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Granger Causality on Spatial Manifolds:
applications to Neuroimaging
Pedro A. Valdés-Sosa
Cuban Neuroscience Centre
Multivariate Autoregressive Model
for EEG/fMRI
1
2
…
t
t-1
…
p
 y1,t   a1,1
 y  a
 2,t    2,1

 

 
 y p ,t   a p ,1
a1,2
a2,2
a p ,2
a1, p   y1,t 1   e1,t 
a2, p   y2,t 1   e2,t 


  

  
a p , p   y p ,t 1   e p ,t 
t =1,…,N
y t  A y t 1  et
t =1,…,Nt
W= {1,L , p}
Point influence Measures
I s® u
is the simple test
H0 : a (s, u)= 0
s, u ОW
Granger Causality must be measured on a
MANIFOLD
W= surface of the brain
r
y (s, t ) =
е т т тa
k
k= 1
W
(s, u ) y (u, t - k )du + e (s, t )
Influence Measures defined on a Manifold
An influence field I s® W is a multiple test H 0 : a (s, u)= 0
for a given s ОW
and all
u ОW
Discretization of the Continuos AR Model -I
r
y (s, t ) =
е т т тa
k
k= 1
yi , t = т т т y (u, t ) du
D (si )
йy1; t щ
к ъ
к Mъ
к ъ
y t = ккyi ; t ъ
ъ
к Mъ
к ъ
кy ъ
кл p ; t ы
ъpґ 1
(s, u ) y (u, t - k )du + e (s, t )
W
aik, j = т L т ak (siў, u ўj ) ds ўdu ў
D (si )ґ D (ui )
r
yt =
е
A k y t-
k= 1
et ~ N (0,  )
k
+ et
Multivariate Regression Formulation
й y Tr ... y1T щ
к
ъ
к .
ъ
.
к
ъ
X = кк .
...
. ъ
ъ
к .
ъ
.
к
ъ
кT
ъ
T
y
...
y
кл N - 1
N- r ъ
ы
T
B = [A1 , K , A r ]
T
щ
Z= й
лy r + 1 , K , y N ы
Z = XB+ E
й1 щ
к ъ
vec (B)   = ккL ъ
ъ
к p ъ
кл ъ
ы
й1i щ
к ъ
i
i
i
 = ккL ъ

=
b
{
k
j ,k }
ъ
к i ъ
кл r ы
ъ
ML Estimation and detection of Influence
fields
ˆ = arg min Z - XB
B
B
ˆ = ( XT X)- 1 XT Z
B
tki , j
bˆki , j
=
SE bˆ i
( )
k, j
2
Σ
= arg min Z - XB
2
B
ˆ i = (XT X)- 1 X T z i
I k , i® W = {tki , j }
1Ј iЈ p
Problemas with the Multivariate Autoregressive
Model for Brain Manifolds
 y1,t   a1,1
 y  a
 2,t    2,1

 

 
 y p ,t   a p ,1
p→∞
a1, p   y1,t 1   e1,t 
a2, p   y2,t 1   e2,t 


  

  
a p , p   y p ,t 1  e p ,t 
a1,2
a2,2
a p ,2
t =1,…,N
y t  A y t 1  et
# of parameters
2
(
p
+ p)
g = r Чp 2 +
2
Prior Model on Influence Fields
M
p (;(P1 ,  1 ),L , (PM ,  M )) = C. Х exp (- Pm (-m1 ))
m= 1
ˆ = arg min Z - X B
B
B
2

M
+ е Pm (-m1 )
m= 1
X  = tr (XT  - 1X)
2
Pm (w ) =
length ( x )
е
l= 1
( )
pm wl
Priors for Influence Fields
I x® B
Are of
minimum
norm, or
maximal
smoothness,
etc.
Valdés-Sosa PA Neuroinformatics (2004) 2:1-12
Valdés-Sosa PA et al. Phil. Trans R. Soc. B (2005) 360: 969-981
Penalty Functions
Estimation via MM algorithm
ˆ ik + 1 = (XT X + D(ˆ ik + 1 ))- 1 XT zi
D( ) =
i
M
е
ў
diag ( pm ( wli ) / wli )
m= 1
pm, e (q) = pm (q) - e т
De ( ) =
i
M
е
ў
|q|
0
pl
dt
e+ t
diag ( pm ( wli ) / ( wli + e))
m= 1
Penalty Covariance combinations
sparseness
Model
(
)
L1, I rp2
(
smoothness
L2, I rp2
)
Known as to
wavleteers as
LASSO
Basis Pursuit
Ridge
Frames
Data Fusion
(L1, L )
rp 2
Spline (“LORETA”)
(L2, L )
rp2
(
)(
L1, I rp2 L2, I rp2
(
)(
L1, I rp2 L1, Drp2
both
Name in statistics
Elastic Net
)
Fused Lasso
)
“Ridge Fusion”
(L2, I )(L2, D )
rp2
rp2
(L1, I )(L1, L )(L2, I )(L2, L )
rp2
rp2
rp 2
rp 2
?
Simulated “fMRI”
yt  A yt 1  et
Correlations of the EEG with the fMRI
3
EEG
fMRI
2.5
2
1.5
1
0.5
0
-0.5
-1
r=-0.62
-1.5
-2
10
20
30
40
50
Martinez et. al Neuroimage July 2004
60
70
80
90
100