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 - 2002 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 www.madic.org

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

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

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

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

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

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

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

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

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

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

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

JB Poline MAD/SHFJ/CEA MLM first component p < 0.0001 03/12/2018 JB Poline MAD/SHFJ/CEA

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

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

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

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

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

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

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

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

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

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

JB Poline MAD/SHFJ/CEA Y 03/12/2018 JB Poline MAD/SHFJ/CEA

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

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

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

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