Principal Component Analysis PCA for Clustering Gene Expression Data

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Principal Component Analysis(PCA) for
ClusteringGene Expression Data

• K. Y. Yeung and W. L. Ruzzo

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Organization

• Association of PCA and this paper
• Approach of this paper
• Data sets
• Clustering algorithms and similarity metrics
• Results and discussion

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The Functions of PCA?

• PCA can reduce the dimensionality of the data
  set.
• Few PCs may capture most of the variation in the
  original data set.
• PCs are uncorrelated and ordered.
• We expect the first few PCs may extract the
  cluster structure in the original data set.

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This Papers Point of View

• A theoretical result shows that the first few PCs
  may not contain cluster information. (Chang,
  1983).
• Changs example.
• A motivating example. (Coming next).

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A Motivating Example

• Data A subset of the sporulation data (477
  genes) were classified into seven temporal
  patterns (Chu et al., 1998)
• The first 2 PCs contains 85.9 of the variation
  in the data. (Figure 1a)
• The first 3 PCs contains 93.2 of the variation
  in the data. (Figure 1b)

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Sporulation Data

• The patterns overlap around the origin in (1a).
• The patterns are much more separated in (1b).

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The Goal

• EMPIRICALLY investigate the effectiveness of
  clustering gene expression data using PCs instead
  of the original variables.

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Outline of Methods

• Genes are to be clustered, and the experimental
  conditions are the variables.
• Effectiveness of clustering with the orginal data
  and with different sets of PCs is determined,
  measured by comparing the clustering results to
  an objective external criterion.
• Assume the number of clusters is known.

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Agreement Between Two Partitions

• The Rand index (Rand, 1971)
• Given a set of n objects S, let U and V be two
  different partitions of S. Let
• a of pairs that are placed in the same
  cluster in U and in the same cluster in V
• d of pairs that are placed in different
  clusters in U and in different clusters in V
• Rand index (ad)/nC2

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Agreement (Contd)

• The adjusted Rand index (ARI, Hubert Arabie,
  1985)
• Note Higher ARI means higher correspondence
  between two partitions.

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Subset of PCs

• Motivated by Changs example, it is possible to
  find other subsets of PCs to preserve the cluster
  structure better than the first few PCs.
• How?
• --- The greedy approach.
• --- The modified greedy approach.

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The Greedy Approach

• Let m0 be the minimum number of PCs to be
  clustered, and p be the number of variables in
  the data.
• Search for a set of m0 PCs with maximum ARI,
  denoted as sm0.
• For each m(m01),p, add another PC to s(m-1)
  and calculate ARI. The PC giving the maximum ARI
  is then added to get sm.

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The Modified Greedy Approach

• In each step of the greedy approach ( of PCs
  m), retain the k best subsets of PCs for the next
  step ( of PCs m1).
• If k1, this is just the greedy approach.
• k3 in this paper.

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The Scheme of the Study

• Given a gene expression data set with n genes
  (subjects) and p experimental conditions
  (variables), apply a clustering algorithm to
• the given data set, ARI w/ external criterion.
• the first m PCs where mm0,p.
• the subset of PCs found by the (modified) greedy
  approach.
• 30 sets of random PCs.
• 30 sets of random orthogonal projections.

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Data Sets

• Class refers to a group in the external
  criterion. Cluster refers to clusters obtained
  by a clustering algorithm.
• There are two real data sets and three synthetic
  data sets in this study.

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The Ovary Data

• The data contains 235 clones and 24 tissue
  samples.
• For the 24 tissue samples, 7 are from normal
  tissues, 4 from blood samples, and 13 from
  ovarian cancers.
• The 235 clones were found to correspond four
  different genes (classes), each having 58, 88, 57
  and 32 clones.
• The data for each clone was normalized across the
  24 experiments to have mean 0 and variance 1.

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The Yeast Cell Cycle Data

• The data set shows the fluctuation of expression
  levels over two cell cycles.
• 380 gene were classified into five phases
  (classes).
• The data for each gene was normalized to have
  mean 0 and variance 1 across each cell cycle.

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Mixture of Normal on Ovary

• In each gene (class), the sample covariance
  matrix and the mean vector are computed.
• Sample (58, 88, 57, 32) clones from the MVN in
  each class. 10 replicates.
• It preserves the mean and covariance of the
  original data, but relies on the MVN assumption.

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Marginal Normality
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Randomly Resample Ovary Data

• For each class c (c1,,4) under experimental
  condition j (j1,,24), resample the expression
  level with replacement. Retain the size of each
  class. 10 replicates.
• No MVN assumption. The independent sampling for
  different experimental conditions is reasonable
  as inspected.

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Cyclic Data

• This data set models cyclic behavior of genes
  over different time points.
• Behavior of genes modeled by the sine function.
• A drawback of this model is the arbitrary choice
  of several parameters.

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Clustering Algorithmsand Similarity Metrics

• Clustering algorithms
• Cluster affinity search technique (CAST)
• Hierarchical average-link algorithm
• K-mean algorithm
• Similarity metrics
• Euclidean distance (m02)
• Correlation coefficient (m03)

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Table 1
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Table 2

• One sided Wilcoxon signed rank test.
• CAST always favorites no PCA.
• The two significances for PCA are not clear
  sucesses.

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Conclusion

• The quality of clustering results on the data
  after PCA is not necessarily higher than that on
  the original data, sometimes lower.
• The first m PCs do not give the highest adjusted
  Rand index, i.E. Another set of PCs gives higher
  ARI.

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Conclusion (Contd)

• There are no clear trends regarding the choice of
  optimal number of PCs over all the data sets and
  over all the clustering algorithms and over the
  different similarity metrics. There is no obvious
  relationship between cluster quality and the
  number or set of PCs used.

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Conclusion (Contd)

• On average, the quality of clusters obtained by
  clustering random sets of PCs tend to be slightly
  lower than those obtained by clustering random
  sets of orthogonal projections, esp. when the
  number of components is small.

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Grand Conclusion

• In general, we recommend AGAINST using PCA to
  reduce dimensionality of the data before applying
  clustering algorithms unless external information
  is available.