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Synergy and Redundancy Dominated Effects in Time Series via Transfer Entropy Decompositions

Jan Østergaard, Payam Boubakani

TL;DR

This work develops a decomposition framework for transfer entropy to quantify how different parts of a conditioning time series $S$ influence the directional information transfer between two other time series $X$ and $Y$ in the presence of $Z$. It introduces computable measures of synergistic and redundancy-dominated information exchanges, including lower bounds that are robust to estimator bias, and demonstrates a theoretical result showing opposite effects for early versus late past segments. The framework is applied to intracranial EEG data from epilepsy patients, revealing that the early past can induce synergistic effects and identifying brain regions with dominant synergistic contributions during seizures. The proposed methods offer a practical tool for disentangling past-dependent information dynamics in neural systems and potentially inform understanding of brain connectivity and seizure propagation.

Abstract

We present a new decomposition of transfer entropy to characterize the degree of synergy- and redundancy-dominated influence a time series has upon the interaction between other time series. We prove the existence of a class of time series, where the early past of the conditioning time series yields a synergistic effect upon the interaction, whereas the late past has a redundancy-dominated effect. In general, different parts of the past can have different effects. Our information theoretic quantities are easy to compute in practice, and we demonstrate their usage on real-world brain data.

Synergy and Redundancy Dominated Effects in Time Series via Transfer Entropy Decompositions

TL;DR

This work develops a decomposition framework for transfer entropy to quantify how different parts of a conditioning time series influence the directional information transfer between two other time series and in the presence of . It introduces computable measures of synergistic and redundancy-dominated information exchanges, including lower bounds that are robust to estimator bias, and demonstrates a theoretical result showing opposite effects for early versus late past segments. The framework is applied to intracranial EEG data from epilepsy patients, revealing that the early past can induce synergistic effects and identifying brain regions with dominant synergistic contributions during seizures. The proposed methods offer a practical tool for disentangling past-dependent information dynamics in neural systems and potentially inform understanding of brain connectivity and seizure propagation.

Abstract

We present a new decomposition of transfer entropy to characterize the degree of synergy- and redundancy-dominated influence a time series has upon the interaction between other time series. We prove the existence of a class of time series, where the early past of the conditioning time series yields a synergistic effect upon the interaction, whereas the late past has a redundancy-dominated effect. In general, different parts of the past can have different effects. Our information theoretic quantities are easy to compute in practice, and we demonstrate their usage on real-world brain data.
Paper Structure (11 sections, 3 theorems, 32 equations, 2 figures)

This paper contains 11 sections, 3 theorems, 32 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. New Synergy and Redundancy Measures
  4. Conditional Mutual Information
  5. Causally Conditioned Transfer Entropy
  6. Synergistic Versus Common Information
  7. Synergy- and Redundancy-Dominated Measures
  8. Example on Brain Data
  9. Effect of Early Versus Late Past
  10. Greatest Synergistic Effect
  11. Conclusions

Key Result

Theorem 1

Let $A,C,C'$ be mutually independent binary random variables, where $C,C'$ are uniformly distributed on $\{0,1\}$, and $A$ is arbitrarily distributed with $p=\mathbb{P}_A(0)=1-\mathbb{P}_A(1)$. Let $B=A \wedge C$ and $D = A \wedge C'$, where $\wedge$ denotes the Boolean logical AND operator. Then, f with equality in $(a)$ and $(b)$ if and only if $p=0$ or $p=1$.

Figures (2)

  • Figure 1: The conditional mutual information exhibits mainly synergistic $I(A;B|C)>I(A;B)$ and redundancy-dominated $I(A;B|D)<I(A;B)$ effects, when conditioning upon $C$ and $D$, respectively.
  • Figure 2: Left: $D^\tau_\phi$ in \ref{['eq:Dtau']} as a function of delay $\tau$ and subset $\mathcal{J}_\phi, \phi=1,2$. Right: The difference $\hat{I}_{E(76)}^{\mathrm{syn}}(i) - \hat{I}_{E(75)}^{\mathrm{syn}}(i)$ for $i=1,\dotsc, 64$. Positive numbers indicate that the effect of electrode $E(76)$ dominates over that of $E(75)$, whereas the opposite is true for negative numbers.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Lemma 1