Three-way Parallel Independent Component Analysis for Imaging - - PowerPoint PPT Presentation

Three-way Parallel Independent Component Analysis for Imaging Genetics Using Multi-Objective Optimization

& Alvaro Ulloa, Jingyu Liu, Victor Vergara, Jiayu Chen, Vince Calhoun, and Marios Pattichis August 30, 2014

3pICA for Imaging Genetics using MOO

Outline Background: Imaging genetics, multimodality Motivation Method

Simulation Framework ICA, pICA, 3pICA

Results Conclusions

3pICA for Imaging Genetics using MOO

Background

Imaging genetics → Multimodal datasets

Magnetic resonance images Single nucleotide polymorphisms, copy number variations, ... behavioral assessments

Vast source of information for a group

• f individuals

Novel multivariate methods are desired to efficiently mine high-dimensional data.

1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

rs539307 rs7213462 rs6894357 rs9897794 rs10853136 rs4470197 rs1057993 rs9913111 rs1531131 rs704716

chromosome SNP component 11

3pICA for Imaging Genetics using MOO

Motivation

Multimodal framework for imaging genetics Early approaches have limitations on their assumptions

joint ICA and linked ICA: Same mixing matrix, same number of components. Principal component regression, sparse reduced-rank regression, sparse partial least squares, sparse canonical correlation. pICA: only for two modalities

Our focus is to extend pICA to three modalities. jICA, Linked ICA pICA

3pICA for Imaging Genetics using MOO

Independent Component Analysis

Matrix decomposition X = AS

S: row sources,the weighted pattern of variables. A: how each source is represented across subjects.

Source assumptions

Non-Gaussian stationary in the statistical sense independent

Maximization of independence

INFOMAX: Maximization of Entropy W = argmax

W

{H(WX)} where, A = W +, S = WX and H(.) denotes the entropy function.

3pICA for Imaging Genetics using MOO

Three-way Parallel ICA

Joint analysis of up to three data modalities. Exploit shared information in latent variables in the form of correlation. max(p,q,r),W (1),W (2),W (3)

• βT ·
• H(W (1)), H(W (2)), H(W (3)), ρ1,2

p,q, ρ2,3 q,r, ρ1,3 p,r

• β : scalarization vector

p, q, r : component indexes of A(1), A(2) and A(3) W (1), W (2), W (3) : unmixing matrices ρi,j

p,q: Corr2(A(i) p , A(i) q , )

3pICA for Imaging Genetics using MOO

Three-way Parallel ICA

Algorithm Input: X (1), X (2), X (3), and s Output: W (1), W (2), and W (3) Initialization: W (i) ← I, i = 1, 2, 3; while {||△W (1)||F , ||△W (2)||F , ||△W (3)||F } > ǫ do A(1) ← W (1)†, A(2) ← W (2)†, A(3) ← W (3)† Solve {p, q, r} ← argmax

{p,q,r}

{ρ1,2

p,q + ρ1,3 p,r + ρ2,3 q,r }

for i = 1, 2, 3, j = 2, 3, 1, x = p, q, r, y = q, r, p do Compute ∇W (i) if

• ρi,j

x,y > s then

Compute ∇ρi,j and ∇ρj,i else ∇ρi,j = 0 and ∇ρj,i = 0 end end ∇A(1) = ∇ρ1,2 + ∇ρ1,3, ∇A(2) = ∇ρ2,1 + ∇ρ2,3 ∇A(3) = ∇ρ3,1 + ∇ρ3,2 for i = 1, 2, 3 do if ||△W (i)||F > ǫ then W (i) ←

• (W (i) + βi ∇W (i))−1 + βi+3∇A(i)−1

if Entropy decreases then λi ← 0.9λi , end end end end 3pICA for Imaging Genetics using MOO

Simulation Framework

Simulation settings Strategy: Fix all but one, then measure component accuracy and link estimation error 6 sources (4 non-Gaussian, 1 Gaussian and 1 from bimodal Gaussian) 200 subjects 10K variables 10 dB Link strength: 0.4 Effect size of 2

3pICA for Imaging Genetics using MOO

Simulation Framework

Table : Simulation settings.

Variable Measure Range Step Default Effect Size Cohen’s d [0, 3] 0.5 2 Correlation ρ1,2

p,q = ρ2,3 q,r = ρ1,3 p,r

[0, 0.6] 0.1 0.4 Noise level SNRdB [0, 10] 2 10 Dimensionality log10

#Variables #Subjects

[1.5, 4] 0.25 1.68

3pICA for Imaging Genetics using MOO

Simulation Framework

Effect size: f (x) = 1 √ 2π

• 0.99e− x2

2 + 0.01e− (x−µe )2 2

• Link Strength: Pearson’s correlation coefficient

Σ =   I Σ12 Σ13 ΣT

12

I Σ23 ΣT

13

ΣT

23

I   , where: Σij =

• ρi,j

1,1

01×(c(j)−1) 0(c(i)−1)×1 0(c(i)−1)×(c(j)−1)

• Noise level:

SNRdB = 10 log10 σ2

X

σ2

η

• ,

Dimension of the problem: log10( variables

subjects )

3pICA for Imaging Genetics using MOO

Simulation results

0.6 0.7 0.8 0.9 1.0 0.00 0.05 0.10 0.15

Accuracy Link Error

1 2 3

Effect Size

Methods ICA 3p−ICA 0.8 0.9 1.0 0.00 0.05 0.10 0.15

Accuracy Link Error

0.0 0.2 0.4 0.6

Correlation

Methods ICA 3p−ICA 0.8 0.9 1.0 0.000 0.025 0.050 0.075 0.100

Accuracy Link Error

2 4 6 8 10

SNR

Methods ICA 3p−ICA 0.7 0.8 0.9 1.0 0.00 0.05 0.10 0.15 0.20

Accuracy Link Error

2 3 4

Log(Nvar/Nsub)

Methods ICA 3p−ICA

3pICA for Imaging Genetics using MOO

SMRI-FMRI-SNP dataset

112 healthy subjects fMRI contrast images during an auditory oddball task Gray matter concentration images from sMRI 65K within coding genes, obtained by the Mind Clinical Imaging Consortium study.

3pICA for Imaging Genetics using MOO

Application to SMRI-FMRI-SNP dataset

3pICA for Imaging Genetics using MOO

Conclusions

Simulation Effect size: The estimation of one modality’s sources has benefited from the other two that contained shared information, and then is able to overcome the impact of µe. Link Strength: The observed jump is due to the threshold set to 0.2. When the imposed correlation increased above the threshold the algorithm was more likely to use the additional enhancement. Noise Level: invariant to noise with SNR varying from 0 to 10 dB. Drops at -15 dB. Dimensionality: diminished shared information when decreased sample size. At least 1 subject per 1000 variables. Application Used as proof of concept, the algorithm provides interpretable results in the three-modal dataset.

3pICA for Imaging Genetics using MOO