Projected normal distribution

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 $$\mathcal{N}_n(\boldsymbol\mu,\, \boldsymbol\Sigma)$$, 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.

Density function
The density of the projected normal distribution $$\mathcal{P N}_n(\boldsymbol\mu, \boldsymbol\Sigma)$$ 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 $$\boldsymbol x = (x_1, \dots, x_n) \in \R^n$$ 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 $$\mathcal{P N}_n(\boldsymbol\mu, \boldsymbol\Sigma)$$ can then be obtained as



p(\boldsymbol \theta | \boldsymbol \mu, \boldsymbol \Sigma) = \int_0^\infty p(r, \boldsymbol \theta | \boldsymbol \mu, \boldsymbol \Sigma) dr. $$

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.