Transcript Principal Component Analysis for Clustering Gene Expression Data

Principal Component Analysis
(PCA) for Clustering
Gene Expression Data
K. Y. Yeung and W. L. Ruzzo
Outline
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Motivation of PCA and this paper
Approach of this paper
Data sets
Clustering algorithms and similarity metrics
Results and discussion
Why Are We Interested in 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.
What Is This Paper for?
• A theoretical result shows that the first few
PCs may not contain cluster information.
(Chang, 1983).
• Chang’s example.
• A motivating example. (Coming next).
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)
Sporulation Data
• The patterns overlap around the origin in (1a).
• The patterns are much more separated in (1b).
The Goal
• EMPIRICALLY investigate the
effectiveness of clustering gene expression
data using PCs instead of the original
variables.
The Setting
• Genes are to be clustered, and the
experimental conditions are the variables.
• Assume there is an objective external
criterion to cluster the data. The quality of
clustering is measured by its distance to
this gold standard here.
• Assume the number of clusters is known.
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
 n
Rand index = (a+d)/  
 2
Agreement (Cont’d)
The adjusted Rand index (ARI, Hubert &
Arabie, 1985):
index  expected index
maximum index - expected index
Note: Higher ARI means higher
correspondence between two partitions.
Subset of PCs
• Motivated by Chang’s 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.
The Greedy Approach
Let m0 be the minimum number of PCs to be
clustered, and p be the number of
variables in the data.
1) Search for a set of m0 PCs with maximum
ARI, denoted as sm .
2) For each m=(m0+1),…p, add another PC
to s(m-1) and calculate ARI. The PC giving
the maximum ARI is then added to get sm.
0
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 = m+1).
• If k=1, this is just the greedy approach.
• k=3 in this paper.
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
1) the given data set.
2) the first m PCs where m=m0,…p.
3) the subset of PCs found by the (modified)
greedy approach.
4) 30 sets of random PCs.
5) 30 sets of random orthogonal projections.
Data Sets
• “Class” refers to a group in the gold
standard. “Cluster” refers to clusters
obtained by a clustering algorithm.
• There are two real data sets and three
synthetic data sets in this study.
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.
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.
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.
Marginal Normality
Randomly Resample Ovary Data
• For each class c (c=1,…,4) under
experimental condition j (j=1,…,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.
Cyclic Data
• This data set models cyclic behavior of
genes over different time points.
• It uses sin function and normal random
numbers.
• A drawback of this model is the arbitrary
choice of several parameters.
Clustering Algorithms
and Similarity Metrics
Clustering algorithms:
– Cluster affinity search technique (CAST)
– Hierarchical average-link algorithm
– K-mean algorithm
Similarity metrics:
– Euclidean distance (m0=2)
– Correlation coefficient (m0=3)
Table 1
Table 2
• One sided Wilcoxon signed rank test.
• CAST always favorites ‘no PCA’.
• The two significances for PCA are not clear sucesses.
Conclusion
1) The quality of clustering results on the
data after PCA is not necessarily higher
than that on the original data, sometimes
lower.
2) The first m PCs do not give the highest
adjusted Rand index, i.E. Another set of
PCs gives higher ARI.
Conclusion (Cont’d)
3) 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.
Conclusion (Cont’d)
4) 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.
Grand Conclusion
In general, we recommend AGAINST using
PCA to reduce dimensionality of the data
before applying clustering algorithms unless
external information is available.