SIMoNe: Statistical Iference for MOdular NEtworks

SIMoNeAn R package for inferring Gausssian networks with latent

clustering

Julien Chiquet (and Camille, Christophe, Gilles, Catherine, Yves)

Laboratoire Statistique et Genome,La genopole - Universite d’Evry

SSB – 13 avril 2010

SIMoNe: inferring Gaussian networks with latent clustering 1

Problem

n ≈ 10s/100s of slides

g ≈ 1000s of genes

O(g2) parameters (edges) !

Inference

Which interactions?

The main statistical issue is the high dimensional setting


Handling the scarcity of data (1)By reducing the number of parameters

AssumptionConnections will only appear between informative genes

differential analysis

select p key genes P

p “reasonable” compared to n

typically, n ∈ [p/5; 5p]

the learning dataset

n size–p vectors of expression

(X1, . . . , Xn) with Xi ∈ Rp

inference


Handling the scarcity of data (2)By collecting as many observations as possible

Multitask learning Go to learning

How should we merge the data?organism

drug 1drug 2

drug 3




by inferring each network independentlyorganism

drug 1drug 2

drug 3

(X(1)1 , . . . , X

(1)n1

), X(1)i ∈ Rp1 (X

(2)1 , . . . , X

(2)n2

), X(2)i ∈ Rp2 (X

(3)1 , . . . , X

(3)n3

), X(3)i ∈ Rp3

inference inference inference




by pooling all the available dataorganism

drug 1drug 2

drug 3

(X1, . . . , Xn), Xi ∈ Rp , with n = n1 + n2 + n3 .

inference




by breaking the separabilityorganism

drug 1drug 2

drug 3

(X(1)1 , . . . , X

(1)n1

), X(1)i ∈ Rp1 (X

(2)1 , . . . , X

(2)n2

), X(2)i ∈ Rp2 (X

(3)1 , . . . , X

(3)n3

), X(3)i ∈ Rp3

inference


Handling the scarcity of data (3)By introducing some prior

Priors should be biologically grounded

1. few genes effectively interact (sparsity),2. networks are organized (latent clustering),3. steady-state or time-course data

(directedness relies on the modelling).

G0 G1

G2

G3

G4

G5

G6

G7

G8

G9


Handling the scarcity of data (3)By introducing some prior

Priors should be biologically grounded

1. few genes effectively interact (sparsity),2. networks are organized (latent clustering),3. steady-state or time-course data

(directedness relies on the modelling).

A1 A2

A3

B1

B2

B3

B4

B5

C1

C2


Outline

Statistical modelsSteady-state dataTime-course dataMultitask learning

Algorithms and methodsOverall viewNetwork inferenceModel selectionLatent structure

Numerical experimentsPerformance on simulated dataR package demo: the breast cancer data set


The graphical models: general settings

AssumptionA microarray can be represented as a multivariate Gaussianvector X = (X(1), . . . , X(p)) ∈ Rp.

Collecting gene expression

1. Steady-state data leads to an i.i.d. sample.2. Time-course data gives a time series.

Graphical interpretationconditional dependency between X(i) and X(j)

ornon null partial correlation between X(i) and X(j)

if and only ifj

i






Graphical interpretationconditional dependency between X(i) and X(j)

ornon null partial correlation between X(i) and X(j)

if and only ifj

i

?






Graphical interpretationconditional dependency between Xt(i) and Xt−1(j)

ornon null partial correlation between Xt(i) and Xt−1(j)

if and only ifj

i

?


The general statistical approach

Let Θ be the parameters to infer (the edges).

A penalized likelihood approach

Θλ = arg maxΘL(Θ; data)− λ pen`1(Θ,Z),

I L is the model log-likelihood,I Z is a latent clustering of the network,I pen`1 is a penalty function tuned by λ > 0.

It performs1. regularization (needed when n� p),2. selection (sparsity induced by the `1-norm),3. model-driven inference (penalty adapted according to Z).


Outline





The Gaussian model for an i.i.d. sample

LetI X ∼ N (0p,Σ) with X1, . . . , Xn i.i.d. copies of X,I X be the n× p matrix whose kth row is Xk,I Θ = (θij)i,j∈P , Σ−1 be the concentration matrix.

Graphical interpretationSince corij|P\{i,j} = −θij/

√θiiθjj for i 6= j,

X(i) ⊥⊥ X(j)|X(P\{i, j})⇔

θij = 0

oredge (i, j) /∈ network.

Θ describes the undirected graph of conditionaldependencies.


Neighborhood selection (1)

LetI Xi be the ith column of X,I X\i be X deprived of Xi.

Xi = X\iβ + ε, where βj = −θijθii.

Meinshausen and Bulhman, 2006Since sign(corij|P\{i,j}) = sign(βj), select the neighbors of i with

arg minβ

1

n

∥∥Xi −X\iβ∥∥22

+ λ ‖β‖`1 .

The sign pattern of Θλ is inferred after a symmetrization step.


Neighborhood selection (2)

The pseudo log-likelihood of the i.i.d Gaussian sample is

Liid(Θ; S) =

p∑i=1

(n∑k=1

logP(Xk(i)|Xk(P\i); Θi)

),

=n

2log det(D)− n

2Trace

(D−1/2ΘSΘD−1/2

)− n

2log(2π),

where D = diag(Θ).

Proposition

Θpseudoλ = arg max

Θ:θij 6=θiiLiid(Θ; S)− λ ‖Θ‖`1

has the same null entries as inferred by neighborhood selection.


The Gaussian likelihood for an i.i.d. sample

Let S = n−1XᵀX be the empirical variance-covariance matrix: Sis a sufficient statistic of Θ.

The log-likelihood

Liid(Θ; S) =n

2log det(Θ)− n

2Trace(SΘ) +

n

2log(2π).

The MLE = S−1 of Θ is not defined for n < p and neversparse.

The need for regularization is huge.


Penalized log-likelihood

Banerjee et al., JMLR 2008

Θλ = arg maxΘ

Liid(Θ; S)− λ‖Θ‖`1 ,

efficiently solved by the graphical LASSO of Friedman et al, 2008.

Ambroise, Chiquet, Matias, EJS 2009Use adaptive penalty parameters for different coefficients

Liid(Θ; S)− λ‖PZ ?Θ‖`1 ,

where PZ is a matrix of weights depending on the underlyingclustering Z.

Works with the pseudo log-likelihood (computationallyefficient).


Outline





The Gaussian model for time-course data (1)

Let X1, . . . , Xn be a first order vector autoregressive process

Xt = ΘXt−1 + b + εt, t ∈ [1, n]

where we are looking for Θ = (θij)i,j∈P andI X0 ∼ N (0p,Σ0),I εt is a Gaussian white noise with covariance σ2Ip,I cov(Xt, εs) = 0 for s > t, so that Xt is markovian.

Graphical interpretationsince

θij =cov (Xt(i), Xt−1(j)|Xt−1(P\j))

var (Xt−1(j)|Xt−1(P\j)),

Xt(i) ⊥⊥ Xt−1(j)|Xt−1(P\j)⇔

θij = 0

oredge (j � i) /∈ network


The Gaussian model for time-course data (2)

LetI X be the n× p matrix whose kth row is Xk,I S = n−1Xᵀ\nX\n be the within time covariance matrix,

I V = n−1Xᵀ\nX\0 be the across time covariance matrix.

The log-likelihood

Ltime(Θ; S,V) = n Trace (VΘ)− n

2Trace (ΘᵀSΘ) + c.

The MLE = S−1V of Θ is still not defined for n < p.


Penalized log-likelihood

Charbonnier, Chiquet, Ambroise, SAGMB 2010

Θλ = arg maxΘ

Ltime(Θ; S,V)− λ‖PZ ?Θ‖`1

where PZ is a (non-symmetric) matrix of weights depending onthe underlying clustering Z.

Major difference with the i.i.d. caseThe graph is directed:

θij =cov (Xt(i), Xt−1(j)|Xt−1(P\j))

var (Xt−1(j)|Xt−1(P\j))

6= cov (Xt(j), Xt−1(i)|Xt−1(P\i))var (Xt−1(i)|Xt−1(P\i))

.


Outline





Coupling related problems

ConsiderI T samples concerning the expressions of the same p genes,

I X(t)1 , . . . , X

(t)nt is the tth sample drawn from N (0p,Σ

(t)), withcovariance matrix S(t).

Multiple samples setup Go to scheme

Ignoring the relationships between the tasks leads to

arg maxΘ(t),t=1...,T

T∑t=1

L(Θ(t); S(t))− λ pen`1(Θ(t),Z).

Breaking the separability

I Either by modifying the objective functionI or the constraints.


Coupling related problems

ConsiderI T samples concerning the expressions of the same p genes,

I X(t)1 , . . . , X

(t)nt is the tth sample drawn from N (0p,Σ

(t)), withcovariance matrix S(t).

Multiple samples setup Go to scheme

Ignoring the relationships between the tasks leads to


T∑t=1

L(Θ(t); S(t))− λ pen`1(Θ(t),Z).

Breaking the separability

I Either by modifying the objective functionI or the constraints.

RemarksI In the sequel, the Z is eluded for clarity (no loss of

generality).I Multitask learning is easily adapted to time-course data yet

only steady state version is presented here.


Coupling problems through the objective function

The Intertwined LASSO

maxΘ(t),t...,T

T∑t=1

L(Θ(t); S(t))− λ‖Θ(t)‖`1

I S = 1n

∑Tt=1 ntS

(t) is an “across-task” covariance matrix.I S(t) = αS(t) + (1− α)S is a mixture between inner/over-tasks

covariance matrices.

setting α = 0 is equivalent to pooling all the data and inferone common network,

setting α = 1 is equivalent to treating T independentproblems.


Coupling problems by grouping variables (1)

Groups definition

I Groups are the T -tuple composed by the (i, j) entries ofeach Θ(t), t = 1, . . . , T .

I Most relationships between the genes are kept or removedacross all tasks simultaneously.

The graphical group-LASSO

maxΘ(t),t...,T

T∑t=1

L(Θ(t); S(t)

)− λ

∑i,j∈Pi 6=j

(T∑t=1

(θ(t)ij

)2)1/2

.


1

1

−1

−1

β(1)2

β(1)1 1

1

−1

−1

β(1)2

β(1)1

1

1

−1

−1

β(1)2

β(1)1 1

1

−1

−1

β(1)2

β(1)1

β(2)

1=

0β(2)

1=

0.3

β(2)2 = 0 β

(2)2 = 0.3

Group-LASSO penaltyAssume

I 2 tasks (T = 2)

I 2 coefficients (p = 2)

Let represent the unit ball

2∑i=1

(2∑

t=1

β(t)i

2

)1/2

≤ 1


Coupling problems by grouping variables (2)

Graphical group-LASSO modification

I Inside a group, value are most likeliky sign consistent.

The graphical cooperative-LASSO

maxΘ(t),t...,T

T∑t=1

L(S(t); Θ(t)

)

− λ∑i,j∈Pi 6=j

(

T∑t=1

[θ(t)ij

]2+

)1/2

+

(T∑t=1

[θ(t)ij

]2−

)1/2 ,

where [u]+ = max(0, u) and [u]− = min(0, u).


1

1

−1

−1

β(1)2

β(1)1 1

1

−1

−1

β(1)2

β(1)1

1

1

−1

−1

β(1)2

β(1)1 1

1

−1

−1

β(1)2

β(1)1

β(2)

1=

0β(2)

1=

0.3

β(2)2 = 0 β

(2)2 = 0.3

Coop-LASSO penaltyAssume

I 2 tasks (T = 2)

I 2 coefficients (p = 2)

Let represent the unit ball

2∑i=1

(2∑

t=1

[β(t)i

]2+

)1/2

+

2∑i=1

(2∑

t=1

[−β(t)

i

]+

)1/2

≤ 1


Outline





The overall strategy

Our basic criteria is of the form

L(Θ; data)− λ ‖PZ ?Θ‖`1 .

What we are looking for

I the edges, through Θ,I the correct level of sparsity λ,I the underlying clustering Z with connectivity matrix πZ.

What SIMoNe does

1. Infer a family of networks G = {Θλ : λ ∈ [λmax, 0]}2. Select G? that maximizes an information criteria3. Learn Z on the selected network G?

4. Infer a family of networks with PZ ∝ 1− πZ

5. Select G?Z that maximizes an information criteria


SIMoNe

SIMoNESuppose you want to recover a clustered network:

Target Adjacency Matrix

Graph

Target Network


SIMoNe

SIMoNEStart with microarray data

Data


SIMoNe

SIMoNE

DataAdjacency Matrix

corresponding to G?

SIMoNE without prior


SIMoNe

SIMoNE


corresponding to G?


πZ

Connectivity matrix

Mixer

Penalty matrix PZ

Decreasing transformation


SIMoNe

SIMoNE


corresponding to G?


πZ

Connectivity matrix

Mixer

Penalty matrix PZ

Decreasing transformation

Adjacency Matrixcorresponding to G?Z

+

SIMoNE


Outline





Monotask framework: problem decomposition

Consider the following reordering of Θ

Θ =

[Θ\i\i Θi\iΘᵀi\i θii

], Θi =

[Θi\iθii

].

Block coordinate descent algorithm

arg maxΘ

L(Θ; data)− λ pen`1(Θ)

relies on p penalized, convex-optimization problems

arg minβ∈Rp−1

f(β; S) + λ pen`1(β), (1)

where f is convex and β = Θi\i for steady-state data.


Monotask framework: problem decomposition

Consider the following reordering of Θ

Θ =

[Θ\i\i Θi\iΘᵀi\i θii

], Θi =

[Θi\iθii

].

Block coordinate descent algorithm

arg maxΘ

L(Θ; data)− λ pen`1(Θ)

relies on p penalized, convex-optimization problems

arg minβ∈Rp

f(β; S,V) + λ pen`1(β), (1)

where f is convex and β = Θi for time-course data.


Monotask framework: algorithms

1. steady-state: Covsel/GLasso (Liid(Θ)− λ‖Θ‖`1)I starts from S + λIp positive definite,I iterates on the columns of Θ−1 until stabilization,I both estimation and selection of Θ.

2. steady-state: neighborhood selection (Liid(Θ)− λ‖Θ‖`1)I select signs patterns of Θi\i with the LASSO,I only one pass per column required,I post-symmetrization needed.

3. time-course: VAR(1) inference (Ltime(Θ)− λ‖Θ‖`1)I select and estimate Θi with the LASSO,I only one pass per column required,I both estimation and selection.


Multitask framework: problem decomposition (1)

Consider the (p T )× (p T ) block-diagonal matrix C composed bythe empirical covariance matrices of each tasks

C =

S(1) 0. . .

0 S(T )

,

and define

C\i\i =

S(1)\i\i 0

. . .

0 S(T )\i\i

, Ci\i =

S(1)i\i...

S(T )i\i

.

The (p − 1)T × (p − 1)T matrix C\i\i is the matrix C where weremoved each line and each column pertaining to variable i.

RemarkLet us consider multitask algorithms in the steady-state frame-work (easily adapted to time-course data)



Consider the (p T )× (p T ) block-diagonal matrix C composed bythe empirical covariance matrices of each tasks

C =

S(1) 0. . .

0 S(T )

,

and define

C\i\i =

S(1)\i\i 0

. . .

0 S(T )\i\i

, Ci\i =

S(1)i\i...

S(T )i\i

.

The (p − 1)T × (p − 1)T matrix C\i\i is the matrix C where weremoved each line and each column pertaining to variable i.



Estimate the ith-columns of the T tasks bind together


T∑t=1

L(Θ(t); S(t))− λ pen`1(Θ(t))

is decomposed into p convex optimization problems

arg minβ∈RT×(p−1)

f(β; C) + λ pen`1(β),

where we set β(t) = Θ(t)i\i and

β =

β(1)

...β(T )

∈ RT×(p−1).


Solving the sub-problem

Subdifferential approach

minβ∈RT×(p−1)

L(β) = f(β) + pen`1(β) ,

β is a minimizer iif 0p ∈ ∂βL(β), with

∂βL(β) = ∇βf(β) + λ∂βpen`1(β).




minβ∈RT×(p−1)

L(β) = f(β) + pen`1(β) ,



For the graphical Intertwined LASSO

pen`1(β) =

T∑t=1

∥∥∥β(t)∥∥∥1,

where the grouping effect is managed by the function f .




minβ∈RT×(p−1)

L(β) = f(β) + pen`1(β) ,



For the graphical Group-LASSO

pen`1(β) =

p−1∑i=1

∥∥∥β[1:T ]i

∥∥∥2,

where β[1:T ]i =

(β(1)i , . . . , β

(T )i

)ᵀ∈ RT is the vector of the ith com-

ponent across tasks.




minβ∈RT×(p−1)

L(β) = f(β) + pen`1(β) ,



For the graphical Coop-LASSO

pen`1(β) =

p−1∑i=1

(∥∥∥∥(β[1:T ]i

)+

∥∥∥∥2

+

∥∥∥∥(−β[1:T ]i

)+

∥∥∥∥2

),

where β[1:T ]i =

(β(1)i , . . . , β

(T )i

)ᵀ∈ RT is the vector of the ith com-

ponent across tasks.


General active set algorithm: , yellow belt// 0. INITIALIZATION

β ← 0,A ← ∅while 0 /∈ ∂βL(β) do

// 1. MASTER PROBLEM: OPTIMIZATION WITH RESPECT TO βAFind a solution h to the smooth problem

∇hf(βA + h) + λ∂hpen`1 (βA + h) = 0, where ∂hpen`1 ={∇hpen`1

}.

βA ← βA + h// 2. IDENTIFY NEWLY ZEROED VARIABLES

A ← A\{i}

// 3. IDENTIFY NEW NON-ZERO VARIABLES

// Select a candidate i ∈ Ac

i← arg maxj∈Ac

vj , where vj = minν∈∂βj gk

∣∣∣ ∂f(β)∂βj

+ λν∣∣∣

endSIMoNe: inferring Gaussian networks with latent clustering 35

General active set algorithm: / orange belt// 0. INITIALIZATION

β ← 0,A ← ∅while 0 /∈ ∂βL(β) do



}.


A ← A\{i}

// 3. IDENTIFY NEW NON-ZERO VARIABLES

// Select a candidate i ∈ Ac which violates the more the optimality

conditions

i← arg maxj∈Ac


∣∣∣ ∂f(β)∂βj

+ λν∣∣∣

if it exists such an i thenA ← A∪ {i}

elseStop and return β, which is optimal

endend


General active set algorithm: M green belt// 0. INITIALIZATION

β ← 0,A ← ∅while 0 /∈ ∂βL(β) do



}.


while ∃i ∈ A : βi = 0 and minν∈∂βigk

∣∣∣ ∂f(β)∂βi

+ λν∣∣∣ = 0 do

A ← A\{i}end// 3. IDENTIFY NEW NON-ZERO VARIABLES

// Select a candidate i ∈ Ac such that an infinitesimal change of βiprovides the highest reduction of L

i← arg maxj∈Ac


∣∣∣ ∂f(β)∂βj

+ λν∣∣∣

if vi 6= 0 thenA ← A∪ {i}

elseStop and return β, which is optimal

endend


Outline





Tuning the penalty parameterWhat does the literature say?

Theory based penalty choices

1. Optimal order of penalty in the p� n framework:√n log p

Bunea et al. 2007, Bickel et al. 2009

2. Control on the probability of connecting two distinctconnectivity sets

Meinshausen et al. 2006, Banerjee et al. 2008, Ambroise et al. 2009

practically much too conservative

Cross-validationI Optimal in terms of prediction, not in terms of selectionI Problematic with small samples:

changes the sparsity constraint due to sample size


Tuning the penalty parameterBIC / AIC

Theorem (Zou et al. 2008)

df(β lassoλ ) =

∥∥∥β lassoλ

∥∥∥0

Straightforward extensions to the graphical framework

BIC(λ) = L(Θλ; X)− df(Θλ)log n

2

AIC(λ) = L(Θλ; X)− df(Θλ)

I Rely on asymptotic approximations, but still relevant forsmall data set

I Easily adapted to Liid, Liid,Ltime and multitask framework.


Outline





MixNetErdos-Renyi Mixture for Networks

The data is now the network itselfConsider A = (aij)i,j∈P , the adjacency matrix associated to Θ:

aij = 1{θij 6=0}.

Latent structure modeling (Daudin et al., 2008)Spread the nodes on a set Q = {1, . . . , q, . . . , Q} of classes withI α a Q–size vector giving αi = P(i ∈ q),I ziq = 1{i∈q} are independent hidden variables Zi ∼M(1,α),I π a Q×Q matrix giving πq` = P(aij = 1|i ∈ q, j ∈ `).

Connexion probabilities depends on the node class belonging:

aij |{ZiqZj` = 1} ∼ B(πq`).


Variational inference

PrincipleApproximate P(Z|A,α,π) by Rτ (Z) chosen to minimize

KL(Rτ (Z);P(Z|A,α,π)),

where Rτ is such as logRτ (Z) =∑

iq Ziq log τiq and τ are thevariational parameters to optimize.

Variational Bayes (Latouche et al.)

I Put appropriate priors on α and π,I Give good performances especially for the choice of Q

and is thus relevant in the SIMoNe context.


Outline





Network generation

Let fixI the number p = card(P) of nodes,I if the graph is directed or not.

Affiliation matrix A = (aij)i,j∈P

1. usual MixNet frameworkI the Q×Q matrix Π, with πq` = P(aij = 1|i ∈ q, j ∈ `),I the Q-size vector α with αq = P(i ∈ q).

2. constraint MixNet versionI the Q×Q matrix Π, with πq` = card{(i, j) ∈ P × P : i ∈ q, j ∈ `},I the Q-size vector α with αq = card({i ∈ P : i ∈ q})/p.


Gaussian data generation

The Θ? matrix1. for undirected case, Θ? is the concentration matrix

I compute the normalized Laplacian of A,I generate a symmetric pattern of random signs.

2. for directed case, Θ? represents the VAR(1) parametersI generate random correlations for aij 6= 0,I normalized by the eigen-value with greatest modulus,I generate a pattern of random signs.

The Gaussian sample X

1. for undirected case,I compute Σ? by pseudo-inversion of Θ?,I generate the multivariate Gaussian sample with Cholesky

decomposition of Σ?.2. for directed case,

I Θ? permits to generate a stable VAR(1) process.


Example 1: time-course data with star-pattern

Simulation settings

1. 50 networks with p = 100 edges, time series of length n = 100,2. two classes, hubs and leaves, with proportions α = (0.1, 0.9),3. P(hub to leaf) = 0.3, P(hub to hub) = 0.1, 0 otherwise.



Simulation settings


precision wocl.BIC precision wocl.AIC

0.2

0.4

0.6

0.8

Boxplot of Precision values, without and with structure inference

precision = TP/(TP+FP)



Simulation settings


recall wocl.BIC recall wcl.BIC recall wocl.AIC recall wcl.AIC

0.2

0.4

0.6

0.8

1.0

Boxplot of Recall values, without and with structure inference

recall = TP/P (power)



Simulation settings


●

●

●

●

●

●

●

●

●

●

●

●

fallout wocl.BIC fallout wcl.BIC fallout wocl.AIC fallout wcl.AIC

0.00

0.01

0.02

0.03

0.04

Boxplot of Fallout values, without and with structure inference

fallout = FP/N (type I error)


Example 2: steady-state, multitask framework

Simulating the tasks

1. generate a “ancestor” with p = 20 node and K = 20 edges,2. generate T = 4 children by adding and deleting δ edges,3. generate T = 4 Gaussian samples.

Figure: ancestor and children with δ perturbations


Multitask: simulation results

Precision/Recall curveprecision = TP/(TP+FP)


ROC curvefallout = FP/N (type I error)




0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

CoopLassoGroupLassoIntertwinedIndependentPooled

penalty: λmax −→ 0

recall

pre

cisi

on

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0



falloutre

ca

ll

Figure: nt = 25, δ = 1



0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0



recall

pre

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Outline





Breast cancerPrediction of the outcome of preoperative chemotherapy

Two types of patientsPatient response can be classified as

1. either a pathologic complete response (PCR),2. or residual disease (not PCR).

Gene expression data

I 133 patients (99 not PCR, 34 PCR)I 26 identified genes (differential analysis)


Pooling the data

cancer data: pooling approach

demo/cancer_pooled.swf


Multitask approach: PCR / not PCR

cancer data: graphical cooperative Lasso

demo/cancer_mtasks.swf


Conclusions

To sum-up

I SIMoNe embeds most state-of-the-art statistical methods forGGM inference based upon `1-penalization,

I both steady-state and time course data can be dealt with,I (hopefully) biologist-friendly R package.

PerspectivesAdding transversal tools such asI network comparison,I bootstrap to limit the number of false positives,I more critieria to choose the penalty parameter,I interface to Gene Ontology.


Publications

Ambroise, Chiquet, Matias, 2009.Inferring sparse Gaussian graphical models with latent structureElectronic Journal of Statistics, 3, 205-238.

Chiquet, Smith, Grasseau, Matias, Ambroise, 2009.SIMoNe: Statistical Inference for MOdular NEtworks Bioinformatics,25(3), 417-418.

Charbonnier, Chiquet, Ambroise, 2010.Weighted-Lasso for Structured Network Inference from TimeCourse Data., SAGMB, 9.

Chiquet, Grandvalet, Ambroise, arXiv preprint.Inferring multiple Gaussian graphical models.


Publications

Ambroise, Chiquet, Matias, 2009.Inferring sparse Gaussian graphical models with latent structureElectronic Journal of Statistics, 3, 205-238.

Chiquet, Smith, Grasseau, Matias, Ambroise, 2009.SIMoNe: Statistical Inference for MOdular NEtworks Bioinformatics,25(3), 417-418.

Charbonnier, Chiquet, Ambroise, 2010.Weighted-Lasso for Structured Network Inference from TimeCourse Data., SAGMB, 9.

Chiquet, Grandvalet, Ambroise, arXiv preprint.Inferring multiple Gaussian graphical models.

Working paper: Chiquet, Charbonnier, Ambroise, Grasseau.SIMoNe: An R package for inferring Gausssian networks withlatent structure, Journal of Statistical Softwares.

Working paper: Chiquet, Grandvalet, Ambroise, Jeanmougin.Biological analysis of breast cancer by multitasks learning.


SIMoNe: Statistical Iference for MOdular NEtworks

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