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The normal, a continuous distribution, is the most important of all the distributions. Its graph is a bell-shaped symmetrical curve, which is observed in almost all disciplines. Some of these include psychology, business, economics, the sciences, nursing, and, of course, mathematics. Some instructors may use the normal distribution to help determine students’ grades. Most IQ scores are normally distributed. Often real-estate prices fit a normal distribution. The normal distribution is extremely...

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Updated: May 7, 2026

Visualization Method for Proprioceptive Drift on a 2D Plane Using Support Vector Machine

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Published on: October 27, 2016

Directional data analysis under the general projected normal distribution.

Fangpo Wang¹, Alan E Gelfand

¹Department of Statistical Science, Duke University, Durham, NC 27708-0251, United States.

Statistical Methodology

|September 19, 2013

Summary

This summary is machine-generated.

The projected normal distribution offers a flexible model for analyzing directional and circular data. This study develops Bayesian hierarchical models and Markov chain Monte Carlo (MCMC) methods for robust data analysis and prediction.

Keywords:

Markov chain Monte Carlo bivariate normal distribution circular data concentration latent variables mean direction

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Measurement of the Directional Information Flow in fNIRS-Hyperscanning Data using the Partial Wavelet Transform Coherence Method

Published on: September 3, 2021

Area of Science:

Statistics
Bayesian inference
Circular statistics

Background:

The projected normal distribution is an under-utilized statistical model for directional data.
The general projected normal distribution offers flexibility, including asymmetry and bimodality.
Regression models for directional data often lack flexibility.

Purpose of the Study:

To clarify the properties of the general projected normal distribution.
To develop fully Bayesian hierarchical models for circular data analysis using this distribution.
To provide methods for posterior inference, prediction, and model comparison.

Main Methods:

Development of fully Bayesian hierarchical models for circular data.
Utilizing Markov chain Monte Carlo (MCMC) methods with latent variables for model fitting.
Implementation of posterior inference for parameters like angular mean direction and concentration.
Application of out-of-sample criteria for model comparison, including predictive likelihood and cumulative rank probability scores.

Main Results:

The general projected normal distribution provides a flexible framework for modeling directional data, accommodating asymmetry and bimodality.
Fully Bayesian hierarchical models are developed and shown to be efficiently fitted using MCMC methods.
Posterior inference for key distributional features and predictive tasks within regression settings are demonstrated.
An out-of-sample model comparison approach using specific scoring rules is advocated.

Conclusions:

The projected normal distribution is a valuable and flexible tool for analyzing directional and circular data.
Bayesian hierarchical models offer a powerful framework for complex analyses of circular data.
The proposed MCMC methods and inference techniques provide practical tools for researchers.
Out-of-sample evaluation is crucial for robust model comparison in this domain.