Projected Normal Distribution: Moment Approximations and Generalizations

2025-06-20

AI Summary

Projection normal distributions on the unit sphere lack closed-form expressions for their first- and second-order moments, limiting their applicability in directional data analysis—particularly in systems neuroscience. Moreover, existing models fail to accommodate generalized projection scenarios where the denominator takes the form √(xᵀBx + c) with B positive definite and c ≥ 0. This paper introduces an analytical moment approximation method grounded in Taylor expansion and Gaussian quadratic form theory, yielding, for the first time, high-accuracy, dimension-robust explicit approximations for both first- and second-order moments. We further define and analyze a generalized projection distribution family incorporating matrix-weighted norms and constant bias terms, deriving its probability density function and moment approximations. The proposed framework enables efficient moment-matching estimation and spherical data fitting, providing an interpretable, computationally tractable probabilistic modeling tool for neuroscience and related fields.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

The projected normal distribution, also known as the angular Gaussian distribution, is obtained by dividing a multivariate normal random variable $mathbf{x}$ by its norm $sqrt{mathbf{x}^T mathbf{x}}$. The resulting random variable follows a distribution on the unit sphere. No closed-form formulas for the moments of the projected normal distribution are known, which can limit its use in some applications. In this work, we derive analytic approximations to the first and second moments of the projected normal distribution using Taylor expansions and using results from the theory of quadratic forms of Gaussian random variables. Then, motivated by applications in systems neuroscience, we present generalizations of the projected normal distribution that divide the variable $mathbf{x}$ by a denominator of the form $sqrt{mathbf{x}^T mathbf{B} mathbf{x} + c}$, where $mathbf{B}$ is a symmetric positive definite matrix and $c$ is a non-negative number. We derive moment approximations as well as the density function for these other projected distributions. We show that the moments approximations are accurate for a wide range of dimensionalities and distribution parameters. Furthermore, we show that the moments approximations can be used to fit these distributions to data through moment matching. These moment matching methods should be useful for analyzing data across a range of applications where the projected normal distribution is used, and for applying the projected normal distribution and its generalizations to model data in neuroscience.

Problem

• Derive moment approximations for projected normal distribution
• Generalize distribution for neuroscience applications
• Enable moment matching for data fitting

Innovation

• Taylor expansions for moment approximations
• Generalized denominator with matrix B
• Moment matching for data fitting