Doc5
T1
0
3
3
0
2
T2
4
1
0
1
2
T3
0
4
0
0
2
T4
0
3
0
3
3
T5
0
1
3
0
1
Cluster Centroids
T6
2
2
0
0
4
CIS 732: Machine Learning and Pattern Recognition
T7
1
0
3
2
0
T8
3
1
0
0
2
C1
4/2
4/2
3/2
1/2
4/2
C2
0/2
7/2
0/2
3/2
5/2
C3
2/2
3/2
3/2
0/2
5/2
Kansas State University
Department of Computing and Information Sciences
Example: K-Means
•
Example (continued)
Now using simple similarity measure, compute the new cluster-term similarity matrix
Class1
Class2
Class3
Assign
T1
T2
T3
T4
T5
T6
T7
T8
29/2
29/2
24/2
27/2
17/2
32/2
15/2
24/2
31/2
20/2
38/2
45/2
12/2
34/2
6/2
17/2
28/2
21/2
22/2
24/2
17/2
30/2
11/2
19/2
to Class2 Class1 Class2 Class2 Class3 Class2 Class1 Class1
Now compute new cluster centroids using the original document-term matrix
Doc1
Doc2
Doc3
Doc4
Doc5
T1
0
3
3
0
2
T2
4
1
0
1
2
T3
0
4
0
0
2
T4
0
3
0
3
3
T5
0
1
3
0
1
T6
2
2
0
0
4
T7
1
0
3
2
0
T8
3
1
0
0
2
C1
8/3
2/3
3/3
3/3
4/3
C2
2/4
12/4
3/4
3/4
11/4
C3
0/1
1/1
3/1
0/1
1/1
The process is repeated until no further changes are made to the clusters
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
K-Means Algorithm
•
Strength of the k-means:
– Relatively efficient: O(tkn), where n is # of objects, k is # of clusters, and t
is # of iterations. Normally, k, t << n
– Often terminates at a local optimum
•
Weakness of the k-means:
– Applicable only when mean is defined; what about categorical data?
– Need to specify k, the number of clusters, in advance
– Unable to handle noisy data and outliers
•
Variations of K-Means usually differ in:
– Selection of the initial k means
– Dissimilarity calculations
– Strategies to calculate cluster means
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Terminology
•
Expectation-Maximization (EM) Algorithm
– Iterative refinement: repeat until convergence to a locally optimal label
– Expectation step: estimate parameters with which to simulate data
– Maximization step: use simulated (“fictitious”) data to update parameters
•
Unsupervised Learning and Clustering
– Constructive induction: using unsupervised learning for supervised learning
• Feature construction: “front end” - construct new x values
• Cluster definition: “back end” - use these to reformulate y
– Clustering problems: formation, segmentation, labeling
– Key criterion: distance metric (points closer intra-cluster than inter-cluster)
– Algorithms
• AutoClass: Bayesian clustering
• Principal Components Analysis (PCA), factor analysis (FA)
• Self-Organizing Maps (SOM): topology preserving transform (dimensionality
reduction) for competitive unsupervised learning
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences
Summary Points
•
Expectation-Maximization (EM) Algorithm
•
Unsupervised Learning and Clustering
– Types of unsupervised learning
• Clustering, vector quantization
• Feature extraction (typically, dimensionality reduction)
– Constructive induction: unsupervised learning in support of supervised learning
• Feature construction (aka feature extraction)
• Cluster definition
– Algorithms
• EM: mixture parameter estimation (e.g., for AutoClass)
• AutoClass: Bayesian clustering
• Principal Components Analysis (PCA), factor analysis (FA)
• Self-Organizing Maps (SOM): projection of data; competitive algorithm
– Clustering problems: formation, segmentation, labeling
•
Next Lecture: Time Series Learning and Characterization
CIS 732: Machine Learning and Pattern Recognition
Kansas State University
Department of Computing and Information Sciences