Granger Causality on Spatial Manifolds: applications to Neuroimaging

1. Granger Causality on Spatial Manifolds: applications to Neuroimaging Pedro A. Valdés-Sosa Cuban Neuroscience Centre

2. Multivariate Autoregressive Model for EEG/fMRI 1 2 … p … t =1,…,N t t-1 t =1,…,Nt

3. Point influence Measures is the simple test

4. Granger Causality must be measured on a MANIFOLD

5. Influence Measures defined on a Manifold An influence field is a multiple test for a given and all

6. Discretization of the Continuos AR Model -I

7. Multivariate Regression Formulation

8. ML Estimation and detection of Influence fields

9. Problemas with the Multivariate Autoregressive Model for Brain Manifolds p→∞ t =1,…,N # of parameters

10. Prior Model on Influence Fields

11. 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

12. Penalty Functions

13. Estimation via MM algorithm

14. Penalty Covariance combinations Model Name in statistics Known as to wavleteers as LASSO Basis Pursuit Ridge Frames Data Fusion Spline (“LORETA”) Elastic Net Fused Lasso “Ridge Fusion” ? sparseness smoothness both

15. Simulated “fMRI”

16. Correlations of the EEG with the fMRI Martinez et. al Neuroimage July 2004