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Related Concept Videos

Entropy02:39

Entropy

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...
Entropy01:18

Entropy

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.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
Entropy Changes Accompanying Specific Processes01:21

Entropy Changes Accompanying Specific Processes

Entropy, a measure of disorder in a system, changes during phase transitions like freezing or boiling. At the transition temperature Ttrs, where two phases are in equilibrium, the phase transition is a reversible process. The entropy change can be calculated from a substance's enthalpy of transition using the equation ΔStrs = ΔtrsH /Ttrs.When a perfect gas expands isothermally from one volume to another, entropy increases logarithmically with volume. Conversely, isothermal compression results...
Entropy and the Second Law of Thermodynamics01:20

Entropy and the Second Law of Thermodynamics

The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
The relation  between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
Entropy and the Second Law of Thermodynamics01:26

Entropy and the Second Law of Thermodynamics

Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.

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Related Experiment Video

Updated: Jun 14, 2026

Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy
11:15

Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy

Published on: June 27, 2013

Kernel entropy component analysis.

Robert Jenssen1

  • 1Department of Physics and Technology, University of Tromsø, Tromsø, Norway. robert.jenssen@uit.no

IEEE Transactions on Pattern Analysis and Machine Intelligence
|March 20, 2010
PubMed
Summary
This summary is machine-generated.

Kernel entropy component analysis (kernel ECA) offers novel data transformation and dimensionality reduction by analyzing Renyi entropy. This method reveals distinct data structures and improves spectral clustering and pattern denoising.

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Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy

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Area of Science:

  • Machine Learning
  • Data Analysis
  • Information Theory

Background:

  • Dimensionality reduction and data transformation are crucial for analyzing complex datasets.
  • Kernel principal component analysis (kernel PCA) is a popular nonlinear dimensionality reduction technique.
  • Existing methods may not fully capture all relevant data structures, particularly those related to information-theoretic properties.

Purpose of the Study:

  • Introduce kernel entropy component analysis (kernel ECA) as a new method for data transformation and dimensionality reduction.
  • Explore the unique data structures revealed by kernel ECA compared to kernel PCA.
  • Develop and evaluate a novel spectral clustering algorithm based on kernel ECA.

Main Methods:

  • Kernel ECA projects data onto entropy-preserving kernel principal component analysis (kernel PCA) axes.
  • Estimates Renyi entropy of input space data using a kernel matrix and Parzen windowing.
  • Utilizes a subset of kernel PCA axes, not necessarily corresponding to top eigenvalues.

Main Results:

  • Kernel ECA reveals structures related to Renyi entropy, distinct from kernel PCA.
  • Transformed datasets exhibit a unique angle-based structure.
  • A new spectral clustering algorithm using kernel ECA shows positive results.
  • Kernel ECA demonstrates utility for pattern denoising.

Conclusions:

  • Kernel ECA provides a novel approach to dimensionality reduction and data transformation.
  • The method captures information-theoretic structures missed by traditional kernel PCA.
  • Kernel ECA enables effective spectral clustering and pattern denoising.