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InfoMat: A Tool for the Analysis and Visualization Sequential Information Transfer

Dor Tsur, Haim Permuter

TL;DR

The paper tackles the lack of intuitive visualization tools for information transfer in sequential data and introduces InfoMat, an $m\times m$ matrix whose entries are $\mathrm{I}^{X,Y}_{i,j} := I(X_i;Y_j|X^{i-1},Y^{j-1})$, with $I(X^m;Y^m) = \sum_{i=1}^m \sum_{j=1}^m \mathrm{I}^{X,Y}_{i,j}$. It develops both Gaussian and neural estimators to populate the matrix, enabling visualization of conservation laws, new transfer relations, and dependence structures; it also connects matrix patterns to classic measures like directed information and instantaneous information. The authors derive and illustrate new information-theoretic relations, such as the transfer entropic decomposition $I(D^k \circ X^m\to Y^m) = \sum_{i=1}^{m-k} T^{X\to Y}_{i+1}(i,i)$, and show how these can be read as color-grouped sums in the InfoMat. Through applications to continuous Gaussian ARMA data and Ising-channel coding schemes, the work demonstrates that InfoMat visually encodes time-series dependence and coding dynamics, offering a practical tool for data exploration in time-series analysis, empowerment, and causal inference.

Abstract

Despite the popularity of information measures in analysis of probabilistic systems, proper tools for their visualization are not common. This work develops a simple matrix representation of information transfer in sequential systems, termed information matrix (InfoMat). The simplicity of the InfoMat provides a new visual perspective on existing decomposition formulas of mutual information, and enables us to prove new relations between sequential information theoretic measures. We study various estimation schemes of the InfoMat, facilitating the visualization of information transfer in sequential datasets. By drawing a connection between visual patterns in the InfoMat and various dependence structures, we observe how information transfer evolves in the dataset. We then leverage this tool to visualize the effect of capacity-achieving coding schemes on the underlying exchange of information. We believe the InfoMat is applicable to any time-series task for a better understanding of the data at hand.

InfoMat: A Tool for the Analysis and Visualization Sequential Information Transfer

TL;DR

The paper tackles the lack of intuitive visualization tools for information transfer in sequential data and introduces InfoMat, an matrix whose entries are , with . It develops both Gaussian and neural estimators to populate the matrix, enabling visualization of conservation laws, new transfer relations, and dependence structures; it also connects matrix patterns to classic measures like directed information and instantaneous information. The authors derive and illustrate new information-theoretic relations, such as the transfer entropic decomposition , and show how these can be read as color-grouped sums in the InfoMat. Through applications to continuous Gaussian ARMA data and Ising-channel coding schemes, the work demonstrates that InfoMat visually encodes time-series dependence and coding dynamics, offering a practical tool for data exploration in time-series analysis, empowerment, and causal inference.

Abstract

Despite the popularity of information measures in analysis of probabilistic systems, proper tools for their visualization are not common. This work develops a simple matrix representation of information transfer in sequential systems, termed information matrix (InfoMat). The simplicity of the InfoMat provides a new visual perspective on existing decomposition formulas of mutual information, and enables us to prove new relations between sequential information theoretic measures. We study various estimation schemes of the InfoMat, facilitating the visualization of information transfer in sequential datasets. By drawing a connection between visual patterns in the InfoMat and various dependence structures, we observe how information transfer evolves in the dataset. We then leverage this tool to visualize the effect of capacity-achieving coding schemes on the underlying exchange of information. We believe the InfoMat is applicable to any time-series task for a better understanding of the data at hand.
Paper Structure (17 sections, 6 theorems, 35 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 6 theorems, 35 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The InfoMat
  3. Capturing Information Conservation Laws
  4. Developing New Information Theoretic Relations
  5. Gaussian Approximation
  6. Proposed Estimator
  7. Beyond Gaussian - Neural Estimation
  8. Visualization of Information Transfer
  9. Continuous Data - Gaussian Approximation
  10. Neural Estimator
  11. Visualizing Coding Schemes
  12. Conclusion
  13. Proof of Proposition \ref{['prop:te_conservation']}
  14. Proof of Proposition \ref{['prop:di_chain_rule']}
  15. Proof of Proposition \ref{['prop:gaussian_estiamtor']}
  16. ...and 2 more sections

Key Result

Proposition 1

For $(X^m,Y^m)\sim P_{X^m,Y^m}$ and $1 \leq k \leq m$, we have

Figures (4)

  • Figure 1: Neural estimation model. Dashed line represents maximum-likelihood (ML) training phase, while the filled lines account for the inference (MI calculation) phase.
  • Figure 2: Visual patterns in the InfoMat for several ARMA processes over continuous spaces.
  • Figure 3: Estimated InfoMat under nonlinearities and cyclic shift.
  • Figure 4: Visualization of information transfer in the Ising channel under various coding schemes.

Theorems & Definitions (7)

  • Proposition 1: Transfer entropic decomposition of directed information
  • proof
  • Proposition 2: Conservation of transfer entropy
  • Proposition 3: Directed information chain rule
  • Lemma 1
  • Proposition 4: Gaussian approximation performance guarantees
  • Proposition 5: Conditional mutual information invariance