Quantifying Redundant Information Transfer in Multivariate Time Series

Core Concepts

The paper proposes a new notion of directed redundancy to quantify the redundant information that a set of relevant source processes provide to a target process, even when the source processes are causally independent.

Abstract

The paper introduces a new concept of "directed redundancy" to quantify the redundant information that a set of relevant source processes provide to a target process. The key ideas are:

The source processes may be causally independent, but can still provide redundant information to the target.
The authors hypothesize the existence of a hidden "redundancy process" that governs the shared information among the relevant source processes.
They define the directed redundancy as the minimum of:

The transfer entropy from the hidden redundancy process to the target
The transfer entropy from the hidden redundancy process to the relevant source processes
The transfer entropy from the relevant source processes to the target

This approach allows identifying the relevant source processes that provide redundant information to the target, even when the source processes are causally independent.
The authors provide an efficient algorithm to identify the hidden redundancy process and the relevant source processes.
They demonstrate the proposed method on real-world intracranial EEG data, showing that it can uncover redundant information patterns that are not evident from just analyzing the pairwise transfer entropies.

Stats

The paper does not provide explicit numerical data, but presents the following key figures:

Transfer entropies between the processes in the example system (7)-(11) (Table I)
Redundancy measures (24)-(25) for the intracranial EEG data (Fig. 1 left)
Histograms of the hidden redundancy processes (Fig. 1 right) and relevant source processes (Fig. 2 left) for the intracranial EEG data

Quotes

The paper does not contain any direct quotes, but the following are key statements:
"Two causally independent sources could provide different (and hence non-redundant) information to the target. Thus, if we only measure the amount of causal information exchange from the sources and to the target, then these sources would wrongly be classified as providing redundant information transfer to the target."
"It is necessary but non-trivial to include the dependency between the sources to guarantee that they actually provide redundant information to the target."

Key Insights Distilled From

Directed Redundancy in Time Series

by Jan ... at arxiv.org 05-02-2024

https://arxiv.org/pdf/2405.00368.pdf

Deeper Inquiries

How can the proposed directed redundancy measure be extended to handle more complex dependencies, such as nonlinear or higher-order interactions, between the source processes and the target process

The proposed directed redundancy measure can be extended to handle more complex dependencies by incorporating nonlinear or higher-order interactions between the source processes and the target process. One approach could involve utilizing nonlinear information-theoretic measures such as mutual information between higher-order terms or conditional mutual information to capture the intricate relationships between variables. By considering not only pairwise interactions but also higher-order interactions, the directed redundancy measure can more accurately quantify the shared and redundant information among multiple processes. Additionally, techniques from nonlinear dynamical systems analysis, such as recurrence plots or phase space reconstruction, could be employed to uncover hidden patterns and dependencies that may not be captured by linear measures alone. This extension would enable a more comprehensive understanding of the information flow and redundancy in complex systems.

What are the potential limitations or drawbacks of relying on transfer entropy as the underlying information-theoretic measure, and how could alternative measures be incorporated into the directed redundancy framework

While transfer entropy is a powerful tool for quantifying directed information flow in time series data, it has certain limitations that should be considered when using it as the underlying information-theoretic measure in the directed redundancy framework. One limitation is that transfer entropy assumes a linear causal relationship between variables, which may not always hold in real-world systems with nonlinear dynamics. To address this limitation, alternative measures such as conditional mutual information, Granger causality, or convergent cross-mapping could be integrated into the framework to capture nonlinear dependencies and causal interactions more effectively. By combining multiple information-theoretic measures, the directed redundancy analysis can provide a more nuanced and accurate assessment of redundant information transfer in complex systems. Additionally, the choice of parameters in transfer entropy estimation, such as the embedding dimension and time delay, can impact the results and introduce biases. Robustness checks and sensitivity analyses should be conducted to ensure the reliability of the findings and mitigate the effects of parameter selection on the results.

Can the insights gained from analyzing the directed redundancy in time series data be leveraged to improve the design of neural networks or other machine learning models that aim to capture and exploit redundant information in multivariate inputs

Insights gained from analyzing directed redundancy in time series data can indeed be leveraged to enhance the design of neural networks and other machine learning models that aim to leverage redundant information in multivariate inputs. By understanding the causal relationships and information flow between different variables, machine learning models can be designed to exploit redundant information effectively for improved performance and robustness. For example, in neural network architectures, incorporating knowledge of directed redundancy can help in designing more efficient and interpretable models by identifying and leveraging shared information among input features. This can lead to the development of more compact and accurate models that generalize well to unseen data. Moreover, by optimizing the network structure based on the insights from directed redundancy analysis, it is possible to enhance the model's ability to capture relevant information while reducing overfitting and improving overall performance. Ultimately, integrating insights from directed redundancy analysis into machine learning frameworks can lead to more effective and adaptive models that make better use of the available information in complex datasets.