Clustering microarray data

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Clustering microarray data

• 09/26/07

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Sub-classes of lung cancer types have signature
genes
(Bhattacharjee 2001)
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Promoter analysis of commonly regulated genes
David J. Lockhart Elizabeth A. Winzeler, NATURE
VOL 405 15 JUNE 2000, p827
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Discovery of new cancer subtype
These classes are unknown at the time of study.
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Overview

• Clustering is an unsupervised learning clustering
  is used to build groups of genes with related
  expression patterns.
• The classes are not known in advance.
• Aim is to discover new patterns from microarray
  data.
• In contrast, supervised learning refers to the
  learning process where classes are known. The aim
  is to define classification rules to separate the
  classes. Supervised learning will be discussed in
  the next lecture.

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Dissimilar function

• To identify clusters, we first need to define
  what close means. There are many choices of
  distances
• Euclidian distance
• 1 Pearson correlation
• Manhattan distance

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Where is the truth?

• In the context of unsupervised learning, there
  is no such direct measure of success. It is
  difficult to ascertain the validity of inference
  drawn from the output of most unsupervised
  learning algorithms. One must often resort to
  heuristic arguments not only for motivating the
  algorithm, but also for judgments as to the
  quality of results. This uncomfortable situation
  has led to heavy proliferation of proposed
  methods, since effectiveness is a matter of
  opinion and cannot be verified directly.

Hastie et al. 2001 ESL
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Clustering Methods

• Partitioning methods
• Seek to optimally divide objects into a fixed
  number of clusters.
• Hierarchical methods
• Produce a nested sequence of clusters

(Speed, Chapter 4)
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Methods

• k-means
• Hierarchical clustering
• Self-organizing maps (SOM)

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k-means

• Divide objects into k clusters.
• Goal is to minimize total intra-cluster variance
• Global minimum is difficult to obtain.

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Algorithm for k-means clustering

• Step 1 Initialization randomly select k
  centroids.
• Step 2 For each object, find its closest
  centroid, assign the object to the corresponding
  cluster.
• Step 3 For each cluster, update its centroid to
  the mean position of all objects in that cluster.
• Repeat Steps 2 and 3 until convergence.

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Shows the initial randomized centers and a number
of points.
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Centers have been associated with the points and
have been moved to the respective centroids.
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Now, the association is shown in more detail,
once the centroids have been moved.
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Again, the centers are moved to the centroids of
the corresponding associated points.
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Properties of k-means

• Achieves local minimum of
• Very fast.

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Practical issues with k-means

• k must be known in advance
• Results are dependent on initial assignment of
  centroids.

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How to choose k?
Milligan Cooper(1985) compared 30 published
rules. 1. Calinski Harabasz (1974) 2.
Hartigan (1975) , Stop when
H(k)lt10 .
W(k) total sum of squares within clusters B(k)
sum of squares between cluster means
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How to choose k (continued)?
Random
(Tibshriani 2001) Estimate log Wk for randomly
data (uniformly distributed in a
rectangle) Choose k so that Gap is largest.
Observed
log WK
Gap
k
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How to select initial centroids

• Repeat the procedure many times with randomly
  chosen initial centroids.
• Alternatively, initialize centroids smartly,
  e.g. by hierarchical clustering

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K-means requires good initial values.
Hierarchical Clustering could be used but
sometimes performs poorly.
with-in sum of Sq. X965.32 O305.09
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Hierarchical clustering
Hierarchical clustering builds a hierarchy of
clusters, represented by a tree (called a
dendrogram). Close clusters are joined together.
Height of a branch represents the dissimilarity
between the two clusters joined by it.
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How to construct a dendrogram

• Bottom-up approach
• Initialization each cluster contains a single
  object
• Iteration merge the closest clusters.
• Stop when all objects are included in a single
  cluster
• Top-down approach
• Starting from a single cluster containing all
  objects, iteratively partition into smaller
  clusters.
• Truncate dendrogram at a similarity threshold
  level, e.g., correlation gt 0.6 or requiring a
  cluster containing at least a minimum number of
  objects.

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Hierarchical Clustering
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Dendrogram can be reordered
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Ordered dendrograms

• 2 n-1 linear orderings of n elements
• (n genes or conditions)
• Maximizing adjacent similarity is impractical.
  So order by
• Average expression level,
• Time of max induction, or
• Chromosome positioning

Eisen98
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Properties of Hierarchical Clustering

• Top-down approach is more favorable when only a
  few clusters are desired.
• Single linkage tends to produce long chains of
  clusters.
• Complete linkage tends to produce compact
  clusters.

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Partitioning clustering vs hierarchical clustering
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k 4
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Partitioning clustering vs hierarchical clustering
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k 3
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Partitioning clustering vs hierarchical clustering
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k 2
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Self-organizing Map

• Impose partial structure on the clusters (in
  contrast to the rigid structure of hierarchical
  clustering, the strong prior hypotheses used in
  Bayesian clus-tering, and the nonstructure of
  k-means clustering)
• easy visualization and interpretation.

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SOM Algorithm

• Initialize prototypes mj on a lattice of p X q
  nodes. Each prototype is a weight vector whose
  dimension is the same as input data.
• Iteration for each observation xi, find the
  closest prototype mj, and for all neighbors of mk
  of mj, move by
• During iterations, reduce learning rate a and
  neighborhood size r gradually.
• May take many iterations before convergence.

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(Hastie 2001)
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(Hastie 2001)
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(Hastie 2001)
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SOM clustering of periodic genes
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Applications to microarray data
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• With only a few nodes, one tends not to see
  distinct patterns and there is large
  within-cluster scatter. As nodes are added,
  distinctive and tight clusters emerge.
• SOM is an incremental learning algorithm
  involving cases by case presentation rather than
  batch presentation.
• As with all exploratory data analysis tools, the
  use of SOMs involves inspection of the data to
  extract insights.

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Other Clustering Methods

• Gene Shaving
• MDS
• Affinity Propagation
• Spectral Clustering
• Two-way clustering

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• Algorithms for unsupervised classification or
  cluster analysis abound. Unfortunately however,
  algorithm development seems to be a preferred
  activity to algorithm evaluation among
  methodologists.
• No consensus or clear guidelines exist to guide
  these decisions. Cluster analysis always produces
  clustering, but whether a pattern observed in the
  sample data characterizes a pattern present in
  the population remains an open question.
  Resampling-based methods can address this last
  point, but results indicate that most clusterings
  in microarray data sets are unlikely to reflect
  reproducible patterns or patterns in the overall
  population.
• -Allison et al. (2006)

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Stability of a cluster

• Motivation Real clusters should be reproducible
  under perturbation adding noise, omission of
  data, etc.
• Procedure
• Perturb observed data by adding noise.
• Apply clustering procedure to cluster the
  perturbed data.
• Repeat the above procedures, generate a sample of
  clusters.
• Global test
• Cluster-specific tests R-index, D-index.

(McShane et al. 2002)
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Global test

• Null hypothesis Data come from a multivariate
  Gaussian distribution.
• Procedure
• Consider a subspace spanned by top principle
  components.
• Estimate distribution of nearest neighbor
  distances
• Compare observed with simulated data.

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R-index

• If cluster i contains ni objects, then it
  contains mi ni(ni 1)/2 of pairs.
• Let ci be the number of pairs that fall in the
  same cluster for the re-clustered perturbed data.
• ri ci/mi measures the robustness of the
  cluster i.
• R-index Si ci / Si mi measures overall
  stability of a clustering algorithm.

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D-index

• For each cluster, determine the closest cluster
  for the perturbed data
• Calculated the average discrepancy between the
  clusters for the original and perturbed data
  omission vs addition.
• D-index is a summation of all cluster-specific
  discrepancy.

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Applications

• 16 prostate cancer 9 benign tumor
• 6500 genes
• Use hierarchical clustering to obtain 2,3, and 4
  clusters.
• Questions are these clusters reliable?

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Issues with calculating R and D indices

• How big is the size of perturbation?
• How to quantify the significance level?
• What about nested consistency?

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Acknowldegment

• Slide sources from
• Cheng Li