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