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 SNP component 11 Vast source of information for a group 7 rs539307 rs7213462 rs6894357 rs10853136 rs9897794 rs4470197 rs9913111 rs1057993 rs1531131 rs704716 6 of individuals 5 Novel multivariate methods are desired to 4 3 efficiently mine high-dimensional data. 2 1 0 chromosome 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 3pICA for Imaging Genetics using MOO

Motivation jICA, Linked ICA Multimodal framework for imaging genetics Early approaches have limitations on their assumptions joint ICA and linked ICA: Same mixing matrix, same number of components. pICA 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 . 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 { H ( WX ) } W 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) � β T · � �� H ( W (1) ) , H ( W (2) ) , H ( W (3) ) , ρ 1 , 2 p , q , ρ 2 , 3 q , r , ρ 1 , 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 p , q : Corr 2 ( A ( i ) p , A ( i ) ρ i , j 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) † { ρ 1 , 2 p , q + ρ 1 , 3 p , r + ρ 2 , 3 Solve { p , q , r } ← argmax q , r } { p , q , r } for i = 1 , 2 , 3 , j = 2 , 3 , 1 , x = p , q , r, y = q , r , p do Compute ∇ W ( i ) � ρ i , j if x , y > s then Compute ∇ ρ i , j and ∇ ρ j , i else ∇ ρ i , j = 0 and ∇ ρ j , i = 0 end end ∇ A (1) = ∇ ρ 1 , 2 + ∇ ρ 1 , 3 , ∇ A (2) = ∇ ρ 2 , 1 + ∇ ρ 2 , 3 ∇ A (3) = ∇ ρ 3 , 1 + ∇ ρ 3 , 2 for i = 1 , 2 , 3 do if ||△ W ( i ) || F > ǫ then ( W ( i ) + β i ∇ W ( i ) ) − 1 + β i +3 ∇ A ( i ) � − 1 W ( i ) ← � if Entropy decreases then λ i ← 0 . 9 λ i , 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 ρ 1 , 2 p , q = ρ 2 , 3 q , r = ρ 1 , 3 Correlation [0 , 0 . 6] 0.1 0.4 p , r [0 , 10] 2 10 Noise level SNR dB # Variables Dimensionality log 10 [1 . 5 , 4] 0.25 1.68 # Subjects 3pICA for Imaging Genetics using MOO

Simulation Framework Effect size: � � 1 2 + 0 . 01 e − ( x − µ e )2 0 . 99 e − x 2 f ( x ) = √ 2 2 π Link Strength: Pearson’s correlation coefficient   I Σ 12 Σ 13  , Σ T Σ = Σ 23 I 12  Σ T Σ T I 13 23 where: � � ρ i , j 0 1 × ( c ( j ) − 1 ) 1 , 1 Σ ij = 0 ( c ( i ) − 1 ) × 1 0 ( c ( i ) − 1 ) × ( c ( j ) − 1 ) Noise level: � σ 2 � X SNR dB = 10 log 10 , σ 2 η Dimension of the problem: log 10 ( variables subjects ) 3pICA for Imaging Genetics using MOO

Simulation results 1.0 1.0 1.0 1.0 0.9 Methods Accuracy Accuracy Accuracy 0.9 Accuracy ICA 0.9 0.9 3p−ICA 0.8 0.8 0.7 0.8 Methods 0.8 ICA 0.7 0.6 3p−ICA 0.15 0.15 0.100 Methods 0.20 Methods ICA ICA 3p−ICA 3p−ICA Link Error Link Error 0.075 Link Error Link Error 0.15 0.10 0.10 0.050 0.10 0.05 0.05 0.025 0.05 0.00 0.00 0.000 0.00 0 1 2 3 0.0 0.2 0.4 0.6 0 2 4 6 8 10 2 3 4 Effect Size Correlation SNR Log(Nvar/Nsub) 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 µ e . 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