Structured Regularization for conditional Gaussian graphical model

Structured Regularization for conditional Gaussian
Graphical Models
Julien Chiquet, St´ephane Robin, Tristan Mary-Huard
MAP5 – April the 4th, 2014
arXiv preprint http://arxiv.org/abs/1403.6168
Application to Multi-trait genomic selection (MLCB 2013 NIPS Workshop)
R-package spring https://r-forge.r-project.org/projects/spring-pkg/.
1

Multivariate regression analysis
Consider n samples and let for individual i
yi be the q-dimensional vector of responses,
xi be the p-dimensional vector of predictors,
B be the p × q matrix of regression coeﬃcients
εi be a noise term with a q-dimensional covariance matrix R.
yi = BT
xi + εi , εi ∼ N(0, R), ∀i = 1, . . . , n,
Matrix notation
Let Y(n × q) and X(n × p) be the data matrices, then
Y = XB + ε, vec(ε) ∼ N(0, Ip ⊗ R).
Remark
If X is a design matrix, this is called the“General Linear Model”(GLM).
2

Motivating example: cookie dough data
Osborne, B.G., Fearn, T., Miller, A.R., and Douglas, S.
Application of near infrared reﬂectance spectroscopy to compositional analys is of
biscuits and biscuit doughs. J. Sci. Food Agr., 1984.
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10
20
30
40
50
fat sucrose dry flour water
variable
fat
sucrose
dry flour
water
responses
0.8
1.2
1.6
1500 1750 2000 2250
position
reflectance
predictors
q = 4 responses related to the composition of biscuit dough.
p = 256 wavelengths equally sampled between 1380nm and 2400nm.
n = 70 biscuit dough samples. 3

From Low to High dimensional setup
Low dimensional setup
Mardia, Kent and Bibby, Multivariate Analysis, Academic Press, 1979.
Mathematics is the same for both GLM and MLR.
Application of maximum likelihood, least squares and generalized
least squares lead to an estimator which is not deﬁned when n < p
ˆB = XT
X
−1
XT
Y.
High dimensional setup: regularization is a popular answer
Biais B towards a given feasible set to enhance both prediction
performance and interpretability.
What features are required for the coeﬃcients? (sparsity, and. . . )
How do we shape the feasible this set?
4

From Low to High dimensional setup
Low dimensional setup
Mardia, Kent and Bibby, Multivariate Analysis, Academic Press, 1979.
Mathematics is the same for both GLM and MLR.
Application of maximum likelihood, least squares and generalized
least squares lead to an estimator which is not deﬁned when n < p
ˆB = XT
X
−1
XT
Y.
High dimensional setup: regularization is a popular answer
Biais B towards a given feasible set to enhance both prediction
performance and interpretability.
What features are required for the coeﬃcients? (sparsity, and. . . )
How do we shape the feasible this set?
Our proposal: SPRINGa
a
Structured selection of Primordial Relationships IN the General linear model
1. account for the dependency structure between the outputs if it exists
by estimating R
2. pay attention to the direct links between predictors and responses
by means of sparse GGM
3. Integrate some prior information about the predictors
by means of graph-regularization
4

Outline
Statistical Model
Regularizing Scheme and optimization
Inference and Optimization
Simulation Studies
Spectroscopy and the cookie dough data
Multi-trait genomic selection for a biparental population (Colza)
5

Outline
Statistical Model
Simulation Studies
6

Connection between multivariate regression and GGM (I)
Multivariate Linear Regression (MLR)
The model writes
yi |xi ∼ N(BT
xi , R).
with negative log-likelihood
− log L(B, R) =
n
2
log |R| +
1
2
tr (Y − XB)R−1
(Y − XB)T
+ cst,
which is only bi-convex in (B, R).
7

Connection between multivariate regression and GGM (II)
Used in Sohn & Kim (2012) and others
Assume that xi , yi are centered and jointly Gaussian such as
xi
yi
∼ N(0, Σ), with Σ =
Σxx Σxy
Σyx Σyy
, Ω Σ−1
=
Ωxx Ωxy
Ωyx Ωyy
.
A Convex loglikelihood
The model writes yi |xi ∼ N −Ω−1
yyΩyxxi , Ω−1
yy and
−
2
n
log L(Ωxy, Ωyy) = − log |Ωyy| + tr (SyyΩyy)
+ 2tr (SxyΩyx) + tr(ΩyxSxxΩxyΩ−1
yy) + cst.
(with Sxx = XT X/n and so on)
8

CGGM: interpretation (I)
Matrix Ω is related to partial correlations (direct links).
corij|i,j =
−Ωij
Ωii Ωjj
, so Ωij = 0 ⇔ Xi ⊥⊥ Xj |i, j.
Linking parameters of MLR to cGGM
The cGGM“splits”the regression coeﬃcients into two parts
B = −ΩxyΩ−1
yy, R = Ω−1
yy.
1. Ωxy describes the direct links between predictors and responses
2. Ωyy is the inverse of the residual covariance R
B entails both direct and indirect links, the latter due to correlation
between responses
9

CGGM: interpretation (II)
Illustrative examples
NO structure along the predictors
Ωxy: p = 40 predictors, q = 5 outcomes.
R: toeplitz scheme Rij = τ|i−j|
B = −ΩxyR.
Direct relationships are masked in B in case of
strong correlations between the responses.
Ωxy
10

R: toeplitz scheme Rij = 0.1|i−j|
B = −ΩxyR.
Rlow
Ωxy Blow
10

B = −ΩxyR.
Rmed
Ωxy Bmed
10

B = −ΩxyR.
Rhigh
Ωxy Bhigh
10

Strong structure along the predictors
R: toeplitz scheme Rij = τ|i−j|
B = −ΩxyR.
R
Ωxy B
10

B = −ΩxyR.
Rlow
Ωxy Blow
10

B = −ΩxyR.
Rmed
Ωxy Bmed
10

B = −ΩxyR.
Rhigh
Ωxy Bhigh
10

along the predictors
R: toeplitz scheme Rij = |i−j|
B = −ΩxyR.
R
Ωxy B
Consequence
Our regularization scheme will be applied on the direct links Ωxy.
Remarks
sparsity on Ωxy does not necessarily induce sparsity on B.
The prior structure on the predictors is identical on Ωxy and B as it
applies on the ”rows”
10

Outline
Statistical Model
Simulation Studies
11

Ball crafting towards structured regularization (1)
Elastic-Net
Grouping eﬀect that catches highly
correlated predictors simultaneously
minimize
β∈Rp
||Xβ − y||2
2
+ λ1 ||β||1
+ λ2 ||β||2
2 .
Zou, H. and Hastie, T.
Regularization and variable selection via the elastic net. JRSS B, 2005.
12

Fused-Lasso
Encourages sparsity and identical
consecutive parameters.
minimize
β∈Rp
||Xβ − y||2
2
+ λ1 ||β||1
+ λ2
p−1
j=1
|βj+1 − βj |.
Tibshirani, R., Saunders, M., Rosset, S., Zhu, J. and Knight, K.
Sparsity and smoothness via the fused lasso. JRSS B, 2005.
13

Fused-Lasso
Encourages sparsity and identical
consecutive parameters.
minimize
β∈Rp
||Xβ − y||2
2
+ λ1 ||β||1
+ λ2 ||Dβ||1 ,
with
D =



1 ... ... p
1 −1 1
...
...
...
p−1 −1 1



13

Structured/Generalized Elastic-Net
A“smooth”version of fused-Lasso
(neighbors should be close, not
identical).
minimize
β∈Rp
||Xβ − y||2
2
+ λ1 ||β||1
+ λ2
p−1
j=1
(βj+1 − βj )2
.
Slawski, zu Castell and Tutz.
Feature selection guided by structural information. Ann. Appl. Stat., 2010.
Hebiri and van De Geer
The smooth-lasso and other l1 + l2 penalized methods EJS, 2011.
14

Structured/Generalized Elastic-Net
A“smooth”version of fused-Lasso
(neighbors should be close, not
identical).
minimize
β∈Rp
||Xβ − y||2
2
+ λ1 ||β||1
+ λ2 βT
DT
Dβ.
L = DT
D =









1 ... ... ... p
1 1 −1
... −1 2 −1
...
...
...
...
...
−1 2 1
p −1 1









.
14

Generalized fused penalty: the univariate case
Graphical interpretation of the fusion penalty
p−1
j=1
|βj+1 − βj |
Fused Lasso
p−1
j=1
(βj+1 − βj )2
Generalized ridge
A chain graph between the successive (ordered) predictors
Generalization via a graphical argument
Let G = (E, V, W) be a graph with weighted edges. Then
(i,j)∈E
wij |βj − βi | = Dβ 1
(i,j)∈E
wij (βj − βi )2
= βT
Lβ,
L = DT D 0 is the graph Laplacian associated o G
15

Adapting this scheme to multivariate settings
Bayesian Interpretation
Suppose the prior structure is encoded in a matrix L.
Univariate case: the conjugate prior for β is N(0, L−1).
Multivariate case: combine with the covariance, then
vec(B) ∼ N(0, R ⊗ L−1
).
Using vec and ⊗ properties, we have for the direct links
vec(Ωxy) ∼ N(0, R−1
⊗ L−1
).
Corresponding regularization term
log P(Ωxy|L, R) =
1
2
tr ΩT
xyLΩxyR + cst.
16

Outline
Statistical Model
Simulation Studies
17

Optimization problem
Penalized criterion
Encourage sparsity with structuring prior on the direct links:
J(Ωxy, Ωyy) =
−
1
n
log L(Ωxy, Ωyy) +
λ2
2
tr ΩyxLΩxyΩ−1
yy + λ1 Ωxy 1.
Proposition
The objective function is jointly convex in (Ωxy, Ωyy) and admits at
least one global minimum which is unique when n ≥ q and (λ2L + Sxx)
is positive deﬁnite.
18

Algorithm
Alternate optimization
ˆΩ
(k+1)
yy = arg min
Ωyy 0
Jλ1λ2 ( ˆΩ
(k)
xy , Ωyy), (1a)
ˆΩ
(k+1)
xy = arg min
Ωxy
Jλ1λ2 (Ωxy, ˆΩ
(k+1)
yy ). (1b)
(1a) boilds down to the diagonalization of a q × q matrix.
O(q3)
(1b) can be recast as generalized Elastic-Net with size pq.
O(npqk) where k is the ﬁnal number of nonzero entries in ˆΩxy
Convergence
Despite nonsmoothness of the objective, the 1 penalty is separable in
(Ωxy, Ωyy) and results of Tseng (2001, 2009) on convergence of
coordinate descent apply.
19

First block: covariance estimation
Analytic resolution of ˆR
If Ωxy = 0, then Ωyy = S−1
yy. Otherwise we rely on the following:
Proposition
Let n > q. Assume that the following eigen decomposition holds
ˆΩyx
ˆΣ
λ2
xx
ˆΩxySyy = Udiag(ζ)U−1
and denote by η = (η1, . . . , ηq) the roots of η2
j − ηj − ζj . Then
ˆΩyy = Udiag(η/ζ)U−1 ˆΩyx
ˆΣ
λ2
xx
ˆΩxy(= ˆR−1
), (2a)
ˆΩ
−1
yy = SyyUdiag(η−1
)U−1
(= ˆR). (2b)
Proof.
Diﬀerentiation of the objective, commuting matrices property,
algebra.
20

Second block: parameters estimation
Reformulation as a Elastic-Net problem
Proposition
Solution ˆΩxy for a ﬁx ˆΩyy is given by vec( ˆΩxy) = ˆω where ˆω solves the
Elastic-Net problem
arg min
ω∈Rpq
1
2
Aω − b 2
2 + λ1 ω 1
λ2
2
ωT ˆΩ
−1
yy ⊗ L ω,
where A and b are deﬁned thanks to the Cholesky decomposition
CT C = ˆΩ
−1
yy, so as
A = C ⊗ X/
√
n , b = −vec YC−1/
√
n
T
.
Proof.
Algebra with bad vec/tr/⊗ properties.
21

Monitoring convergence
Example on the cookies dough data
0 50 100 150 200 250
−5−4−3−2−10
iteration
objective
3e−01
2e−01
1e−01
1e−01
7e−02
4e−02
3e−02
2e−02
1e−02
9e−03
6e−03
4e−03
3e−03
2e−03
1e−03
8e−04
5e−04
4e−04
2e−04
2e−04
λ1
Figure: monitoring the objective along the whole path of λ1
22

Monitoring convergence
Example on the cookies dough data
0 50 100 150 200 250
50100150200
iteration
loglikelihood
3e−01
2e−01
1e−01
1e−01
7e−02
4e−02
3e−02
2e−02
1e−02
9e−03
6e−03
4e−03
3e−03
2e−03
1e−03
8e−04
5e−04
4e−04
2e−04
2e−04
λ1
Figure: monitoring the likelihood along the whole path of λ1
22

Tuning the penalty parameters
K−fold cross-Validation
Computationally intensive, but works! For κ : {1, . . . , n} → {1, . . . , K},
(λcv
1 , λcv
2 ) = arg min
(λ1,λ2)∈Λ1×Λ2
1
n
n
i=1
xT
i
ˆBλ1,λ2
−κ(i) − yi
2
2
.
Information criteria adapted to regularized methods
(λpen
1 , λpen
2 ) = arg min
λ1,λ2
−2 log L( ˆΩ
λ1,λ2
xy , ˆΩ
λ1,λ2
yy ) + pen (dfλ1,λ2 ) .
Proposition (Generalized degrees of freedom)
dfλ1,λ2 = card(A) − λ2tr ( ˆR ⊗ L)AA( ˆR ⊗ (Sxx + λ2L))−1
AA ,
where A = j : vec ˆΩ
λ1,λ2
xy = 0 , the set of active guys.
23

Tuning the penalty parameters
K−fold cross-Validation
Computationally intensive, but works! For κ : {1, . . . , n} → {1, . . . , K},
(λcv
1 , λcv
2 )
?
= arg min
(λ1,λ2)∈Λ1×Λ2
1
n
n
i=1
log L( ˆΩ
λ1,λ2
xy , ˆΩ
λ1,λ2
yy ; xi , yi ).
Information criteria adapted to regularized methods
(λpen
1 , λpen
2 ) = arg min
λ1,λ2
−2 log L( ˆΩ
λ1,λ2
xy , ˆΩ
λ1,λ2
yy ) + pen (dfλ1,λ2 ) .
Proposition (Generalized degrees of freedom)
dfλ1,λ2 = card(A) − λ2tr ( ˆR ⊗ L)AA( ˆR ⊗ (Sxx + λ2L))−1
AA ,
where A = j : vec ˆΩ
λ1,λ2
xy = 0 , the set of active guys.
23

Outline
Statistical Model
Simulation Studies
24

Assessing gain brought by covariance estimation
Simulation settings
Parameters
p = 40 predictors, q = 5 outcomes.
Ωxy: 25 non null entries in {−1, 1}
no particular structure along the predictors
R: toeplitz scheme
Rij = τ|i−j| with τ ∈ {0.1, 0.5, 0.9}.
B = −ΩxyR.
Data generation
Draw ntrain = 50 + ntest = 1000 samples from
yi = BT
xi +εi , with xi ∼ N(0, I) and εi ∼ N(0, R).
Evaluating performance
Compare prediction error on 100 runs between
Lasso, group-Lasso and SPRING.
Ωxy
25

Simulation settings
Parameters
R: toeplitz scheme
Rij = τ|i−j| with τ ∈ {0.1, 0.5, 0.9}.
B = −ΩxyR.
Data generation
yi = BT
Rlow
Ωxy Blow
25

Simulation settings
Parameters
R: toeplitz scheme
Rij = τ|i−j| with τ ∈ {0.1, 0.5, 0.9}.
B = −ΩxyR.
Data generation
yi = BT
Rmed
Ωxy Bmed
25

Simulation settings
Parameters
R: toeplitz scheme
Rij = τ|i−j| with τ ∈ {0.1, 0.5, 0.9}.
B = −ΩxyR.
Data generation
yi = BT
Rhigh
Ωxy Bhigh
25

Simulation settings
Parameters
R: toeplitz scheme
Rij = τ|i−j| with τ ∈ {0.1, 0.5, 0.9}.
B = −ΩxyR.
Data generation
yi = BT
R
Ωxy B
25

Results
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q
1
2
3
4
low medium high
PredictionError
estimator
spring (oracle)
spring
lasso
group−lasso
Figure: Prediction error for 100 runs illustrates the inﬂuence of correlations
between outcomes. Scenarios {low, med, high} map to τ ∈ {.1, .5, .9}
26

Assessing gain brought by structure integration
Simulation settings
Parameters
p = 100, q = 1 to remove covariance eﬀect.
Ωxy ωxy: a vector with two successive bumps
ωj =



−((30 − j)2 − 100)/200 j = 21, . . . 39,
((70 − j)2 − 100)/200 j = 61, . . . 80,
0 otherwise.
−0.50
−0.25
0.00
0.25
0.50
0 25 50 75 100
27

Assessing gain brought by structure integration
Simulation settings
Parameters
p = 100, q = 1 to remove covariance eﬀect.
Ωxy ωxy: a vector with two successive bumps
ωj =



−((30 − j)2 − 100)/200 j = 21, . . . 39,
((70 − j)2 − 100)/200 j = 61, . . . 80,
0 otherwise.
R ρ = 5: a residual (scalar) variance.
β = −ωxy/ρ.
Data generation
yi = βT
xi + εi , with xi ∼ N(0, I) and εi ∼ N(0, ρ).
27

Assessing gain brought by structure information
Results (1): predictive performance
40
80
120
160
0.001 0.100
penalty level λ1 (scaled)
PredictionError
estimator
spring (λ2 = .01)
spring (λ2 = .00)
lasso
Figure: Mean PE + standard error for 100 runs on a grid of λ1 – SPRING with
and without structural regularization (L = D D) and Lasso 28

Assessing gain brought by structure information
Results (2): robustness
What if we introduce a“wrong”structure?
Evaluate performance with the same settings but
randomly swap all elements in ωxy to remove any structure.
keep exactly the same xi , εi ,
draw yi with swapped an unswapped parameters,
use the same folds for cross-validation,
then replicate 100 times.
Method Scenario MSE PE
LASSO – .336 (.096) 58.6 (10.2)
E-Net (L = I) – .340 (.095) 59 (10.3)
SPRING (L = I) – .358 (.094) 60.7 (10)
S. E-net unswapped .163 (.036) 41.3 ( 4.08)
(L = DT
D) swapped .352 (.107) 60.3 (11.42)
SPRING unswapped .062 (.022) 31.4 ( 2.99)
(L = DT
D) swapped .378 (.123) 62.9 (13.15)
29

Outline
Statistical Model
Simulation Studies
30

Cookie dough data: performance
Method fat sucrose ﬂour water
Step. MLR .044 1.188 .722 .221
Decision th. .076 .566 .265 .176
PLS .151 .583 .375 .105
PCR .160 .614 .388 .106
Bayes. Reg. .058 .819 .457 .080
LASSO .045 .860 .376 .104
grp LASSO .127 .918 .467 .102
str E-net .039 .666 .365 .100
MRCE .151 .821 .321 .081
SPRING (CV) .065 .397 .237 .083
SPRING (BIC) .048 .389 .243 .066
Table: Test error
Brown, P.J., Fearn, T., and Vannucci, M.
Bayesian wavelet regression on curves with applications to a spectroscopic
calibration problem. JASA, 2001.
31

Step. MLR .044 1.188 .722 .221
Decision th. .076 .566 .265 .176
PLS .151 .583 .375 .105
PCR .160 .614 .388 .106
Bayes. Reg. .058 .819 .457 .080
LASSO .045 .860 .376 .104
grp LASSO .127 .918 .467 .102
str E-net .039 .666 .365 .100
MRCE .151 .821 .321 .081
SPRING (CV) .065 .397 .237 .083
SPRING (BIC) .048 .389 .243 .066
Table: Test error
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q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
qq
q
qq
q
qqq
q
qqqqq
q
q
q
q
qqqqq
q
qqq
q
qqq
q
qqqq
q
qqqq
−100
0
100
1500 1750 2000 2250
The Lasso induces sparsity on B
No structure along the predictors.
No structure between responses.
31

Step. MLR .044 1.188 .722 .221
Decision th. .076 .566 .265 .176
PLS .151 .583 .375 .105
PCR .160 .614 .388 .106
Bayes. Reg. .058 .819 .457 .080
LASSO .045 .860 .376 .104
grp LASSO .127 .918 .467 .102
str E-net .039 .666 .365 .100
MRCE .151 .821 .321 .081
SPRING (CV) .065 .397 .237 .083
SPRING (BIC) .048 .389 .243 .066
Table: Test error
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
qqqqqqqqqqqqqqqqqqq
q
q
q
qqqqqqqqqqqqqq
qqqqqqqqqqqq
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
qqqqqqqqqqqqqq
q
q
qqqqqqqqqq
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqq
q
q
qqqqqqqqqq
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqq
q
q
qqqqqqqqqq
q
q
q
q
qqqqq
−100
−50
0
50
100
150
1500 1750 2000 2250
The Group-Lasso induces sparsity on B group-wise across the responses
(Too) strong structure between responses.
31

Step. MLR .044 1.188 .722 .221
Decision th. .076 .566 .265 .176
PLS .151 .583 .375 .105
PCR .160 .614 .388 .106
Bayes. Reg. .058 .819 .457 .080
LASSO .045 .860 .376 .104
grp LASSO .127 .918 .467 .102
str E-net .039 .666 .365 .100
MRCE .151 .821 .321 .081
SPRING (CV) .065 .397 .237 .083
SPRING (BIC) .048 .389 .243 .066
Table: Test error
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
qqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
qqqqqqqq
q
q
q
q
q
qqqqqqqqqqqqqqqq
q
q
qq
q
q
qqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
qqqqqqqqqqqqqq
q
qqqqqq
q
q
q
qqqq
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
qq
q
q
q
qqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
qqqqq
q
q
qq
q
q
qqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
qqqqqqqqq
q
qqq
q
q
q
q
qq
q
q
qqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
qqqqqqqqqqqq
q
qqq
q
q
qq
q
qqq
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqq
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
−20
−10
0
10
20
30
1500 1750 2000 2250
The Structured Elastic-Net induces sparsity on B with a smooth
neighborhood prior along the predictors (L = DT D)
Structure along the predictors.
No structure between responses. 31

Step. MLR .044 1.188 .722 .221
Decision th. .076 .566 .265 .176
PLS .151 .583 .375 .105
PCR .160 .614 .388 .106
Bayes. Reg. .058 .819 .457 .080
LASSO .045 .860 .376 .104
grp LASSO .127 .918 .467 .102
str E-net .039 .666 .365 .100
MRCE .151 .821 .321 .081
SPRING (CV) .065 .397 .237 .083
SPRING (BIC) .048 .389 .243 .066
Table: Test error
qqq
q
q
q
q
qqqqqq
q
q
q
q
q
q
q
q
qqqqqqqqqqq
q
q
q
qqqq
q
q
q
qq
q
q
q
q
q
q
qq
q
q
q
qqq
q
q
q
q
q
q
q
q
qqqq
q
qq
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
qq
q
q
q
qq
q
q
qq
qqqqqqqqqq
q
q
q
q
q
q
qq
q
q
qqqqqqqqqqqqqq
q
q
q
q
q
qq
q
q
q
qq
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
qqqqqqqqq
q
q
q
q
q
q
q
q
q
qq
q
qqq
q
q
q
q
qqqqqqqqq
q
q
q
q
q
q
q
q
q
qq
q
qqqqqqq
q
q
q
q
qqq
qq
q
q
q
qqqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
qqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
qqqq
q
q
q
q
q
q
qq
qq
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
qqqqq
q
q
qqqqq
q
q
q
q
qqqqqqqqqqq
q
q
q
q
q
q
qqqqqqqq
q
q
q
qqqqqqqq
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqq
q
q
q
qqqqq
q
qq
q
qq
qqqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
qqqqqqqqqqq
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
qq
q
q
q
qqqq
q
q
q
q
q
q
q
qqq
q
q
q
qq
qqqqqqqqqqqqqqq
q
q
q
qqq
q
q
q
qqqq
qq
q
q
q
qqqqqqq
q
q
q
q
q
q
q
qqqqqqqqq
q
q
q
q
qq
q
q
q
qq
q
qq
q
q
q
q
q
qq
q
q
q
q
qqq
q
qq
q
q
qqqqqqqqq
q
q
qq
qq
q
q
q
qqqqq
q
q
q
q
qqqqqq
q
q
qq
q
q
qq
q
qq
q
q
q
qqqqqqqqqq
q
q
q
q
q
q
q
qq
q
q
q
q
q
qqq
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
qqqqqqqqq
q
q
qqqqqqqqqqqqqqqqq
q
q
q
q
qqqq
−20
−10
0
10
20
30
1500 1750 2000 2250
MRCE induces sparsity on B and sparsity on R−1
(Supposed to add) Structure between responses.
31

Step. MLR .044 1.188 .722 .221
Decision th. .076 .566 .265 .176
PLS .151 .583 .375 .105
PCR .160 .614 .388 .106
Bayes. Reg. .058 .819 .457 .080
LASSO .045 .860 .376 .104
grp LASSO .127 .918 .467 .102
str E-net .039 .666 .365 .100
MRCE .151 .821 .321 .081
SPRING (CV) .065 .397 .237 .083
SPRING (BIC) .048 .389 .243 .066
Table: Test error
q
q
q
q
q
q
q
q
qqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
qqqqqqqqqqqqqq
q
q
qqqqqqqqqqqqq
qqqqqqqqqqq
q
q
q
q
q
q
q
qqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
qqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqq
q
q
q
q
q
qqqqqq
q
q
qqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
qqqq
q
q
qq
q
q
qqqqqq
q
qqq
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
qqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqq
q
q
qqqqqqq
qqq
q
q
q
q
qqq
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
qqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
qqqqq
q
qqq
−20
−10
0
10
20
30
1500 1750 2000 2250
SPRING
Use GGM to induce structured sparsity on the direct links between
the responses and the predictors + smooth neighborhood prior via
L = DT D.
31

Cookie dough data: parameters
ˆB − ˆΩxy
ˆR
q
q
q
q
q
q
q
q
qqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
qqqqqqqqqqqqqq
q
q
qqqqqqqqqqqqq
qqqqqqqqqqq
q
q
q
q
q
q
q
qqqqqqqq
q
q
q
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q
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q
q
q
q
q
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q
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qqqqqq
q
q
q
q
q
q
q
q
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q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqq
q
q
q
q
q
qqqqqq
q
q
qqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
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q
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q
q
q
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q
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q
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q
q
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q
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qqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
qqqq
q
q
qq
q
q
qqqqqq
q
qqq
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
qqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqq
q
q
qqqqqqq
qqq
q
q
q
q
qqq
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
qqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
qqqqq
q
qqq
−20
−10
0
10
20
30
1500 1750 2000 2250
q
q
q
q
q
q
q
q
q
q
qqqq
q
q
q
q
q
q
q
q
q
qqqq
q
q
q
q
q
q
q
q
q
q
qqqq
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
qqqqqqqqqq
q
q
q
q
q
qq
q
q
q
q
q
qqqqqqqqqq
qqq
qqqqqqqqqq
qqqqq
q
q
q
q
qqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqq
q
q
q
q
q
q
q
q
qqqqqqqqqq
q
q
qqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqq
q
q
q
q
q
qqq
q
q
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q
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q
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q
q
qqqq
q
q
q
qqq
q
q
q
q
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q
q
q
q
q
q
qq
q
q
q
q
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q
q
q
q
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q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqq
q
q
qqqq
q
q
q
qq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
qqq
q
q
qqq
q
q
q
q
qqqq
q
q
q
q
q
q
q
q
qqqqqqqq
q
q
q
q
q
qqqqqqq
q
q
q
q
q
q
q
q
q
q
q
qqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqq
q
q
qq
q
q
q
qq
q
q
qq
q
qq
q
q
qqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
qqqq
q
q
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
qq
q
q
q
q
q
q
q
qqqqqqqqqqqqqqqqq
q
q
q
q
qq
q
q
q
q
q
q
q
−20
−10
0
10
20
30
1500 1750 2000 2250
dry flour
fat
sucrose
water
dryflour
fat
sucrose
water
−0.25
0.00
0.25
0.50
value
32

Cookie dough data: model selection with BIC
q
q
q
q
q
q
q
q
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qq
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q
−700
−600
−500
−400
0.001 0.100
log10(λ1)
criterion'svalue
λ2
q
q
q
q
0.01
0.1
1
10
BIC
33

Outline
Statistical Model
Simulation Studies
34

Quantitative Trait Loci (QTL) study in Colza
Doubled haployd samples
n = 103 homozygous lines of Brassica napus by crossing ‘Stellar’
and ‘Major’ cultivars.
Bi-parental markers
p = 300 markers with known loci dispatched on the 19 chromosomes
with value in {Major, Stellar, Missing} → {1, −1, 0}.
Traits
Consider q = 8 traits including
survival traits (% survival in winter)
surv92, surv93, surv94, surv97, surv99
ﬂowering traits (no vernalization, 4 weeks or 8 weeks vernalization)
flower0, flower4, flower8
35

Include genetic linkage information
Genetic distance between markers A1 and A2
Let r12 be the recombination rate between A1 and A2, then
d12 = −
1
2
log {1 − 2r12} .
Linkage disequilibrium as covariance between the markers
In a biparental population with independent recombination events, one
has
cor(A1, A2) = ρd13
= ρd12+d13
, with ρ = e−2
.
Proposition (Including LD information in the model)
The matrix L is given by inverting the covariance matrix, which can be
done analytically.
36

Analytical form of L as a precision matrix
Usually met in AR(1) processes
L is given by the inverse of the correlation between the markers
L = UT
ΛU
with
U =











1 −ρd12 0 . . . 0 0
0 1 −ρd23 . . . 0 0
0 0 1
... 0 0
...
...
...
...
...
...
0 0 0 . . . 1 −ρdm−1m
0 0 0 . . . 0 1











Λ =











(1 − ρ2d12 )−1 0 . . . . . . . . . 0
0 (1 − ρ2d23 )−1 0 . . . . . . 0
0 0
...
...
...
...
...
...
0 0 0 . . . (1 − ρ2dm−1m )−1 0
0 0 0 . . . 0 1











.
37

Predictive performance
1. Split the data into training/test sets (n1 = 70, n2 = 33),
2. Adjust each procedure using 5-fold CV for model selection,
3. Compute test (prediction) error.
Method surv92 surv93 surv94 surv97 surv99 Mean PE
LASSO .79 .98 .90 1.02 1.00 .938
group-LASSO .90 1.00 .92 .99 .92 .946
Enet (no LD) .87 1.01 .97 1.03 1.03 .983
Gen-Enet LD) .75 .98 .89 1.03 1.02 .934
our proposal (LD) .77 .96 .84 1.00 1.02 .918
Table: Survival traits
Method flower0 flower4 flower8 Mean PE
LASSO .58 .53 .74 .616
group-LASSO .59 .55 .74 .626
Enet (no LD) .55 .54 .69 .593
Gen-Enet (LD) .55 .50 .74 .596
our proposal (LD) .48 .46 .68 .54
Table: Flowering traits
38

Estimated Residual Covariance ˆR
flower0
flower4
flower8
surv92
surv93
surv94
surv97
surv99
flower0 flower4 flower8 surv92 surv93 surv94 surv97 surv99
−1.0
−0.5
0.0
0.5
1.0
correlation
39

Estimated Regression Coeﬃcients ˆB
−0.1
0.0
0.1
0 500 1000 1500
position of the markers
outcomes
surv92
surv93
surv94
surv97
surv99
flower0
flower4
flower8
40

Estimated Direct Eﬀects ˆΩxy
−0.1
0.0
0.1
0 500 1000 1500
position of the markers
outcomes
surv92
surv93
surv94
surv97
surv99
flower0
flower4
flower8
41

QTL Mapping (chr. 2, 8, 10), regression coeﬃcients ˆB
ec2e5a
E33M49.117
ec3b12wg2d11b
wg1g8a
E32M49.73
ec3a8
wg7f3a
E33M59.59
E35M62.117wg6b10
wg8g1b
wg5a5
tg6a12
Aca1E38M50.133
E35M59.117
ec2d1a
wg1a10tg2h10b
tg2f12
wg4b6b
wg6g9E33M59.147eru1ec4h3E33M62.99E38M50.157
wg6d9
E38M62.189tg3c1ec3d3bE33M49.175E33M48.268E35M62.80E35M48.143
wg1g4a
E33M47.182b
E38M50.119wg7b3
E33M59.64
ec3g3c
ec2h2bE32M48.212
q
q
q
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q
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q
0
40
80
120
2 8 10
loci
outcomes
q
q
q
q
q
q
q
q
surv92
surv93
surv94
surv97
surv99
flower0
flower4
flower8
42

QTL Mapping (chr. 2, 8, 10), direct links ˆΩxy
ec2e5a
E33M49.117
ec3b12wg2d11b
wg1g8a
E32M49.73
ec3a8
wg7f3a
E33M59.59
E35M62.117wg6b10
wg8g1b
wg5a5
tg6a12
Aca1E38M50.133
E35M59.117
ec2d1a
wg1a10tg2h10b
tg2f12
wg4b6b
wg6g9E33M59.147eru1ec4h3E33M62.99E38M50.157
wg6d9
E38M62.189tg3c1ec3d3bE33M49.175E33M48.268E35M62.80E35M48.143
wg1g4a
E33M47.182b
E38M50.119wg7b3
E33M59.64
ec3g3c
ec2h2bE32M48.212
q
q
q
q
q
q
q
q
q
qqq
q
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q q
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qq
q
q q
q
q
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qq
q
q
0
40
80
120
2 8 10
loci
outcomes
q
q
q
q
q
q
q
surv92
surv94
surv97
surv99
flower0
flower4
flower8
42

QTL Mapping (all chromosomes), ˆB
wg1h4c
wg1g5c
tg5e11b
E35M47.262
tg6c3aE32M48.249E33M50.252
E35M48.120
ec5d5
wg1g10a
E33M48.369E33M50.451
tg1f8
wg7b6a
isoACO
ec4f1
ec2e5a
E33M49.117
ec3b12wg2d11b
wg1g8a
E32M49.73wg3g11
ec3a8
wg7f3a
E33M59.59
E35M62.117wg6b10
wg8g1b
wg5a5
tg6a12
Aca1E38M50.133
E35M59.117
ec2d1a
wg1a10tg2h10b
tg2f12
E33M49.491E38M62.229wg1g10b
ec4h7
wg4d10
E32M50.409
E38M50.159
E33M47.338
ec2d8a
E33M49.165
wg4f4cE35M48.148
wg6c6ec4g7bwg7a11
wg5b1a
ec4e7a
E32M61.218
wg4a4b
E33M50.183E33M62.140
wg9c7
wg6b2
E33M49.211
wg2e12b
isoIdh
wg3f7c
ec3b4
E33M62.206wg5b2
E32M59.302
wg6a12
wg4d7c
ec4c5b
E32M61.166
E32M47.136
E32M62.107
wg6f10
ec5e12c
E38M62.358
E35M62.256
ec5a7b
wg3c9
E33M47.154E35M59.581
E32M47.460
ec4g7aec6b2
E35M62.111wg1g6
E35M62.201
ec4c5aec5a7awg1h5
wg6a10
E33M50.120ec4e8
E33M48.191
E32M47.168
E35M62.225E35M62.340wg1g8cE32M62.75E32M49.473E32M59.330
wg7e10
wg6h1b
wg2c1
tg5h12
wg3b6
wg7d9awg1g3b
wg7h2
wg9d5
E32M59.359
E33M59.353
E32M61.137
ec3h4
wg8g3wg2a11tg2b4
E35M47.367
ec2e4bE32M47.512ec2h2aLemtg5d9a
wg7f5a
wg5a1a
ec3e12a
wg4b6b
wg6g9
E33M59.147eru1ec4h3
E33M62.99
E38M50.157
wg6d9
E33M60.50
wg4h5a
wg3h8ec3d3aec2c7
wg4d11tg1h12
ec2e5bE38M62.461
wg3f7aE35M60.312
E38M62.189
tg3c1
ec3d3b
E33M49.175
E33M48.268E35M62.80
E35M48.143
wg1g4a
E33M47.182b
E38M50.119wg7b3
E33M59.64
ec3g3c
ec2h2b
E32M48.212
wg1g5b
tg6c3bec5a1wg6f3
E32M62.115
E33M62.250
E32M62.186
wg2b7
wg8h5
wg3h4
tg2h10a
tg5e11a
E32M50.90
ec2d1b
E32M50.77
wg1g4c
wg8g1a
wg2c3
wg7f3b
wg4h1
ec4e7bwg5a6
ec2c12
wg2d11a
ec2e12a
wg7a8a
isoLapE33M62.176
E35M48.84
E33M49.293
E35M62.136
eru2E38M50.186
ec4f11
E32M50.252
E32M59.107
wg1g8b
wg2g9
E33M50.282
E35M48.123
wg1e3
wg6d6wg4f4a
ec5c4E35M48.198
E35M62.135wg1a4
ec2e12bwg3h6
wg4d5a
wg5b1b
E33M61.54
ec3d2
E32M48.191E33M59.333
wg6e3b
ec4d11
E32M59.88
ec4g4
wg1g4b
ec3g12
ec3g3a
wg9f2
E35M62.222
wg4a4a
E33M59.234E33M61.84
wg4d7b
ec5e12bec4c11
wg6e3a
E32M48.69
ec3b2b
E32M47.186
ec4d9
wg4d5c
E33M48.67
E35M60.329
wg1h4b
E38M62.188
E32M50.261
E33M50.118
wg1g3a
E35M60.230
wg6a11wg6h1a
E32M62.241
E32M47.288
E33M48.316
E33M59.225ec2e4c
ec3e12b
wg5a1bwg2a3c
ec5e12a
wg7f5bE32M47.159
tg5d9bslg6E35M59.85
ec2d8b
E35M62.132
E35M47.337
E35M47.257
wg9e9
ec2b3E33M60.229
E32M50.325
wg6c1ec3b2a
E35M47.170
wg2d5a
E33M60.120
E33M47.115
wg1g10c
E33M62.130
E32M47.344
E32M50.255
wg3g9
E32M50.424
pr2
tg5b2
E33M59.165
E35M60.125
E33M62.196
ec2h2c
wg3f7b
ec3f1
E35M60.107
wg1g2
E33M48.346
E33M50.371
E33M47.138
tg4d2b
E32M62.394E33M47.189
E32M49.409
wg7b6b
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0
50
100
150
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
chromosomes
loci
outcomes
q
q
q
q
q
q
q
q
surv92
surv93
surv94
surv97
surv99
flower0
flower4
flower8
43

QTL Mapping (all chromosomes), ˆΩxy
wg1h4c
wg1g5c
tg5e11b
E35M47.262
tg6c3aE32M48.249E33M50.252
E35M48.120
ec5d5
wg1g10a
E33M48.369E33M50.451
tg1f8
wg7b6a
isoACO
ec4f1
ec2e5a
E33M49.117
ec3b12wg2d11b
wg1g8a
E32M49.73wg3g11
ec3a8
wg7f3a
E33M59.59
E35M62.117wg6b10
wg8g1b
wg5a5
tg6a12
Aca1E38M50.133
E35M59.117
ec2d1a
wg1a10tg2h10b
tg2f12
E33M49.491E38M62.229wg1g10b
ec4h7
wg4d10
E32M50.409
E38M50.159
E33M47.338
ec2d8a
E33M49.165
wg4f4cE35M48.148
wg6c6ec4g7bwg7a11
wg5b1a
ec4e7a
E32M61.218
wg4a4b
E33M50.183E33M62.140
wg9c7
wg6b2
E33M49.211
wg2e12b
isoIdh
wg3f7c
ec3b4
E33M62.206wg5b2
E32M59.302
wg6a12
wg4d7c
ec4c5b
E32M61.166
E32M47.136
E32M62.107
wg6f10
ec5e12c
E38M62.358
E35M62.256
ec5a7b
wg3c9
E33M47.154E35M59.581
E32M47.460
ec4g7aec6b2
E35M62.111wg1g6
E35M62.201
ec4c5aec5a7awg1h5
wg6a10
E33M50.120ec4e8
E33M48.191
E32M47.168
E35M62.225E35M62.340wg1g8cE32M62.75E32M49.473E32M59.330
wg7e10
wg6h1b
wg2c1
tg5h12
wg3b6
wg7d9awg1g3b
wg7h2
wg9d5
E32M59.359
E33M59.353
E32M61.137
ec3h4
wg8g3wg2a11tg2b4
E35M47.367
ec2e4bE32M47.512ec2h2aLemtg5d9a
wg7f5a
wg5a1a
ec3e12a
wg4b6b
wg6g9
E33M59.147eru1ec4h3
E33M62.99
E38M50.157
wg6d9
E33M60.50
wg4h5a
wg3h8ec3d3aec2c7
wg4d11tg1h12
ec2e5bE38M62.461
wg3f7aE35M60.312
E38M62.189
tg3c1
ec3d3b
E33M49.175
E33M48.268E35M62.80
E35M48.143
wg1g4a
E33M47.182b
E38M50.119wg7b3
E33M59.64
ec3g3c
ec2h2b
E32M48.212
wg1g5b
tg6c3bec5a1wg6f3
E32M62.115
E33M62.250
E32M62.186
wg2b7
wg8h5
wg3h4
tg2h10a
tg5e11a
E32M50.90
ec2d1b
E32M50.77
wg1g4c
wg8g1a
wg2c3
wg7f3b
wg4h1
ec4e7bwg5a6
ec2c12
wg2d11a
ec2e12a
wg7a8a
isoLapE33M62.176
E35M48.84
E33M49.293
E35M62.136
eru2E38M50.186
ec4f11
E32M50.252
E32M59.107
wg1g8b
wg2g9
E33M50.282
E35M48.123
wg1e3
wg6d6wg4f4a
ec5c4E35M48.198
E35M62.135wg1a4
ec2e12bwg3h6
wg4d5a
wg5b1b
E33M61.54
ec3d2
E32M48.191E33M59.333
wg6e3b
ec4d11
E32M59.88
ec4g4
wg1g4b
ec3g12
ec3g3a
wg9f2
E35M62.222
wg4a4a
E33M59.234E33M61.84
wg4d7b
ec5e12bec4c11
wg6e3a
E32M48.69
ec3b2b
E32M47.186
ec4d9
wg4d5c
E33M48.67
E35M60.329
wg1h4b
E38M62.188
E32M50.261
E33M50.118
wg1g3a
E35M60.230
wg6a11wg6h1a
E32M62.241
E32M47.288
E33M48.316
E33M59.225ec2e4c
ec3e12b
wg5a1bwg2a3c
ec5e12a
wg7f5bE32M47.159
tg5d9bslg6E35M59.85
ec2d8b
E35M62.132
E35M47.337
E35M47.257
wg9e9
ec2b3E33M60.229
E32M50.325
wg6c1ec3b2a
E35M47.170
wg2d5a
E33M60.120
E33M47.115
wg1g10c
E33M62.130
E32M47.344
E32M50.255
wg3g9
E32M50.424
pr2
tg5b2
E33M59.165
E35M60.125
E33M62.196
ec2h2c
wg3f7b
ec3f1
E35M60.107
wg1g2
E33M48.346
E33M50.371
E33M47.138
tg4d2b
E32M62.394E33M47.189
E32M49.409
wg7b6b
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0
50
100
150
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
chromosomes
loci
outcomes
q
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surv92
surv93
surv94
surv97
surv99
flower0
flower4
flower8
43

Some concluding remarks
Perspectives
1. Modelling
Generalized Fused-Lasso penalty
Automatic inference of L
Environment? Multiparental aspect, Multi-population ?
2. Technical algorithmic points
active set strategy in the alternating algorithm
smart screening of irrelevant predictors
full C++ implementation
3. Applications to regulatory motifs discovery
Y is a matrix of q microarrays for n genes (the individuals),
X is the matrice of motif counts in the promotor of each gene.
L is a matrix based upon editing distance between motifs.
A ﬁrst attempt is made in the paper but we would like to consider
large scale problems (10s/100s of q, 1000s of n, 10,000s of p).
44

Thanks
Hiring! We are looking for a post-doc with strong background in
Optimization and Statistics.
Thanks to you for your patience and to my co-workers
45

Structured Regularization for conditional Gaussian graphical model

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