Feature selection from gene expression data
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Feature selection from gene expression data
Molecular signatures for breast cancer prognosis and inference of gene
regulatory networks.
Anne-Claire Haury
Centre for Computational Biology
Mines ParisTech, INSERM U900, Institut Curie
PhD Defense, December 14, 2012
1
Introduction
2
Gene expression
Figure: Central Dogma of Molecular Biology (Source: Wikipedia)
=> RNA as a proxy to measure gene expression.
3
Microarrays: measuring gene expression
Microarrays: one option.
RNA hybridized onto a chip.
Quantification of gene activity in different conditions at the
genome scale.
Resulting data: gene expression data.
High expression
samples
Low expression
genes
4
Feature selection: extract relevant information
Genome ≈ 25,000 genes
From gene expression, explain a phenomenon
Feature selection: deciding which variables/genes are relevant.
5Feature selection: extract relevant information
Genome ≈ 25,000 genes
From gene expression, explain a phenomenon
Feature selection: deciding which variables/genes are relevant.
Mathematically:
Explain response Y using variables (Xj )j=1...p
Y = f (X1 , X2 , X3 , X4 , . . . , Xp ) + ε
5Feature selection: extract relevant information
Genome ≈ 25,000 genes
From gene expression, explain a phenomenon
Feature selection: deciding which variables/genes are relevant.
Mathematically:
Explain response Y using relevant variables amongst (Xj )j=1...p
Y = f (X1 , X2 , X3 , X4 . . . , Xp ) + ε
Objective 1: more accurate predictors
Objective 2: more interpretable predictors
Objective 3: faster algorithms
5Contributions of this thesis
Gene Regulatory Network Inference
TIGRESS: new method based on local feature selection.
Ranked 3rd/29 at DREAM5 challenge.
Linear method, competitive with more complex algorithms
Molecular signatures for breast cancer prognosis
Select biomarkers to predict metastasis/relapse in breast cancer
patients.
Complete benchmark of feature selection methods.
Investigation of the stability issue.
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7Gene Regulatory Networks
Gene regulation: control gene expression.
Transcription factors (TF) activate or repress target genes (TG).
Gene Regulatory Network (GRN): representation of the regulatory
interactions between genes.
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Figure: E. coli regulatory network
8DREAM network inference challenge
Network inference challenge:
DREAM5 results:
Method Network 1 Network 3 Network 4 Overall
AUPR AUROC AUPR AUROC AUPR AUROC
GENIE31 0.291 0.815 0.093 0.617 0.021 0.518 40.28
ANOVerence2 0.245 0.780 0.119 0.671 0.022 0.519 34.02
Naive TIGRESS 0.301 0.782 0.069 0.595 0.020 0.517 31.1
1
Huynh-Thu et al., 2010
2
Kueffner et al., 2012
9Purposes
Introduce TIGRESS: Trustful Inference of Gene REgulation using
Stability Selection.
Assess the impact of the parameters.
Test and benchmark TIGRESS on several datasets.
10Outline
1 Methods
Regression-based inference
TIGRESS
Material
2 Results
In silico network results
In vitro networks results
Undirected case: DREAM4
3 Conclusion
11Regression-based inference: hypotheses
Notations
ntf transcription factors (TF), ntg target genes (TG), nexp experiments
Expression data: X (nexp × (ntf + ntg )).
Xg : expression levels of gene g.
XG : expression levels of genes in G.
Tg : candidate TFs for gene g.
Hypotheses
1 The expression level Xg of a TG g is a function of the expression
levels XTg of Tg :
Xg = fg (XTg ) + ε.
2 A score sg (t) can be derived from fg , for all t ∈ Tg to assess the
probability of the interaction (t, g).
12Regression-based inference: main steps
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
TF 1
TF 2
...
TF ntf
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
13Regression-based inference: main steps
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
TF 1
TF 2
...
TF ntf
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
13Regression-based inference: main steps
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
TF 1 -
TF 2 0.97
... ...
TF ntf 0
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
13Regression-based inference: main steps
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
TF 1 - 0.23
TF 2 0.97 -
... ... ...
TF ntf 0 0
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
13Regression-based inference: main steps
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
TF 1 - 0.23 0
TF 2 0.97 - 0.03
... ... ... ...
TF ntf 0 0 0
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
13Regression-based inference: main steps
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
TF 1 - 0.23 0 ... 0.11
TF 2 0.97 - 0.03 ... 0
... ... ... ... ... ...
TFntf 0 0 0 ... 0.76
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
13Regression-based inference: main steps
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
TF 1 - 0.23 0 ... 0.11
TF 2 0.97 - 0.03 ... 0
... ... ... ... ... ...
TF ntf 0 0 0 ... 0.76
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
13Regression-based inference: main steps
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
TF 1 - 0.23 0 ... 0.11
TF 2 0.97 - 0.03 ... 0
... ... ... ... ... ...
TF ntf 0 0 0 ... 0.76
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
13Regression-based inference: main steps
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
TF 1 - 0.23 0 ... 0.11
TF 2 0.97 - 0.03 ... 0
... ... ... ... ... ...
TF ntf 0 0 0 ... 0.76
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
13Adding a linearity assumption
TIGRESS’ first hypothesis: regulations are linear
Xg = fg (XTg ) + ε = XTg ω g + ε
g
Consequence: if ωt = 0, no edge between g and t.
14Adding a sparsity assumption
TIGRESS’ second hypothesis: few TFs regulate each TG
g
∀g, ]{ωt 6= 0} L TFs selected.
15Stability Selection
Problem: LARS efficiency is limited:
bad response to correlation;
no confidence score for each TF.
Solution: Stability Selection with randomized LARS (Bach, 2008 ;
Meinshausen and Bühlmann, 2009):
Resample the experiments: run LARS many (e.g. 1, 000) times
with different training sets.
“Resample” the variables: also weight the variables
Xit ← Wt Xit (1)
where Wj ; U([α, 1]) for all t = 1...ntf . The smaller α, the more
randomized the variables.
Get a frequency of selection for each TF.
16Stability Selection
Problem: LARS efficiency is limited:
bad response to correlation;
no confidence score for each TF.
Solution: Stability Selection with randomized LARS (Bach, 2008 ;
Meinshausen and Bühlmann, 2009):
Resample the experiments: run LARS many (e.g. 1, 000) times
with different training sets.
“Resample” the variables: also weight the variables
Xit ← Wt Xit (1)
where Wj ; U([α, 1]) for all t = 1...ntf . The smaller α, the more
randomized the variables.
Get a frequency of selection for each TF.
16Stability Selection path
1
0.9
Frequency of selection of each TF
0.8
0.7
over the subsamplings
0.6
0.5
0.4
0.3
0.2
0.1
0
0 2 4 6 8 10 12 14 16 18 20
Number L of LARS steps
(example for one target gene)
17Scoring
1
0.9
Frequency of selection of each TF
0.8
0.7
over the subsamplings
0.6
0.5
0.4
0.3
0.2
0.1
0
0 2 4 6 8 10 12 14 16 18 20
Number L of LARS steps
18Scoring
Choose L, then:
Original scoring
Area scoring (contribution)
18Scoring
Let Ht be the rank of TF t. Then,
score = E[φ(Ht )]
with
Original: φ(h) = 1 if h ≤ L , 0 otherwise
Area: φ(h) = L + 1 − h if h ≤ L , 0 otherwise
=> Area scoring takes the value of the rank into account.
18TIGRESS Summary
Idea: consider as many problems as TGs (ntg subproblems)
subproblem g ⇔ find regulators TFs(g) of gene g
1 For each TG, score all ntf candidate interactions:
TG 1 TG 2 TG 3 ... TG ntg
LARS TF 1 - 0.23 0 ... 0.11
+ Stab. Selection TF 2 0.97 - 0.03 ... 0
=>
+ Choose L ... ... ... ... ... ...
+ Score TF ntf 0 0 0 ... 0.76
2 Rank the scores altogether:
TF 12 → TG 17 1
TF 23 → TG 5 0.99
TF 2 → TG 1 0.97
... ... ... ...
3 Threshold to a value or a given number N of edges.
19Parameters
TIGRESS needs four parameters to be set:
scoring method (original, area, ...);
number of runs R: large;
randomization level α: between 0 and 1;
number of LARS steps L: not obvious.
20Data
Network ] TF ] Genes ] Chips ] Edges
DREAM5 Net 1 (in-silico) 195 1643 805 4012
DREAM5 Net 3 (E. coli) 334 4511 805 2066
DREAM5 Net 4 (S. cerevisiae) 333 5950 536 3940
E. coli Net from Faith et al., 2007 180 1525 907 3812
DREAM4 Multifactorial Net 1 100 100 100 176
DREAM4 Multifactorial Net 2 100 100 100 249
DREAM4 Multifactorial Net 3 100 100 100 195
DREAM4 Multifactorial Net 4 100 100 100 211
DREAM4 Multifactorial Net 5 100 100 100 193
21Outline
1 Methods
Regression-based inference
TIGRESS
Material
2 Results
In silico network results
In vitro networks results
Undirected case: DREAM4
3 Conclusion
22Impact of the parameters: results on in silico network
Area (1,000 runs) Original (1,000 runs)
20 20
15 15
L
10 10
5 5
0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9
20
Area (4,000 runs)
20
Original (4,000 runs) Area less sensitive than
original to α and L.
15 15
Area systematically
L
10 10
5 5
outperforms original.
0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9
The more runs, the better
Area (10,000 runs) Original (10,000 runs)
20 20
Best values:
15 15
α = 0.4, L = 2, R = 10, 000.
L
10 10
5 5
0.1 0.3 0.5 0.7 0.9 0.1 0.3 0.5 0.7 0.9
α α
40 60 80 100
Overall Score
231000
200
How
500
to choose L? 100
0 0
0 0 2 4 6 8 0 5 10 15
0 2 4 6 8
First 50,000
First 100,000 edges
edges First 100,000 edges
800 200 150
600 150
100
400 100
50
200 50
0 0 0
0 10 1020 3020 40 30 50 0 50 100 150
L=2: number of TFs/TG smaller L=20: greater number of TFs/TG,
and more variable. less sparsity.
=> L should depend on the expected network’s topology
24TIGRESS vs state-of-the-art
1 1
0.8 0.8
Precision
0.6 GENIE3 0.6
TPR
ANOVerence
0.4 CLR 0.4
ARACNE
0.2 Naive TIGRESS 0.2
TIGRESS
0 0
0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15
FPR Recall
TIGRESS is competitive with state-of-the-art.
25Results on E. coli network
1 1
0.8 0.8
Precision
0.6 0.6
TPR
GENIE3
0.4 0.4
CLR
0.2 ARACNE
0.2
TIGRESS
0 0
0 0.5 1 0 0.05 0.1 0.15
FPR Recall
Random Forests-based and DREAM winner GENIE3 overperforms all
methods.
26False discovery analysis on E. coli
Main false positive pattern found by TIGRESS:
Should find Does find
Good news: spuriously inferred edges close to true edges
Bad news: confusion (due to linear model?)
27Undirected case: DREAM4 challenge
DREAM4: undirected networks (TFs not known in advance)
A posteriori comparison of Default TIGRESS and GENIE3:
Method Network 1 Network 2 Network 3 Network 4 Network 5
AUPR AUROC AUPR AUROC AUPR AUROC AUPR AUROC AUPR AUROC
GENIE3 0.154 0.745 0.155 0.733 0.231 0.775 0.208 0.791 0.197 0.798
TIGRESS 0.165 0.769 0.161 0.717 0.233 0.781 0.228 0.791 0.234 0.764
Overall scores:
GENIE3: 37.48
TIGRESS: 38.85
28Outline
1 Methods
Regression-based inference
TIGRESS
Material
2 Results
In silico network results
In vitro networks results
Undirected case: DREAM4
3 Conclusion
29Conclusion
Contributions:
Automatization and adaptation of Stability Selection to GRN
inference.
Area scoring setting: better results and less elasticity to parameters.
Insights on network’s behavior
TIGRESS is:
Linear
Competitive
Parallelizable
Available (http://cbio.ensmp.fr/tigress)
But outperformed in some cases by random forests: limits of
linearity?
Perspectives:
Adaptive/changeable value for L.
Group selection of TFs.
Use of time series/knock-out/replicates information.
30Molecular signatures for breast
cancer prognosis
ACH, Gestraud, P. and Vert, J.-P., The Influence of Feature Selection
Methods on Accuracy, Stability and Interpretability of Molecular
Signatures, 2011, PLoS ONE
ACH, Jacob, L. and Vert, J.-P., Improving Stability and Interpretability of
Molecular Signatures, 2010, arXiv 1001.3109
31Motivation
Prediction of breast cancer outcome
Assist breast cancer prognosis based on gene expression.
Avoid adjuvant/preventive chemotherapy when not needed.
Gene expression signature
Data: primary site tumor expression arrays.
Among the genome, find the few (50-100) genes sufficient to
predict metastasis/relapse.
Main challenge: high-dimensional data (few samples, many
variables).
32Background
2002: Van’t Veer et al. publish 70-gene signature MammaPrint.
Since then: at least 47 published signatures (Venet et al., 2011).
Little overlap, if any (Fan et al., 2006; Thomassen et al., 2007).
Many gene sets are equally predictive (Michiels et al., 2005;
Ein-Dor et al., 2005), even within the same dataset.
Prediction discordances (Reyal et al., 2008)
Stability at the functional level? Not sure.
Seeking stability
Accuracy is not enough to choose the right genes.
=> Stability as a confidence indicator.
33Background
2002: Van’t Veer et al. publish 70-gene signature MammaPrint.
Since then: at least 47 published signatures (Venet et al., 2011).
Little overlap, if any (Fan et al., 2006; Thomassen et al., 2007).
Many gene sets are equally predictive (Michiels et al., 2005;
Ein-Dor et al., 2005), even within the same dataset.
Prediction discordances (Reyal et al., 2008)
Stability at the functional level? Not sure.
Seeking stability
Accuracy is not enough to choose the right genes.
=> Stability as a confidence indicator.
33Background
2002: Van’t Veer et al. publish 70-gene signature MammaPrint.
Since then: at least 47 published signatures (Venet et al., 2011).
Little overlap, if any (Fan et al., 2006; Thomassen et al., 2007).
Many gene sets are equally predictive (Michiels et al., 2005;
Ein-Dor et al., 2005), even within the same dataset.
Prediction discordances (Reyal et al., 2008)
Stability at the functional level? Not sure.
Seeking stability
Accuracy is not enough to choose the right genes.
=> Stability as a confidence indicator.
33Background
2002: Van’t Veer et al. publish 70-gene signature MammaPrint.
Since then: at least 47 published signatures (Venet et al., 2011).
Little overlap, if any (Fan et al., 2006; Thomassen et al., 2007).
Many gene sets are equally predictive (Michiels et al., 2005;
Ein-Dor et al., 2005), even within the same dataset.
Prediction discordances (Reyal et al., 2008)
Stability at the functional level? Not sure.
Seeking stability
Accuracy is not enough to choose the right genes.
=> Stability as a confidence indicator.
33Background
2002: Van’t Veer et al. publish 70-gene signature MammaPrint.
Since then: at least 47 published signatures (Venet et al., 2011).
Little overlap, if any (Fan et al., 2006; Thomassen et al., 2007).
Many gene sets are equally predictive (Michiels et al., 2005;
Ein-Dor et al., 2005), even within the same dataset.
Prediction discordances (Reyal et al., 2008)
Stability at the functional level? Not sure.
Seeking stability
Accuracy is not enough to choose the right genes.
=> Stability as a confidence indicator.
33Background
2002: Van’t Veer et al. publish 70-gene signature MammaPrint.
Since then: at least 47 published signatures (Venet et al., 2011).
Little overlap, if any (Fan et al., 2006; Thomassen et al., 2007).
Many gene sets are equally predictive (Michiels et al., 2005;
Ein-Dor et al., 2005), even within the same dataset.
Prediction discordances (Reyal et al., 2008)
Stability at the functional level? Not sure.
Seeking stability
Accuracy is not enough to choose the right genes.
=> Stability as a confidence indicator.
33Background
2002: Van’t Veer et al. publish 70-gene signature MammaPrint.
Since then: at least 47 published signatures (Venet et al., 2011).
Little overlap, if any (Fan et al., 2006; Thomassen et al., 2007).
Many gene sets are equally predictive (Michiels et al., 2005;
Ein-Dor et al., 2005), even within the same dataset.
Prediction discordances (Reyal et al., 2008)
Stability at the functional level? Not sure.
Seeking stability
Accuracy is not enough to choose the right genes.
=> Stability as a confidence indicator.
33Contributions
1 Systematic comparison of feature selection methods in terms of:
predictive performance;
stability;
biological stability and interpretability.
2 Group selection of genes:
with predefined groups (Graph Lasso);
with latent groups (k-overlap norm).
3 Evaluation of Ensemble methods
34Evaluation
Accuracy: how well selected genes + classifier predict metastatic
events on test data.
Stability: how similar two lists of genes are.
Interpretability: how much biological sense selected genes make.
35Data
Four public breast cancer datasets from the same technology (Affymetrix
U133A):
GEO Reference ] genes ] samples ] positives
GSE1456 12,065 159 40
GSE2034 12,065 286 107
GSE2990 12,065 125 49
GSE4922 12,065 249 89
36Outline
1 A simple start
2 An attempt at enforcing stability: Ensemble methods
3 Using prior knowledge: Graph Lasso
4 Acknowledging latent team work
5 Conclusion
37Classical feature selection/feature ranking methods
Type Characteristics Examples
Univariate, fast T-test, KL Divergence
Filters Only depend on the data Wilcoxon rank-sum test
Do not use loss function
Learning machine as a criterion SVM RFE, Greedy FS
Wrappers
Computationally expensive
Learning + selecting Lasso, Elastic Net
Embedded
Possible use of prior knowledge
Each of these algorithms returns a ranked list of genes to be
thresholded.
38First results
100 genes over four databases.
0.16
Random
Ttest
0.14
Entropy
0.12 Bhattacharryya
Wilcoxon
0.1 SVM RFE
GFS
Stability
0.08 Lasso
E−Net
0.06
0.04
0.02
0
0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65
Accuracy
39First conclusions
0.16
Random
0.14
Ttest
Entropy
Random better than tossing a
coin.
0.12 Bhattacharryya
Wilcoxon
0.1 SVM RFE
GFS
Stability
0.08 Lasso
E−Net
Elastic Net neither more stable
0.06
0.04
nor more accurate than Lasso.
0.02
Accuracy/Stability trade-off
0
0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65
Accuracy
T-test: both simplest and best.
Next step:
Can we have a better stability without decreasing accuracy?
40First conclusions
0.16
Random
0.14
Ttest
Entropy
Random better than tossing a
coin.
0.12 Bhattacharryya
Wilcoxon
0.1 SVM RFE
GFS
Stability
0.08 Lasso
E−Net
Elastic Net neither more stable
0.06
0.04
nor more accurate than Lasso.
0.02
Accuracy/Stability trade-off
0
0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65
Accuracy
T-test: both simplest and best.
Next step:
Can we have a better stability without decreasing accuracy?
40Outline
1 A simple start
2 An attempt at enforcing stability: Ensemble methods
3 Using prior knowledge: Graph Lasso
4 Acknowledging latent team work
5 Conclusion
41Ensemble Methods
Run each algorithm R times on subsamples.
Get R ranked lists of genes (r b )b=1...R .
Aggregate and get a score for each gene:
R
1X b
S(g) = f (rg ).
R
b=1
average : f (r ) = (p − r )/p
exponential : f (r ) = exp(−αr )
stability selection : f (r ) = δ(r ≤ k )
Sort S by decreasing order and threshold to get final signature
42Results
100 genes over four databases.
Random
0.2
Ttest
0.18 Entropy
Bhattacharryya
0.16
Wilcoxon
0.14 SVM RFE
GFS
Stability
0.12
Lasso
0.1 E−Net
Single−run
0.08
Stab. sel
0.06
0.04
0.02
0
0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65
Accuracy
43Functional stability
100 genes over four databases.
Random
0.3 Ttest
Entropy
0.25 Bhattacharryya
Wilcoxon
Functional stability
0.2 SVM RFE
GFS
Lasso
0.15
E−Net
Single−run
0.1
Stab. sel
0.05
0
0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65
Accuracy
44Ensemble methods: conclusions
Random Random
0.2
Ttest 0.3 Ttest
0.18 Entropy Entropy
Bhattacharryya Bhattacharryya
0.25
0.16
Wilcoxon
Wilcoxon
Functional stability
0.14 SVM RFE
0.2 SVM RFE
GFS
GFS
Stability
0.12
Lasso
Lasso
0.15
0.1 E−Net
E−Net
Single−run
0.08 Single−run
Stab. sel 0.1
Stab. sel
0.06
0.05
0.04
0.02 0
0 0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65
0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65 Accuracy
Accuracy
Expected improvement in stability not happening.
Slight improvement in accuracy in some cases.
Loss in functional stability.
T-test: still the preferred method.
Next step:
Can we do better by incorporating prior knowledge?
45Ensemble methods: conclusions
Random Random
0.2
Ttest 0.3 Ttest
0.18 Entropy Entropy
Bhattacharryya Bhattacharryya
0.25
0.16
Wilcoxon
Wilcoxon
Functional stability
0.14 SVM RFE
0.2 SVM RFE
GFS
GFS
Stability
0.12
Lasso
Lasso
0.15
0.1 E−Net
E−Net
Single−run
0.08 Single−run
Stab. sel 0.1
Stab. sel
0.06
0.05
0.04
0.02 0
0 0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65
0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65 Accuracy
Accuracy
Expected improvement in stability not happening.
Slight improvement in accuracy in some cases.
Loss in functional stability.
T-test: still the preferred method.
Next step:
Can we do better by incorporating prior knowledge?
45Outline
1 A simple start
2 An attempt at enforcing stability: Ensemble methods
3 Using prior knowledge: Graph Lasso
4 Acknowledging latent team work
5 Conclusion
46Data
Expression data: Van’t Veer et al.,
2002; Wang et al., 2005.
PPI network with 8141 genes (Chuang
et al., 2007)
Assumption: genes close on the graph behave similarly
Idea: instead of selecting single genes, select edges
47Selecting groups of genes: `1 methods
Lasso : selects single genes (Tibshirani, 1996)
Why groups?
selecting similar genes: improving stability and interpretability
smoothing out noise by "averaging": improving accuracy
Group Lasso (Yuan & Lin, 2006): implies group sparsity for groups
of covariates that form a partition of {1...p}
Overlapping group Lasso (Jacob et al., 2009): selects a union of
potentially overlapping groups of covariates (e.g. gene pathways).
Graph Lasso: uses groups induced by the graph (e.g. edges)
48Is group sparsity enough?
`1 methods work well, but face serious stability issues when
groups are correlated.
Solution: randomization through stability selection.
49Accuracy results
Test on data from Wang et al., 2005.
Signature learnt on Van’t Veer dataset
0.8
Signature learnt on Wang dataset
Balanced Accuracy
0.6
0.4
0.2
0
Lasso Lasso + stab. sel. Graph Lasso Graph Lasso + stab. sel.
Neither prior knowledge nor stability selection bring any
improvement!
50Stability results
7
Number of genes in the overlap
6 Lasso
Graph Lasso with stability selection
5 Lasso with stability selection
Graph Lasso
4
3
2
1
0
0 20 40 60 80 100 120
Number of genes in the signatures
Graph Lasso slightly improves stability.
51Interpretability
Signature obtained using Lasso:
52Interpretability
Signature obtained using Graph Lasso + Stability Selection:
52Graph Lasso conclusion
Graphical prior seems to increase stability and interpretability.
However: no change in accuracy.
Next step:
Grouping increases stability. Now on to accuracy!
53Outline
1 A simple start
2 An attempt at enforcing stability: Ensemble methods
3 Using prior knowledge: Graph Lasso
4 Acknowledging latent team work
5 Conclusion
54Latent grouping
Grouping genes makes sense.
Let the data tell which genes to select together.
The k-support norm
Introduced by Argyriou et al., 2012
A trade-off between `1 and `2 .
Equivalent to overlapping group Lasso (Jacob et al., 2009) with all
possible groups of size k .
Results in selecting groups that are not predefined.
55Extreme randomization
Following Breiman’s random forests
Sample both the examples and the covariates.
Less variables = less correlation.
Give each gene a chance to be selected.
Extreme randomization
For each of the R runs:
Bootstrap samples (classical Ensemble method)
Sample the covariates: randomly choose 10% of them.
Run FS procedure on the restricted data.
=> Compute frequency of selection: P(g selected |g preselected)
56Accuracy or stability?
0.05
T−test
Random
0.045
Ttest
0.04
Lasso
0.035 ENet
kSupport (k=2)
0.03
kSupport (k=10)
Stability
0.025 kSupport (k=20)
Single−run
0.02
Extreme Rand. + SS
0.015
0.01
0.005
0
0.62 0.625 0.63 0.635 0.64 0.645 0.65 0.655
Accuracy
57Stability or redundancy?
0.05
Random
0.045
Ttest
Lasso
0.04
ENet
0.035 kSupport (k=2)
kSupport (k=10)
Stability
0.03
kSupport (k=20)
0.025 Single−run
Extreme Rand. + SS
0.02
0.015
0.01
0.005
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Correlation
58Conclusions
0.05 0.05
T−test Random
Random
0.045 0.045
Ttest
Ttest
0.04 Lasso
Lasso 0.04
ENet
0.035 ENet
0.035 kSupport (k=2)
kSupport (k=2)
0.03
kSupport (k=10)
kSupport (k=10)
Stability
0.03
Stability
0.025 kSupport (k=20)
kSupport (k=20)
0.025 Single−run
Single−run
0.02
Extreme Rand. + SS
Extreme Rand. + SS 0.02
0.015
0.015
0.01
0.005 0.01
0 0.005
0.62 0.625 0.63 0.635 0.64 0.645 0.65 0.655 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Accuracy Correlation
Extreme Randomization improves accuracy.
Grouping improves stability.
But: both effects do not add up that well (redundancy)
T-test still the best trade-off?
59Outline
1 A simple start
2 An attempt at enforcing stability: Ensemble methods
3 Using prior knowledge: Graph Lasso
4 Acknowledging latent team work
5 Conclusion
60Signature selection: conclusion
Contributions:
Step-by-step study of FS methods behavior on several breast
cancer datasets.
Systematic analysis of accuracy, stability, interpretability.
Insights on the accuracy/stability trade-off.
What have we learned?
Best methods: simple t-test or complex black box
Grouping improves gene and functional stability.
Randomization improves accuracy (sometimes) but has unwanted
effects on stability.
Accuracy/Stability Trade-off: Stability => redundancy => lower
accuracy.
61Signature selection: perspectives
One unique signature?
single breast cancer subtype
many samples
larger signature
Is expression data sufficient?
probably not all information is there
clinical data: same accuracy (same information?)
possibly look at genotype, methylation, clinical and expression
Is stability important?
not as important as accuracy + prediction concordance
possibly not even achievable
62Conclusion
63Conclusion
Gene expression data:
High-dimensional, noisy
Possibly contains important information
Feature selection:
Find the needle in the haystack.
Output relevant genes to be studied further.
Main issues:
Results are not necessarily transferable across datasets.
Models rely on hypotheses!
Fixing:
Testing on many databases.
Keeping model hypotheses in mind / not being afraid of black boxes.
64Conclusion
Gene expression data:
High-dimensional, noisy
Possibly contains important information
Feature selection:
Find the needle in the haystack.
Output relevant genes to be studied further.
Main issues:
Results are not necessarily transferable across datasets.
Models rely on hypotheses!
Fixing:
Testing on many databases.
Keeping model hypotheses in mind / not being afraid of black boxes.
64Conclusion
Gene expression data:
High-dimensional, noisy
Possibly contains important information
Feature selection:
Find the needle in the haystack.
Output relevant genes to be studied further.
Main issues:
Results are not necessarily transferable across datasets.
Models rely on hypotheses!
Fixing:
Testing on many databases.
Keeping model hypotheses in mind / not being afraid of black boxes.
64Conclusion
Gene expression data:
High-dimensional, noisy
Possibly contains important information
Feature selection:
Find the needle in the haystack.
Output relevant genes to be studied further.
Main issues:
Results are not necessarily transferable across datasets.
Models rely on hypotheses!
Fixing:
Testing on many databases.
Keeping model hypotheses in mind / not being afraid of black boxes.
64Acknowledgements
Fantine Paola Pierre Laurent
Mordelet Vera-Licona Gestraud Jacob
65The k-support norm
It can be shown that:
k −r −1 d
!2 12
1
Ωsp (|ω|↓i )2 + |ω|↓i
X X
k (ω) =
r +1
i=1 i=k −r
where r is the only integer in {0, . . . , k − 1} statisfying
d
1 X
|ω|↓k −r −1 > |ω|↓i ≥ |ω|↓k −r .
r +1
i=k −r
and |ω|↓i is the i-th largest value of |ω| (|ω|↓0 = +∞).
66Link with overlapping Group Lasso
The k-support norm is equivalent to the overlapping group Lasso
norm
X
Ωsp
X
k (ω) = min ||v ||
I 2 : supp(vI ) ⊆ I, vI = ω
v ∈Rp×Gk
I∈Gk I∈Gk
where Gk denotes all subsets of {1, . . . , d} of cardinality k .
Remark 1: it selects at least k variables.
Remark 2: the first selected group consists of the k variables most
correlated with the response
67ADMM - applied to k-support problem
Our problem
minω,β R̂l (ω) + λ2 Ωsp 2
k (β)
s.t. ω − β = 0
Augmented Lagrangian
Lρ (ω, β, µ) = R̂l (ω) + λ2 Ωsp 2 0 ρ
k (β) + µ (ω − β) + 2 ||ω − β||
2
Algorithm
1 Initialize: β (1) , ω (1) , µ(1)
2 for t = 1, 2, ..., do
n o
w (t+1) = arg minw R̂l (w) + µ(t)T w + ρ2 ||w − β (t) ||2
(t)
β (t+1) = prox λ Ωsp (.)2 w (t+1) + µρ
2ρ k
µ(t+1) = µ(t) + ρ(w (t+1) − β (t+1) )
68ADMM - optimality conditions
Three first order conditions:
Primal condition: ω ∗ − β ∗ = 0
Dual condition 1: ∇R̂l (ω ∗ ) + µ∗ = 0
Dual condition 2: 0 ∈ λ2 ∂Ωsp ∗ 2
k (β ) + µ
∗
Resulting in a definition for the residuals at step t + 1:
Primal residuals: r (t+1) = ω (t+1) − β (t+1)
Dual residuals: s(t+1) = ρ(β (t+1) − β (t) )
As the algorithm converges, the (norm of the) residuals tend to zero.
69ADMM - choice of parameter
Parameter ρ is critical in ADMM: it controls how much variables change.
It can be seen as a step size.
How to choose it?
Adaptive ADMM
One solution
is to let(t)it adapt to the problem:
(1 + τ )ρ if ||r (t+1) ||2 > η||s(t+1) ||2 and t ≤ tmax
ρ(t+1) = ρ(t) /(1 + τ ) if η||r (t+1) ||2 < ||s(t+1) ||2 and t ≤ tmax
(t)
ρ otherwise
In practice, we use τ = 1, η = 10 and tmax = 100.
=> Adaptive ADMM forces the primal and dual residuals to be of a
similar amplitude.
70Comparison
k=1 k=5
5 5
0 0
RDG
RDG
−5 −5
−10 −10
−15 −15
0 10 20 0 10 20
Time (Seconds) Time (Seconds)
ADMM − adaptive
k = 10 ADMM − rho=1 k = 100
5 ADMM − rho=10 5
ADMM − rho=100
0 FISTA
0
RDG
RDG
−5 −5
−10 −10
−15 −15
0 10 20 0 20 40 60
Time (Seconds) Time (Seconds)
71Accuracy vs size of the signature
Single−run Ensemble−Mean
0.65 0.65
0.6
AUC
0.6
AUC
0.55 0.55
0.5 0.5
0.45 0.45
0 20 40 60 80 100 0 20 40 60 80 100
Ensemble−Exponential Ensemble−Stability Selection
0.65 0.65
0.6
AUC
0.6
AUC
0.55 0.55
0.5 0.5
0.45 0.45
0 20 40 60 80 100 0 20 40 60 80 100
Random T−test Entropy Bhatt. Wilcoxon SVM RFE GFS Lasso E−Net
72Stability vs size of the signature
0.8
Single-run 0.8
Ensemble-average
Stability
0.6
Stability
0.6
0.4 0.4
0.2 0.2
0 0
0 20 40 60 80 100 0 20 40 60 80 100
0.8
Ensemble-exponential Ensemble-stability selection
0.8
Stability
Stability
0.6 0.6
0.4 0.4
0.2 0.2
0 0
0 20 40 60 80 100 0 20 40 60 80 100
Random T test Entropy Bhatt. Wilcoxon RFE GFS Lasso E−Net
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