Transcript Independent Component Analysis.

Independent components
analysis of starch deficient
pgm mutants
GCB 2004
M. Scholz, Y. Gibon, M. Stitt, J. Selbig
Matthias Maneck - Journal Club WS 04/05
Overview
Introduction
 Methods

– Principal Component Analysis
 ICA – Independent Component Analysis
 Kurtosis
 PCA
Results
 Summary

Matthias Maneck - Journal Club WS 04/05
Introduction – techniques

visualization techniques
 supervised

biological background information
 unsupervised
present major global information
 General questions about the underlying data
structure.
 Detect relevant components independent from
background knowledge.

Matthias Maneck - Journal Club WS 04/05
Introduction – techniques

PCA
 dimensionality
reduction
 extracts relevant information related to the
highest variance

ICA
 Optimizes
independence condition
 Components represent different nonoverlapping information
Matthias Maneck - Journal Club WS 04/05
Introduction - experiments

Micro plate assays of
enzymes form
Arabidopsis thaliana.


data
j Samples
i Enzymes

pgm mutant vs. wild type
continuous night
Matthias Maneck - Journal Club WS 04/05
Introduction – workflow
Data
PCA
ICA
ICs
1st IC
j Samples
Matthias Maneck - Journal Club WS 04/05
j Samples
ICs
PC’s
j Samples
i Enzymes
Kurtosis
2nd IC
PCA – principal component analysis
4
3
2
Enzyme 2
1
0
-1
-2
-3
-4
-4
-3
-2
-1
0
Enzyme 1
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2
3
4
PCA – principal component analysis
4
3
2
1. Principal Component
Enzyme 2
1
0
-1
-2
2. Principal Component
-3
-4
-4
-3
-2
-1
0
Enzyme 1
Matthias Maneck - Journal Club WS 04/05
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2
3
4
PCA – principal component analysis
4
3
2
2. PC
1
0
-1
-2
-3
-4
-4
-3
-2
-1
0
1. PC
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2
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4
PCA – calculation
Eigenvalues
λ1
Cov-Matrix
- mean
- mean
i Enzymes
i Enzymes
- mean
...
i Enzymes
j Samples
...
Data-Matrix
λi
Eigenvectors
- mean
x1 ... ... xi
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PCA – dimensionality reduction
Selected Components
Data Matrix
Reduced Data Matrix
j Samples
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j Samples
=
PCs
i Enzymes
PCs
i Enzymes
PCA – principal component analysis
4
3
2
1. Principal Component
Enzyme 2
1
0
-1
-2
2. Principal Component
-3
-4
-4
-3
-2
-1
0
Enzyme 1
Matthias Maneck - Journal Club WS 04/05
1
2
3
4
PCA – principal component analysis
4
3
2
1
0
-1
-2
-3
-4
-4
-3
-2
-1
0
1. PC
Matthias Maneck - Journal Club WS 04/05
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2
3
4
PCA – principal component analysis




Minimizes correlation between components.
Components are orthogonal to each other.
Delivers transformation matrix, that gives the influence of
the enzymes on the principal components.
PCs ordered by size of eigenvalues of cov-matrix
Reduced Data Matrix
Selected Components
Data Matrix
j Samples
PCs
PCs
=
Matthias Maneck - Journal Club WS 04/05
i Enzymes
i Enzymes
j Samples
ICA – independent component analysis

Mike 1
Person 1
Person 2
microphone signals are
mixed speech signals
x1 (t )  a11 s1 (t )  a12 s 2 (t )  a13 s3 (t )
x2 (t )  a 21 s1 (t )  a 22 s 2 (t )  a 23 s3 (t )
Mike 2
x3 (t )  a31 s1 (t )  a32 s 2 (t )  a33 s3 (t )
Person 3
Mike 3
Matthias Maneck - Journal Club WS 04/05
ICA – independent component analysis
Microphone Signals X
Mixing Matrix A
mixing
speech
Microphone signals X
Speech signals S
time t
time t
microphone
signals
demixing
speech
Matthias Maneck - Journal Club WS 04/05
=
speech
signals
Demixing matrix A-1
speech
signals
=
time t
microphone
microphone
signals
time t
speaker
Speech Signals S
ICA – independent component analysis
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The sum of distribution of the same
time is more Gaussian.
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25
20
15
60
10
35
5
30
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25
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ICA – independent component analysis


ICs

ICs
Demixing Matrix
Data Matrix
j Samples
PCs
j Samples
=
Matthias Maneck - Journal Club WS 04/05
PCs

Maximizes independence (non Gaussianity) between
components.
ICA doesn’t work with purely Gaussian distributed data.
Components are not orthogonal to each other.
Delivers transformation matrix, that gives the influence of the PCs
on the independent components.
ICs are unordered
ICs

Kurtosis – significant components

measure of non Gaussianity
n
4
– random variable (IC)
(
z


)

i
 μ – mean
kurtosis( z )  i 1
3
4
(n  1)
 σ – standard deviation
z

positive kurtosis  super Gaussian

negative kurtosis  sub Gaussian
Matthias Maneck - Journal Club WS 04/05
Kurtosis – significant components
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Influence Values

Which enzymes have most influence on ICs?
Reduced Data Matrix
Selected Components
Data Matrix
j Samples
i Enzymes
i Enzymes
PCs
=
Demixing Matrix
Data Matrix
j Samples
PCs
j Samples
=
Matthias Maneck - Journal Club WS 04/05
PCs
ICs
ICs
ICs
PCs
j Samples
Influence Values
Influence Matrix
Demixing Matrix
i Enzymes
Selected Components
ICs
i Enzymes
PCs
=
ICs
ICs
PCs
Influence Matrix
Data Matrix
i Enzymes
=
Matthias Maneck - Journal Club WS 04/05
ICs
ICs
j Samples
i Enzymes
j Samples
Results

pgm mutant
 compares
wild type and pgm mutant
 17 enzymes,125 samples


wild type, pgm mutant
continuous night
 response
to carbon starvation
 17 enzymes, 55 samples

+0, +2, +4, +8, +24, +48, +72, +148 h
Matthias Maneck - Journal Club WS 04/05
Results – pgm mutant
Matthias Maneck - Journal Club WS 04/05
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Results – continuous night
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Results – combined
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Results – combined
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Results – combined
Matthias Maneck - Journal Club WS 04/05
Summary





ICA in combination with PCA has higher
discriminating power than only PCA.
Kurtosis is used for selection optimal PCA
dimension and ordering of ICs.
pgm experiment, 1st IC discriminates between
mutant and wild type.
Continuous night, 2nd IC represents time
component.
The two most strongly implicated enzymes are
identical.
Matthias Maneck - Journal Club WS 04/05
References




Scholz M., Gibon Y., Stitt M., Selbig J.: Independent
components analysis of starch deficient pgm mutants.
Scholz M., Gatzek S., Sterling A., Fiehn O., Selbig J.:
Metabolite fingerprinting: an ICA approach.
Blaschke, T., Wiskott, L.: CuBICA: Independent
Component Analysis by Simultaneous Third- and FourthOrder Cumulant Diagonalization. IEEE Transactions on
Signal Processing, 52(5):1250-1256.
http://itb.biologie.hu-berlin.de/~blaschke/
Hyvärinen A., Karhunen J., Oja E.: Independent
Component Analysis. J. Wiley. 2001.
Matthias Maneck - Journal Club WS 04/05