CIS732-Lecture-36-20070418

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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/CIS
732 Readings Section 7.1 7.3, Han Kamber 2e
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EM AlgorithmExample 3
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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 agents (hidden) plans
• Hidden changes in context e.g., aviation
  computer security MUDs

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Unsupervised LearningAutoClass 1
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Unsupervised LearningAutoClass 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 Bayess 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

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Unsupervised LearningAutoClass 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/projec
  ts/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-partitio
  ning.com/cluster.html

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Unsupervised LearningCompetitive 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

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Unsupervised LearningKohonens 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 gtgt 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

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Unsupervised LearningKohonens Self-Organizing
Map (SOM) 2
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Unsupervised LearningKohonens Self-Organizing
Map (SOM) 3
j
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Unsupervised LearningSOM and Other Projections
for Clustering
Cluster Formation and Segmentation Algorithm
(Sketch)
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Unsupervised LearningOther 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 xis (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

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Clustering MethodsDesign Choices
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Clustering Applications
Information Retrieval Text Document Categorizatio
n
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Unsupervised Learning andConstructive Induction

• Unsupervised Learning in Support of Supervised
  Learning
• Given D ? labeled vectors (x, y)
• Return D ? transformed training examples (x,
  y)
• Solution approach constructive induction
• Feature construction generic term
• Cluster definition
• Feature Construction Front End
• Synthesizing new attributes
• Logical x1 ? ? x2, arithmetic x1 x5 / x2
• Other synthetic attributes f(x1, x2, , xn),
  etc.
• Dimensionality-reducing projection, feature
  extraction
• 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

x / (x1, , xp)
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ClusteringRelation 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

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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
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Today

• Clustering
• Distance Measures
• Graph-based Techniques
• K-Means Clustering
• Tools and Software for Clustering

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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

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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

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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

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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

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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

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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

Term Frequencies for Documents
An Employee DB
Which objects are more similar?
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Distance or Similarity Measures

• Properties of Distance Measures
• for all objects A and B, dist(A, B) ³ 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
• Manhattan distance
• Euclidean distance
• Cosine similarity

Can be normalized to make values fall between 0
and 1.
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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.,
• Nominal (categorical) Attributes
• can use simple matching distance1 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

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Distance or Similarity Measures

• Example
• max distance for age 100000-19000 79000
• max distance for age 52-27 25
• 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

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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
• natural distance functions may exist 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, if first digits are different
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
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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

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 1s
(similarity) or all 0s (distance)
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Example Term Similarities in Documents
Term-Term Similarity Matrix
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Similarity (Distance) Thresholds

• A similarity (distance) threshold may be used to
  mark pairs that are sufficiently similar

Using a threshold value of 10 in the previous
example
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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)

If no threshold is used, then matrix can be
represented as a weighted graph
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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

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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

T3
T1
Maximal cliques (and therefore the clusters) in
the previous example are T1, T3, T4,
T6 T2, T4, T6 T2, T6, T8 T1,
T5 T7 Note that, for example, T1, T3, T4
is also a clique, but is not maximal.
T5
T4
T2
T7
T6
T8
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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

T3
T1
In this case the single link method produces only
two clusters 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.
T5
T4
T2
T7
T6
T8
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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

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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
Cluster Centroids
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Example K-Means

• Example (continued)

Now using simple similarity measure, compute the
new cluster-term similarity matrix
Now compute new cluster centroids using the
original document-term matrix
The process is repeated until no further changes
are made to the clusters
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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 ltlt 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

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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

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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