Projected Normal Distribution: Moment Approximations and Generalizations

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.