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A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
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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 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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Double-Granule Conditional-Entropies Based on Three-Level Granular Structures.

Taopin Mu1,2, Xianyong Zhang1,2, Zhiwen Mo1,2

  • 1School of Mathematical Sciences, Sichuan Normal University, Chengdu 610066, China.

Entropy (Basel, Switzerland)
|December 3, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces novel double-granule conditional-entropies for data mining, enhancing uncertainty measurement in rough set theory. These new measures offer improved hierarchical granulation for more effective information processing and knowledge reasoning.

Keywords:
conditional entropygranular computinginformation theoryrough set theorythree-level granular structuresuncertainty

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Area of Science:

  • Information Theory
  • Data Mining
  • Rough Set Theory

Background:

  • Existing local conditional-entropies in rough set theory possess limitations.
  • There is a need for improved uncertainty measures in data mining and information processing.

Purpose of the Study:

  • To establish and investigate the properties of double-granule conditional-entropies.
  • To enhance uncertainty measurements using hierarchical granulation within rough set theory.

Main Methods:

  • Developed double-granule conditional-entropies based on three-level granular structures (micro-bottom, meso-middle, macro-top).
  • Utilized hierarchical granulation and successive granular summation integrations.
  • Proposed measures at the micro-bottom level using a dual condition-granule system.

Main Results:

  • Acquired number distribution, calculation algorithms, three bounds, and granulation non-monotonicity for the new measures at three levels.
  • Demonstrated the second-order characteristic and hierarchical granulation of the proposed entropies.
  • Verified the effectiveness of hierarchical constructions and properties through decision table examples and data set experiments.

Conclusions:

  • Double-granule conditional-entropies deepen the classical entropy system and local conditional-entropies.
  • These novel measures provide enhanced uncertainty quantification for information processing and knowledge reasoning in data mining.