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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Geometric Partition Entropy: Coarse-Graining a Continuous State Space.

Christopher Tyler Diggans1, Abd AlRahman R AlMomani2

  • 1Information Directorate, Air Force Research Laboratory, Rome, NY 13441, USA.

Entropy (Basel, Switzerland)
|July 8, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a novel geometric partition entropy to quantify ignorance in continuous data. This new measure offers a more consistent and informative approach than traditional methods for complex datasets.

Keywords:
entropy estimatorignoranceinformation theoryuncertainty

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Area of Science:

  • Information Theory
  • Statistical Mechanics
  • Time Series Analysis

Background:

  • Traditional entropy estimators (thermodynamic, Shannon) are discrete and problematic for continuous data.
  • Differential entropy definitions face limitations similar to thermodynamic challenges.
  • Sampled data represents microstates, with underlying macrostates often unknown.

Purpose of the Study:

  • Re-examine entropy as a measure of ignorance in continuous phenomena predictability.
  • Develop a novel entropy estimator suitable for sampled, continuous data.
  • Address limitations of existing discrete and continuous entropy measures.

Main Methods:

  • Defined macrostates using data sample quantiles.
  • Developed an ignorance density distribution based on quantile distances.
  • Calculated geometric partition entropy as the Shannon entropy of this distribution.

Main Results:

  • The geometric partition entropy is more consistent and informative than histogram-binning.
  • This method excels with complex distributions, outliers, and limited sampling.
  • It is computationally efficient and avoids negative values, outperforming k-nearest neighbors estimators.

Conclusions:

  • Geometric partition entropy provides a robust quantification of ignorance for continuous phenomena.
  • The method is applicable to time series analysis and approximating ergodic symbolic dynamics.
  • This estimator offers unique advantages for handling real-world, limited observational data.