The density of the projected normal distribution
can be constructed from the density of its generator n-variate normal distribution
by re-parametrising to n-dimensional spherical coordinates and then integrating over the radial coordinate.
In spherical coordinates with radial component
and angles
, a point
can be written as
, with
. 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 }})}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a7040e802713d76c38ed0cafce315905e44bcdb5)
and the density of
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.}](//wikimedia.org/api/rest_v1/media/math/render/svg/744c14cc013b404e2ea0b6b9182464135c8b6c56)
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
, the density function can be written with respect to the parameters
and
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 )}](//wikimedia.org/api/rest_v1/media/math/render/svg/1cda3281d7c5f7d391b25bfc7e71fa5ca2691bf6)
where
and
are the density and cumulative distribution of a standard normal distribution,
, and
is the indicator function.[3]
In the circular case, if the mean vector
is parallel to the eigenvector associated to the largest eigenvalue of the covariance, the distribution is symmetric and has a mode at
and either a mode or an antimode at
, where
is the polar angle of
. 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
and an antimode at
.[6]
Spherical distribution
Parametrising the position on the unit sphere in spherical coordinates as
where
are the azimuth
and inclination
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})}](//wikimedia.org/api/rest_v1/media/math/render/svg/5d60267ebfb589e2c48e1da746b839559e6a164f)
where
,
,
, and
have the same meaning as the circular case.[7]