Cluster Analysis 1

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Cluster Analysis (1)
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What is Cluster Analysis?

• Finding groups of objects such that the objects
  in a group will be similar (or related) to one
  another and different from (or unrelated to) the
  objects in other groups

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Applications of Cluster Analysis

• Clustering for Understanding
• Group related documents for browsing
• Group genes and proteins that have similar
  functionality
• Group stocks with similar price fluctuations
• Segment customers into a small number of groups
  for additional analysis and marketing activities.
  
• Clustering for Summarization
• Reduce the size of large data sets

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Similarity and Dissimilarity

• Similarity
• Numerical measure of how alike two data objects
  are.
• Higher when objects are more alike.
• Can be transformed to fall in interval 0,1 by
  doing
• s (s min_s)/(max_s min_s)
• Dissimilarity
• Numerical measure of how different are two data
  objects
• Lower when objects are more alike
• Minimum dissimilarity is often 0
• Can be transformed to fall in interval 0,1 by
  doing
• d (d min_d)/(max_d min_d)
• These proximity measures for objects with a
  number of attributes are defined by combining the
  proximities of individual attributes.

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Similarity/Dissimilarity for Simple Attributes

• p and q are the attribute values for two data
  objects.
• Nominal
• E.g. province attribute of an address with
  values
• BC, AB, ON, QC,
• Order not important.
• Dissimilarity
• d0 if pq
• d1 if p?q
• Similarity
• s1 if pq
• s0 if p?q

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Similarity/Dissimilarity for Simple Attributes

• p and q are the attribute values for two data
  objects.
• Ordinal
• E.g. quality attribute of a product with values
• poor, fair, OK, good, wonderful
• Order is important, but exact difference between
  values is undefined or not important.
• Map the values of the attribute to successive
  integers
• poor0, fair1, OK2, good3, wonderful4
• Dissimilarity
• d(p,q) p q / (max_d min_d)
• e.g. d(wonderful, fair) 4-1 / (4-0) .75
• Similarity
• s(p,q) 1 d(p,q) e.g. d(wonderful, fair)
  .25

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Similarity/Dissimilarity for Simple Attributes

• p and q are the attribute values for two data
  objects.
• Continuous (or Interval)
• E.g. weight attribute of a product
• Dissimilarity
• d(p,q) p q
• Similarity
• s(p,q) d(p,q)
• Of course, we can transform them in the 0,1
  scale.

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Combining Similarities

• Sometimes attributes are of many different types,
  but an overall similarity/dissimilarity is
  needed.
• For the k-th attribute, compute a similarity sk
  in the range 0,1.
• Then,

• Similar formula for dissimilarity

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Euclidean Distance

• When all the attributes are continuous we can use
  the Euclidean Distance
• Where n is the number of dimensions
  (attributes) and pk and qk are, respectively, the
  kth attributes (components) or data objects p and
  q.
• Standardization is necessary, if scales differ
• E.g. weight, salary have different scales

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Euclidean Distance
Distance Matrix
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Minkowski Distance

• Minkowski Distance is a generalization of
  Euclidean Distance
• Where r is a parameter, n is the number of
  dimensions (attributes) and pk and qk are,
  respectively, the kth attributes (components) or
  data objects p and q.
• Examples
• r 1. City block (Manhattan, taxicab, L1 norm)
  distance.
• r 2. Euclidean distance
• r ? ?. supremum (Lmax norm, L? norm) distance.
  
• This is the maximum difference between any
  component of the vectors

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Minkowski Distance
Distance Matrix
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Similarity Between Binary Vectors

• Common situation is that objects, p and q, have
  only binary attributes
• Compute similarities using the following
  quantities
• M01 the number of attributes where p was 0 and
  q was 1
• M10 the number of attributes where p was 1 and
  q was 0
• M00 the number of attributes where p was 0 and
  q was 0
• M11 the number of attributes where p was 1 and
  q was 1
• Simple Matching and Jaccard Coefficients
• SMC number of matches / number of attributes
• (M11 M00) / (M01 M10 M11
  M00)
• J number of M11 matches / number of
  not-both-zero attributes values
• (M11) / (M01 M10 M11)

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SMC versus Jaccard Example

• p 1 0 0 0 0 0 0 0 0 0
• q 0 0 0 0 0 0 1 0 0 1
• M01 2 (the number of attributes where p was 0
  and q was 1)
• M10 1 (the number of attributes where p was 1
  and q was 0)
• M00 7 (the number of attributes where p was 0
  and q was 0)
• M11 0 (the number of attributes where p was 1
  and q was 1)
• SMC (M11 M00)/(M01 M10 M11 M00) (07)
  / (2107) 0.7
• J (M11) / (M01 M10 M11) 0 / (2 1 0)
  0

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Cosine Similarity

• If D1 and D2 are two document vectors, then
• cos( D1, D2 ) (D1 ? D2) / D1.D2 ,
• where ? indicates vector dot product and D
  is the length of vector D.
• Example
• D1 ? D2
• .40 .330 0.33 01 .17.33 .0561
• D1 sqrt(.402 .332 .172) .55
• D2 sqrt(.332 12 .332) 1.1
• cos( D1, D2 ) .0561 / (.55 1.1) .093

If the cosine similarity is 1, the angle between
D1 and D2 is 0o, and D1 and D2 are the same
except for the magnitude. If the cosine
similarity is 0, then the angle between D1 and D2
is 90o, and they dont share any terms (words).
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What is Cluster Analysis?

• Finding groups of objects such that the objects
  in a group will be similar (or related) to one
  another and different from (or unrelated to) the
  objects in other groups

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Types of Clusters Well-Separated

• Well-Separated Clusters
• Any point in a cluster is closer (or more
  similar) to every other point in the cluster than
  to any point not in the cluster.

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Types of Clusters Center-Based

• Center-based
• An object in a cluster is closer (more similar)
  to the center of a cluster, than to the center
  of any other cluster
• The center of a cluster is often a centroid, the
  average of all the points in the cluster, or a
  medoid, the most representative point of a
  cluster

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Types of Clusters Contiguity-Based

• Contiguous Cluster (Nearest neighbor or
  Transitive)
• A point in a cluster is closer (or more similar)
  to one or more other points in the cluster than
  to any point not in the cluster.

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Types of Clusters Density-Based

• Density-based
• A cluster is a dense region of points, which is
  separated by low-density regions, from other
  regions of high density.
• Used when the clusters are irregular or
  intertwined, and when noise and outliers are
  present.

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K-means Clustering

• Each cluster is associated with a centroid
  (center point)
• Each point is assigned to the cluster with the
  closest centroid
• Number of clusters, K, must be specified
• Basic algorithm is very simple

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Example
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K-means Clustering Details

• Initial centroids may be chosen randomly.
• Clusters produced vary from one run to another.
• The centroid is (typically) the mean of the
  points in the cluster.
• Closeness is measured by Euclidean distance,
  cosine similarity, etc.
• Most of the convergence happens in the first few
  iterations.
• Often the stopping condition is changed to Until
  relatively few points change clusters
• Complexity is O(I K n d )
• n number of points, K number of clusters, I
  number of iterations, d number of attributes

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Evaluating K-means Clusters

• Most common measure is Sum of Squared Error (SSE)
• For each point, the error is the distance to the
  nearest cluster
• To get SSE, we square these errors and sum them
  up.

x is a data point in cluster Ci and mi is the
representative point for cluster Ci
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Reducing SSE with Post-processing

• Obvious way to reduce the SSE is to find more
  clusters, i.e., to use a larger K.
• However, in many cases, we would like to improve
  the SSE, but don't want to increase the number of
  clusters.
• Various techniques are used to fix up the
  resulting clusters in order to produce a
  clustering that has lower SSE.
• Commonly used approach Use alternate cluster
  splitting and merging phases.
• Split a cluster
• split the cluster with the largest SSE
• Merge two clusters
• merge the two clusters that result in the
  smallest increase in total SSE.

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Limitations of K-means

• K-means has problems when clusters are of
• Differing Sizes
• Differing Densities
• Non-globular shapes

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Limitations of K-means Differing Sizes
K-means (3 Clusters)
Original Points
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Limitations of K-means Differing Density
K-means (3 Clusters)
Original Points
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Limitations of K-means Non-globular Shapes
Original Points
K-means (2 Clusters)
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Overcoming K-means Limitations
Original Points K-means Clusters
One solution is to use many clusters. Find parts
of clusters. Apply merge strategy
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Overcoming K-means Limitations
Original Points K-means Clusters
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Overcoming K-means Limitations
Original Points K-means Clusters
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Importance of Choosing Initial Centroids
Starting with two initial centroids in one
cluster of each pair of clusters
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Importance of Choosing Initial Centroids
Starting with two initial centroids in one
cluster of each pair of clusters
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Importance of Choosing Initial Centroids
Starting with some pairs of clusters having three
initial centroids, while other have only one.
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Importance of Choosing Initial Centroids
Starting with some pairs of clusters having three
initial centroids, while other have only one.
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Problems with Selecting Initial Points

• Of course, the ideal would be to choose initial
  centroids, one from each true cluster. However,
  this is very difficult.
• If there are K real clusters then the chance of
  selecting one centroid from each cluster is
  small.
• Chance is relatively small when K is large
• If clusters are the same size, n, then

• For example, if K 10, then probability
  10!/1010 0.00036
• Sometimes the initial centroids will readjust
  themselves in the right way, and sometimes they
  dont.
• Consider an example of five pairs of clusters

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Solutions to Initial Centroids Problem

• Multiple runs
• Helps, but probability is not on your side
• Bisecting K-means
• Not as susceptible to initialization issues

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Bisecting Kmeans

• Straightforward extension of the basic Kmeans
  algorithm. Simple idea
• To obtain K clusters, split the set of points
  into two clusters, select one of these clusters
  to split, and so on, until K clusters have been
  produced.
• Algorithm
• Initialize the list of clusters to contain the
  cluster consisting of all points.
• repeat
• Remove a cluster from the list of clusters.
• //Perform several trial bisections of the
  chosen cluster.
• for i 1 to number of trials do
• Bisect the selected cluster using basic Kmeans
  (i.e. 2-means).
• end for
• Select the two clusters from the bisection with
  the lowest total SSE.
• Add these two clusters to the list of clusters.
• until the list of clusters contains K clusters.

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Bisecting K-means Example