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04/19/23 JB Poline MAD/SHFJ/CEA
SPM course - 2002SPM course - 2002The Multivariate ToolBox (The Multivariate ToolBox (F. KherifF. Kherif, JBP et al.), JBP et al.)
The RFT
Hammering a Linear Model
Use forNormalisation
T and F tests : (orthogonal projections)
Jean-Baptiste PolineOrsay SHFJ-CEAwww.madic.org
Multivariate tools (PCA, PLS, MLM ...)
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04/19/23 JB Poline MAD/SHFJ/CEA 2From Ferath KherifMADIC-UNAF-CEA
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04/19/23 JB Poline MAD/SHFJ/CEA 3
SVD : the basic conceptSVD : the basic conceptSVD : the basic conceptSVD : the basic concept
A time-series of 1D A time-series of 1D imagesimages128 scans of 40 128 scans of 40 “voxels”“voxels”
Expression of 1st 3 Expression of 1st 3 “eigenimages”“eigenimages”
Eigenvalues and Eigenvalues and spatial “modes”spatial “modes”
The time-series The time-series ‘reconstituted’‘reconstituted’
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04/19/23 JB Poline MAD/SHFJ/CEA 4
Y Y (DATA)(DATA)
timetime
voxelsvoxels
Y = USVY = USVTT = = ss11UU11VV11TT + + ss22UU22VV22
T T + ... + ...
APPROX. APPROX. OF YOF Y
UU11
==APPROX. APPROX.
OF YOF YAPPROX. APPROX.
OF YOF Y
+ + ss22 + + ss33 + ...+ ...ss11
UU22 UU33
VV11 VV22 VV33
Eigenimages and SVDEigenimages and SVDEigenimages and SVDEigenimages and SVD
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04/19/23 JB Poline MAD/SHFJ/CEA 5
Linear model : recall ...Linear model : recall ...Linear model : recall ...Linear model : recall ...
e
= +Y X
data matrix
des
ign
mat
rix
+=
voxelsvoxels
scansscans
^
residualsparameterestimates
Variance(e) =
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04/19/23 JB Poline MAD/SHFJ/CEA 6
SVD of Y (corresponds to PCA...)SVD of Y (corresponds to PCA...)SVD of Y (corresponds to PCA...)SVD of Y (corresponds to PCA...)
Y
voxelsvoxels
scansscans
e
= +Y X
data matrix
desi
gn m
atri
x
+= voxelsvoxels
scansscans
^
residuals
parameterestimates
Variance(e) =
= APPROX. APPROX.
OF YOF Yss11
VV11
UU11
+ APPROX. APPROX. OF YOF Y
ss22
VV22
UU22
+ ...
[U S V] = SVD (Y)
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04/19/23 JB Poline MAD/SHFJ/CEA 7
SVD of SVD of (corresponds to PLS...)(corresponds to PLS...)SVD of SVD of (corresponds to PLS...)(corresponds to PLS...)
e
= +Y X
data matrix
desi
gn m
atri
x
+= voxelsvoxels
scansscans
^
residuals
parameterestimates
Variance(e) =
= APPROX. APPROX.
OF YOF Yss11
VV11
UU11
+ APPROX. APPROX. OF YOF Y
ss22
VV22
UU22
+ ...
parameterestimates
[U S V] = SVD (X’Y)
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04/19/23 JB Poline MAD/SHFJ/CEA 8
SVD of residuals : a tool for model SVD of residuals : a tool for model checkingchecking
SVD of residuals : a tool for model SVD of residuals : a tool for model checkingchecking
E
voxelsvoxels
scansscans
e
= +Y X
data matrix
desi
gn m
atri
x
+= voxelsvoxels
scansscans
^
residuals
parameterestimates
Variance(e) =
= APPROX. APPROX.
OF YOF Yss11
VV11
UU11
+ APPROX. APPROX. OF YOF Y
ss22
VV22
UU22
+ ...
/
E / std = normalised residuals
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04/19/23 JB Poline MAD/SHFJ/CEA 9
Normalised residuals :first component
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04/19/23 JB Poline MAD/SHFJ/CEA 10
Normalised residuals :first component of a language
study
Temporal pattern difficult to interpret
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04/19/23 JB Poline MAD/SHFJ/CEA 11
SVD of normalised SVD of normalised (MLM ...)(MLM ...)SVD of normalised SVD of normalised (MLM ...)(MLM ...)
e
= +Y X
data matrix
desi
gn m
atri
x
+= voxelsvoxels
scansscans
^
residuals
parameterestimates
Variance(e) =
=
APPROX. APPROX. OF YOF Y
ss11
VV11
UU11
+ APPROX. APPROX.
OF YOF Yss22
VV22
UU22
+ ...
parameterestimates
[U S V] = SVD ((X’ CX)-1/2 X’Y )
(X’ VX)-1/2 X’
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04/19/23 JB Poline MAD/SHFJ/CEA 12
MLM : some good pointsMLM : some good pointsMLM : some good pointsMLM : some good points
• Takes into account the temporal and spatial structure without Takes into account the temporal and spatial structure without withening withening
• Provides a testProvides a test– sum of q last eigenvalues Ssum of q last eigenvalues Sii for q = n, n-1, ..., 1 for q = n, n-1, ..., 1
– find a distribution for this sum under the null hypothesis (Worsley et al)find a distribution for this sum under the null hypothesis (Worsley et al)
• Temporal and spatial responses : Temporal and spatial responses : – Yt = Y V’Yt = Y V’ Temporal OBSERVED responseTemporal OBSERVED response
– Xt = X(X’X)Xt = X(X’X)-1 -1 (X’ C(X’ CX)X)1/2 1/2 U’SU’S Temporal PREDICTED responseTemporal PREDICTED response
– Sp = (X’ CSp = (X’ CX)X)-1/2 -1/2 X’Y U SX’Y U S-1-1 Spatial responseSpatial response
• Takes into account the temporal and spatial structure without Takes into account the temporal and spatial structure without withening withening
• Provides a testProvides a test– sum of q last eigenvalues Ssum of q last eigenvalues Sii for q = n, n-1, ..., 1 for q = n, n-1, ..., 1
– find a distribution for this sum under the null hypothesis (Worsley et al)find a distribution for this sum under the null hypothesis (Worsley et al)
• Temporal and spatial responses : Temporal and spatial responses : – Yt = Y V’Yt = Y V’ Temporal OBSERVED responseTemporal OBSERVED response
– Xt = X(X’X)Xt = X(X’X)-1 -1 (X’ C(X’ CX)X)1/2 1/2 U’SU’S Temporal PREDICTED responseTemporal PREDICTED response
– Sp = (X’ CSp = (X’ CX)X)-1/2 -1/2 X’Y U SX’Y U S-1-1 Spatial responseSpatial response
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04/19/23 JB Poline MAD/SHFJ/CEA 13
MLMfirst component
p < 0.0001
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04/19/23 JB Poline MAD/SHFJ/CEA 14
MLM : more general and computations MLM : more general and computations improved ... improved ...
MLM : more general and computations MLM : more general and computations improved ... improved ...
• From X’Y to XFrom X’Y to XGG’Y’YGG
XXGG = X - G(G’G) = X - G(G’G)++G’XG’X
YYGG = Y - G(G’G) = Y - G(G’G)++G’YG’Y
• X and XX and XGG used to need to be of full rank : used to need to be of full rank :
– not any morenot any more
• G is chosen through an « F-contrast » that defines a G is chosen through an « F-contrast » that defines a space of interestspace of interest
• From X’Y to XFrom X’Y to XGG’Y’YGG
XXGG = X - G(G’G) = X - G(G’G)++G’XG’X
YYGG = Y - G(G’G) = Y - G(G’G)++G’YG’Y
• X and XX and XGG used to need to be of full rank : used to need to be of full rank :
– not any morenot any more
• G is chosen through an « F-contrast » that defines a G is chosen through an « F-contrast » that defines a space of interestspace of interest
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04/19/23 JB Poline MAD/SHFJ/CEA 15
MLM : implementationMLM : implementationMLM : implementationMLM : implementation
• Computation through betasComputation through betas
• Several subjects Several subjects
• IN : IN : – An SPM analysis directory (the model has An SPM analysis directory (the model has
been estimated) IN GENERAL, GET A been estimated) IN GENERAL, GET A FLEXIBLE MODEL FOR MLMFLEXIBLE MODEL FOR MLM
– A CONTRAST defining a space of interest A CONTRAST defining a space of interest or of no interest … (here G) IN GENERAL, or of no interest … (here G) IN GENERAL, GET A FLEXIBLE CONTRAST FOR GET A FLEXIBLE CONTRAST FOR MLM MLM
– Output directoryOutput directory
– names for eigenimagesnames for eigenimages
• OUT : eigenimages, MLM.mat (stat, OUT : eigenimages, MLM.mat (stat, …) observed and predicted temporal …) observed and predicted temporal responses; Y’Yresponses; Y’Y
• Computation through betasComputation through betas
• Several subjects Several subjects
• IN : IN : – An SPM analysis directory (the model has An SPM analysis directory (the model has
been estimated) IN GENERAL, GET A been estimated) IN GENERAL, GET A FLEXIBLE MODEL FOR MLMFLEXIBLE MODEL FOR MLM
– A CONTRAST defining a space of interest A CONTRAST defining a space of interest or of no interest … (here G) IN GENERAL, or of no interest … (here G) IN GENERAL, GET A FLEXIBLE CONTRAST FOR GET A FLEXIBLE CONTRAST FOR MLM MLM
– Output directoryOutput directory
– names for eigenimagesnames for eigenimages
• OUT : eigenimages, MLM.mat (stat, OUT : eigenimages, MLM.mat (stat, …) observed and predicted temporal …) observed and predicted temporal responses; Y’Yresponses; Y’Y
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04/19/23 JB Poline MAD/SHFJ/CEA 16
Re-inforcement in space ...Re-inforcement in space ...Re-inforcement in space ...Re-inforcement in space ...
= ss11
VV11
UU11
+ ss22
VV22
UU22
+ ...Y
Subjet 1
Subjet 2
Subjet n
voxelsvoxels
APPROX. APPROX. OF YOF Y
APPROX. APPROX. OF YOF Y
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04/19/23 JB Poline MAD/SHFJ/CEA 17
... or time ... or time ... or time ... or time
= ss11
VV11
UU11
+ + ...
Y
Subjet 1
Subjet nvoxelsvoxels
APPROX. APPROX. OF YOF Y
Subjet 2
ss22
VV22
UU22
APPROX. APPROX. OF YOF Y
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04/19/23 JB Poline MAD/SHFJ/CEA 18
SVD : implementationSVD : implementationSVD : implementationSVD : implementation
• Choose or not to divide by the sd of residual fields (ResMS)Choose or not to divide by the sd of residual fields (ResMS)– removes components due to large blood vesselsremoves components due to large blood vessels
• Choose or not to apply a temporal filter (stored in xX)Choose or not to apply a temporal filter (stored in xX)• Choose a projector that can be either « in » X or in a space Choose a projector that can be either « in » X or in a space
orthogonal to itorthogonal to it• study the residual field by choosing a contrast that define the all spacestudy the residual field by choosing a contrast that define the all space
• study the data themselves by choosing a null contrast (we need to generalise spm_conman a little)study the data themselves by choosing a null contrast (we need to generalise spm_conman a little)
– to detect non modeled sources of variance that may lead to non valid or non optimal data to detect non modeled sources of variance that may lead to non valid or non optimal data analyses.analyses.
– to rank the different source of variance with decreasing importance.to rank the different source of variance with decreasing importance.
• Possibility of several subjectsPossibility of several subjects
• Choose or not to divide by the sd of residual fields (ResMS)Choose or not to divide by the sd of residual fields (ResMS)– removes components due to large blood vesselsremoves components due to large blood vessels
• Choose or not to apply a temporal filter (stored in xX)Choose or not to apply a temporal filter (stored in xX)• Choose a projector that can be either « in » X or in a space Choose a projector that can be either « in » X or in a space
orthogonal to itorthogonal to it• study the residual field by choosing a contrast that define the all spacestudy the residual field by choosing a contrast that define the all space
• study the data themselves by choosing a null contrast (we need to generalise spm_conman a little)study the data themselves by choosing a null contrast (we need to generalise spm_conman a little)
– to detect non modeled sources of variance that may lead to non valid or non optimal data to detect non modeled sources of variance that may lead to non valid or non optimal data analyses.analyses.
– to rank the different source of variance with decreasing importance.to rank the different source of variance with decreasing importance.
• Possibility of several subjectsPossibility of several subjects
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04/19/23 JB Poline MAD/SHFJ/CEA 19
SVD : implementationSVD : implementationSVD : implementationSVD : implementation
• Computation through the svd(PY’YP’) = v s v’Computation through the svd(PY’YP’) = v s v’– compute Y ’Y once, reuse it for an other projectorcompute Y ’Y once, reuse it for an other projector– Y can be filtered or not; divided by the res or notY can be filtered or not; divided by the res or not– to get the spatial signal, reread the data and compute Yvsto get the spatial signal, reread the data and compute Yvs -1-1
• TAKES A LONG TIME …TAKES A LONG TIME …
• possibility of several subjects (in that case, sums the possibility of several subjects (in that case, sums the individual Y’Y)individual Y’Y)
• (near) future implementation : use the betas when P (near) future implementation : use the betas when P projects in the space of Xprojects in the space of X
• Computation through the svd(PY’YP’) = v s v’Computation through the svd(PY’YP’) = v s v’– compute Y ’Y once, reuse it for an other projectorcompute Y ’Y once, reuse it for an other projector– Y can be filtered or not; divided by the res or notY can be filtered or not; divided by the res or not– to get the spatial signal, reread the data and compute Yvsto get the spatial signal, reread the data and compute Yvs -1-1
• TAKES A LONG TIME …TAKES A LONG TIME …
• possibility of several subjects (in that case, sums the possibility of several subjects (in that case, sums the individual Y’Y)individual Y’Y)
• (near) future implementation : use the betas when P (near) future implementation : use the betas when P projects in the space of Xprojects in the space of X
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04/19/23 JB Poline MAD/SHFJ/CEA 20
SVD : implementationSVD : implementationSVD : implementationSVD : implementation
• IN : IN : – Liste of images (possibly several « subjects »)Liste of images (possibly several « subjects »)
– Input SPM directory (this is not always theoretically necessary but it Input SPM directory (this is not always theoretically necessary but it is in the current implementation)is in the current implementation)
– A CONTRAST defining a space of interest or of no interest … A CONTRAST defining a space of interest or of no interest …
– in the residual space of that contrast or not ?in the residual space of that contrast or not ?
– Output directory (general, per subject …)Output directory (general, per subject …)
– names for eigenimagesnames for eigenimages
• OUT : eigenimages, SVD.mat, observed temporal responses; OUT : eigenimages, SVD.mat, observed temporal responses; Y’Y;Y’Y;
• IN : IN : – Liste of images (possibly several « subjects »)Liste of images (possibly several « subjects »)
– Input SPM directory (this is not always theoretically necessary but it Input SPM directory (this is not always theoretically necessary but it is in the current implementation)is in the current implementation)
– A CONTRAST defining a space of interest or of no interest … A CONTRAST defining a space of interest or of no interest …
– in the residual space of that contrast or not ?in the residual space of that contrast or not ?
– Output directory (general, per subject …)Output directory (general, per subject …)
– names for eigenimagesnames for eigenimages
• OUT : eigenimages, SVD.mat, observed temporal responses; OUT : eigenimages, SVD.mat, observed temporal responses; Y’Y;Y’Y;
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04/19/23 JB Poline MAD/SHFJ/CEA 21
Multivariate Toolbox : An application for model Multivariate Toolbox : An application for model specification in neuroimagingspecification in neuroimaging
(F. Kherif et al., NeuroImage 2002 ) (F. Kherif et al., NeuroImage 2002 )
Multivariate Toolbox : An application for model Multivariate Toolbox : An application for model specification in neuroimagingspecification in neuroimaging
(F. Kherif et al., NeuroImage 2002 ) (F. Kherif et al., NeuroImage 2002 )
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04/19/23 JB Poline MAD/SHFJ/CEA 22From Ferath KherifMADIC-UNAF-CEA
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Y
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04/19/23 JB Poline MAD/SHFJ/CEA 26
From Ferath KherifMAD-UNAF-CEA
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04/19/23 JB Poline MAD/SHFJ/CEA 27
MODEL SELECTION
Subject 1
Selected model
Subject 2 +Subject 3 -Subject 4 +Subject 5 +Subject 6 +Subject 7 +Subject 8 +Subject 9 +
RESULTS
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04/19/23 JB Poline MAD/SHFJ/CEA 28
Group Homogeneity Analysis
Z1=M-1/2 X’Y1
Z2=M-1/2 X’Y2
…
Zk=M-1/2 X’Yk
W1=Z1 Z1’
W2= Z2 Z2’
…
Wk= Z2 Z2’
RVij =Tr(WiWj)
Sqrt[Tr(Wi2) Tr(Wj
2)]
D = 1- Rvij , 1 < i,j < k
Similarity measure
Distance matrix
Subjects classification (multi-dimensionnal scaling)