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The Dangers of Granger: What They Didn’t Tell you about Granger Causality (in fMRI) Victor Solo†,∗ † School of Electrical Engineering University of New South Wales Sydney, AUSTRALIA ∗ Martinos Center for Biomedical Imaging MGH/HMS, Boston, MA, USA HBM 2011 Quebec City, CANADA, June 2011 – p. 1/1 Topics 1 What is Granger Causality? 2 fMRI Critical Heuristics: Subsampling and HRF Filtering 3 From Heuristics to Theory: Forward, Inverse and Computational Problems 4 Computational Solutions via State Space 5 Theory of Subsampling re Granger Causality 6 Theory of Filtering re Granger Causality 7 Illustrations: Spurious Causality 8 Conclusions Martinos Center Collaborators: Dr M Hamalainen, Dr Fa-Hsuan Lin, Dr M Vangel. Grants NCRR 2P41RR014075-11 and NIH Shared Instrumentation Grants. Views expressed here are not necessarily endorsed by NCRR or NIH – p. 2/1 What is Granger Causality? Gedanken Experiment on Dynamics: Rainfall → Catchment → River Level Rt Lt FR→L R pushes L = ln 2 σL|L − FL→R 2 σL|R − ,L− L pushes R = ln 2 σR|R − 2 σR|R − ,L− GEMs: FRoL = FR→L + FL→R + FL.R An Aside: But can one discern (temporal) causal relations from observational (time series) data? – p. 3/1 fMRI Heuristics Subsampling Neuronal Processes are on a ∼30ms-50ms time scale whereas fMRI is on a ∼1 sec time scale. It is too slow. Filtering The fMRI hemodynamic response differentially filters and so messes up the dynamical interactions. (And then subsampling makes this even worse). Other Issues Measurement Noise Nonlinearity Data Reduction Omitted Third Variable – p. 4/1 From Heuristics to Theory Forward Problems Does A ⇒ B? If GC relations exist on a fast time-scale, are they preserved under subsampling? What does HRF filtering do to GC relations? Inverse Problems Does B ⇒ A? If GC relations are found on a slow time-scale do the same GC relations exist at a faster time-scale? Can effects of HRF filtering be undone? Computational Problems → – p. 5/1 Computational Solutions How to get GEMs Reliably? h i Need induced submodels e.g. Rt model from RLtt model Also e.g. submodel of VAR is VARMA ≡ state space. Solution: State Space Models + Ricatti Equations How to get Subsampled Models reliably and so GEMs? Solution: State Space Models + Ricatti Equations GEMs can be decomposed by frequency and above state space solution method ensures reliable computation of the frequency domain GEMs. – p. 6/1 Theorems: Subsampling Forward Problem A ⇒ B Strong unidirectional Granger causality (FL→R = 0, FR.L = 0) is preserved under subsampling Inverse Problem B ; A Granger causal relations can be manipulated nearly arbitarily under subsampling. NB. This is not so simple to do since FL→R , FR→L depend nonlinearly on model parameters. – p. 7/1 Theorems: Filtering Minimum-phase filtering preserves GC Nonminimum-phase filtering does not preserve GC But HRFs are non-minimum phase! HRF 0.2 0 −0.2 0 5 10 15 20 25 20 25 ln|roots| 5 0 −5 0 5 10 15 Double Gamma Hemodynamic Response Function & Roots – p. 8/1 Spurious Causality I Unidirectional GC is preserved: but degrades semi-regularly GEMs x and y t t 1.4 F 8 y−>x 1.2 (−)standardised x t F x−>y 6 1 4 0.8 2 0.6 0 0.4 −2 0.2 0 0 10 −4 1 (−.)standardised y t 50 2 10 10 100 GEMs 200 250 200 250 x and y t 2.5 F 6 F 5 y−>x 2 150 x−>y t (−)standardised x t 4 3 1.5 2 1 1 0 0.5 −1 −2 0 0 10 1 10 GC degrades irregularly 2 10 (−.)standardised yt 50 100 150 – p. 9/1 Spurious Causality II GEMs x and y t t 2 1.5 Fy−>x 6 Fx−>y 5 (−)standardised x t 4 3 2 1 1 0 0.5 −1 −2 0 0 10 −3 1 10 2 10 (−.)standardised y t 50 100 150 200 250 Equal GEMs Reverse Strongly – p. 10/1 Spurious Causality III Near equal GEMs become nearly unidirectional GEMs x and y t t 2 F y−>x 6 (−)standardised x t F x−>y 1.5 4 2 1 0 0.5 0 0 10 −2 −4 1 (−.)standardised y t 50 2 10 10 100 GEMs 200 250 200 250 x and y t 3 F 7 F 6 y−>x 2.5 150 t (−)standardised x t x−>y 5 2 4 3 1.5 2 1 1 0 −1 0.5 −2 0 0 10 −3 1 10 2 10 (−.)standardised y t 50 100 Near unidirectional GEMs become nearly equal 150 – p. 11/1 Subsampled MEG Source Signals Subsampling Reconstructed MEG Sourcce Signals – p. 12/1 Conclusions • Subsampling irretrievably destroys possibility of inverse GC recovery in fMRI. • Non-minimum phase filtering, hence HRF filtering, destroys possibility of GC recovery in fMRI. Although higher order methods could help. • New state space based computational methods provide reliable computation of submodels, subsampled models, GEMs and frequency domain GEMs. • Need ms time-scale measurements e.g. MEG/EEG to pursue dynamic causality. – p. 13/1