Projected normal distribution

In directional statistics, the projected normal distribution (also known as offset normal distribution, angular normal distribution or angular Gaussian distribution)Template:SfnTemplate:Sfn 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 $𝑿\in {ℝ}^n$ that follows a multivariate normal distribution ${𝒩}_n(\mathit{\mu },\mathrm{\Sigma })$, the projected normal distribution ${𝒫𝒩}_n(\mathit{\mu },\mathrm{\Sigma })$ represents the distribution of the random variable $𝒀=\frac{𝑿}{\Vert 𝑿\Vert }$ obtained projecting $𝑿$ 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.Template:Sfn The first version of such distribution was introduced in Pukkila and Rao (1988).Template:Sfn

Density function

The density of the projected normal distribution ${𝒫𝒩}_n(\mathit{\mu },\mathrm{\Sigma })$ 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 $𝒙=(x_1,\dots ,x_n)\in {ℝ}^n$ can be written as $𝒙=r𝒗$, with $\Vert 𝒗\Vert =1$. The joint density becomes

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

and the density of ${𝒫𝒩}_n(\mathit{\mu },\mathrm{\Sigma })$ can then be obtained asTemplate:Sfn

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

The same density had been previously obtained in Pukkila and Rao (1988, Eq. (2.4))Template:Sfn using a different notation.

Circular distribution

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

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

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

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 $\theta =\alpha +\pi$.Template:Sfn

Spherical distribution

Parametrising the position on the unit sphere in spherical coordinates as $𝒗=(\cos {\theta }_1\sin {\theta }_2,\sin {\theta }_1\sin {\theta }_2,\cos {\theta }_2)$ 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

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

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

See also

• Directional statistics
• Multivariate normal distribution

Sources

• Template:Cite journal
• Template:Cite journal
• Template:Cite journal