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Projected Normal Distribution: Moment Approximations and Generalizations.

Daniel Herrera-Esposito1, Johannes Burge1

  • 1Department of Psychology, University of Pennsylvania, Hamilton Walk, Philadelphia, 19104, Pennsylvania, United States.

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|September 22, 2025
PubMed
Summary
This summary is machine-generated.

This study derives analytic approximations for the moments of the projected normal distribution and its generalizations. These methods enable accurate data fitting for applications in systems neuroscience and beyond.

Keywords:
Angular GaussianDirectional statisticsDivisive normalizationMoments approximationProjected normal distributionQuadratic forms

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Area of Science:

  • Statistics
  • Computational Neuroscience

Background:

  • The projected normal distribution, or angular Gaussian distribution, models variables on a unit sphere.
  • Existing methods lack closed-form formulas for its moments, limiting applications.

Purpose of the Study:

  • Derive analytic approximations for the first and second moments of the projected normal distribution.
  • Generalize the distribution and derive its density and moment approximations.
  • Provide tools for data analysis in neuroscience and other fields.

Main Methods:

  • Utilized Taylor expansions and quadratic forms of Gaussian random variables.
  • Developed generalized projected normal distributions with novel denominators.
  • Derived density functions and moment approximations for generalized distributions.

Main Results:

  • Obtained accurate analytic approximations for moments of projected normal distributions.
  • Demonstrated accuracy across various dimensionalities and parameters.
  • Validated moment matching for fitting distributions to data.

Conclusions:

  • The derived moment approximations are accurate and broadly applicable.
  • Moment matching provides a robust method for analyzing data.
  • The generalized distributions and fitting methods are valuable for systems neuroscience and statistical modeling.