We propose a new measure to estimate the direction of information flux in multivariate time series from complex systems. This measure, based on the slope of the phase spectrum (Phase Slope Index) has invariance properties that are important for applications in real physical or biological systems: (a) it is strictly insensitive to mixtures of arbitrary independent sources, (b) it gives meaningful results even if the phase spectrum is not linear, and (c) it properly weights contributions from different frequencies. Simulations of a class of coupled multivariate random data show that for truly unidirectional information flow without additional noise contamination our measure detects the correct direction as good as the standard Granger causality. For random mixtures of independent sources Granger Causality erroneously yields highly significant results whereas our measure correctly becomes non-significant. An application of our novel method to EEG data (88 subjects in eyes-closed condition) reveals a strikingly clear front-to-back information flow in the vast majority of subjects and thus contributes to a better understanding of information processing in the brain.
Deep Dive into Robustly estimating the flow direction of information in complex physical systems.
We propose a new measure to estimate the direction of information flux in multivariate time series from complex systems. This measure, based on the slope of the phase spectrum (Phase Slope Index) has invariance properties that are important for applications in real physical or biological systems: (a) it is strictly insensitive to mixtures of arbitrary independent sources, (b) it gives meaningful results even if the phase spectrum is not linear, and (c) it properly weights contributions from different frequencies. Simulations of a class of coupled multivariate random data show that for truly unidirectional information flow without additional noise contamination our measure detects the correct direction as good as the standard Granger causality. For random mixtures of independent sources Granger Causality erroneously yields highly significant results whereas our measure correctly becomes non-significant. An application of our novel method to EEG data (88 subjects in eyes-closed condition
To understand interacting systems it is of fundamental importance to distinguish the driver from the recipient, and hence to be able to estimate the direction of information flow. If one cannot interfere with the system, the direction can be estimated with a temporal argument: the driver is earlier than the recipient from which it follows that the driver contains information about the future of the recipient not contained in the past of the recipient while the reverse is not the case. This argument is the conceptual basis of Granger Causality [1,2] which is probably the most prominent method to estimate the direction of causal influence in time series analysis. Granger Causality was originally developed in econometry, but is applied to many different problems in physics, geosciences (cause of climate change), social sciences, and biology with special emphasis on neural system [3,4,5,6,7].
The difficulty in realistic measurements in complex systems is that asymmetries in detection power may as well arise due to other factors, specifically independent background ac-tivity having nontrivial spectral properties and eventually being measured in unknown superposition in the channels. In this case the interpretation of the asymmetry as a direction of information flow can lead to significant albeit false results [8]. The purpose of this paper is to propose a novel estimate of flux direction which is highly robust against false estimates caused by confounding factors of very general nature.
More formally, we are interested in statistical dependencies in complex physical systems and especially in causal relations between a signal of interest consisting of two sources with time series x i (t) for i = 1, 2. The measured data y(t) are assumed to be a superposition of these sources of interest and additive noise η(t) in the form
where η(t) is a set of M independent noise sources which are mixed into the measurement channels by an unknown 2 × M mixing matrix B.
The proposed method is based on the slope of the phase of cross-spectra between two time series. A fixed time delay for an interaction between two systems will affect different frequency components in different ways. This is most easily seen if we assume that the interaction is merely a delay by a time τ , i.e. y 2 (t) = ay 1 (t -τ ) with a being some constant. In the Fourier-domain this relation reads ŷ2 (f ) = a exp(-i2πf τ )ŷ 1 (f ). For the cross-spectrum S ij (f ) between the two channels i and j one has
where • denotes expectation value. The phase-spectrum Φ(f ) = 2πf τ is linear and proportional to the time delay τ . The slope of Φ(f ) can be estimated, and the causal direction is estimated to go from y 1 to y 2 (y 2 to y 1 ) if it is positive (negative).
The idea here is now to define an average phase slope in such a way that a) this quantity properly represents relative time delays of different signals and especially coincides with the classical definition for linear phase spectra , b) it is insensitive to signals which do not interact regardless of spectral content and superpositions of these signals, and c) it properly weights different frequency regions according to the statistical relevance. This quantity is termed ‘Phase Slope Index’ (PSI) and is defined as
where
is the complex coherency, S is the cross-spectral matrix, δf is the frequency resolution, and (•) denotes taking the imaginary part. F is the set of frequencies over which the slope is summed.
To see that the definition of Ψij corresponds to a meaningful estimate of the average slope it is convenient to rewrite it as
For smooth phase spectra, sin(Φ(f + δf ) -Φ(f )) ≈ Φ(f + δf ) -Φ(f ) and hence Ψ corresponds to a weighted average of the slope. We emphasize that since Ψ vanishes if the imaginary part of coherency vanishes it will be insensitive to mixtures of non-interacting sources [9,10].
Finally, it is convenient to normalize Ψ by an estimate of its standard deviation
with std( Ψ) being estimated by the Jackknife method.
In the examples below we always show normalized measures of directionality, and we consider absolute values larger than 2 as significant.
Estimations of cross-spectra is standard [9,11] but technical details may differ. Here, we first divide the whole data into epochs containing continuous data (4 seconds duration), then we divide each epoch further into segments of time T , here of 2 seconds duration corresponding to a frequency resolution of δf = 0.5 Hz, multiply the data for each segment with a Hanning window, Fourier-transform the data, and estimate the cross-spectra according to Eq.2 as an average over all segments. The segments have 50% overlap within each epoch but not across epochs. To apply the Jackknife method, for each pair of channels we calculate Ψk from data with the k.th epoch removed for all k. The standard deviation of Ψ is finally estimated for K epochs as √ Kσ where σ is the standard deviation of the set of Ψk .
Our new method is compared t
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