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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