Projected normal distribution

In directional statistics, the projected normal distribution (also known as offset normal distribution, angular normal distribution or angular Gaussian distribution)^[1][2] 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.

Projected normal distribution

Notation	${\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}$
Parameters	${\displaystyle {\boldsymbol {\mu }}\in \mathbb {R} ^{n}}$ (location) ${\displaystyle {\boldsymbol {\Sigma }}\in \mathbb {R} ^{n\times n}}$ (scale)
Support	${\displaystyle {\boldsymbol {\theta }}\in [0,\pi ]^{n-2}\times [0,2\pi )}$
PDF	complicated, see text

Definition and properties

Given a random variable ${\displaystyle {\boldsymbol {X}}\in \mathbb {R} ^{n}}$ that follows a multivariate normal distribution ${\displaystyle {\mathcal {N}}_{n}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})}$, the projected normal distribution ${\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}$ represents the distribution of the random variable ${\displaystyle {\boldsymbol {Y}}={\frac {\boldsymbol {X}}{\lVert {\boldsymbol {X}}\rVert }}}$ obtained projecting ${\displaystyle {\boldsymbol {X}}}$ over the unit sphere. In the general case, the projected normal distribution can be asymmetric and multimodal. In case ${\displaystyle {\boldsymbol {\mu }}}$ is orthogonal to an eigenvector of ${\displaystyle {\boldsymbol {\Sigma }}}$, the distribution is symmetric.^[3] The first version of such distribution was introduced in Pukkila and Rao (1988).^[4]

Density function

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

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

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

and the density of ${\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}$ can then be obtained as^[5]

${\displaystyle p({\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})=\int _{0}^{\infty }p(r,{\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})dr.}$

The same density had been previously obtained in Pukkila and Rao (1988, Eq. (2.4))^[4] using a different notation.

Circular distribution

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

${\displaystyle p(\theta |{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {e^{-{\frac {1}{2}}{\boldsymbol {\mu }}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{2\pi {\sqrt {|{\boldsymbol {\Sigma }}|}}{\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {v}}}}\left(1+T(\theta ){\frac {\Phi (T(\theta ))}{\phi (T(\theta ))}}\right)I_{[0,2\pi )}(\theta )}$

where ${\displaystyle \phi }$ and ${\displaystyle \Phi }$ are the density and cumulative distribution of a standard normal distribution, ${\displaystyle T(\theta )={\frac {{\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}{\sqrt {{\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {v}}}}}}$, and ${\displaystyle I}$ is the indicator function.^[3]

In the circular case, if the mean vector ${\displaystyle {\boldsymbol {\mu }}}$ is parallel to the eigenvector associated to the largest eigenvalue of the covariance, the distribution is symmetric and has a mode at ${\displaystyle \theta =\alpha }$ and either a mode or an antimode at ${\displaystyle \theta =\alpha +\pi }$, where ${\displaystyle \alpha }$ is the polar angle of ${\displaystyle {\boldsymbol {\mu }}=(r\cos \alpha ,r\sin \alpha )}$. 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 ${\displaystyle \theta =\alpha }$ and an antimode at ${\displaystyle \theta =\alpha +\pi }$.^[6]

Spherical distribution

Parametrising the position on the unit sphere in spherical coordinates as ${\displaystyle {\boldsymbol {v}}=(\cos \theta _{1}\sin \theta _{2},\sin \theta _{1}\sin \theta _{2},\cos \theta _{2})}$ where ${\displaystyle {\boldsymbol {\theta }}=(\theta _{1},\theta _{2})}$ are the azimuth ${\displaystyle \theta _{1}\in [0,2\pi )}$ and inclination ${\displaystyle \theta _{2}\in [0,\pi ]}$ angles respectively, the density function becomes

${\displaystyle p({\boldsymbol {\theta }}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }})={\frac {e^{-{\frac {1}{2}}{\boldsymbol {\mu }}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {\mu }}}}{{\sqrt {|{\boldsymbol {\Sigma }}|}}\left(2\pi {\boldsymbol {v}}^{\top }{\boldsymbol {\Sigma }}^{-1}{\boldsymbol {v}}\right)^{\frac {3}{2}}}}\left({\frac {\Phi (T({\boldsymbol {\theta }}))}{\phi (T({\boldsymbol {\theta }}))}}+T({\boldsymbol {\theta }})\left(1+T({\boldsymbol {\theta }}){\frac {\Phi (T({\boldsymbol {\theta }}))}{\phi (T({\boldsymbol {\theta }}))}}\right)\right)I_{[0,2\pi )}(\theta _{1})I_{[0,\pi ]}(\theta _{2})}$

where ${\displaystyle \phi }$, ${\displaystyle \Phi }$, ${\displaystyle T}$, and ${\displaystyle I}$ have the same meaning as the circular case.^[7]

See also

• Directional statistics
• Multivariate normal distribution