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Related Concept Videos

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A phase diagram combines plots of pressure versus temperature for the liquid-gas, solid-liquid, and solid-gas phase-transition equilibria of a substance. These diagrams indicate the physical states that exist under specific conditions of pressure and temperature and also provide the pressure dependence of the phase-transition temperatures (melting points, sublimation points, boiling points). Regions or areas labeled solid, liquid, and gas represent single phases, while lines or curves represent...
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A toroid is a closely wound donut-shaped coil constructed using a single  conducting wire. In general, it is assumed that a toriod consists of  multiple circular loops perpendicular to its axis.
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The phase of a given substance depends on the pressure and temperature. Thus, plots of pressure versus temperature showing the phase in each region provide considerable insights into the thermal properties of substances. Such plots are known as phase diagrams. For instance, in the phase diagram for water (Figure 1), the solid curve boundaries between the phases indicate phase transitions (i.e., temperatures and pressures at which the phases coexist).
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Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the resultant vector or, for short, the resultant.
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Updated: Aug 15, 2025

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
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TORUS GRAPHS FOR MULTIVARIATE PHASE COUPLING ANALYSIS.

Natalie Klein1, Josue Orellana1, Scott L Brincat2

  • 1Department of Statistics and Data Science, Carnegie Mellon University.

The Annals of Applied Statistics
|January 6, 2023
PubMed
Summary
This summary is machine-generated.

Torus graphs, a novel graphical model, effectively identify complex phase coupling in brain oscillations. This method surpasses traditional analyses for understanding coordinated activity across brain regions during memory tasks.

Keywords:
Graphical modelscircular statisticsnetwork analysis

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Area of Science:

  • Computational neuroscience
  • Statistical modeling
  • Time series analysis

Background:

  • Angular data, common in neuroscience, are modeled as circular random variables.
  • Multivariate circular data can be represented on a torus, posing unique analytical challenges.
  • Identifying coordinated activity (phase coupling) in brain oscillations is crucial for understanding neural communication.

Purpose of the Study:

  • To introduce a new class of graphical models, termed torus graphs, for analyzing multivariate circular data.
  • To address the limitations of existing methods like phase locking value in capturing complex phase interactions.
  • To apply torus graphs to identify phase coupling in brain oscillatory signals during a memory task.

Main Methods:

  • Development of torus graphs, a graphical model based on the exponential family with pairwise interactions on a torus.
  • Simulation studies to compare torus graphs with standard methods like phase locking value.
  • Application of torus graphs to analyze 24 phase angles from 4 brain regions across 840 trials during a memory task.

Main Results:

  • Torus graphs accurately identify conditional associations in multivariate circular data, outperforming traditional phase locking value.
  • Simulations demonstrated the superior ability of torus graphs to capture multivariate phase interactions.
  • Analysis of brain data revealed intuitive results regarding phase coupling across distinct brain regions.

Conclusions:

  • Torus graphs provide a unified framework for multivariate analysis of circular data, particularly in neuroscience.
  • This novel approach effectively models phase coupling in oscillatory signals, offering insights into brain coordination.
  • Torus graphs represent a significant advancement with potential for broad applications in analyzing circular data.