Bayesian Modeling of Complex-valued fMRI Signals forBrain Activation

Cheng-Han Yu, Raquel Prado, Hernando Ombao, and Daniel B. Rowe

Joint Statistical Meetings, August 1, 2017

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Motivation of using a complex-valued model

• Many kinds of data sets are complex-valued (CV), for example,imaging, radar, and sonar. fMRI data are complex-valued after FTand IFT image reconstruction. But most fMRI studies usemagnitude-only (MO) data and phase information is discarded.

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Goal and Result

We propose a model that takes real and imaginary parts into account,then fast detect which voxels are activated. We find that the proposedBayesian complex-valued model has higher power and lower type I errorrate than traditional magnitude models.

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Complex-valued linear regression: Rowe (2005)

• In fMRI studies, given time t = 1, . . . ,T and at voxel v = 1, . . . ,N,we have (Rowe-Logan constant phase)

y vt = ρvt cos(φv ) + iρvt sin(φv ) + εvt ,

ρvt = αv0 + αv

1x1,t + αv2x2,t + · · ·+ αv

pxp,t ,

•

yv1...y vT

=

1 x11 · · · xp1...

.... . .

...1 x1T · · · xpT

αv0 cos(φv ) + iαv

0 sin(φv )...

αvp cos(φv )︸ ︷︷ ︸

βvRe

+i αvp sin(φv )︸ ︷︷ ︸

βvIm

+

εv1...εvT

• yv = Xβv

Re + iXβvIm + εv , or yv = Xβv + εv

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Complex-valued linear regression: yv = Xβv + εv

• X is a designed matrix formed by Blood Oxygen Level Dependentcontrasts (BOLD), which is convolution of an stimulus function froman experiment and a hemodynamic response function (HRF).

Source: Martin Lindquist (2008)

• In general, noise εvt = εvt,re + iεvt,im ∼ CN1(0, σ2, τ2)

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Approach to identifying activations for yv = Xβv + εv

• Variable Selection: βvj 6= 0 iff voxel v at task j is activated.

• Complex normal spike-and-slab prior on βv :

βvj ∼ (1− γvj )CN(0, ω0, λ0)︸ ︷︷ ︸spike

+γvj CN(0, ω1, λ1)︸ ︷︷ ︸slab

, ω0 < ω1, λ0 < λ1.

• If γvj = 0, treat βvj as zero, and if γvj = 1, βvj is non-zero.

• CN(µ, σ2, 0): Circular normal that real and imaginary parts areindependent.

• Activation is inferred by borrowing information across voxels througha Bernoulli prior on γvj ∼ Ber(θvj = θj) with a common probability ofactivation for all voxels.

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Model setup

We develop an complex-valued EM variable selection (C-EMVS)algorithm for fast detecting activation at the lowest voxel level.

yv = Xβv + εv , εv ∼ CNT (0, 2σ2v I, 0), v = 1, ...,N

βvj |γvjindep∼ (1− γvj )CN1(0, djσ

2vΓj , ejσ

2vCj) + γvj CN1(0, σ2vΓj , σ

2vCj),

dj << 1, ej << 1, j = 1, ..., p

γvj |θjIID∼ Ber(θj)

θjIID∼ Beta(aθ, bθ)

σ2v ∼ IG (aσ, bσ)

We determine if the real and imaginary parts of βvj are zero jointly:βvj 6= 0 if Pr(γvj = 1 | β∗, θ∗, σ∗, y) > 0.5, where ∗ means the posteriormode and 0.5 is the threshold value.

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Simulation study: data generating

y vt,re = (β0 + βv

1 xbold,t) cos(π/4) + εvt,re , εvt,reIID∼ N(0, 0.25)

y vt,im = (β0 + βv

1 xbold,t) sin(π/4) + εvt,im, εvt,imIID∼ N(0, 0.25)

εvt ∼ CN(0, 0.5, 0)

y vt,mag =

√(y v

t,re)2 + (y vt,im)2, SNR = β0/σ, CNRv = βv

1/σ

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Simulation study: modeling

• The model:

yv = Xβv + εv , εv ∼ CNT (0, 2σ2I , 0), v = 1, ...,V

βvj | γvjindep∼ (1− γvj )CN1(0, v02σ2, 0) + γvj CN1(0, 2σ2, 0),

σ2 ∝ 1/σ2

γvj | θjIID∼ Ber(θj)

θjIID∼ Beta(aθ = 1, bθ = 1)

• Voxel v is active if Pr(γvBOLD = 1 | β∗, θ∗, σ∗) > 0.5

SNR 0.5 1 5 10

CNR 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5

Data No. 1 2 3 4 5 6 7 8 9 10 11 12

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• The CV model detects more true positives than the MO when theSNR is small, leading to higher sensitivity, precision and accuracy.

• The CV model performs better in estimating strength.

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SNR 0.5 1 5 10

CNR 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5

Data No. 1 2 3 4 5 6 7 8 9 10 11 12

• EM performs better than lasso and adaptive lasso.• CV-EM performs consistently across different SNRs.

• CV-EM dominates when SNR is small; MO-EM is better when SNR is large.

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SNR 0.5 1 5 10

CNR 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5 0.5 1 1.5

Data No. 1 2 3 4 5 6 7 8 9 10 11 12

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How to choose v0?

• From Rockova and George (2014) and Wang et al. (2015), choose v∗0that maximizes the Pr(γ | y) that evaluates γ containing only thosevariables for which γv = 1.

• Suggestion: Create a grid between 1/√

100Tp and 1/√

10Tp

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How to choose v0?

• From Rockova and George (2014) and Wang et al. (2015), choose v∗0that maximizes the Pr(γ | y) that evaluates γ containing only thosevariables for which γv = 1.

• Suggestion: Create a grid between 1/√

100Tp and 1/√

10Tp

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Comparison to traditional GLM and different HRFs

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Extension 1: AR(1) noises to human subject data

εvt,k = ϕv εvt−1,k + zvt,k , zvt,k ∼ N(0, σ2), k = Re, Im

p(ϕv ) ∼ Unif (−1, 1)

Figure : Data from Karaman, Bruce and Rowe (KBR, 2014) Left to right:KBR-CV, KBR-MO, DeTeCT-ING, EM-CV

• removes false positives outside the brain area, comparing to KBR-CV.• has higher detecting power than KBR-MO.• is comparable to the nonlinear sophisticated DeTeCT-ING model.

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Extension 2: IID to Spatial dependence (Bezener (2015))

• The spatial dependence is governed by an underlying areal model.

• Parcellating the images into clusters of voxels.

• A spatial hierarchical prior that allow prior anatomical information isused to model the spatial dependence.

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Figure : Left: Spatial MCMC. Right: Non-spatial EMVS

1. The spatial model removes false positives outside the brain and iscomparable to the nonlinear sophisticated DeTeCT-ING model.2. The spatial model further eliminates single isolated active voxels, andencourage grouping active voxels, comparing to the non-spatial models.

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Summary

• C-EMVS fast detects activations at the voxel level.

• Bayesian modeling does not have multiple testing issues, for example,Bonferroni or FDR correction.

• Using complex-valued data detects more true positives (active voxels)and/or less false positives, especially when SNR is small, whilemagnitude data can be used if SNR is large.

• C-EMVS is based on linear model and does not use sophisticatedspatio-temporal or nonlinear models, but its detecting performance iscomparable to those models.

• The CV algorithm can be applied to any complex-valued data.

• Thank you! cheyu@soe.ucsc.edu

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Summary

• C-EMVS fast detects activations at the voxel level.

• Bayesian modeling does not have multiple testing issues, for example,Bonferroni or FDR correction.

• Using complex-valued data detects more true positives (active voxels)and/or less false positives, especially when SNR is small, whilemagnitude data can be used if SNR is large.

• C-EMVS is based on linear model and does not use sophisticatedspatio-temporal or nonlinear models, but its detecting performance iscomparable to those models.

• The CV algorithm can be applied to any complex-valued data.

• Thank you! cheyu@soe.ucsc.edu

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Complex Normal Distribution

• Z = X + iY ∼ CNn(µz ,Γ,C) iff(XY

)∼ N2n

((µx

µy

),Σ =

(ΣX ΣXY

ΣYX ΣY

)). With µz = 0,

Γ := E(ZZ̄′) = ΣX + ΣY + i(ΣXY − ΣYX ),

C := E(ZZ′) = ΣX − ΣY + i(ΣXY + ΣYX ).

• Γ: covariance matrix. C: relation matrix

• C = 0 iff ΣX = ΣY and ΣXY = −ΣYX , and Z is called proper orsecond order circular. Its real and imaginary parts have the samecovariance, and are uncorrelated.

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Distribution of Magnitude and Phase

• Given a complex normal r.v. Z = X + iY(1) the amplitude R =

√X 2 + Y 2 follows (marginal) Rician

distribution,(2) the phase φ conditional on R, (φ | R), follows Tikhonovdistribution (ODonoughue and Moura (IEEE 2012)).

• The model yv = Xβv + εv , with βv = βvRe + iβv

Im is linear and easyto work with.

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Work in process and future work

• Develop a corresponding approximate inference, say variationalinference that includes spatio-temporal structure and fast detectactivations.

• Find a way to adaptively determine the region shapes and sizes ofimages.

• Examine the effects of different shrinkage priors, for example,spike-slab lasso, horseshoe, etc.

• Build a more general model that includes connectivity issue.

• Study the noncircular complex normal behavior in detail.

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