Projected normal distribution

Short description: Probability distribution

Projected normal distribution

Notation	${{𝒫}{𝒩}}_n(\mathit{\mu },\mathrm{\Sigma })$
Parameters	$\mathit{\mu }\in {\mathbb{ℝ}}^n$ (location) $\mathrm{\Sigma }\in {\mathbb{ℝ}}^{n\times n}$ (scale)
Support	$\mathit{\theta }\in [0,\pi {]}^{n-2}\times [0,2\pi )$
PDF	complicated, see text

In directional statistics, the projected normal distribution (also known as offset normal distribution or angular normal distribution) is a probability distribution over directions that describes the radial projection of a random variable with n-variate normal distribution over the unit (n-1)-sphere.

Definition and properties

Given a random variable $\mathit{X}\in {\mathbb{ℝ}}^n$ that follows a multivariate normal distribution ${{𝒩}}_n(\mathit{\mu },\mathrm{\Sigma })$, the projected normal distribution ${{𝒫}{𝒩}}_n(\mathit{\mu },\mathrm{\Sigma })$ represents the distribution of the random variable $\mathit{Y}=\frac{\mathit{X}}{\Vert \mathit{X}\Vert }$ obtained projecting $\mathit{X}$ over the unit sphere. In the general case, the projected normal distribution can be asymmetric and multimodal. In case $\mathit{\mu }$ is orthogonal to an eigenvector of $\mathrm{\Sigma }$, the distribution is symmetric.

Density function

The density of the projected normal distribution ${{𝒫}{𝒩}}_n(\mathit{\mu },\mathrm{\Sigma })$ can be constructed from the density of its generator n-variate normal distribution ${{𝒩}}_n(\mathit{\mu },\mathrm{\Sigma })$ by re-parametrising to n-dimensional spherical coordinates and then integrating over the radial coordinate.

In spherical coordinates with radial component $r\in [0,\mathrm{\infty })$ and angles $\mathit{\theta }=({\theta }_1,\dots ,{\theta }_{n-1})\in [0,\pi {]}^{n-2}\times [0,2\pi )$, a point $\mathit{x}=(x_1,\dots ,x_n)\in {\mathbb{ℝ}}^n$ can be written as $\mathit{x}=r\mathit{v}$, with $\Vert \mathit{v}\Vert =1$. The joint density becomes

$p(r,\mathit{\theta }|\mathit{\mu },\mathrm{\Sigma })=\frac{r^{n-1}}{\sqrt{|\mathrm{\Sigma }|}(2\pi {)}^{\frac{n}{2}}}{e}^{-\frac{1}{2}(r\mathit{v}-\mathit{\mu }{)}^{\mathrm{\top }}{\mathrm{\Sigma }}^{-1}(r\mathit{v}-\mathit{\mu })}$

and the density of ${{𝒫}{𝒩}}_n(\mathit{\mu },\mathrm{\Sigma })$ can then be obtained as

$p(\mathit{\theta }|\mathit{\mu },\mathrm{\Sigma })=\underset{0}{\overset{\mathrm{\infty }}{\int }}p(r,\mathit{\theta }|\mathit{\mu },\mathrm{\Sigma })dr.$

Circular distribution

Parametrising the position on the unit circle in polar coordinates as $\mathit{v}=(\cos \theta ,\sin \theta )$, the density function can be written with respect to the parameters $\mathit{\mu }$ and $\mathrm{\Sigma }$ of the initial normal distribution as

$p(\theta |\mathit{\mu },\mathrm{\Sigma })=\frac{e^{-\frac{1}{2}{\mathit{\mu }}^{\mathrm{\top }}{\mathrm{\Sigma }}^{-1}\mathit{\mu }}}{2\pi \sqrt{|\mathrm{\Sigma }|}{\mathit{v}}^{\mathrm{\top }}{\mathrm{\Sigma }}^{-1}\mathit{v}}(1+T(\theta )\frac{\mathrm{\Phi }(T(\theta ))}{\varphi (T(\theta ))})I_{[0,2\pi )}(\theta )$

where $\varphi$ and $\mathrm{\Phi }$ are the density and cumulative distribution of a standard normal distribution, $T(\theta )=\frac{{\mathit{v}}^{\mathrm{\top }}{\mathrm{\Sigma }}^{-1}\mathit{\mu }}{\sqrt{{\mathit{v}}^{\mathrm{\top }}{\mathrm{\Sigma }}^{-1}\mathit{v}}}$, and $I$ is the indicator function.

In the circular case, if the mean vector $\mathit{\mu }$ is parallel to the eigenvector associated to the largest eigenvalue of the covariance, the distribution is symmetric and has a mode at $\theta =\alpha$ and either a mode or an antimode at $\theta =\alpha +\pi$, where $\alpha$ is the polar angle of $\mathit{\mu }=(r\cos \alpha ,r\sin \alpha )$. If the mean is parallel to the eigenvector associated to the smallest eigenvalue instead, the distribution is also symmetric but has either a mode or an antimode at $\theta =\alpha$ and an antimode at $\theta =\alpha +\pi$.

Spherical distribution

Parametrising the position on the unit sphere in spherical coordinates as $\mathit{v}=(\cos {\theta }_1\sin {\theta }_2,\sin {\theta }_1\sin {\theta }_2,\cos {\theta }_2)$ where $\mathit{\theta }=({\theta }_1,{\theta }_2)$ are the azimuth ${\theta }_1\in [0,2\pi )$ and inclination ${\theta }_2\in [0,\pi ]$ angles respectively, the density function becomes

$p(\mathit{\theta }|\mathit{\mu },\mathrm{\Sigma })=\frac{e^{-\frac{1}{2}{\mathit{\mu }}^{\mathrm{\top }}{\mathrm{\Sigma }}^{-1}\mathit{\mu }}}{\sqrt{|\mathrm{\Sigma }|}{\left(2\pi {\mathit{v}}^{\mathrm{\top }}{\mathrm{\Sigma }}^{-1}\mathit{v}\right)}^{\frac{3}{2}}}(\frac{\mathrm{\Phi }(T(\mathit{\theta }))}{\varphi (T(\mathit{\theta }))}+T(\mathit{\theta })(1+T(\mathit{\theta })\frac{\mathrm{\Phi }(T(\mathit{\theta }))}{\varphi (T(\mathit{\theta }))}))I_{[0,2\pi )}({\theta }_1)I_{[0,\pi ]}({\theta }_2)$

where $\varphi$, $\mathrm{\Phi }$, $T$, and $I$ have the same meaning as the circular case.

See also

• Directional statistics
• Multivariate normal distribution

Sources

• Hernandez-Stumpfhauser, Daniel; Breidt, F. Jay; van der Woerd, Mark J. (2017). "The General Projected Normal Distribution of Arbitrary Dimension: Modeling and Bayesian Inference". Bayesian Analysis 12 (1): 113–133. 
• Wang, Fangpo; Gelfand, Alan E (2013). "Directional data analysis under the general projected normal distribution". Statistical methodology (Elsevier) 10 (1): 113–127.