Projected normal distribution explained

Projected normal distribution
Type:	density
Notation:	l{P N} _{n(\boldsymbol\mu,} \boldsymbol\Sigma)
Parameters:	\boldsymbol\mu\in\R ^{n} (location) \boldsymbol\Sigma\in\R ^{n} (scale)
Support:	\boldsymbol \theta \in [ 0 , \pi] ^{n} x [ 0 , 2 \pi) | pdf = complicated, see text

In directional statistics, the projected normal distribution (also known as offset normal distribution or angular normal 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 \R ^{n} that follows a multivariate normal distribution l{N} _{n(\boldsymbol\mu,} \boldsymbol\Sigma) , the projected normal distribution l{PN} _{n(\boldsymbol\mu,} \boldsymbol\Sigma) represents the distribution of the random variable \boldsymbol Y = (\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.

Density function

The density of the projected normal distribution

l{P N} _{n(\boldsymbol\mu,} \boldsymbol\Sigma) can be constructed from the density of its generator n-variate normal distribution l{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 , inf ty) and angles \boldsymbol \theta = (\theta _{1,} ... , \theta _{n-1} ) \in [ 0 , \pi] ^{n} x [ 0 , 2 \pi) , a point \boldsymbol x = (x _{1,} ... , x _{n)} \in \R ^{n} can be written as \boldsymbol x = r \boldsymbol v , with \lVert \boldsymbol v \rVert = 1 . The joint density becomesp(r, \boldsymbol \theta | \boldsymbol \mu, \boldsymbol \Sigma) = (r ^{n-1})/(\sqrt{|\boldsymbol \Sigma|) ( 2 \pi) ^{(n)/(2)} }e^

and the density of

l{P N} _{n(\boldsymbol\mu,} \boldsymbol\Sigma) can then be obtained asp(\boldsymbol \theta | \boldsymbol \mu, \boldsymbol \Sigma) = \int ^{inf ty} _{0} p(r, \boldsymbol \theta | \boldsymbol \mu, \boldsymbol \Sigma) dr .

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 asp(\theta | \boldsymbol\mu, \boldsymbol\Sigma) = (e ^{- (1)/(\boldsymbol \mu ^{\top} \boldsymbol \Sigma ^{-1} \boldsymbol \mu 2)})/(2 \pi \sqrt{|\boldsymbol \Sigma|) \boldsymbol v ^{\top} \boldsymbol \Sigma ^{-1} \boldsymbol v} \left( 1 + T(\theta) (\Phi(T(\theta)))/(\phi(T(\theta))) \right) I _{[0,} (\theta)

where

\phi and \Phi are the density and cumulative distribution of a standard normal distribution, T(\theta) = (\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 becomesp(\boldsymbol \theta | \boldsymbol\mu, \boldsymbol\Sigma) = (e ^{- (1)/(\boldsymbol \mu ^{\top} \boldsymbol \Sigma ^{-1} \boldsymbol \mu 2)})/(\sqrt{|\boldsymbol \Sigma|) \left( 2 \pi \boldsymbol v ^{\top} \boldsymbol \Sigma ^{-1} \boldsymbol v \right) ^{(3)/(2)} }\left(\frac + T(\boldsymbol \theta) \left(1 + T(\boldsymbol \theta) \frac \right) \right)I_