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Granger Causality on Spatial Manifolds: applications to Neuroimaging Pedro A. Valdés-Sosa Cuban Neuroscience Centre
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Multivariate Autoregressive Model for EEG/fMRI 12…p12…p … t t-1 t =1,…,Nt t =1,…,N
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Point influence Measures is the simple test
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Granger Causality must be measured on a MANIFOLD
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Influence Measures defined on a Manifold An influence field is a multiple test and all for a given
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Discretization of the Continuos AR Model -I
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Multivariate Regression Formulation
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ML Estimation and detection of Influence fields
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Problemas with the Multivariate Autoregressive Model for Brain Manifolds p→∞p→∞ t =1,…,N # of parameters
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Prior Model on Influence Fields
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Priors for Influence Fields Are of minimum norm, or maximal smoothness, etc. Valdés-Sosa PA Neuroinformatics (2004) 2:1-12 Valdés-Sosa PA et al. Phil. Trans R. Soc. B (2005) 360: 969-981
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Penalty Functions
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Estimation via MM algorithm
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Penalty Covariance combinations ? “Ridge Fusion” Fused Lasso Elastic Net Spline (“LORETA”) Data Fusion FramesRidge Basis PursuitLASSO Known as to wavleteers as Name in statisticsModel sparseness smoothness both
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Simulated “fMRI”
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Correlations of the EEG with the fMRI Martinez et. al Neuroimage July 2004
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