Important clustering methods used in microarray data analysis

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Important clustering methods used in microarray
data analysis

• Steve Horvath
• Human Genetics and Biostatistics
• UCLA

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Contents

• Multidimensional scaling plots
• Related to principal component analysis
• k-means clustering
• hierarchical clustering

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Introduction to clustering
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MDS plot of clusters
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MDS plot of clusters
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2 references for clustering

• T. Hastie, R. Tibshirani, J. Friedman (2002) The
  elements of Statistical Learning. Springer Series
• L. Kaufman, P. Rousseeuw (1990) Finding groups in
  data. Wiley Series in Probability
•  

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Introduction to clustering
Cluster analysis aims to group or segment a
collection of objects into subsets or "clusters",
such that those within each cluster are more
closely related to one another than objects
assigned to different clusters.   An object can
be described by a set of measurements (e.g.
covariates, features, attributes) or by its
relation to other objects.   Sometimes the goal
is to arrange the clusters into a natural
hierarchy, which involves successively grouping
or merging the clusters themselves so that at
each level of the hierarchy clusters within the
same group are more similar to each other than
those in different groups.  

•  

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Proximity matrices are the input to most
clustering algorithms

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Proximity between pairs of objects similarity or
dissimilarity. If the original data were
collected as similarities, a monotone-decreasing
function can be used to convert them to
dissimilarities. Most algorithms use
(symmetric) dissimilarities (e.g. distances) But
the triangle inequality does not have to hold.
Triangle inequality  
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Different intergroup dissimilarities
Let G and H represent 2 groups.
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Agglomerative clustering, hierarchical clustering
and dendrograms
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Hierarchical clustering plot
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Agglomerative clustering

• Agglomerative clustering algorithms begin with
  every observation representing a singleton
  cluster.
• At each of the N-1 the closest 2 (least
  dissimilar) clusters are merged into a single
  cluster.
• Therefore a measure of dissimilarity between 2
  clusters must be defined.
•  

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Comparing different linkage methods

•  If there is a strong clustering tendency, all 3
  methods produce similar results.
• Single linkage has a tendency to combine
  observations linked by a series of close
  intermediate observations ("chaining). Good for
  elongated clusters
• Bad Complete linkage may lead to clusters where
  observations assigned to a cluster can be much
  closer to members of other clusters than they are
  to some members of their own cluster. Use for
  very compact clusters (like perls on a string).
• Group average clustering represents a compromise
  between the extremes of single and complete
  linkage. Use for ball shaped clusters

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Dendrogram

• Recursive binary splitting/agglomeration can be
  represented by a rooted binary tree.
• The root node represents the entire data set.
• The N terminal nodes of the trees represent
  individual observations.
• Each nonterminal node ("parent") has two daughter
  nodes.
• Thus the binary tree can be plotted so that the
  height of each node is proportional to the value
  of the intergroup dissimilarity between its 2
  daughters.
• A dendrogram provides a complete description of
  the hierarchical clustering in graphical format.

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Comments on dendrograms

• Caution different hierarchical methods as well
  as small changes in the data can lead to
  different dendrograms.
• Hierarchical methods impose hierarchical
  structure whether or not such structure actually
  exists in the data.
• In general dendrograms are a description of the
  results of the algorithm and not graphical
  summary of the data.
• Only valid summary to the extent that the
  pairwise observation dissimilarities obey the
  ultrametric inequality

for all i,i,k
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Figure 1
average
complete
single