Directed Redundancy in Time Series
Jan Østergaard
TL;DR
The paper introduces a formal notion of directed redundancy to quantify how a set of source time series redundantly informs a target via causal interactions. It proposes a TE-based measure that accounts for a potentially hidden redundancy process $\phi$ driving shared information among relevant sources, and derives an upper bound using $\mathbb{TE}$ terms from $\phi$ to the target and sources. An efficient pipeline identifies target-relevant sources, isolates the hidden redundancy, and bounds the minimal redundant information by $\min( R_{\phi\to Z}, R_{\phi\to \mathcal{R}}, R_{\mathcal{R}\to Z} )$. The authors validate the framework on intracranial EEG data, revealing concentrated, depth-strip–localized redundancy and highlighting a frequently selected hidden driver, thereby providing a practical tool for analyzing redundancy in complex networks with potential hidden drivers. These contributions enable more accurate attribution of redundancy in multivariate time-series and have potential implications for understanding information processing in brain networks and other causal systems.
Abstract
We quantify the average amount of redundant information that is transferred from a subset of relevant random source processes to a target process. To identify the relevant source processes, we consider those that are connected to the target process and in addition share a certain proportion of the total information causally provided to the target. Even if the relevant processes have no directed information exchange between them, they can still causally provide redundant information to the target. This makes it difficult to identify the relevant processes. To solve this issue, we propose the existence of a hidden redundancy process that governs the shared information among the relevant processes. We bound the redundancy by the minimal average directed redundancy from the relevant processes to the target, from the hidden redundancy process to the target, and from the hidden redundancy process to the relevant processes.