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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
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The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
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Transfer entropy for finite data.

Alec Kirkley1

  • 1University of Hong Kong, University of Hong Kong, University of Hong Kong, Institute of Data Science, Hong Kong SAR, China; Department of Urban Planning and Design, Hong Kong SAR, China; and Urban Systems Institute, Hong Kong SAR, China.

Physical Review. E
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This study introduces a new transfer entropy measure for discrete data, overcoming bias and significance issues in small or high-cardinality time series analysis. It enables reliable information flow assessment without simulations.

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Area of Science:

  • Complex systems analysis
  • Information theory
  • Time series analysis

Background:

  • Transfer entropy quantifies directed information flow but faces challenges with continuous data.
  • For discrete data, it suffers from positive bias with sparse counts and lacks statistical significance assessment.
  • Existing methods struggle with finite data streams of small size or high cardinality.

Purpose of the Study:

  • To develop a novel transfer entropy measure for finite discrete data streams.
  • To address the limitations of existing estimators, specifically bias and lack of significance testing.
  • To enable nonparametric statistical significance assessment without relying on simulations.

Main Methods:

  • Derived a new transfer entropy measure by computing information content in finite data streams.
  • Avoided explicit consideration of symbols as random variables.
  • Ensured asymptotic equivalence to the standard plug-in estimator.

Main Results:

  • The new measure is asymptotically equivalent to the standard plug-in estimator.
  • It effectively remedies positive bias issues associated with sparse bin counts.
  • It permits a fully nonparametric assessment of statistical significance for finite time series.
  • The method is suitable for time series of small size and/or high cardinality.

Conclusions:

  • The proposed transfer entropy measure offers a robust solution for analyzing information flow in discrete finite data.
  • It overcomes critical limitations of traditional methods, enhancing reliability and interpretability.
  • Enables rigorous statistical validation of directed information transfer in complex systems.