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

Disease Normal

Template

General Linear Model for Group Analysis

RegistrationRegistration

Disease Normal

All subjectsAGE

M D

AGE

M D

General Linear Model for Group Analysis

P-value map

Disease Normal

All subjectsAGE

M D

General Linear Model for Group Analysis

: Group (patient or normal)

General Linear Model (GLM)

f : R ! R

: Group (patient or normal): Age

General Linear Model (GLM)

f : R2 ! R

: Group (patient or normal)

: Gender: Age

General Linear Model (GLM)

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

GLM on scalar valued summaries

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

Ax

TDx > 0, x 6= 0

DTI P-value ?

Response Y is manifold-valued

GLM on scalar valued summaries

• 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

Euclidean model for manifolds

Training

Test

TrainingTest

Euclidean model for manifolds

Predictions need to be projected onto manifoldsPredictions in ambient space

Problem 1: Model in ambient space

Euclidean model for manifolds

Euclidean model for manifolds

Problem 2: Distance metric in ambient space

Euclidean model for manifolds

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

Single covariateE(p, v) =

1

2

NX

i=1

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

M y3

p

TpM

v

y2y4y1

Single covariateE(p, v) =

1

2

NX

i=1

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

M y3

p

TpMx1x2

x3x4

v

y1y2

y4

Single covariateE(p, v) =

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

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

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

Optimization (iterative method)

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

Optimization (iterative method)

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

Optimization (iterative method)

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

Optimization (iterative method)

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

Optimization (iterative method)

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

Optimization (iterative method)

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

Optimization (iterative method)

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

Optimization (iterative method)

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

Optimization (iterative method)

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

Optimization (iterative method)

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)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 25 / 37

Experiments Synthetic data

Synthetic data (ODFs)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 25 / 37

Experiments (synthetic ODF)

Experiments Synthetic data

Synthetic data (ODFs)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 25 / 37

Experiments (synthetic ODF)

Experiments Synthetic data

Synthetic data (ODFs)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 25 / 37

Experiments (synthetic ODF)

Experiments Synthetic data

Synthetic data (ODFs)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 25 / 37

Experiments (synthetic ODF)

Experiments Synthetic data

Synthetic data (ODFs)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 25 / 37

Experiments (synthetic ODF)

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)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 25 / 37

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

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

• 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

!

Experiments Neuroimaging studies

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)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 28 / 37

Experiments (neuroimaging study)

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

!

!

!

!

!

Experiments Neuroimaging studies

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

GLM

Full

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

Gender+ v 3Age)

GLM

Group

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

Gender)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 30 / 37

Neuroimaging study 1

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

!

!

!

!

!

Experiments Neuroimaging studies

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

GLM

Full

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

Gender+ v 3Age)

GLM

Age

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

Age)

Group 2 {LTM,WLC}.

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 31 / 37

Neuroimaging study 1

!

!

!

!

!

!

Experiments Neuroimaging studies

Age e↵ect of AD data set (DTI)

GLM

Full

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

Gender+ v 3Age)

GLM

Group

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

Gender)

Hyunwoo Kim (UW-Madison) MGLM on manifolds June 12, 2014 33 / 37

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

Poster session ID: O-3C-384

Thank you

Poster session ID: O-3C-384

Research supported in part by !NSF CAREER RI 1252725!

NIH R01 AG040396