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On the Monotonicity of Information Aging

MD Kamran Chowdhury Shisher, Yin Sun

TL;DR

The monotonicity of information aging and the divergence of from being a Markov chain are analyzed in a remote estimation system, where historical observations of a Gaussian autoregressive AR(p) process are used to predict its future values.

Abstract

In this paper, we analyze the monotonicity of information aging in a remote estimation system, where historical observations of a Gaussian autoregressive AR(p) process are used to predict its future values. We consider two widely used loss functions in estimation: (i) logarithmic loss function for maximum likelihood estimation and (ii) quadratic loss function for MMSE estimation. The estimation error of the AR(p) process is written as a generalized conditional entropy which has closed-form expressions. By using a new information-theoretic tool called $ε$-Markov chain, we can evaluate the divergence of the AR(p) process from being a Markov chain. When the divergence $ε$ is large, the estimation error of the AR(p) process can be far from a non-decreasing function of the Age of Information (AoI). Conversely, for small divergence $ε$, the inference error is close to a non-decreasing AoI function. Each observation is a short sequence taken from the AR(p) process. As the observation sequence length increases, the parameter $ε$ progressively reduces to zero, and hence the estimation error becomes a non-decreasing AoI function. These results underscore a connection between the monotonicity of information aging and the divergence of from being a Markov chain.

On the Monotonicity of Information Aging

TL;DR

The monotonicity of information aging and the divergence of from being a Markov chain are analyzed in a remote estimation system, where historical observations of a Gaussian autoregressive AR(p) process are used to predict its future values.

Abstract

In this paper, we analyze the monotonicity of information aging in a remote estimation system, where historical observations of a Gaussian autoregressive AR(p) process are used to predict its future values. We consider two widely used loss functions in estimation: (i) logarithmic loss function for maximum likelihood estimation and (ii) quadratic loss function for MMSE estimation. The estimation error of the AR(p) process is written as a generalized conditional entropy which has closed-form expressions. By using a new information-theoretic tool called -Markov chain, we can evaluate the divergence of the AR(p) process from being a Markov chain. When the divergence is large, the estimation error of the AR(p) process can be far from a non-decreasing function of the Age of Information (AoI). Conversely, for small divergence , the inference error is close to a non-decreasing AoI function. Each observation is a short sequence taken from the AR(p) process. As the observation sequence length increases, the parameter progressively reduces to zero, and hence the estimation error becomes a non-decreasing AoI function. These results underscore a connection between the monotonicity of information aging and the divergence of from being a Markov chain.
Paper Structure (18 sections, 5 theorems, 48 equations, 2 figures, 1 table)

This paper contains 18 sections, 5 theorems, 48 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. Monotonicity of Estimation Error with AoI: An $\epsilon$-Markov Chain Approach
  4. Information-theoretic Metrics for Estimation Error
  5. Evaluating $L$-conditional Entropy
  6. Logarithmic Loss (log loss)
  7. Quadratic Loss
  8. $L$-conditional Entropy vs. AoI
  9. Characterizing the Parameter $\epsilon$ of An $\epsilon$-Markov Chain
  10. Numerical Results
  11. Evaluating $L$-conditional Entropy Using Propositions \ref{['prop1']}-\ref{['prop2']}
  12. Evaluating $\epsilon(l)$ Using Proposition \ref{['prop3']}
  13. Analysis of the Numerical Results
  14. Conclusion
  15. Proof of Lemma \ref{['lemma1']}
  16. ...and 3 more sections

Key Result

Lemma 1

Estimation error $\mathrm{err}_{\mathrm{estimation}}(\delta, l)$ is equal to $L$-conditional entropy of $Y_t$ given $\mathbf X^l_{t-\delta}$, i.e.,

Figures (2)

  • Figure 1: A remote estimation system.
  • Figure 2: $L$-conditional entropy vs. AoI with (a) quadratic loss function and (b) log loss function (base 2). The $L$-conditional entropy is not always a monotonic function of AoI. An AR($4$) model as discussed in Section \ref{['simulation']} is considered for this simulation.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 1: $\epsilon$-Markov Chain
  • Lemma 2
  • Proposition 3
  • proof