Multivariate General Linear Models (MGLM) on Riemannian...

Hyunwoo J. Kim, Nagesh Adluru, Maxwell D. Collins, Moo K. Chung, !Barbara B. Bendlin, Sterling C. Johnson, Richard J. Davidson, Vikas Singh

http://pages.cs.wisc.edu/~hwkim/projects/riem-mglm/

Multivariate General Linear Models (MGLM) on Riemannian Manifolds

Poster session ID: O-3C-384

http://pages.cs.wisc.edu/~hwkim/projects/riem-mglm/

Disease Normal

Template

General Linear Model for Group Analysis

RegistrationRegistration

Disease Normal

All subjectsAGE

M D

AGE

M D


P-value map

Disease Normal

All subjectsAGE

M D


: Group (patient or normal)

General Linear Model (GLM)

f : R ! R

: Group (patient or normal): Age


f : R2 ! R

: Group (patient or normal)

: Gender: Age


f : R3 ! R

DTI P-value map

GLM on scalar valued summaries

D =

0

@1.53 1.38 0.651.38 1.33 0.700.65 0.70 1.06

1

A

DTI P-value mapFA

MD


DTI P-value map

?


D =

0

@1.53 1.38 0.651.38 1.33 0.700.65 0.70 1.06

1

Ax

TDx > 0, x 6= 0

DTI P-value ?

Response Y is manifold-valued


• Unit sphere, quotient spaces of spheres

• SPD matrix i.e covariance matrix, diffusion tensors

• Probability density functions (PDFs), orientation density functions (ODFs)

• Kendall shape manifolds

• Lie groups i.e. O(n), SO(n), GL(n), SL(n)

Manifold -valued data

Can we directly use the ordinary general linear model with manifold

data?

Euclidean model for manifolds


Training

Test

TrainingTest


Predictions need to be projected onto manifoldsPredictions in ambient space

Problem 1: Model in ambient space


Problem 2: Distance metric in ambient space


Geodesic distance is needed.Problem 2: Distance metric in ambient space

Model on manifolds

Regression with a Single Covariate

Fletcher, P. Thomas. "Geodesic regression and the theory of least squares on Riemannian manifolds.” IJCV, 2013.f : R ! M

Model on manifoldsJia Du, Alvina Goh, Sergey Kushnareva, Anqi Qiu, "Geodesic regression on orientation distribution functions with its application to an aging study." NeuroImage, 2014.

f : R ! M

Model on manifoldsJia Du, Alvina Goh, Sergey Kushnareva, Anqi Qiu, "Geodesic regression on orientation distribution functions with its application to an aging study." NeuroImage, 2014.

f : R ! M

f : Rn ! M Our MGLM

Single covariateE(p, v) =

1

2

NX

i=1

d(Exp(p, xiv), yi)2

M

y2

y3

y4y1

Single covariate E(p, v) =

1

2

NX

i=1

d(Exp(p, xiv), yi)2

M

y1y2

y3

y4

p

Single covariate

y2y4y1

E(p, v) =

1

2

NX

i=1

d(Exp(p, xiv), yi)2

M y3

p

TpM

v

y2y4y1


1

2

NX

i=1

d(Exp(p, xiv), yi)2

M y3

p

TpM

v

y2y4y1


1

2

NX

i=1

d(Exp(p, xiv), yi)2

M y3

p

TpMx1x2

x3x4

v

y1y2

y4


1

2

NX

i=1

d(Exp(p, xiv), yi)2

M y3

p

TpMx1x2

x3x4

v

Multiple covariates

M

p

TpMx

11

v1x

21

v2

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2

Multiple covariates

M

p

TpMx

11

x

21

x

12

x

22

v1

v2

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2

Multiple covariates

p

TpMx

11

x

21

x

12

x

22

v1

v2

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2

Optimization (iterative method)

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2

pk+1 Exp (pk,↵⌃i�yi!pkLog(yi, yi))Step 1 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


pk+1 Exp (pk,↵⌃i�yi!pkLog(yi, yi)) Error term

pkv1

v2y1

y1

Step 1 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


Error term

pk+1 Exp (pk,↵⌃i�yi!pkLog(yi, yi))Parallel transported error

v1

v2

pk

y1y1

Step 1 :

Optimization (iterative method)E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2

Parallel transported error

pk+1 Exp (pk,↵⌃i�yi!pkLog(yi, yi))

pk

y1

pk+1

v1

v2

Step 1 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


pk+1 Exp (pk,↵⌃i�yi!pkLog(yi, yi))

y1y1

v1

v2

pkpk+1

Step 1 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


v

jk+1/2 = v

jk + ↵

X

i

x

ji�yi!pLog(yi, yi) Error term

pk

y1 y1v2k

v1k

Step 2 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


Error term

Parallel transported Error

pk

y1 y1

v1k

v2k

v

jk+1/2 = v

jk + ↵

X

i

x

ji�yi!pLog(yi, yi)Step 2 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


y1 y1

pk

v2kv2k+ 12

v1k+ 12

v1k

v

jk+1/2 = v

jk + ↵

X

i

x

ji�yi!pLog(yi, yi)Step 2 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


V k+1 = �pk!pk+1(V k+1/2)

pk

pk+1

v2k+ 12

v1k+ 12

v1k+1

v2k+1

Step 3 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


V k+1 = �pk!pk+1(V k+1/2)

pk

pk+1

v2k+ 12

v1k+ 12

v1k+1

v2k+1

Step 3 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


Must we solve for p or can we approximate it?

V k+1 = �pk!pk+1(V k+1/2)

pk

pk+1

v2k+ 12

v1k+ 12

v1k+1

v2k+1

Step 3 :

E(p,V ) =

1

2

NX

i=1

d(Exp(p,

X

j

x

jiv

j), yi)

2


Must we solve for p or can we approximate it?

Xie, Yuchen, Baba C. Vemuri, and Jeffrey Ho “Statistical analysis of tensor fields”,MICCAI’10

ExperimentsExperiments Neuroimaging studies

Full model of Long-term meditators (ODFs)

GLM

Full

: y = Exp(p, v 1Group+ v 2

Gender+ v 3Age)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 29 / 37

Synthetic data Neuroimaging dataExperiments Synthetic data

Synthetic data (ODFs)


Experiments Synthetic data



Experiments (synthetic ODF)

ExperimentsExperiments Neuroimaging studies


GLM

Full


Gender+ v 3Age)


Synthetic data Neuroimaging dataExperiments Synthetic data



LTM (ODF) AD risk (DTI)

Subjects 49 343

Group LTM WLC APOE4+ APOE4-

Gender Female Male Female Male

Age 28 - 65 43 - 75

Experiments (neuroimaging study)

GLMAge : y = Exp(p, v2Gender + v3Age)

GLMFull : y = Exp(p, v1Group + v2Gender + V 3Age)

GLMGroup

: y = Exp(p, v1Group + v2Gender

Experiments Neuroimaging studies


GLM

Full


Gender+ v 3Age)


• p-values maps and histograms for effect of age and group computed from simulating the Null distribution of the F ratio statistic using 20,000 permutations.

• F ratio statistic is defined for a pair of nested GLMs as

!


Permutation Test

1p-values computed using the 20,000 permutations to characterize theNull distribution of the R

2 values.

2p-value maps and histograms for e↵ect of age and group computedfrom simulating the Null distribution of the F ratio statistic using20,000 permutations.

3F ratio statistic is defined for a pair of nested GLMs as,

F =RSS1�RSS2

p2�p1

RSS2N�p2

(13)


Experiments (neuroimaging study)

• Age effect in study 1 (long-term meditators, ODFs)

!

!

!

!

!


Age e↵ect of Long-term meditators (ODFs)

GLM

Full


Gender+ v 3Age)

GLM

Group


Gender)


Neuroimaging study 1

• Group effect in study 1 (long-term meditators, ODFs)

!

!

!

!

!


Group e↵ect of Long-term meditators (ODFs)

GLM

Full


Gender+ v 3Age)

GLM

Age

: y = Exp(p, v 2Gender+ v 3

Age)

Group 2 {LTM,WLC}.


Neuroimaging study 1

!

!

!

!

!

!


Age e↵ect of AD data set (DTI)

GLM

Full


Gender+ v 3Age)

GLM

Group


Gender)


Neuroimaging study 2• Age model of study 2 (AD, DTI)

!

!

!

!

!

• Generalization of multivariate general linear model (MGLM) to Riemannian manifolds

• Especially useful when response is manifold valued and we want to control for one or more covariates. Here, the analysis obtains significantly improved statistical power

• Applicable to other manifold-valued statistical inference problems

• Code is available. See me at the posters!

Conclusion


Thank you


Research supported in part by !NSF CAREER RI 1252725!

NIH R01 AG040396

Multivariate General Linear Models (MGLM) on Riemannian...

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