Cluster analysis for microarray data

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Cluster analysis for microarray data

• Anja von Heydebreck

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Aim of clustering Group objects according to
their similarity
Cluster a set of objects that are similar to
each other and separated from the
other objects. Example green/ red data
points were generated from two different normal
distributions
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Clustering microarray data
gene expression data matrix

• Genes and experiments/samples are given as the
  row and column vectors of a gene expression data
  matrix.
• Clustering may be applied either to genes or
  experiments (regarded as vectors in Rp or Rn).

n experiments
p genes
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Why cluster genes?

• Identify groups of possibly co-regulated genes
  (e.g. in conjunction with sequence data).
• Identify typical temporal or spatial gene
  expression patterns (e.g. cell cycle data).
• Arrange a set of genes in a linear order that is
  at least not totally meaningless.

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Why cluster experiments/samples?

• Quality control Detect experimental
  artifacts/bad hybridizations
• Check whether samples are grouped according to
  known categories (though this might be better
  addressed using a supervised approach
  statistical tests, classification)
• Identify new classes of biological samples (e.g.
  tumor subtypes)

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Alizadeh et al., Nature 403503-11, 2000
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Cluster analysis

• Generally, cluster analysis is based on two
  ingredients
• Distance measure Quantification of
  (dis-)similarity of objects.
• Cluster algorithm A procedure to group objects.
  Aim small within-cluster distances, large
  between-cluster distances.

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Some distance measures

• Given vectors x (x1, , xn), y (y1, , yn)
• Euclidean distance
• Manhattan distance
• Correlation
• distance

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Which distance measure to use?

• The choice of distance measure should be based
  on the application area. What sort of
  similarities would you like to detect?
• Correlation distance dc measures trends/relative
  differences
• dc(x, y) dc(axb, y) if a gt 0.

x (1, 1, 1.5, 1.5) y (2.5, 2.5, 3.5, 3.5)
2x 0.5 z (1.5, 1.5, 1, 1)
dc(x, y) 0, dc(x, z) 2. dE(x, z) 1, dE(x,
y) 3.54.
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Which distance measure to use?

• Euclidean and Manhattan distance both measure
  absolute differences between vectors. Manhattan
  distance is more robust against outliers.
• May apply standardization to the observations
• Subtract mean and divide by standard
  deviation
• After standardization, Euclidean and correlation
  distance are equivalent

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K-means clustering

• Input N objects given as data points in Rp
• Specify the number k of clusters.
• Initialize k cluster centers. Iterate until
  convergence
• - Assign each object to the cluster with the
  closest center (wrt Euclidean distance).
• - The centroids/mean vectors of the obtained
  clusters are taken as new cluster centers.
• K-means can be seen as an optimization problem
• Minimize the sum of squared within-cluster
  distances,
• Results depend on the initialization. Use several
  starting points and choose the best solution
  (with minimal W(C)).

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K-means/PAM How to choose K (the number of
clusters)?

• There is no easy answer.
• Many heuristic approaches try to compare the
  quality of clustering results for different
  values of K (for an overview see Dudoit/Fridlyand
  2002).
• The problem can be better addressed in
  model-based clustering, where each cluster
  represents a probability distribution, and a
  likelihood-based framework can be used.

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Hierarchical clustering

• Similarity of objects is represented in a tree
  structure (dendrogram).
• Advantage no need to specify the number of
  clusters in advance. Nested clusters can be
  represented.

Golub data different types of leukemia.
Clustering based on the 150 genes with highest
variance across all samples.
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Agglomerative hierarchical clustering

• Bottom-up algorithm (top-down (divisive) methods
  are less common).
• Start with the objects as clusters.
• In each iteration, merge the two clusters with
  the minimal distance from each other - until you
  are left with a single cluster comprising all
  objects.
• But what is the distance between two clusters?

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Distances between clusters used for hierarchical
clustering

• Calculation of the distance between two
  clusters is based on the pairwise distances
  between members of the clusters.
• Complete linkage largest distance
• Average linkage average distance
• Single linkage smallest distance

Complete linkage gives preference to
compact/spherical clusters. Single linkage can
produce long stretched clusters.
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Hierarchical clustering

• The height of a node in the dendrogram represents
  the distance of the two children clusters.
• Loss of information n objects have n(n-1)/2
  pairwise distances, tree has n-1 inner nodes.
• The ordering of the leaves is not uniquely
  defined by the dendrogram 2n-2 possible choices.

Golub data different types of leukemia.
Clustering based on the 150 genes with highest
variance across all samples.
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Alternative direct visualization of
similarity/distance matrices
Useful if one wants to investigate a specific
factor (advantage no loss of information). Sort
experiments according to that factor.
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Clustering of time course data

• Suppose we have expression data from different
  time points t1, , tn, and want to identify
  typical temporal expression profiles by
  clustering the genes.
• Usual clustering methods/distance measures dont
  take the ordering of the time points into account
  the result would be the same if the time points
  were permuted.
• Simple modification Consider the difference
  xi(j1) xij between consecutive timepoints as
  an additional observation yij. Then apply a
  clustering algorithm such as K-means to the
  augmented data matrix.

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Biclustering

• Usual clustering algorithms are based on global
  similarities of rows or columns of an expression
  data matrix.
• But the similarity of the expression profiles of
  a group of genes may be restricted to certain
  experimental conditions.
• Goal of biclustering identify homogeneous
  submatrices.
• Difficulties computational complexity, assessing
  the statistical significance of results
• Example Tanay et al. 2002.

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The role of feature selection

• Sometimes, people first select genes that appear
  to be differentially expressed between groups of
  samples. Then they cluster the samples based on
  the expression levels of these genes. Is it
  remarkable if the samples then cluster into the
  two groups?
• No, this doesnt prove anything, because the
  genes were selected with respect to the two
  groups! Such effects can even be obtained with a
  matrix of i.i.d. random numbers.

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Classification Additional class information
given
object-space
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Classification methods

• Linear Discriminant Analysis (LDA, Fisher)
• Nearest neighbor procedures
• Neural nets
• Support vector machines

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Embedding methods

• Attempt to represent high-dimensional objects as
  points in a low-dimensional space (2 or 3
  dimensions)
• Principal Component Analysis
• Correspondence Analysis
• (Multidimensional Scaling represents objects for
  which distances are given)

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Principal Component Analysis

• Finds a low-dimensional projection such that the
  sum-of-squares of the residuals is minimized

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Contingency Tables Results from a poll among
people of different nationality
nationality
Ge
Au
It
favorite dish
Pasta
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Wiener Schnitzel
23
11
18
Sauerbraten
7
9
19
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Microarray experiment
Genes
Cdc14
Gle2
Gal1
Experimental condition
Wt yeast
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12
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wt yeast galactose
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11
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Yeast transgene
7
9
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Correspondence Analysis and Principal Component
Analysis

• Objects are depicted as points in a plane or in
  three-dimensional space, trying to maintain their
  proximity
• CA is a variant of PCA, although based on another
  distance measure ?2-distance
• Embedding tries to preserve ?2-distance

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Correspondence analysis Interpretation

• Shows both gene-vectors and condition-vectors as
  dots in the plane
• Genes that are nearby are similar
• Conditions that are nearby are similar
• When genes and conditions point in the same
  direction then the gene is up-regulated in that
  condition.

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?2-distance
2
(fkj?fk - flj ?fl)
dkl ?
fj ?n
j1..n
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Correspondence analysis Interpretation

• Shows both gene-vectors and condition-vectors as
  dots in the plane
• Genes that are nearby are similar
• Conditions that are nearby are similar
• When genes and conditions point in the same
  direction then the gene is up-regulated in that
  condition.

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G1 S G2
M
Spellman et al took several samples per
time-point and hybridized the RNA to a glass
chips with all yeast genes
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