1. SPM course - 2002 The Multivariate ToolBox (F. Kherif, JBP et al.)
In particular, I wish to give you enough information such that you may be able to undezrstand some difficulties that may arise when using linear model.
Among the various tools that SPM uses to anlyse brain images, I will concentrate on two aspects : the linear model that is fitted to the data and the test.
SPM course The Multivariate ToolBox (F. Kherif, JBP et al.)
T and F tests :
(orthogonal projections)
Hammering a Linear Model
The RFT
Multivariate tools
(PCA, PLS, MLM ...)
Use for
Normalisation
Jean-Baptiste Poline
Orsay SHFJ-CEA

2. JB Poline MAD/SHFJ/CEA
From Ferath Kherif
MADIC-UNAF-CEA
03/12/2018
JB Poline MAD/SHFJ/CEA

3. JB Poline MAD/SHFJ/CEA
SVD : the basic concept
A time-series of 1D images
128 scans of 40 “voxels”
Expression of 1st 3 “eigenimages”
Eigenvalues and spatial “modes”
The time-series ‘reconstituted’
03/12/2018
JB Poline MAD/SHFJ/CEA

4. JB Poline MAD/SHFJ/CEA
Eigenimages and SVD V1 V2 V3
voxels
APPROX.
OF Y U1 U2
APPROX.
OF Y U3
APPROX.
OF Y s1
+ s2
+ s3
Y (DATA) =
+ ...
time
Y = USVT = s1U1V1T s2U2V2T
03/12/2018
JB Poline MAD/SHFJ/CEA

5. JB Poline MAD/SHFJ/CEA
Linear model : recall ...
voxels
parameter estimates =  +
residuals
data matrix
design matrix
scans  ^ e = + Y X
Variance(e) =
03/12/2018
JB Poline MAD/SHFJ/CEA

6. SVD of Y (corresponds to PCA...)= + Y X
data matrix
design matrix 
voxels
scans  ^
residuals
parameter estimates
Variance(e) = V1 V2 U1 U2
voxels
APPROX.
OF Y
APPROX.
OF Y s2 = s1 +
+ ... Y
scans
[U S V] = SVD (Y)
03/12/2018
JB Poline MAD/SHFJ/CEA

7. SVD of  (corresponds to PLS...)= + Y X
data matrix
design matrix 
voxels
scans  ^
residuals
parameter estimates
Variance(e) = V1 V2 U1 U2
APPROX.
OF Y
parameter estimates
APPROX.
OF Y s2 = s1 +
+ ...
[U S V] = SVD (X’Y)
03/12/2018
JB Poline MAD/SHFJ/CEA

8. SVD of residuals : a tool for model checking= + Y X
data matrix
design matrix 
voxels
scans  ^
residuals
parameter estimates
Variance(e) = V1 V2
voxels U1 U2
APPROX.
OF Y
APPROX.
OF Y E s2
scans = s1 +
+ ... /
E / std = normalised residuals
03/12/2018
JB Poline MAD/SHFJ/CEA

9. Normalised residuals :
first component
03/12/2018
JB Poline MAD/SHFJ/CEA

10. Normalised residuals : first component of a language study
Temporal pattern difficult to interpret
03/12/2018
JB Poline MAD/SHFJ/CEA

11. SVD of normalised  (MLM ...)= + Y X
data matrix
design matrix 
voxels
scans  ^
residuals
parameter estimates
Variance(e) = V1 V2
parameter estimates U1 U2
APPROX.
OF Y
APPROX.
OF Y
(X’ V X)-1/2 X’ = s1 + s2
+ ...
[U S V] = SVD ((X’ C X)-1/2 X’Y )
03/12/2018
JB Poline MAD/SHFJ/CEA

12. JB Poline MAD/SHFJ/CEA
MLM : some good points
Takes into account the temporal and spatial structure without withening
Provides a test
sum of q last eigenvalues Si for q = n, n-1, ..., 1
find a distribution for this sum under the null hypothesis (Worsley et al)
Temporal and spatial responses :
Yt = Y V’ Temporal OBSERVED response
Xt = X(X’X)-1 (X’ C X)1/2 U’S Temporal PREDICTED response
Sp = (X’ C X)-1/2 X’Y U S-1 Spatial response
03/12/2018
JB Poline MAD/SHFJ/CEA

13. JB Poline MAD/SHFJ/CEA
MLM
first component
p <
03/12/2018
JB Poline MAD/SHFJ/CEA

14. MLM : more general and computations improved ...
From X’Y to XG’YG
XG = X - G(G’G)+G’X
YG = Y - G(G’G)+G’Y
X and XG used to need to be of full rank :
not any more
G is chosen through an « F-contrast » that defines a space of interest
03/12/2018
JB Poline MAD/SHFJ/CEA

15. JB Poline MAD/SHFJ/CEA
MLM : implementation
Computation through betas
Several subjects
IN :
An SPM analysis directory (the model has been estimated) IN GENERAL, GET A FLEXIBLE MODEL FOR MLM
A CONTRAST defining a space of interest or of no interest … (here G) IN GENERAL, GET A FLEXIBLE CONTRAST FOR MLM
Output directory
names for eigenimages
OUT : eigenimages, MLM.mat (stat, …) observed and predicted temporal responses; Y’Y
03/12/2018
JB Poline MAD/SHFJ/CEA

16. Re-inforcement in space ...V1 V2
voxels U1 U2
Subjet 1
Subjet 2
APPROX.
OF Y
APPROX.
OF Y Y s2 = s1 +
+ ...
Subjet n
03/12/2018
JB Poline MAD/SHFJ/CEA

17. JB Poline MAD/SHFJ/CEA
... or time
Subjet 1 V1
voxels
Subjet 2
Subjet n U1
APPROX.
OF Y s1 Y = V2 U2 +
APPROX.
OF Y
+ ... s2
03/12/2018
JB Poline MAD/SHFJ/CEA

18. JB Poline MAD/SHFJ/CEA
SVD : implementation
Choose or not to divide by the sd of residual fields (ResMS)
removes components due to large blood vessels
Choose or not to apply a temporal filter (stored in xX)
Choose a projector that can be either « in » X or in a space orthogonal to it
study 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)
to detect non modeled sources of variance that may lead to non valid or non optimal data analyses.
to rank the different source of variance with decreasing importance.
Possibility of several subjects
03/12/2018
JB Poline MAD/SHFJ/CEA

19. JB Poline MAD/SHFJ/CEA
SVD : implementation
Computation through the svd(PY’YP’) = v s v’
compute Y ’Y once, reuse it for an other projector
Y can be filtered or not; divided by the res or not
to get the spatial signal, reread the data and compute Yvs-1
TAKES A LONG TIME …
possibility of several subjects (in that case, sums the individual Y’Y)
(near) future implementation : use the betas when P projects in the space of X
03/12/2018
JB Poline MAD/SHFJ/CEA

20. JB Poline MAD/SHFJ/CEA
SVD : implementation
IN :
Liste of images (possibly several « subjects »)
Input SPM directory (this is not always theoretically necessary but it is in the current implementation)
A CONTRAST defining a space of interest or of no interest …
in the residual space of that contrast or not ?
Output directory (general, per subject …)
names for eigenimages
OUT : eigenimages, SVD.mat, observed temporal responses; Y’Y;
03/12/2018
JB Poline MAD/SHFJ/CEA

21. JB Poline MAD/SHFJ/CEA
Multivariate Toolbox : An application for model specification in neuroimaging (F. Kherif et al., NeuroImage 2002 )
03/12/2018
JB Poline MAD/SHFJ/CEA

22. JB Poline MAD/SHFJ/CEA
From Ferath Kherif
MADIC-UNAF-CEA
03/12/2018
JB Poline MAD/SHFJ/CEA

23. JB Poline MAD/SHFJ/CEA
03/12/2018
JB Poline MAD/SHFJ/CEA

24. JB Poline MAD/SHFJ/CEAY
03/12/2018
JB Poline MAD/SHFJ/CEA

25. JB Poline MAD/SHFJ/CEA
03/12/2018
JB Poline MAD/SHFJ/CEA

26. JB Poline MAD/SHFJ/CEA
From Ferath Kherif
MAD-UNAF-CEA
03/12/2018
JB Poline MAD/SHFJ/CEA

27. JB Poline MAD/SHFJ/CEA
RESULTS
Subject 1
Selected model
Subject 2 +
Subject 3 -
Subject 4 +
Subject 5 +
Subject 6 +
Subject 7 +
Subject 8 +
Subject 9 +
MODEL SELECTION
03/12/2018
JB Poline MAD/SHFJ/CEA

28. Subjects classification (multi-dimensionnal scaling)
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’
Similarity measure
RVij =
Tr(WiWj)
Sqrt[Tr(Wi2) Tr(Wj2)]
Distance matrix
D = 1- Rvij
, 1 < i,j < k
Subjects classification
(multi-dimensionnal scaling)
Group Homogeneity Analysis
03/12/2018
JB Poline MAD/SHFJ/CEA