Multivariate Granger causality between macro variables and ...
Granger Causality on Spatial Manifolds: applications to Neuroimaging
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Transcript of Granger Causality on Spatial Manifolds: applications to Neuroimaging
Granger Causality on Spatial Manifolds: applications to Neuroimaging
Pedro A. Valdés-SosaCuban Neuroscience
Centre
Multivariate Autoregressive Model for EEG/fMRI
1
2
…
p…
t t-1
t =1,…,Nt
{ }1, , pW= L
1, 1,1 1,2 1, 1, 1 1,
2, 2,1 2,2 2, 2, 1 2,
, ,1 ,2 , , 1 ,
t p t t
t p t t
p t p p p p p t p t
y a a a y e
y a a a y e
y a a a y e
1t t ty A y e t =1,…,N
Granger Causality must be measured on a MANIFOLD
( ) ( ) ( ) ( )1
, , , ,r
kk
y s t a s u y u t k du e s t= W
= - +å òòò
surface of the brainW=
Influence Measures defined on a Manifold
sI ®W0 :H ( ), 0a s u =
s Î W u Î W
An influence field is a multiple test and all for a given
( ) ( ) ( ) ( )1
, , , ,r
kk
y s t a s u y u t k du e s t= W
= - +å òòò
1;
;
; 1
t
i tt
p t p
y
y
y´
é ùê úê úê úê ú= ê úê úê úê úê úë û
y
M
M
( )( ), ,
i
i ts
y y u t duD
= òòò
1
r
t k t k tk
-=
= +åy A y e
Discretization of the Continuos AR Model -I
( ) ( )( ), ,
i i
ki j k i j
s ua a s u ds du¢ ¢
D ´ D¢ ¢= ò òL
( )0,t N~e
1
1
...
. .
. ... .
. .
...
T Tr
T TN N r- -
é ùê úê úê úê ú= ê úê úê úê úê úë û
y y
X
y y
= +Z XB E
Multivariate Regression Formulation
[ ]1
1
, ,
, ,
T
r
T
r N+
=
é ù= ë û
B A A
Z y y
K
K
( ) { }1
1
,
i
i i ik j k
p ir
vec b
é ù é ùê ú ê úê ú ê ú= = =ê ú ê úê ú ê úê ú ê úë û ë û
B
L L
2 2ˆ arg min arg min= - = -Σ
B BB Z XB Z XB
1ˆ ( )T T-=B X X X Z 1ˆ ( ) ii T T-= X X z X
( ),
,
,
ˆ
ˆ
ik ji
k j ik j
tSE
b
b= { }, , 1
ik i k j i pI t®W £ £
=
ML Estimation and detection of Influence fields
Problemas with the Multivariate Autoregressive Model for Brain Manifolds
1, 1,1 1,2 1, 1, 1 1,
2, 2,1 2,2 2, 2, 1 2,
, ,1 ,2 , , 1 ,
t p t t
t p t t
p t p p p p p t p t
y a a a y e
y a a a y e
y a a a y e
1t t ty A y e p→∞ t =1,…,N
22 ( )
2
p pg r p
+= × +# of parameters
( ) ( )( ) ( )( )11 1
1; , , , , . exp
M
M M m mm
P P C Pp -
== Õ - L
( )2 1
1
ˆ arg minM
m mm
P -
== - + å
BB Z X B
( )2 1Ttr -=X X X
( ) ( )( )
1 l
length
m ml
P p w=
= åx
w
Prior Model on Influence Fields
Priors for Influence Fields
x BI ® Are of minimum norm, or maximal smoothness, etc.
Valdés-Sosa PA Neuroinformatics (2004) 2:1-12Valdés-Sosa PA et al. Phil. Trans R. Soc. B (2005) 360: 969-981
11 1
ˆ ˆ( ( ))i T i Tk k i
-+ += +X X D X z
1
( ) ( ( ) / )M
i i im l l
m
diag p w w¢
=
=åD
| |
,0
( ) ( )m m
pp p dt
t
ql
e q q ee
= -+ò
( )1
( ) ( ( ) / )M
i i im l l
m
diag p w we e¢
=
= +åD
Estimation via MM algorithm
Penalty Covariance combinations
( )21,rp
L I
( )22,rp
L I
( )21,rp
L L
( )22,rp
L L
( )( )2 21, 2,rp rp
L LI I
( )( )2 21, 1,rp rp
L LI D
( )( )2 22, 2,rp rp
L LI D
( )( )( )( )2 2 2 21, 1, 2, 2,rp rp rp rp
L L L LI L I L ?
“Ridge Fusion”
Fused Lasso
Elastic Net
Spline (“LORETA”)
Data Fusion
FramesRidge
Basis PursuitLASSO
Known as to wavleteers as
Name in statisticsModel
spa
rsen
ess
smo
oth
nes
sb
oth
10 20 30 40 50 60 70 80 90 100
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
EEGfMRI
r=-0.62
Correlations of the EEG with the fMRI
Martinez et. al Neuroimage July 2004