L16: Micro-array analysis

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L16 Micro-array analysis

• Dimension reduction
• Unsupervised clustering

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PCA motivating example

• Consider the expression values of 2 genes over 6
  samples.
• Clearly, the expression of g1 is not informative,
  and it suffices to look at g2 values.
• Dimensionality can be reduced by discarding the
  gene g1

g1
g2
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PCA Ex2

• Consider the expression values of 2 genes over 6
  samples.
• Clearly, the expression of the two genes is
  highly correlated.
• Projecting all the genes on a single line could
  explain most of the data.

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PCA

• Suppose all of the data were to be reduced by
  projecting to a single line ? from the mean.
• How do we select the line ??

?
m
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PCA contd

• Let each point xk map to xkmak?. We want to
  minimize the error
• Observation 1 Each point xk maps to xk m
  ?T(xk-m)?
• (ak ?T(xk-m))

xk
?
xk
m
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Proof of Observation 1
Differentiating w.r.t ak
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Minimizing PCA Error

• To minimize error, we must maximize ?TS?
• By definition, ? ?TS? implies that ? is an
  eigenvalue, and ? the corresponding eigenvector.
• Therefore, we must choose the eigenvector
  corresponding to the largest eigenvalue.

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PCA

• The single best dimension is given by the
  eigenvector of the largest eigenvalue of S
• The best k dimensions can be obtained by the
  eigenvectors ?1, ?2, , ?k corresponding to the
  k largest eigenvalues.
• To obtain the k dimensional surface, take BTM

?1T
BT
M
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Clustering

• Suppose we are not given any classes.
• Instead, we are asked to partition the samples
  into clusters that make sense.
• Alternatively, partition genes into clusters.
• Clustering is part of unsupervised learning

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Microarray Data

• Microarray data are usually transformed into an
  intensity matrix (below)
• The intensity matrix allows biologists to make
  correlations between different genes (even if
  they are
• dissimilar) and to understand how genes
  functions might be related
• Clustering comes into play

Time 1 Time i Time N
Gene 1 10 8 10
Gene 2 10 0 9
Gene 3 4 8.6 3
Gene 4 7 8 3
Gene 5 1 2 3
Intensity (expression level) of gene at measured
time
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Clustering of Microarray Data

• Plot each gene as a point in N-dimensional space
• Make a distance matrix for the distance between
  every two gene points in the N-dimensional space
• Genes with a small distance share the same
  expression characteristics and might be
  functionally related or similar
• Clustering reveals groups of functionally related
  genes

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Graphing the intensity matrix inmulti-dimensional
space
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The Distance Matrix, d
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Homogeneity and Separation Principles

• Homogeneity Elements within a cluster are close
  to each other
• Separation Elements in different clusters are
  further apart from each other
• clustering is not an easy task!

Given these points a clustering algorithm might
make two distinct clusters as follows
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Bad Clustering
This clustering violates both Homogeneity and
Separation principles
Close distances from points in separate clusters
Far distances from points in the same cluster
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Good Clustering
This clustering satisfies both Homogeneity and
Separation principles
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Clustering Techniques

• Agglomerative Start with every element in its
  own cluster, and iteratively join clusters
  together
• Divisive Start with one cluster and iteratively
  divide it into smaller clusters
• Hierarchical Organize elements into a tree,
  leaves represent genes and the length of the
  paths between leaves represents the distances
  between genes. Similar genes lie within the same
  subtrees.

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Hierarchical Clustering

• Initially, each element is its own cluster
• Merge the two closest clusters, and recurse
• Key question What is closest?
• How do you compute the distance between clusters?

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Hierarchical Clustering Computing Distances

• dmin(C, C) min d(x,y) for all
  elements x in C and y in C
• Distance between two clusters is the smallest
  distance between any pair of their elements
• davg(C, C) (1 / CC) ? d(x,y) for all
  elements x in C
• and y in C
• Distance between two clusters is the average
  distance between all pairs of their elements

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Computing Distances (continued)

• However, we still need a base distance metric
  for pairs of gene
• Euclidean distance
• Manhattan distance
• Dot Product
• Mutual information

What are some qualitative differences between
these?
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Geometrical interpretation of distances

• The distance measures are all related.
• In some cases, the magnitude of the vector is
  important, in other cases it is not.

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Comparison between metrics

• Euclidean and Manhattan tend to perform similarly
  and emphasize the overall magnitude of
  expression.
• The dot-product is very useful if the shape of
  the expression vector is more important than its
  magnitude.
• The above metrics are less useful for identifying
  genes for which the expression levels are
  anti-correlated. One might imagine an instance in
  which the same transcription factor can cause
  both enhancement and repression of expression.
  In this case, the squared correlation (r2) or
  mutual information is sometimes used.

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But how many orderings can we have?
1
2
4
5
3
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• For n leaves there are n-1 internal nodes
• Each flip in an internal node creates a new
  linear ordering of the leaves
• There are therefore 2n-1 orderings

E.g., flip this node
1
2
4
5
3
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Bar-Joseph et al. Bioinformatics (2001)
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Computing an Optimal Ordering

• Define LT(u,v) as the optimum score of all
  orderings for the subtree rooted at T where
• u is the left node, and
• v is the right node
• Is it sufficient to compute LT(u,v) for all T,u,v
  ?

T
u
v
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T
T1
T2
v
m
k
u
LT(u,v) max k,m LT1(u,k) LT2(u,m)
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Time complexity of the algorithm?
T

• The recursion LT(u,w) is applied for each T,u,v.
  Each recursion takes O(n2) time.

• Each pair of nodes has a unique Least common
  ancestor.
• LT(u,w) only needs to be computed if LCA(u,w)
  T
• Total time O(n4)

w
u
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Speed Improvements

• For all m in LT1(u,R)
• If LT1(u,m)LT2(k0,w) C(T1,T2) lt CurrMax
• Exit loop
• For all k in LT1(w,L)
• If LT1(u,m)LT2(k,w)C(T1,T2) lt CurrMax
• Exit loop
• Else recompute CurrMax.
• In practice, this leads to great speed
  improvements
• 1500 genes, 7 hrs. changes to 7 min.

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