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Identification of Disease-related Spatial Covariance Patterns using Neuroimaging Data
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Published on: June 26, 2013

Linear transformations of variance/covariance matrices.

Pascal Parois1, Martin Lutz

  • 1Bijvoet Center for Biomolecular Research, Crystal and Structural Chemistry, Faculty of Science, Utrecht University, The Netherlands.

Acta Crystallographica. Section A, Foundations of Crystallography
|June 23, 2011
PubMed
Summary
This summary is machine-generated.

Transforming crystallographic parameters and their uncertainties requires careful handling of the variance-covariance matrix. A new method simplifies uncertainty transformation for second-rank tensors, aiding crystallographic software development.

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Area of Science:

  • Crystallography
  • Materials Science
  • Computational Chemistry

Background:

  • Linear transformations are crucial for crystallographic parameter analysis.
  • Standard uncertainty transformation in crystallography is complex, often requiring the full variance-covariance matrix.

Purpose of the Study:

  • To present a simplified method for transforming standard uncertainties of second-rank tensors in crystallography.
  • To facilitate the implementation of these transformations in computational crystallographic tools.

Main Methods:

  • Representing 3x3 matrices as 9x1 vectors for tensor transformation.
  • Applying the variance-covariance matrix for straightforward uncertainty propagation.
  • Developing a method applicable to anisotropic displacement parameters and eigenvalue uncertainties.

Main Results:

  • A straightforward and implementable method for transforming standard uncertainties of second-rank tensors.
  • Successful application in calculating equivalent isotropic displacement parameters.
  • Facilitation of comparisons between crystallographic refinements in different space-group settings.

Conclusions:

  • The proposed vector-based method simplifies the complex task of uncertainty transformation in crystallography.
  • This approach enhances the accuracy and efficiency of crystallographic data analysis and software.
  • The method is broadly applicable to various crystallographic calculations involving tensor transformations.