SPM course - 2002 The Multivariate ToolBox ( F. Kherif , JBP et al.)

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 ...)

04/19/23 JB Poline MAD/SHFJ/CEA 2From Ferath KherifMADIC-UNAF-CEA

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’


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


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) =


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)


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


ss22

VV22

UU22

+ ...

parameterestimates

[U S V] = SVD (X’Y)


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


ss22

VV22

UU22

+ ...

/

E / std = normalised residuals


Normalised residuals :first component


Normalised residuals :first component of a language

study

Temporal pattern difficult to interpret


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) =

=


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’


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


MLMfirst component

p < 0.0001


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


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


• OUT : eigenimages, MLM.mat (stat, OUT : eigenimages, MLM.mat (stat, …) observed and predicted temporal …) observed and predicted temporal responses; Y’Yresponses; Y’Y


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




... or time ... or time ... or time ... or time

= ss11

VV11

UU11

+ + ...

Y

Subjet 1

Subjet nvoxelsvoxels


Subjet 2

ss22

VV22

UU22



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



• 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



• 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 …)


• 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 …)


• OUT : eigenimages, SVD.mat, observed temporal responses; OUT : eigenimages, SVD.mat, observed temporal responses; Y’Y;Y’Y;


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 )

04/19/23 JB Poline MAD/SHFJ/CEA 22From Ferath KherifMADIC-UNAF-CEA



Y



From Ferath KherifMAD-UNAF-CEA


MODEL SELECTION

Subject 1

Selected model

Subject 2 +Subject 3 -Subject 4 +Subject 5 +Subject 6 +Subject 7 +Subject 8 +Subject 9 +

RESULTS


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)

SPM course - 2002 The Multivariate ToolBox ( F. Kherif , JBP et al.)

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