[PPT] - Curve Clustering and Functional Mixed Models. Modeling, variable PowerPoint Presentation

SLIDE 1

Curve Clustering and Functional Mixed Models.

Modeling, variable selection and application to Genomics Franck Picard, LBBE - Lyon

Madison Giacofci LJK (Grenoble) Sophie Lambert-Lacroix (TIMC - Grenoble) Guillemette Marot (Univ. Lille 2) Carlos Correa-Shokiche, (LBBE - Lyon)

F. Picard (LBBE)

JSF - June 2012 1 / 35

SLIDE 2

Introduction

Outline

1

Introduction

2

Functional Clustering Model with random effects

3

Estimation and model selection

4

Applications

5

Dimension Reduction for FANOVA

6

Conclusions & Perspectives

F. Picard (LBBE)

JSF - June 2012 2 / 35

SLIDE 3

Introduction

The Genomic RevolutionS

Genomics is the field that investigates biological processes at the scale of Genomes. It started in the 70s-80s with the development of Molecular Biology techniques (sequencing, transcripts quantification). Genomics (and Post-Genomics) exploded in the 90s-2000s thanks to the miniaturization and industrialization of quantification processes. Sequencing the Human Genome ? took ∼ 10 years and can be done within a week now.

F. Picard (LBBE)

JSF - June 2012 3 / 35

SLIDE 4

Introduction

Towards Population-Based Genomic Studies

Quantification mainly concern : Copy Number Variations, messenger RNAs, and proteins mostly using microarrays and Mass Spectrometry For long the task has been to extract signal from noise for one individual experiment (sometimes with replicates !) Prices decreasing, these technologies are now used at the population levels: this is the rise of Population Genomics Statistical Tasks remain standard Differential Analysis, Clustering, Discrimination but the dimensionality of the data is overwhelming

F. Picard (LBBE)

JSF - June 2012 4 / 35

SLIDE 5

Introduction

Example with Mass Spectrometry data

Aim: characterize the content of a mixture of peptides by mass-spect One peak corresponds to one peptite (signature) Each spectra contains 15154 ionised peptides defined by a m/z ratio. 253 ovarian cancer samples: 91 Controls, 162 Cases [10]

10 20 30 40

control

10 20 30 40

cancer

Figure: MALDI-TOF Spectra. http://home.ccr.cancer.gov/ncifdaproteomics/ppatterns.asp

F. Picard (LBBE)

JSF - June 2012 5 / 35

SLIDE 6

Introduction

Example with array CGH data

Aim: characterize copy number variations between 2 genomes Segments with positive mean corresponds to regions that are amplified (negative/deleted) 55 aCGH profile from Breast Cancer patients Subgroup discovery: hierarchical clustering based on segmentation [11].

−2 −1 1 2

1q16q

−2 −1 1 2

ther

Figure: Breast Cancer CGH profiles [8] (log scale)

F. Picard (LBBE)

JSF - June 2012 6 / 35

SLIDE 7

Introduction

Towards Functional Models

Proteomic Data : records are sampled on a very fine grid (m/z) and spectra have long been modeled using FDA Genomic Data are mapped on a reference genome and show a spatial (1D?) structure Functional models can account for this kind of structure, and working on curves should be more efficient than working on peaks or segments

F. Picard (LBBE)

JSF - June 2012 7 / 35

SLIDE 8

Introduction

Towards Functional Mixed Models

Subject specific fluctuations are known to be the largest source of variability in Mass-Spec data [6] Inter-Individual variability is the “curse” of biological data ! (Technical / Biological Variabilities), and often under-estimated Mixed Linear Models: well known in Genetics to structure the variance according to experimental design and pedigrees We propose to analyze genomic data using functional mixed models

F. Picard (LBBE)

JSF - June 2012 8 / 35

SLIDE 9

Functional Clustering Model with random effects

Outline

1

Introduction

2

Functional Clustering Model with random effects

3

Estimation and model selection

4

Applications

5

Dimension Reduction for FANOVA

6

Conclusions & Perspectives

F. Picard (LBBE)

JSF - June 2012 9 / 35

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Functional Clustering Model with random effects

Functional ANOVA Model

We observe N replicates of a noisy version of function µ over a fine grid t = {t1, . . . , tM}, tj ∈ [0, 1], such that: Yi(tm) = µ(tm) + Ei(tm), Ei(t) ∼ N(0, σ2), with i = 1, . . . , N, m = 1, . . . , M = 2J In the following we use notations Yi(t) = [Yi(t1), . . . , Yi(tM)] , µ(t) = [µ(t1), . . . , µ(tM)] We propose to use wavelets to analyse such data:

Modelling curves with irregularities
Computationaly efficiency (the DWT is in O(M))
Dimension Reduction
F. Picard (LBBE)

JSF - June 2012 10 / 35

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Functional Clustering Model with random effects

Definition of wavelets and wavelet coefficients

Wavelets provide an orthonormal basis of L2([0, 1]) with a scaling function φ and a mother wavelet ψ such that:

φj0k(t), k = 0, . . . , 2j0 − 1; ψjk(t), j ≥ j0, k = 0, . . . , 2j − 1
Any function Y ∈ L2([0, 1]) is then expressed in the form:

Yi(t) =

2j0−1

k=0

c∗

i,j0kφj0k(t) +

j≥j0

2j−1

k=0

d∗

i,jkψjk(t)

where c∗

i,j0k = Yi, φj0k and d∗ i,jk = Yi, ψjk are the theorical

scaling and wavelet coefficients.

F. Picard (LBBE)

JSF - June 2012 11 / 35

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Functional Clustering Model with random effects

The DWT and empirical wavelet coefficients

Denote by W an orthogonal matrix of filters (wavelet specific), The Discrete Wavelet Transform is given by W

[M×M]Yi(t) [M×1]

= ci di

(ci, di) are empirical scaling and wavelet coefficients

Once the data are in the coefficient domain we retrieve a linear model such that: (α = [αj0,k]k=0,...,2j0−1, β = [βjk]k=0,...,2j−1

j=j0,...,2J

) WYi(t) = Wµ(t) + WEi(t) ci di

=

α β

+ εi, εi ∼ N(0M, σ2

εIM)

F. Picard (LBBE)

JSF - June 2012 12 / 35

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Functional Clustering Model with random effects

Functional Clustering Model (FCM)

The idea is to cluster individuals based on functional observations We suppose that the cluster structure concerns the fixed effects of the model When using a mixture model we introduce the label variable ζiℓ ∼ M (1, π = (π1, . . . , πL)) such that given {ζiℓ = 1} Yi(tm) = µℓ(tm) + Ei(tm) In the coefficient domain, we retrieve a Multivariate Gaussian Mixture such that given {ζiℓ = 1} [3]: ci di

=

αℓ βℓ

+ εi.
F. Picard (LBBE)

JSF - June 2012 13 / 35

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Functional Clustering Model with random effects

Functional Clustering Mixed Models

Functional Mixed models are considered to introduce inter-individual functional variability such that given {ζiℓ = 1}: Yi(tm) = µℓ(tm) + Ui(tm) + Ei(tm) Ui(t)|{ζiℓ = 1} ∼ N(0, Kℓ(t, t′)), Ui(t) ⊥ Ei(t) In the wavelet domain, and given {ζiℓ = 1} the model resumes to ci di

=

αℓ βℓ

+

νi θi

+ εi, εi ∼ N(0M, σ2

εIM)

νi θi

∼

N

0M,

Gν Gθ

νi

θi

⊥

εi

F. Picard (LBBE)

JSF - June 2012 14 / 35

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Functional Clustering Model with random effects

Specification of the covariance of random effects

Suppose Gθ is diagonal by the whitening property of wavelets [7] The fixed and random effects should lie in the same Besov space. Introduce parameter η related to the regularity of process Ui Theorem Abramovich & al. [1] Suppose µ(t) ∈ Bs

p,q and V(θi,jk) = 2−jηγ2 θ then

Ui(t) ∈ Bs

p,q[0, 1] a.s.

↔ η = 2s + 1, if 1 ≤ p < ∞ and q = ∞ η > 2s + 1, otherwise.

The structure of the random effect can also vary wrt position and scale (γ2

θ,jk), and/or group membership (γ2 θ,jkℓ)

F. Picard (LBBE)

JSF - June 2012 15 / 35

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Estimation and model selection

Outline

1

Introduction

2

Functional Clustering Model with random effects

3

Estimation and model selection

4

Applications

5

Dimension Reduction for FANOVA

6

Conclusions & Perspectives

F. Picard (LBBE)

JSF - June 2012 16 / 35

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Estimation and model selection

Using the EM algorithm

In the coefficient domain, the model is a Gaussian mixture with structured variance Both label variables ζ and random effects (ν, θ) are unobserved The complete data log-likelihood can be written such that: log L

c, d, ν, θ, ζ; π, α, β, G, σ2

ε

=

log L

c, d|ν, θ, ζ; π, α, β, σ2

ε

+

log L (ν, θ|ζ; G) + log L (ζ; π) . This likelihood can be easily computed thanks to the properties of mixed linear models such that: ci di

νi

θi

, {ζiℓ = 1} ∼ N

αℓ + νi βℓ + θi

, σ2

εI

.
F. Picard (LBBE)

JSF - June 2012 17 / 35

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Estimation and model selection

Predictions of hidden variables

The EM algorithm provides posterior probabilities of membership: τ [h+1]

iℓ

= π[h]

ℓ f

ci, di; α[h]

ℓ , β[h] ℓ , G[h] + σ2[h] ε

I

p π[h]

p f

ci, di; α[h]

p , β[h] p , G[h] + σ2[h] ε

I . The E-step also provides the BLUP of random effects:

ν[h+1]

iℓ

=

ci − α[h]

ℓ

/
1 + λ[h]

ν

, λν = σ2

ε/γ2 ν,

θ

[h+1] iℓ

=

di − β[h]

ℓ

/
1 + 2jηλ[h]

θ

, λθ = σ2

ε/γ2 θ.

F. Picard (LBBE)

JSF - June 2012 18 / 35

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Estimation and model selection

ML estimates for fixed effects & variances

The M-step provides the estimators of the mean curve coefficients and of the variance of random effects α[h+1]

ℓ

=

n

i=1

τ [h]

iℓ

ci −

ν[h]

iℓ

/N[h]

ℓ ,

β[h+1]

ℓ

=

n

i=1

τ [h]

iℓ

di −

θ

[h] iℓ

/N[h]

ℓ ,

γ2[h+1]

θ

= 1 n(M − 1)

ijkℓ

2jητ [h]

iℓ

θ2

ijkℓ [h] +

σ2[h]

ε

1 + 2jηλ[h]

θ

,

γ2[h+1]

ν

= 1 n

iℓ
ν2

i00ℓ [h] +

σ2[h]

ε

1 + λ[h]

ν

.

Parameter η can be estimated by numerical optimization

F. Picard (LBBE)

JSF - June 2012 19 / 35

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Estimation and model selection

Model selection using a BIC

mL stands for a clustering model with L clusters We select the dimension that maximizes BIC(mL) = log L

c, d;

π, α, β, G, σ2

ε, mL

− |mL|

2 × log(N). |mL| = |α| + |β| + |G| + |π| − 1 + |σ2

ε|

= (M + 1)L + |G|. The dimension of G depends on the variance structure of the random effects. |G| = 2 is the case of constant variances (γ2

ν, γ2 θ), and |G| = ML

when variances depend on group, scale and position.

F. Picard (LBBE)

JSF - June 2012 20 / 35

SLIDE 21

Applications

Outline

1

Introduction

2

Functional Clustering Model with random effects

3

Estimation and model selection

4

Applications

5

Dimension Reduction for FANOVA

6

Conclusions & Perspectives

F. Picard (LBBE)

JSF - June 2012 21 / 35

SLIDE 22

Applications

Back to Mass Spectrometry data

91 controls 162 cases [10] Pre-treatment (baseline correction, peak alignment) Results (EER %) on a window

f 512

model global align. group align. m2 38 21 m2[γ2] 24 21 m2[γ2

ℓ]

24 22 m2[γ2

jk]

23 0.4 m2[γ2

jkℓ]

23 36

10 20 30 40

control

10 20 30 40

cancer

Inaccuracy in spectra-alignment is lethal for clustering !

F. Picard (LBBE)

JSF - June 2012 22 / 35

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Applications

Application to array CGH data

3 main subtypes identified [8] Not reproduced by others [11] but one group is associated to the best patient outcome. We were able to identify the 1q/16p subtype on the complete dataset (with 1 mismatch).

cluster ID

SNR2

µ

λU

1 2.1e-4 3.9e-04 2 2.3e-3 3.8e-05 3 1.3e-3 6.4e-04 4 (1q/16p) 1.5e-3 1.3e-04 5 9.3e-4 4.3e-05

−2 −1 1 2

1q16q

−2 −1 1 2

ther

Figure: Array CGH profiles from [8]

F. Picard (LBBE)

JSF - June 2012 23 / 35

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Dimension Reduction for FANOVA

Outline

1

Introduction

2

Functional Clustering Model with random effects

3

Estimation and model selection

4

Applications

5

Dimension Reduction for FANOVA

6

Conclusions & Perspectives

F. Picard (LBBE)

JSF - June 2012 24 / 35

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Dimension Reduction for FANOVA

The Case of One Curve

When only one curve is observed, classical procedures consist in selecting a reduced number of coefficients while controling for reconstruction properties Among classical procedures, soft thresholding is well known[5]

βjk = sign (djk) (|djk| − λ)+

λ is usually chosen as the “universal” threshold σ√2 log M, σ is estimated by the MAD estimator Soft thresholding has good reconstruction properties and attains a near-minimax rate of convergence

F. Picard (LBBE)

JSF - June 2012 25 / 35

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Dimension Reduction for FANOVA

The LASSO and penalized regression

Considering a regression model Y = Xβ + E, the LASSO performs shrinkage and variable selection by solving a penalized estimation problem Denoting by J(β) = 1

2 Y − Xβ2, the LASSO consists in solving

β = arg min

β

{J(β) + λβ1} It is well known that in the case of orthogonal design, the LASSO resumes to soft thresholding. Aim How to use penalization techniques to propose an estimation framework for FANOVA and perform dimension reduction simultaneously ?

F. Picard (LBBE)

JSF - June 2012 26 / 35

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Dimension Reduction for FANOVA

1 fixed effect, N replicates

The first simple model is given by: Yi(tm) = µ(tm) + Ei(tm) First attempts [2] propose to average and shrink coefficients The LASSO gives the appropriate framework by solving J(β) + λpen(β) = 1 2

n

i=1

di − β2 + λβ1

βjk(λ) = sign (d•,jk) (|d•,jk| − λ)+

λ can be estimated using a BIC: BIC(λ) = log L(d, β(λ), σ2) − 1 2 log(N) × β(λ)0

F. Picard (LBBE)

JSF - June 2012 27 / 35

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Dimension Reduction for FANOVA

L fixed effects, N replicates

The Functional Clustering Model is (given {ζiℓ = 1}), Yi(tm) = µℓ(tm) + Ei(tm) The LASSO can be used in the context of mixtures as well JL(ζ; β, π) + λpen(β) = 1 2

n

i=1

ζiℓ di − βℓ2 + λ

L

ℓ=1

πℓβℓ1 MLE is performed by using a penalized EM [9] algorithm with JL(β; π) = −

n

i=1

log L

ℓ=1

πℓf (di; βℓ, σ2)

F. Picard (LBBE)

JSF - June 2012 28 / 35

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Dimension Reduction for FANOVA

1 fixed effect, N replicates, N functional Random Effects

The Functional Mixed Model is Yi(tm) = µ(tm) + Ui(tm) + Ei(tm) ci di

=

α β

+

νi θi

+ εi,

νi θi

∼ N
0,

Gν Gθ

.

Dimension reduction is performed

On Fixed effects β
On random effects through a spare representation of Kernel

K(t, t′) = cov (Ui(t), Ui(t′))

Gθ has general (diagonal) term V(θi,jk) = 2−jηγ2

θ,jk

The LASSO can be used to shrink terms γ2

θ,jk

F. Picard (LBBE)

JSF - June 2012 29 / 35

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Dimension Reduction for FANOVA

The LASSO for Mixed Linear Models

Perform MLE estimation using a hidden variable representation of Mixed Linear Models Use the EM algorithm to optimize [4] J(β, G, σ2) + λβpen(β) + λγpen(γ) = − log L(d; β, G, σ2) + λββ1 + λγγ1 The Maximization is performed indirectly by using the conditional expectation of the complete-data log-likelihood E

log L(d, θ; β, G, σ2)|d
F. Picard (LBBE)

JSF - June 2012 30 / 35

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Dimension Reduction for FANOVA

Reparametrization and M-step

For convexification, use the following reparametrization di = β + G−1/2

θ

θ∗

i + εi, θ∗ i ∼ N(0M, IM)

−2E

log L(d, θ; β, G, σ2)|d
=

Mn log σ2

ε

+ 1 σ2

ε

2d − β − G−1/2

θ

θ

∗ i

+ tr

G−1/2

θ T V {θ∗|d} G−1/2 θ

+
θ

∗ i T

θ

∗ i + cst

For dimensionality purposes, use Conditional M-steps

F. Picard (LBBE)

JSF - June 2012 31 / 35

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Dimension Reduction for FANOVA

Solutions of the LASSO for Functional Mixed Models

The penalized estimator of fixed effects β is based on

di,jk = di,jk − 2−jη/2

γjk θ∗

i,jk

βjk(λβ) = sign
d•,jk

| d•,jk| − λβσ2

ε

N

+

The penalized estimator of γ is based on ρijk(λβ) = 2−jη θ∗

i,jk ×

di,jk −

βjk(λβ)

γjk(λβ, λγ) ∝
|ρ•,jk(λβ)| − λγσ2

ε

N

+
F. Picard (LBBE)

JSF - June 2012 32 / 35

SLIDE 33

Conclusions & Perspectives

Outline

1

Introduction

2

Functional Clustering Model with random effects

3

Estimation and model selection

4

Applications

5

Dimension Reduction for FANOVA

6

Conclusions & Perspectives

F. Picard (LBBE)

JSF - June 2012 33 / 35

SLIDE 34

Conclusions & Perspectives

Conclusions & perspectives

We developed a model for functional clustering with random effects All the codes are available with the R package curvclust http://cran.r-project.org/ Main challenge now concerns thresholding of wavelet coefficients in multiple contexts using the LASSO machinery Mixture Models, Mixed Models, Mixture + Mixed Models ! What are the reconstruction properties of the predicted random effects Ui(t) = E (Ui(t)|Yi(t)) (functional). Do we have a Best-Predictor Property ?

F. Picard (LBBE)

JSF - June 2012 34 / 35

SLIDE 35

Conclusions & Perspectives

References

F. Abramovich, T. Sapatinas, and B.W. Silverman.

Wavelet thresholding via a bayesian approach. Journal of the Royal Statistical Society Series B Stat Methodol, 60:725–749, 1998.

U. Amato and T. Sapatinas.

Wavelet shrinkage approaches to baseline signal estimation from repeated noisy measurements. Advances and Applications in Statistics, 51:21–50, 2005.

A. Antoniadis, J. Bigot, and R. von Sachs.

A multiscale approach for statistical characterization of functional images. Journal of Computational and Graphical Statistics, 18(1):216–237, 2008. H.D. Bondell, A. Krishna, and S.K. Ghosh. Joint variable selection for fixed and random effects in linear mixed-effects models. Biometrics, 66(4):1069–1077, 2010. D.L. Donoho and I.M. Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81(3):425–455, 1994.

J. E. Eckel-Passow, A. L. Oberg, T. M. Therneau, and H. R. Bergen.

An insight into high-resolution mass-spectrometry data. Biostatistics, 10:481–500, Jul 2009.

M. Frazier, B. Jawerth, and G. Weiss.

Littlewood-Paley Theory and the Study of function Spaces. Number 79. American Mathematical Society, 1991. J Fridlyand and al. Breast tumor copy number aberration phenotypes and genomic instability. BMC Cancer, 6:96, 2006.

A. Khalili and J. Chen.
F. Picard (LBBE)

JSF - June 2012 35 / 35

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Conclusions & Perspectives

Model selection using a ICL

It is likely that predictions of random effects provide information regarding L. The ICL criterion is based on the integrated likelihood of the complete data: log L(c, d, ν, θ, ζ|mL[γ2

ℓ ])

Need to derive the integrated log-likelihood of the random effects and for the label variables.

− 2 N × ICL(mL[γ2

ℓ])

= M log RSS(c, d| ν, θ, τ) +

ℓ
πℓ
log RSSℓ(

ν, τ) + (M − 1) log RSSℓ( θ, τ)

−

2 N

ℓ
log Γ

Nℓ 2

+ log Γ

Nℓ(M − 1) 2

−

2

L

ℓ=1
πℓ log(

πℓ) + (M + 1)L N × log(N).

SLIDE 37

Conclusions & Perspectives

Model selection BIC vs ICL

2 4 6

ICL

0.1 1 3 5

2 4 6

BIC

0.1 1 3 5

Haar

2 4 6

0.1 1 3 5

2 4 6

0.1 1 3 5

Bumps

2 4 6

0.1 1 3 5

2 4 6

0.1 1 3 5

Heavisine

2 4 6

0.1 1 3 5

SNR 2 4 6

0.1 1 3 5

SNR

Doppler

lambda=0.25 lambda=1 lambda=4

SLIDE 38

Simulations

Fine Definition of a simulation framework

We properly define the power of the signal: lim

T→∞

1 T − T

2 T 2

ℓ

πℓE

|µℓ(t) + Ui(t)|

2dt We need to control two terms: SNR2

µ

= 1 Mσ2

E L

ℓ=1

πℓ  

2j0−1

k=0

α2

j0kℓ +

j≥j0

2j−1

k=0

β2

jkℓ

  , λU = σ2

E/

γ2

ν +

γ2

θ

1 − 2(1−η)

,

SLIDE 39

Simulations

Simulated data with a low random effect λU = 4

SNR=1

Haar

SNR=3 SNR=5

Bumps

SLIDE 40

Simulations

Simulated data with a strong random effect λU = 1/4

SNR=1

Haar

SNR=3 SNR=5

Bumps

SLIDE 41

Simulations

Aim & design of the simulation study

What is the gain when using a functional random effect in terms of clustering (FCM/FCMM)? What is the performance of splines ? Is dimension reduction appropriate ? n = 50, M = 512, L = 2, SNRµ ∈ {0.1; 1; 3; 5; 7}, λU ∈ {0.25, 1, 4} Fixed effects can be Haar, Bumps, Heavisine, Doppler Study the Empirical Error Rate: EER = 1 N

N

i=1

I{ ζiℓ = ζiℓ} Development of a package curvclust

SLIDE 42

Simulations

Empirical Error Rates (2 clusters)

lambda=0.25

0.5

EER

0.1 1 3 5 7

lambda=1

0.25 0.5 0.1 1 3 5 7

lambda=4

0.25 0.5 0.1 1 3 5 7

Haar

0.5

EER

0.1 1 3 5 7 0.25 0.5 0.1 1 3 5 7 0.25 0.5 0.1 1 3 5 7

Bumps

0.5

EER

0.1 1 3 5 7 0.25 0.5 0.1 1 3 5 7 0.25 0.5 0.1 1 3 5 7

Heavisine

0.5

EER

0.1 1 5 7

SNR

0.25 0.5 0.1 1 5 7

SNR

0.25 0.5 0.1 1 5 7

SNR

Doppler

FCM FCMunion FCMM FCMMunion Spline

SLIDE 43

Simulations

Empirical Error Rates (4 clusters)

lambda=0.25

0.5

EER

0.1 1 3 5 7

lambda=1

0.5 0.1 1 3 5 7

lambda=4

0.5 0.1 1 3 5 7

Haar

0.5

EER

0.1 1 3 5 7 0.5 0.1 1 3 5 7 0.5 0.1 1 3 5 7

Bumps

0.5

EER

0.1 1 3 5 7 0.5 0.1 1 3 5 7 0.5 0.1 1 3 5 7

Heavisine

0.5

EER

0.1 1 5 7

SNR

0.5 0.1 1 5 7

SNR

0.5 0.1 1 5 7

SNR

Doppler

FCM FCMunion FCMM FCMMunion Spline

SLIDE 44

Simulations

Union-set Dimension Reduction performance

FPR FNR % of selected coef SNR2

µ / λU

0.25 1 4 0.25 1 4 0.25 1 4 0.1 68.7 81.4 90.3 2.8 1.4 1.1 7.5 4.2 2.5 1 68.4 78.1 82.9 3.8 2.6 2.2 8.4 5.8 4.6 Haar 3 67.8 75.5 77.2 7.7 6.8 6.7 11.7 9.7 9.4 5 69.1 75.0 75.8 8.6 7.9 7.8 12.3 10.7 10.5 7 70.0 75.2 75.7 8.8 8.2 8.0 12.3 10.9 10.7 0.1 91.3 94.1 96.7 2.3 2.3 2.3 7.0 4.9 3.1 1 88.8 91.8 92.6 2.3 2.3 2.3 8.9 6.7 6.1 Bumps 3 88.6 89.6 90.5 1.5 2.3 2.3 8.9 8.3 7.7 5 88.8 89.6 90.5 1.5 1.5 1.8 8.7 8.1 7.6 7 88.9 89.2 89.9 1.5 1.5 1.5 8.7 8.4 7.9 Table: FPR (non-thresholded among true null coefficients), FNR (thresholded among non null coefficients) and percentage of selected wavelet coefficients

SLIDE 45

Simulations

Time of execution

SNRµ 0.1 1 3 5 7 Haar 2.3 2.4 2.3 2.4 2.3 FCM Bumps 2.6 2.5 2.6 2.5 2.5 Haar 0.4 0.4 0.5 0.5 0.5 FCMunion Bumps 0.5 0.5 0.5 0.5 0.5 Haar 16.0 16.1 15.6 15.8 16.0 FCMM Bumps 16.1 16.3 15.2 15.3 15.4 Haar 6.9 7.1 7.6 7.6 7.6 FCMMunion Bumps 6.7 6.7 6.8 6.7 6.7 Haar 25.5 26.2 23.0 23.6 22.3 Spline Bumps 23.3 26.6 22.0 21.2 21.7