1 CS/CBB 545 - Data Mining Spectral Methods (PCA,SVD) #2 - Application Mark Gerstein, Yale University gersteinlab.org/courses/545 (class 2007,03.08 14:30-15:45)
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CS/CBB 545 - Data MiningSpectral Methods (PCA,SVD)
#2 - Application
Mark Gerstein, Yale Universitygersteinlab.org/courses/545
(class 2007,03.08 14:30-15:45)
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Intuition on interpretation of SVD in terms of genes
and conditions
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SVD for microarray data(Alter et al, PNAS 2000)
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Notation• m=1000 genes
– row-vectors
– 10 eigengene (vi) of dimension 10 conditions
• n=10 conditions (assays)– column vectors
– 10 eigenconditions (ui) of dimension 1000 genes
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Understanding Eigengenes (vi) in terms PCA on (large) gene-gene correlation matrix
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Understanding Eigenconditions (ui) in terms of PCA on (small) condition-condition correlation matrix
Bra - ket notation
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Plotting Experiments in Low Dimension Subspace
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Close up on Eigengenes
10Copyright ©2000 by the National Academy of Sciences
Alter, Orly et al. (2000) Proc. Natl. Acad. Sci. USA 97, 10101-10106
Genes sorted by correlation with top 2 eigengenes
11Copyright ©2000 by the National Academy of Sciences
Alter, Orly et al. (2000) Proc. Natl. Acad. Sci. USA 97, 10101-10106
Same thing different experiment: Genes sorted by relative correlation with first two eigengenes for alpha-factor experiment
12Copyright ©2000 by the National Academy of Sciences
Alter, Orly et al. (2000) Proc. Natl. Acad. Sci. USA 97, 10101-10106
Normalized elutriation
expression in the subspace
associated with the cell cycle
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Biplot Applied to Genes and Conditions
See grouping of arrays and genes on same plot
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Spectral Biclustering
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Biclustering to associate particular genes with certain phenotypes
Conditions
Reo
rder
ed G
enes
(Sor
ted
acco
rdin
g to
a cl
assi
ficat
ion
vect
or)
?
Matrix of raw data
Gen
es
Reordered Conditions(Sorted according to
a classification vector)
Shuffled Matrix(containing checkerboard
“biclusters” of conditions with marker genes)
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Pomeroy et. al. , Nature 415 (2002) 436Prediction of central nervous system embryonal tumor outcome based on gene expression
5 types of brain tumors
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Intuition on Identification of Blocky Matrices
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6 6 6 6
6 6
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Ax yGene partition vector
tumor 1 tumor 2
Gen
e cl
uste
r 1
Gen
e cl
uste
r 2
tumor 3
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5
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'TA y x
Tissue partition vector
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6 6 6 6
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'TA Ax x
Ax y 'TA y x
Biclustering by SVD
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6 6 6 6
6 6
4 4 4 4
4 4 4 4
5 5 5
5 5 56 6
6 6 6
8 8 8 8
8 8 8 8
8 8 8
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7
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Identify checkerboard matrices by their action
on classification vectors: Formulation as “eigenproblem”
Checkerboard Matrix A
Condition Classification Vect. x
Conditions
Gen
es
Gene Classification Vector y
A A x = x’T
A A y = y’T
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4 4 4
4 4 4
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Genes
Con
ditio
ns
x’
yAT
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SVD to Solve Eigenproblem
[Botstein]
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Yuval Kluger et al. Genome Res. 2003; 13: 703-716
Figure 1. Overview of important parts of the biclustering process
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4 8 9 3
8 5
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Gene partition with noisy data
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1R Ax y
Normalization Rescales Rows and Columns to Same Means
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.18 .09 .0.16 .32 .16
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1 'TC A y x Rescale columns
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Representative Cancer Data set
• Lymphoma Data from Dalla-Favera et al. at Columbia
• Informatics from Stolovitzky & Califano at IBM
• Supervised learning some identified characteristic genes associated with different types of lymphoma
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Patients (samples) sorted according to projection onto blocky classification eigenvector
(u2)
Gen
es s
orte
d ac
cord
ing
to
proj
ectio
n on
to b
lock
y cl
assi
ficat
ion
eige
nvec
tor
(v2)
Matrix values represent outer products of two blocky
classification eigenvectors
Results on Representative Cancer Data set
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Actual Data with Normalization and Sorting
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Actual Data just with Sorting
(no normalization)
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Actual Data (no normalization
or sorting)
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Actual Data just with Sorting
(no normalization)
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Actual Data with Normalization and Sorting
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Patients (samples) sorted according to projection onto blocky classification eigenvector
(u2)
Gen
es s
orte
d ac
cord
ing
to
proj
ectio
n on
to b
lock
y cl
assi
ficat
ion
eige
nvec
tor
(v2)
Matrix values represent outer products of two blocky
classification eigenvectors
Just signal from top classification
eigenvectors
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Low Dimension Representation
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Patients (samples) sorted according to projection onto blocky classification eigenvector
(u2)
Gen
es s
orte
d ac
cord
ing
to
proj
ectio
n on
to b
lock
y cl
assi
ficat
ion
eige
nvec
tor
(v2)
Actual Values of Projections onto
Classification Eigenvectors
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Classification of Cancers Based on Projection onto two top classification
eigenvectors: Better with Normalization
Normalized (“bistochastization”)
CLL DLCLFL DLCL
Straight SVD
Four types of Cancer in Della Favera dataset
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Golub, TR et. al., Molecular classification of cancer: Class discovery and class prediction by gene expression monitoring. Science, 1999 286
biclustering bistochastization
SVD bi-normalization Normalized cuts
ALL (B) ALL (T)AML
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