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Estimation of causal direction from time series in the presence of mixed and colored noise
Fraunhofer FIRST, Berlin
- G. Nolte
Estimation of causal direction from time series in the presence of - - PowerPoint PPT Presentation
Estimation of causal direction from time series in the presence of - - PowerPoint PPT Presentation
Estimation of causal direction from time series in the presence of mixed and colored noise G. Nolte Fraunhofer FIRST, Berlin EEG Typical Properties of Data > 10 minutes measurement > 100 samples/sec 19-128 channels Some millions of
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- 1. Divide data into epochs (e.g. of 1 sec)
- 2. Make an analysis in each epoch and average over epochs
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The Problem of volume conduction T1 T2 T1<< T2
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∑
= ∗
=
K k j i ij
k f z k f z K f S
1
) , ( ) , ( 1 ) (
Cross-spectrum:
) ( ) ( ) ( ) ( f S f S f S f C
jj ii ij ij
=
Coherency: Data:
) , ( k f zi
Channel Frequency Epoch
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)) ( Re( ) (
12 12
f S f S
=
)) ( Im(
12 f
S i
+
Independent sources do not contribute to the imaginary part of the cross-spectrum
T M
L P P P L S
=
2 1
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independent
Imaginary parts
Interaction
PISA sPCA MOCA PSI
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Granger causality
x y E
1
=
→
2 1
log E E F
x y
x y y x
F F G
→ →
− =
ˆ
) ˆ ( ˆ G std G G =
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Source Channel A Channel B Noise
Motivation
- Many measurements like EEG/MEG/fMRI are extremely noisy
Source 1 Channel A Channel B Noise Source 2
Mixtures of independent sources: Do we estimate fake direction? Additive noise: Do we estimate wrong direction?
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Source Channel A Channel B Granger Causality Noise
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Observations: A Independent sources do not contribute to the imaginary part of the cross-spectrum B Slope of phase of cross-spectrum indicates direction Phase Slope Index (PSI)(ψ)
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Slope of phase-spectrum indicates temporal ordering
B
Data Decomposition
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Combining A and B: Average of Phase-Slope such that it is insensitive to mixtures
Nolte, et.al., Phys Rev Let., 2008
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White Noise Source 1 Source 2
=
5 . 95 . 95 . ) 1 ( A
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Simulation: (1-γ) true flow + γ mixed noise Comparison with Granger Causality
γ
No Noise Only Noise No Noise Only Noise
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“I know that I don’t know anything” Sokrates
No Noise Only Noise
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wrong wrong correct correct wrong wrong correct correct
Granger causality
wrong wrong correct correct
Phase Slope Index
I don’t know
Correct wrong Total points 638 6 578 Correct wrong Total points 736 100
- 264
Challenge
- correct: +1 point
- wrong: -10
points
- “I don ‘t know”: 0 points
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What matters:
Simulated challenge data:
- problem is generic (details are open to discussion)
- evidence is weighted
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Nonlinearity of order k Granger Causality
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Nonlinearity of order k PSI
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FAQ
Nonlinear systems? Direct vs. indirect ? Bidirectional flux? Question alright with exceptions In general; it possible but difficult to construct counterexamples partialing (rarely) possible PSI is (trivially) correct, (impossible to resolve completely) Answer Estimate delay? Not in the presence of noise, Results are really binary
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Alpha rhythm, Eyes closed, 88 subjects
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Surrogate Data to test for artefacts of volume conduction
( )
) ( , ), ( ) ( Data
1
t x t x t x
n
=
) ( ) ( t x W t s
=
ICA Demix with 1. T * 1)
- (i
by component i.th Delay 2.
) 2 ( ) ( ) ( ) ( ) ( ) (
3 3 2 2 1 1
T t s t v T t s t v t s t v
+ = + = =
) ( ) ( x
1 surr
t v W t
−
=
Remix 3.
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Granger Causality, Data vs. Surrogates
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Phase Slope Index, Data vs. Surrogates
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Summary
- Imaginary parts of cross-spectra is not affected by
non-interacting sources ⇒ valuable quantity to study interactions
- Direction with “Phase Slope Index” (PSI)
- Surrogates with ICA
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