Inferring networks from multiple samples with consensus LASSO
description
Transcript of Inferring networks from multiple samples with consensus LASSO
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Joint network inference with the consensualLASSO
Nathalie Villa-VialaneixJoint work with Matthieu Vignes, Nathalie Viguerie
and Magali San Cristobal
Séminaire de Statistique de Montpellier6 octobre 2014
http://www.nathalievilla.org
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Outline
1 Short overview on biological background
2 Network inference and GGM
3 Inference with multiple samples
4 Simulations
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
DNA
DNA (DeoxyriboNucleic Acid (DNA)
molecule that encodes the genetic instructions used in the developmentand functioning of all known living organisms and many virusesdouble helix made with only four nucleotides (Adenine, Cytosine,Thymine, Guanine): A binds with T and C with G.
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Transcription
Transcription of DNA
transcription is the process by which a particular segment of DNA (calledgene) is copied into a single-strand RNA (which is called message RNA)
A is transcripted into U, T into A, Cinto G and G into C
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
What is mRNA used for?
mRNA then moves out of the cell nucleus and is translated into proteinswhich are made of 20 different amino-acids (using an alphabet: 3 lettersof mRNA→ 1 amino-acid)
Proteins are used by the cell to function.
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Gene expression
In a given cell, at a given time, all the genes are not translated or nottranslated at the same level.
Gene expression
is a process by which a given gene is transcripted or/and traductedIt depends on the type of cell, the environment, ...
Gene expression can be measured either by:
“counting” the number of copies (mRNA) of a given gene in the cell:transcriptomic data;
“counting” the quantity of a given protein in the cell: proteomic data.
Even though a given mRNA is translated into a unique protein, there isno simple relationship between transcriptomic and proteomic data.
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Gene expression
In a given cell, at a given time, all the genes are not translated or nottranslated at the same level.
Gene expression
is a process by which a given gene is transcripted or/and traductedIt depends on the type of cell, the environment, ...
Gene expression can be measured either by:
“counting” the number of copies (mRNA) of a given gene in the cell:transcriptomic data;
“counting” the quantity of a given protein in the cell: proteomic data.
Even though a given mRNA is translated into a unique protein, there isno simple relationship between transcriptomic and proteomic data.
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Transcriptomic data
How are transcriptomic data obtained?
DNA spots associated to targetgenes (probes) are attached to asolid surface (array)
When the binding between mRNAand the probes is good, the spotbecomes fluorescent.
Expression of a given gene is quantified by the intensity of thefluorescent signal (read by a scanner).
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Transcriptomic data
How are transcriptomic data obtained?
RNA material extracting for cells ofinterest (blood, lipid tissue, muscle,urine...) are labeled fluorescentlyand applied on the array
When the binding between mRNAand the probes is good, the spotbecomes fluorescent.
Expression of a given gene is quantified by the intensity of thefluorescent signal (read by a scanner).
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Typical microarray data
Data: large scale gene expression data
individualsn ' 30/50
X =
. . . . . .
. . X ji . . .
. . . . . .
︸ ︷︷ ︸variables (genes expression), p'103/4
Typical design: two (or more) conditions (treated/control for instance)More and more complicated designs: crossed conditions (treated/controland different breeds), longitudinal data...
Typical issues: find differentially expressed genes (i.e., genes whoseexpression is significantly different between the conditions)
Nathalie Villa-Vialaneix | Consensus Lasso 8/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Typical microarray data
Data: large scale gene expression data
individualsn ' 30/50
X =
. . . . . .
. . X ji . . .
. . . . . .
︸ ︷︷ ︸variables (genes expression), p'103/4
Typical design: two (or more) conditions (treated/control for instance)More and more complicated designs: crossed conditions (treated/controland different breeds), longitudinal data...
Typical issues: find differentially expressed genes (i.e., genes whoseexpression is significantly different between the conditions)
Nathalie Villa-Vialaneix | Consensus Lasso 8/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Typical microarray data
Data: large scale gene expression data
individualsn ' 30/50
X =
. . . . . .
. . X ji . . .
. . . . . .
︸ ︷︷ ︸variables (genes expression), p'103/4
Typical design: two (or more) conditions (treated/control for instance)More and more complicated designs: crossed conditions (treated/controland different breeds), longitudinal data...
Typical issues: find differentially expressed genes (i.e., genes whoseexpression is significantly different between the conditions)
Nathalie Villa-Vialaneix | Consensus Lasso 8/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Outline
1 Short overview on biological background
2 Network inference and GGM
3 Inference with multiple samples
4 Simulations
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Systems biology
Instead of being used to produce proteins, some genes’ expressionsactivate or repress other genes’ expressions⇒ understanding the wholecascade helps to comprehend the global functioning of living organisms1
1Picture taken from: Abdollahi A et al., PNAS 2007, 104:12890-12895. c© 2007 byNational Academy of Sciences
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Model framework
Data: large scale gene expression data
individualsn ' 30/50
X =
. . . . . .
. . X ji . . .
. . . . . .
︸ ︷︷ ︸variables (genes expression), p'103/4
What we want to obtain: a graph/network with
nodes: (selected) genes;
edges: strong links between gene expressions.
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Advantages of network inference
1 over raw data: focuses on the strongest direct relationships:irrelevant or indirect relations are removed (more robust) and thedata are easier to visualize and understand (track transcriptionrelations).
Expression data are analyzed all together and not by pairs (systemsmodel).
2 over bibliographic network: can handle interactions with yetunknown (not annotated) genes and deal with data collected in aparticular condition.
Nathalie Villa-Vialaneix | Consensus Lasso 12/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Advantages of network inference
1 over raw data: focuses on the strongest direct relationships:irrelevant or indirect relations are removed (more robust) and thedata are easier to visualize and understand (track transcriptionrelations).Expression data are analyzed all together and not by pairs (systemsmodel).
2 over bibliographic network: can handle interactions with yetunknown (not annotated) genes and deal with data collected in aparticular condition.
Nathalie Villa-Vialaneix | Consensus Lasso 12/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Advantages of network inference
1 over raw data: focuses on the strongest direct relationships:irrelevant or indirect relations are removed (more robust) and thedata are easier to visualize and understand (track transcriptionrelations).Expression data are analyzed all together and not by pairs (systemsmodel).
2 over bibliographic network: can handle interactions with yetunknown (not annotated) genes and deal with data collected in aparticular condition.
Nathalie Villa-Vialaneix | Consensus Lasso 12/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Using correlationsRelevance network [Butte and Kohane, 1999]
First (naive) approach: calculate correlations between expressions for allpairs of genes, threshold the smallest ones and build the network.
Correlations Thresholding Graph
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Using partial correlations
strong indirect correlationy z
x
set.seed(2807); x <- rnorm(100)
y <- 2*x+1+rnorm(100,0,0.1); cor(x,y) [1] 0.998826
z <- 2*x+1+rnorm(100,0,0.1); cor(x,z) [1] 0.998751
cor(y,z) [1] 0.9971105
] Partial correlation
cor(lm(x∼z)$residuals,lm(y∼z)$residuals) [1] 0.7801174
cor(lm(x∼y)$residuals,lm(z∼y)$residuals) [1] 0.7639094
cor(lm(y∼x)$residuals,lm(z∼x)$residuals) [1] -0.1933699
Nathalie Villa-Vialaneix | Consensus Lasso 14/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Using partial correlations
strong indirect correlationy z
x
set.seed(2807); x <- rnorm(100)
y <- 2*x+1+rnorm(100,0,0.1); cor(x,y) [1] 0.998826
z <- 2*x+1+rnorm(100,0,0.1); cor(x,z) [1] 0.998751
cor(y,z) [1] 0.9971105
] Partial correlation
cor(lm(x∼z)$residuals,lm(y∼z)$residuals) [1] 0.7801174
cor(lm(x∼y)$residuals,lm(z∼y)$residuals) [1] 0.7639094
cor(lm(y∼x)$residuals,lm(z∼x)$residuals) [1] -0.1933699
Nathalie Villa-Vialaneix | Consensus Lasso 14/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Using partial correlations
strong indirect correlationy z
x
set.seed(2807); x <- rnorm(100)
y <- 2*x+1+rnorm(100,0,0.1); cor(x,y) [1] 0.998826
z <- 2*x+1+rnorm(100,0,0.1); cor(x,z) [1] 0.998751
cor(y,z) [1] 0.9971105
] Partial correlation
cor(lm(x∼z)$residuals,lm(y∼z)$residuals) [1] 0.7801174
cor(lm(x∼y)$residuals,lm(z∼y)$residuals) [1] 0.7639094
cor(lm(y∼x)$residuals,lm(z∼x)$residuals) [1] -0.1933699
Nathalie Villa-Vialaneix | Consensus Lasso 14/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Partial correlation and GGM
(Xi)i=1,...,n are i.i.d. Gaussian random variables N(0,Σ) (geneexpression); then
j ←→ j′(genes j and j′ are linked)⇔ Cor(X j ,X j′ |(Xk )k,j,j′
), 0
If (concentration matrix) S = Σ−1,
Cor(X j ,X j′ |(Xk )k,j,j′
)= −
Sjj′√SjjSj′ j′
⇒ Estimate Σ−1 to unravel the graph structure
Problem: Σ: p-dimensional matrix and n � p ⇒ (Σ̂n)−1 is a poorestimate of S)!
Nathalie Villa-Vialaneix | Consensus Lasso 15/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Partial correlation and GGM
(Xi)i=1,...,n are i.i.d. Gaussian random variables N(0,Σ) (geneexpression); then
j ←→ j′(genes j and j′ are linked)⇔ Cor(X j ,X j′ |(Xk )k,j,j′
), 0
If (concentration matrix) S = Σ−1,
Cor(X j ,X j′ |(Xk )k,j,j′
)= −
Sjj′√SjjSj′ j′
⇒ Estimate Σ−1 to unravel the graph structure
Problem: Σ: p-dimensional matrix and n � p ⇒ (Σ̂n)−1 is a poorestimate of S)!
Nathalie Villa-Vialaneix | Consensus Lasso 15/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Partial correlation and GGM
(Xi)i=1,...,n are i.i.d. Gaussian random variables N(0,Σ) (geneexpression); then
j ←→ j′(genes j and j′ are linked)⇔ Cor(X j ,X j′ |(Xk )k,j,j′
), 0
If (concentration matrix) S = Σ−1,
Cor(X j ,X j′ |(Xk )k,j,j′
)= −
Sjj′√SjjSj′ j′
⇒ Estimate Σ−1 to unravel the graph structure
Problem: Σ: p-dimensional matrix and n � p ⇒ (Σ̂n)−1 is a poorestimate of S)!
Nathalie Villa-Vialaneix | Consensus Lasso 15/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Various approaches for inferringnetworks with GGM
Graphical Gaussian Model
seminal work:[Schäfer and Strimmer, 2005a, Schäfer and Strimmer, 2005b](with shrinkage and a proposal for a Bayesian test of significance)
estimate Σ−1 by (Σ̂n + λI)−1
use a Bayesian test to test which coefficients are significantly non zero.
sparse approaches:[Meinshausen and Bühlmann, 2006, Friedman et al., 2008]:
∀ j, estimate the linear model:
with ‖βj‖L1 =∑
j′ |βjj′ |
L1 penalty yields to βjj′ = 0 for most j′ (variable selection)
Nathalie Villa-Vialaneix | Consensus Lasso 16/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Various approaches for inferringnetworks with GGM
Graphical Gaussian Model
seminal work:[Schäfer and Strimmer, 2005a, Schäfer and Strimmer, 2005b](with shrinkage and a proposal for a Bayesian test of significance)
estimate Σ−1 by (Σ̂n + λI)−1
use a Bayesian test to test which coefficients are significantly non zero.
relation with LM
sparse approaches:[Meinshausen and Bühlmann, 2006, Friedman et al., 2008]:
∀ j, estimate the linear model:
X j = βTj X−j + ε ; arg max
(βjj′ )j′(log MLj)
because βjj′ = −Sjj′
Sjj.
with ‖βj‖L1 =∑
j′ |βjj′ |
L1 penalty yields to βjj′ = 0 for most j′ (variable selection)
Nathalie Villa-Vialaneix | Consensus Lasso 16/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Various approaches for inferringnetworks with GGM
Graphical Gaussian Model
seminal work:[Schäfer and Strimmer, 2005a, Schäfer and Strimmer, 2005b](with shrinkage and a proposal for a Bayesian test of significance)
estimate Σ−1 by (Σ̂n + λI)−1
use a Bayesian test to test which coefficients are significantly non zero.
relation with LM
sparse approaches:[Meinshausen and Bühlmann, 2006, Friedman et al., 2008]:
∀ j, estimate the linear model:
X j = βTj X−j + ε ; arg min
(βjj′ )j′
n∑i=1
(Xij − β
Tj X−j
i
)2
because βjj′ = −Sjj′
Sjj.
with ‖βj‖L1 =∑
j′ |βjj′ |
L1 penalty yields to βjj′ = 0 for most j′ (variable selection)
Nathalie Villa-Vialaneix | Consensus Lasso 16/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Various approaches for inferringnetworks with GGM
Graphical Gaussian Model
seminal work:[Schäfer and Strimmer, 2005a, Schäfer and Strimmer, 2005b](with shrinkage and a proposal for a Bayesian test of significance)
estimate Σ−1 by (Σ̂n + λI)−1
use a Bayesian test to test which coefficients are significantly non zero.
sparse approaches:[Meinshausen and Bühlmann, 2006, Friedman et al., 2008]:∀ j, estimate the linear model:
X j = βTj X−j + ε ; arg min
(βjj′ )j′
n∑i=1
(Xij − β
Tj X−j
i
)2+λ‖βj‖L1
with ‖βj‖L1 =∑
j′ |βjj′ |
L1 penalty yields to βjj′ = 0 for most j′ (variable selection)
Nathalie Villa-Vialaneix | Consensus Lasso 16/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Outline
1 Short overview on biological background
2 Network inference and GGM
3 Inference with multiple samples
4 Simulations
Nathalie Villa-Vialaneix | Consensus Lasso 17/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Motivation for multiple networksinference
Pan-European project Diogenes2 (with Nathalie Viguerie, INSERM):gene expressions (lipid tissues) from 204 obese women before and aftera low-calorie diet (LCD).
Assumption: A commonfunctioning existsregardless thecondition;
Which genes are linkedindependentlyfrom/depending on thecondition?
2http://www.diogenes-eu.org/; see also [Viguerie et al., 2012]
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Naive approach: independentestimations
Notations: p genes measured in k samples, each corresponding to aspecific condition: (Xc
j )j=1,...,p ∼ N(0,Σc), for c = 1, . . . , k .For c = 1, . . . , k , nc independent observations (Xc
ij )i=1,...,nc and∑c nc = n.
Independent inference
Estimation ∀ c = 1, . . . , k and ∀ j = 1, . . . , p,
Xcj = Xc
\j βcj + εc
j
are estimated (independently) by maximizing pseudo-likelihood:
L(S |X) =k∑
c=1
p∑j=1
nc∑i=1
logP(Xc
ij |Xci,\j , Sc
j
)
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Related papers
Problem: previous estimation does not use the fact that the differentnetworks should be somehow alike!Previous proposals
[Chiquet et al., 2011] replace Σc by Σ̃c = 12 Σc + 1
2 Σ and add asparse penalty;
[Chiquet et al., 2011] LASSO and Group-LASSO type penalties toforce identical or sign-coherent edges between conditions[Danaher et al., 2013] add the penalty
∑c,c′ ‖Sc − Sc′‖L1 ⇒ (sparse
penalty over the concentration matrix entries forces to strongconsistency between conditions);[Mohan et al., 2012] add a group-LASSO like penalty∑
c,c′∑
j ‖Scj − Sc′
j ‖L2 that focuses on differences due to a fewnumber of nodes only.
Nathalie Villa-Vialaneix | Consensus Lasso 20/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Related papers
Problem: previous estimation does not use the fact that the differentnetworks should be somehow alike!Previous proposals
[Chiquet et al., 2011] replace Σc by Σ̃c = 12 Σc + 1
2 Σ and add asparse penalty;[Chiquet et al., 2011] LASSO and Group-LASSO type penalties toforce identical or sign-coherent edges between conditions:∑
jj′
√∑c
(Scjj′)
2 or∑
jj′
√∑
c
(Scjj′)
2+ +
√∑c
(Scjj′)
2−
⇒ Sc
jj′ = 0 ∀ c for most entries OR Scjj′ can only be of a given sign
(positive or negative) whatever c
[Danaher et al., 2013] add the penalty∑
c,c′ ‖Sc − Sc′‖L1 ⇒ (sparsepenalty over the concentration matrix entries forces to strongconsistency between conditions);[Mohan et al., 2012] add a group-LASSO like penalty∑
c,c′∑
j ‖Scj − Sc′
j ‖L2 that focuses on differences due to a fewnumber of nodes only.
Nathalie Villa-Vialaneix | Consensus Lasso 20/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Related papers
Problem: previous estimation does not use the fact that the differentnetworks should be somehow alike!Previous proposals
[Chiquet et al., 2011] replace Σc by Σ̃c = 12 Σc + 1
2 Σ and add asparse penalty;[Chiquet et al., 2011] LASSO and Group-LASSO type penalties toforce identical or sign-coherent edges between conditions[Danaher et al., 2013] add the penalty
∑c,c′ ‖Sc − Sc′‖L1 ⇒ (sparse
penalty over the concentration matrix entries forces to strongconsistency between conditions);
[Mohan et al., 2012] add a group-LASSO like penalty∑c,c′
∑j ‖Sc
j − Sc′j ‖L2 that focuses on differences due to a few
number of nodes only.
Nathalie Villa-Vialaneix | Consensus Lasso 20/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Related papers
Problem: previous estimation does not use the fact that the differentnetworks should be somehow alike!Previous proposals
[Chiquet et al., 2011] replace Σc by Σ̃c = 12 Σc + 1
2 Σ and add asparse penalty;[Chiquet et al., 2011] LASSO and Group-LASSO type penalties toforce identical or sign-coherent edges between conditions[Danaher et al., 2013] add the penalty
∑c,c′ ‖Sc − Sc′‖L1 ⇒ (sparse
penalty over the concentration matrix entries forces to strongconsistency between conditions);[Mohan et al., 2012] add a group-LASSO like penalty∑
c,c′∑
j ‖Scj − Sc′
j ‖L2 that focuses on differences due to a fewnumber of nodes only.
Nathalie Villa-Vialaneix | Consensus Lasso 20/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Consensus LASSO
Proposal
Infer multiple networks by forcing them toward a consensual network: i.e.,explicitly keeping the differences between conditions under control butwith a L2 penalty (allow for more differences than Group-LASSO typepenalties).
Original optimization:
max(βc
jk )k,j,c=1,...,C
∑c
log MLcj − λ
∑k,j
|βcjk |
.
Add a constraint to force inference toward a “consensus” βcons
12βT
j Σ̂\j\jβj + βTj Σ̂j\j + λ‖βj‖L1 + µ
∑c
wc‖βcj − β
consj ‖2L2
with:wc : real number used to weight the conditions;µ regularization parameter;βcons
j whatever you want...?
Nathalie Villa-Vialaneix | Consensus Lasso 21/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Consensus LASSO
Proposal
Infer multiple networks by forcing them toward a consensual network: i.e.,explicitly keeping the differences between conditions under control butwith a L2 penalty (allow for more differences than Group-LASSO typepenalties).
Original optimization:
max(βc
jk )k,j,c=1,...,C
∑c
log MLcj − λ
∑k,j
|βcjk |
.[Ambroise et al., 2009, Chiquet et al., 2011]: is equivalent to minimizep problems having dimension k(p − 1):
12βT
j Σ̂\j\jβj + βTj Σ̂j\j + λ‖βj‖L1 , βj = (β1
j , . . . , βkj )
with Σ̂\j\j : block diagonal matrix Diag(Σ̂1\j\j , . . . , Σ̂
k\j\j
)and similarly for Σ̂j\j .
Add a constraint to force inference toward a “consensus” βcons
12βT
j Σ̂\j\jβj + βTj Σ̂j\j + λ‖βj‖L1 + µ
∑c
wc‖βcj − β
consj ‖2L2
with:wc : real number used to weight the conditions;µ regularization parameter;βcons
j whatever you want...?
Nathalie Villa-Vialaneix | Consensus Lasso 21/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Consensus LASSO
Proposal
Infer multiple networks by forcing them toward a consensual network: i.e.,explicitly keeping the differences between conditions under control butwith a L2 penalty (allow for more differences than Group-LASSO typepenalties).
Add a constraint to force inference toward a “consensus” βcons
12βT
j Σ̂\j\jβj + βTj Σ̂j\j + λ‖βj‖L1 + µ
∑c
wc‖βcj − β
consj ‖2L2
with:wc : real number used to weight the conditions;µ regularization parameter;βcons
j whatever you want...?
Nathalie Villa-Vialaneix | Consensus Lasso 21/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Choice of a consensus: set one...
Typical case:
a prior network is known (e.g., from bibliography);
with no prior information, use a fixed prior corresponding to (e.g.)global inference
⇒ given (and fixed) βcons
Proposition
Using a fixed βconsj , the optimization problem is equivalent to minimizing
the p following standard quadratic problem in Rk(p−1) with L1-penalty:
12βT
j B1(µ)βj + βTj B2(µ) + λ‖βj‖L1 ,
where
B1(µ) = Σ̂\j\j + 2µIk(p−1), with Ik(p−1) the k(p − 1)-identity matrix
B2(µ) = Σ̂j\j − 2µIk(p−1)βcons with βcons =
((βcons
j )T , . . . , (βconsj )T
)T
Nathalie Villa-Vialaneix | Consensus Lasso 22/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Choice of a consensus: set one...
Typical case:
a prior network is known (e.g., from bibliography);
with no prior information, use a fixed prior corresponding to (e.g.)global inference
⇒ given (and fixed) βcons
Proposition
Using a fixed βconsj , the optimization problem is equivalent to minimizing
the p following standard quadratic problem in Rk(p−1) with L1-penalty:
12βT
j B1(µ)βj + βTj B2(µ) + λ‖βj‖L1 ,
where
B1(µ) = Σ̂\j\j + 2µIk(p−1), with Ik(p−1) the k(p − 1)-identity matrix
B2(µ) = Σ̂j\j − 2µIk(p−1)βcons with βcons =
((βcons
j )T , . . . , (βconsj )T
)T
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Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Choice of a consensus: adapt oneduring training...
Derive the consensus from the condition-specific estimates:
βconsj =
∑c
nc
nβc
j
Proposition
Using βconsj =
∑kc=1
ncn β
cj , the optimization problem is equivalent to
minimizing the following standard quadratic problem with L1-penalty:
12βT
j Sj(µ)βj + βTj Σ̂j\j + λ‖βj‖L1
where Sj(µ) = Σ̂\j\j + 2µAT (µ)A(µ) where A(µ) is a[k(p − 1) × k(p − 1)]-matrix that does not depend on j.
Nathalie Villa-Vialaneix | Consensus Lasso 23/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Choice of a consensus: adapt oneduring training...
Derive the consensus from the condition-specific estimates:
βconsj =
∑c
nc
nβc
j
Proposition
Using βconsj =
∑kc=1
ncn β
cj , the optimization problem is equivalent to
minimizing the following standard quadratic problem with L1-penalty:
12βT
j Sj(µ)βj + βTj Σ̂j\j + λ‖βj‖L1
where Sj(µ) = Σ̂\j\j + 2µAT (µ)A(µ) where A(µ) is a[k(p − 1) × k(p − 1)]-matrix that does not depend on j.
Nathalie Villa-Vialaneix | Consensus Lasso 23/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Computational aspects: optimization
Common framework
Objective function can be decomposed into:
convex part C(βj) = 12β
Tj Q
1j (µ) + βT
j Q2j (µ)
L1-norm penalty P(βj) = ‖βj‖L1
optimization by “active set” [Osborne et al., 2000, Chiquet et al., 2011]
1: repeat(λ given)2: Given A and βjj′ st: βjj′ , 0, ∀ j′ ∈ A, solve (over h) the smooth
minimization problem restricted to A
C(βj + h) + λP(βj + h) ⇒ βj ← βj + h
3: Update A by adding most violating variables, i.e., variables st:
abs∣∣∣∂C(βj) + λ∂P(βj)
∣∣∣ > 0
with [∂P(βj)]j′ ∈ [−1, 1] if j′ < A
4: until
all first order conditions satisfied
Repeat:
large λ
↓
small λusing previous βas prior
Nathalie Villa-Vialaneix | Consensus Lasso 24/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Computational aspects: optimization
Common framework
Objective function can be decomposed into:
convex part C(βj) = 12β
Tj Q
1j (µ) + βT
j Q2j (µ)
L1-norm penalty P(βj) = ‖βj‖L1
optimization by “active set” [Osborne et al., 2000, Chiquet et al., 2011]
1: repeat(λ given)2: Given A and βjj′ st: βjj′ , 0, ∀ j′ ∈ A, solve (over h) the smooth
minimization problem restricted to A
C(βj + h) + λP(βj + h) ⇒ βj ← βj + h
3: Update A by adding most violating variables, i.e., variables st:
abs∣∣∣∂C(βj) + λ∂P(βj)
∣∣∣ > 0
with [∂P(βj)]j′ ∈ [−1, 1] if j′ < A
4: until
all first order conditions satisfied
Repeat:
large λ
↓
small λusing previous βas prior
Nathalie Villa-Vialaneix | Consensus Lasso 24/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Computational aspects: optimization
Common framework
Objective function can be decomposed into:
convex part C(βj) = 12β
Tj Q
1j (µ) + βT
j Q2j (µ)
L1-norm penalty P(βj) = ‖βj‖L1
optimization by “active set” [Osborne et al., 2000, Chiquet et al., 2011]
1: repeat(λ given)2: Given A and βjj′ st: βjj′ , 0, ∀ j′ ∈ A, solve (over h) the smooth
minimization problem restricted to A
C(βj + h) + λP(βj + h) ⇒ βj ← βj + h
3: Update A by adding most violating variables, i.e., variables st:
abs∣∣∣∂C(βj) + λ∂P(βj)
∣∣∣ > 0
with [∂P(βj)]j′ ∈ [−1, 1] if j′ < A4: until
all first order conditions satisfied
Repeat:
large λ
↓
small λusing previous βas prior
Nathalie Villa-Vialaneix | Consensus Lasso 24/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Computational aspects: optimization
Common framework
Objective function can be decomposed into:
convex part C(βj) = 12β
Tj Q
1j (µ) + βT
j Q2j (µ)
L1-norm penalty P(βj) = ‖βj‖L1
optimization by “active set” [Osborne et al., 2000, Chiquet et al., 2011]
1: repeat(λ given)2: Given A and βjj′ st: βjj′ , 0, ∀ j′ ∈ A, solve (over h) the smooth
minimization problem restricted to A
C(βj + h) + λP(βj + h) ⇒ βj ← βj + h
3: Update A by adding most violating variables, i.e., variables st:
abs∣∣∣∂C(βj) + λ∂P(βj)
∣∣∣ > 0
with [∂P(βj)]j′ ∈ [−1, 1] if j′ < A4: until all first order conditions satisfied
Repeat:
large λ
↓
small λusing previous βas prior
Nathalie Villa-Vialaneix | Consensus Lasso 24/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Computational aspects: optimization
Common framework
Objective function can be decomposed into:
convex part C(βj) = 12β
Tj Q
1j (µ) + βT
j Q2j (µ)
L1-norm penalty P(βj) = ‖βj‖L1
optimization by “active set” [Osborne et al., 2000, Chiquet et al., 2011]
1: repeat(λ given)2: Given A and βjj′ st: βjj′ , 0, ∀ j′ ∈ A, solve (over h) the smooth
minimization problem restricted to A
C(βj + h) + λP(βj + h) ⇒ βj ← βj + h
3: Update A by adding most violating variables, i.e., variables st:
abs∣∣∣∂C(βj) + λ∂P(βj)
∣∣∣ > 0
with [∂P(βj)]j′ ∈ [−1, 1] if j′ < A4: until all first order conditions satisfied
Repeat:
large λ
↓
small λusing previous βas prior
Nathalie Villa-Vialaneix | Consensus Lasso 24/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Bootstrap estimation ' BOLASSO
x x
xx
x
x
x
x
subsample n observations with replacement
cLasso estimation−−−−−−−−−−−−→ for varying λ, (βλ,bjj′ )jj′
↓ threshold
keep the first T1 largest coefficients
B×
Frequency table(1, 2) (1, 3) ... (j, j′) ...130 25 ... 120 ...
−→ Keep the T2 most frequent pairs
Nathalie Villa-Vialaneix | Consensus Lasso 25/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Bootstrap estimation ' BOLASSO
x x
xx
x
x
x
x
subsample n observations with replacement
cLasso estimation−−−−−−−−−−−−→ for varying λ, (βλ,bjj′ )jj′
↓ threshold
keep the first T1 largest coefficients
B×
Frequency table(1, 2) (1, 3) ... (j, j′) ...130 25 ... 120 ...
−→ Keep the T2 most frequent pairs
Nathalie Villa-Vialaneix | Consensus Lasso 25/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Bootstrap estimation ' BOLASSO
x x
xx
x
x
x
x
subsample n observations with replacement
cLasso estimation−−−−−−−−−−−−→ for varying λ, (βλ,bjj′ )jj′
↓ threshold
keep the first T1 largest coefficients
B×
Frequency table(1, 2) (1, 3) ... (j, j′) ...130 25 ... 120 ...
−→ Keep the T2 most frequent pairs
Nathalie Villa-Vialaneix | Consensus Lasso 25/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Bootstrap estimation ' BOLASSO
x x
xx
x
x
x
x
subsample n observations with replacement
cLasso estimation−−−−−−−−−−−−→ for varying λ, (βλ,bjj′ )jj′
↓ threshold
keep the first T1 largest coefficients
B×
Frequency table(1, 2) (1, 3) ... (j, j′) ...130 25 ... 120 ...
Frequency table(1, 2) (1, 3) ... (j, j′) ...130 25 ... 120 ...
−→ Keep the T2 most frequent pairs
Nathalie Villa-Vialaneix | Consensus Lasso 25/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Bootstrap estimation ' BOLASSO
x x
xx
x
x
x
x
subsample n observations with replacement
cLasso estimation−−−−−−−−−−−−→ for varying λ, (βλ,bjj′ )jj′
↓ threshold
keep the first T1 largest coefficients
B×
Frequency table(1, 2) (1, 3) ... (j, j′) ...130 25 ... 120 ...
−→ Keep the T2 most frequent pairs
Nathalie Villa-Vialaneix | Consensus Lasso 25/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Outline
1 Short overview on biological background
2 Network inference and GGM
3 Inference with multiple samples
4 Simulations
Nathalie Villa-Vialaneix | Consensus Lasso 26/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Simulated data
Expression data with known co-expression network
original network (scale free) taken fromhttp://www.comp-sys-bio.org/AGN/data.html (100 nodes,∼ 200 edges, loops removed);
rewire a ratio r of the edges to generate k “children” networks(sharing approximately 100(1 − 2r)% of their edges);
generate “expression data” with a random Gaussian process fromeach chid:
use the Laplacian of the graph to generate a putative concentrationmatrix;use edge colors in the original network to set the edge sign;correct the obtained matrix to make it positive;invert to obtain a covariance matrix...;... which is used in a random Gaussian process to generateexpression data (with noise).
Nathalie Villa-Vialaneix | Consensus Lasso 27/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Simulated data
Expression data with known co-expression network
original network (scale free) taken fromhttp://www.comp-sys-bio.org/AGN/data.html (100 nodes,∼ 200 edges, loops removed);
rewire a ratio r of the edges to generate k “children” networks(sharing approximately 100(1 − 2r)% of their edges);generate “expression data” with a random Gaussian process fromeach chid:
use the Laplacian of the graph to generate a putative concentrationmatrix;use edge colors in the original network to set the edge sign;correct the obtained matrix to make it positive;invert to obtain a covariance matrix...;... which is used in a random Gaussian process to generateexpression data (with noise).
Nathalie Villa-Vialaneix | Consensus Lasso 27/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
An example with k = 2, r = 5%
mother network3 first child second child
3actually the parent network. My co-author wisely noted that the mistake wasunforgivable for a feminist...
Nathalie Villa-Vialaneix | Consensus Lasso 28/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Choice for T2
Data: r = 0.05, k = 2 and n1 = n2 = 20100 bootstrap samples, µ = 1, T1 = 250 or 500
●● 30 selections at least30 selections at least
0.00
0.25
0.50
0.75
1.00
0.00 0.25 0.50 0.75precision
reca
ll
●●
40 selections at least43 selections at least
0.00
0.25
0.50
0.75
1.00
0.00 0.25 0.50 0.75 1.00precision
reca
ll
Dots correspond to best F = 2 × precision×recallprecision+recall
⇒ Best F corresponds to selecting a number of edges approximatelyequal to the number of edges in the original network.
Nathalie Villa-Vialaneix | Consensus Lasso 29/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Choice for T1 and µ
µ T1 % of improvement0.1/1 {250, 300, 500} of bootstrapping
network sizes rewired edges: 5%20-20 1 500 30.6920-30 0.1 500 11.8730-30 1 300 20.1550-50 1 300 14.3620-20-20-20-20 1 500 86.0430-30-30-30 0.1 500 42.67network sizes rewired edges: 20%20-20 0.1 300 -17.8620-30 0.1 300 -18.3530-30 1 500 -7.9750-50 0.1 300 -7.8320-20-20-20-20 0.1 500 10.2730-30-30-30 1 500 13.48
Nathalie Villa-Vialaneix | Consensus Lasso 30/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Comparison with other approaches
Method compared (direct and bootstrap approaches)
independant Graphical LASSO estimation gLasso
methods implementated in the R package simone and described in[Chiquet et al., 2011]: intertwinned LASSO iLasso, cooperativeLASSO coopLasso and group LASSO groupLasso
fused graphical LASSO as described in [Danaher et al., 2013] asimplemented in the R package fgLasso
consensus Lasso withfixed prior (the mother network) cLasso(p)fixed prior (average over the conditions of independant estimations)cLasso(2)adaptative estimation of the prior cLasso(m)
Parameters set to: T1 = 500, B = 100, µ = 1
Nathalie Villa-Vialaneix | Consensus Lasso 31/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Comparison with other approaches
Method compared (direct and bootstrap approaches)
independant Graphical LASSO estimation gLasso
methods implementated in the R package simone and described in[Chiquet et al., 2011]: intertwinned LASSO iLasso, cooperativeLASSO coopLasso and group LASSO groupLasso
fused graphical LASSO as described in [Danaher et al., 2013] asimplemented in the R package fgLassoconsensus Lasso with
fixed prior (the mother network) cLasso(p)fixed prior (average over the conditions of independant estimations)cLasso(2)adaptative estimation of the prior cLasso(m)
Parameters set to: T1 = 500, B = 100, µ = 1
Nathalie Villa-Vialaneix | Consensus Lasso 31/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Comparison with other approaches
Method compared (direct and bootstrap approaches)
independant Graphical LASSO estimation gLasso
methods implementated in the R package simone and described in[Chiquet et al., 2011]: intertwinned LASSO iLasso, cooperativeLASSO coopLasso and group LASSO groupLasso
fused graphical LASSO as described in [Danaher et al., 2013] asimplemented in the R package fgLassoconsensus Lasso with
fixed prior (the mother network) cLasso(p)fixed prior (average over the conditions of independant estimations)cLasso(2)adaptative estimation of the prior cLasso(m)
Parameters set to: T1 = 500, B = 100, µ = 1
Nathalie Villa-Vialaneix | Consensus Lasso 31/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Selected results (best F)
rewired edges: 5% - conditions: 2 - sample size: 2 × 30direct version
Method gLasso iLasso groupLasso coopLasso0.28 0.35 0.32 0.35
Method fgLasso cLasso(m) cLasso(p) cLasso(2)0.32 0.31 0.86 0.30
bootstrap versionMethod gLasso iLasso groupLasso coopLasso
0.31 0.34 0.36 0.34Method fgLasso cLasso(m) cLasso(p) cLasso(2)
0.36 0.37 0.86 0.35
Conclusions
bootstraping improves performance (except for iLasso and for larger)
joint inference improves performance
using a good prior is (as expected) very efficient
adaptive approach for cLasso is better than naive approach
Nathalie Villa-Vialaneix | Consensus Lasso 32/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Real data
204 obese women ; expression of 221 genes before and after a LCDµ = 1 ; T1 = 1000 (target density: 4%)
Distribution of the number of times an edge is selected over 100bootstrap samples
0
500
1000
1500
0 25 50 75 100Counts
Fre
quen
cy
(70% of the pairs of nodes are never selected)⇒ T2 = 80
Nathalie Villa-Vialaneix | Consensus Lasso 33/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Networks
Before diet
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AZGP1
CIDEA
FADS1
FADS2
GPD1L
PCK2
After diet
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AZGP1
CIDEA
FADS1
FADS2
GPD1L
PCK2
densities about 1.3% - some interactions (both shared and specific)make sense to the biologist
Nathalie Villa-Vialaneix | Consensus Lasso 34/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Thank you for your attention...
Programs available in the R package therese (on R-Forge)4. Joint workwith
Magali SanCristobal Matthieu Vignes(GenPhySe, INRA Toulouse) (Massey University, NZ)
Nathalie Viguerie(I2MC, INSERM Toulouse)
4https://r-forge.r-project.org/projects/therese-pkg
Nathalie Villa-Vialaneix | Consensus Lasso 35/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
Questions?
Nathalie Villa-Vialaneix | Consensus Lasso 36/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
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Inferring sparse Gaussian graphical models with latent structure.Electronic Journal of Statistics, 3:205–238.
Butte, A. and Kohane, I. (1999).
Unsupervised knowledge discovery in medical databases using relevance networks.In Proceedings of the AMIA Symposium, pages 711–715.
Chiquet, J., Grandvalet, Y., and Ambroise, C. (2011).
Inferring multiple graphical structures.Statistics and Computing, 21(4):537–553.
Danaher, P., Wang, P., and Witten, D. (2013).
The joint graphical lasso for inverse covariance estimation accross multiple classes.Journal of the Royal Statistical Society Series B.Forthcoming.
Friedman, J., Hastie, T., and Tibshirani, R. (2008).
Sparse inverse covariance estimation with the graphical lasso.Biostatistics, 9(3):432–441.
Meinshausen, N. and Bühlmann, P. (2006).
High dimensional graphs and variable selection with the lasso.Annals of Statistic, 34(3):1436–1462.
Mohan, K., Chung, J., Han, S., Witten, D., Lee, S., and Fazel, M. (2012).
Structured learning of Gaussian graphical models.In Proceedings of NIPS (Neural Information Processing Systems) 2012, Lake Tahoe, Nevada, USA.
Osborne, M., Presnell, B., and Turlach, B. (2000).
Nathalie Villa-Vialaneix | Consensus Lasso 36/36
Short overview on biological background Network inference and GGM Inference with multiple samples Simulations
On the LASSO and its dual.Journal of Computational and Graphical Statistics, 9(2):319–337.
Schäfer, J. and Strimmer, K. (2005a).
An empirical bayes approach to inferring large-scale gene association networks.Bioinformatics, 21(6):754–764.
Schäfer, J. and Strimmer, K. (2005b).
A shrinkage approach to large-scale covariance matrix estimation and implication for functional genomics.Statistical Applications in Genetics and Molecular Biology, 4:1–32.
Viguerie, N., Montastier, E., Maoret, J., Roussel, B., Combes, M., Valle, C., Villa-Vialaneix, N., Iacovoni, J., Martinez, J., Holst, C., Astrup,
A., Vidal, H., Clément, K., Hager, J., Saris, W., and Langin, D. (2012).Determinants of human adipose tissue gene expression: impact of diet, sex, metabolic status and cis genetic regulation.PLoS Genetics, 8(9):e1002959.
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