Projected normal distribution

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

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^[3]

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

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.^[2]

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 ${\displaystyle \theta =\alpha +\pi }$ .^[4]

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.^[5]

See also

• Directional statistics
• Multivariate normal distribution

References

1. Wang & Gelfand 2013.
2. Hernandez-Stumpfhauser, Breidt & van der Woerd 2017, p. 115.
3. Hernandez-Stumpfhauser, Breidt & van der Woerd 2017, p. 117.
4. Hernandez-Stumpfhauser, Breidt & van der Woerd 2017, Supplementary material, p. 1.
5. Hernandez-Stumpfhauser, Breidt & van der Woerd 2017, p. 123.

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. 10 (1). Elsevier: 113–127.