# Projected normal distribution

In directional statistics, the projected normal distribution (also known as offset normal distribution, angular normal distribution or angular Gaussian 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 ${\boldsymbol {X}}\in \mathbb {R} ^{n}$ that follows a multivariate normal distribution ${\mathcal {N}}_{n}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})$, the projected normal distribution ${\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})$ represents the distribution of the random variable ${\boldsymbol {Y}}={\frac {\boldsymbol {X}}{\lVert {\boldsymbol {X}}\rVert }}$ obtained projecting ${\boldsymbol {X}}$ over the unit sphere. In the general case, the projected normal distribution can be asymmetric and multimodal. In case ${\boldsymbol {\mu }}$ is orthogonal to an eigenvector of ${\boldsymbol {\Sigma }}$, the distribution is symmetric. The first version of such distribution was introduced in Pukkila and Rao (1988).

## Density function

The density of the projected normal distribution ${\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})$ can be constructed from the density of its generator n-variate normal distribution ${\mathcal {N}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})$ by re-parametrising to n-dimensional spherical coordinates and then integrating over the radial coordinate.
In spherical coordinates with radial component $r\in [0,\infty )$ and angles ${\boldsymbol {\theta }}=(\theta _{1},\dots ,\theta _{n-1})\in [0,\pi ]^{n-2}\times [0,2\pi )$, a point ${\boldsymbol {x}}=(x_{1},\dots ,x_{n})\in \mathbb {R} ^{n}$ can be written as ${\boldsymbol {x}}=r{\boldsymbol {v}}$, with $\lVert {\boldsymbol {v}}\rVert =1$. The joint density becomes
    $p(r,{\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {r^{n-1}}{{\sqrt {|{\boldsymbol {\Sigma }}|}}(2\pi )^{\frac {n}{2}}}}e^{-{\frac {1}{2}}(r{\boldsymbol {v}}-{\boldsymbol {\mu }})^{\top }\Sigma ^{-1}(r{\boldsymbol {v}}-{\boldsymbol {\mu }})}$
and the density of ${\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})$ can then be obtained as
    $p({\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})=\int _{0}^{\infty }p(r,{\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})dr.$
The same density had been previously obtained in Pukkila and Rao (1988, Eq. (2.4)) using a different notation.

### Circular distribution

Parametrising the position on the unit circle in polar coordinates as ${\boldsymbol {v}}=(\cos \theta ,\sin \theta )$, the density function can be written with respect to the parameters ${\boldsymbol {\mu }}$ and ${\boldsymbol {\Sigma }}$ of the initial normal distribution as
    $p(\theta |{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {e^{-{\frac {1}{2}}{\boldsymbol {\mu }}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{2\pi {\sqrt {|{\boldsymbol {\Sigma }}|}}{\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {v}}}}\left(1+T(\theta ){\frac {\Phi (T(\theta ))}{\phi (T(\theta ))}}\right)I_{[0,2\pi )}(\theta )$
where $\phi$ and $\Phi$ are the density and cumulative distribution of a standard normal distribution, $T(\theta )={\frac {{\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}{\sqrt {{\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {v}}}}}$, and $I$ is the indicator function.
In the circular case, if the mean vector ${\boldsymbol {\mu }}$ 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 ${\boldsymbol {\mu }}=(r\cos \alpha ,r\sin \alpha )$. 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 ${\boldsymbol {v}}=(\cos \theta _{1}\sin \theta _{2},\sin \theta _{1}\sin \theta _{2},\cos \theta _{2})$ where ${\boldsymbol {\theta }}=(\theta _{1},\theta _{2})$ are the azimuth $\theta _{1}\in [0,2\pi )$ and inclination $\theta _{2}\in [0,\pi ]$ angles respectively, the density function becomes
    $p({\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {e^{-{\frac {1}{2}}{\boldsymbol {\mu }}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{{\sqrt {|{\boldsymbol {\Sigma }}|}}\left(2\pi {\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {v}}\right)^{\frac {3}{2}}}}\left({\frac {\Phi (T({\boldsymbol {\theta }}))}{\phi (T({\boldsymbol {\theta }}))}}+T({\boldsymbol {\theta }})\left(1+T({\boldsymbol {\theta }}){\frac {\Phi (T({\boldsymbol {\theta }}))}{\phi (T({\boldsymbol {\theta }}))}}\right)\right)I_{[0,2\pi )}(\theta _{1})I_{[0,\pi ]}(\theta _{2})$
where $\phi$, $\Phi$, $T$, and $I$ have the same meaning as the circular case.