Projected normal distribution
| Notation | [math]\displaystyle{ \mathcal{P N}_n(\boldsymbol\mu, \boldsymbol\Sigma) }[/math] | ||
|---|---|---|---|
| Parameters |
[math]\displaystyle{ \boldsymbol\mu\in\R^n }[/math] (location) [math]\displaystyle{ \boldsymbol\Sigma\in\R^{n \times n} }[/math] (scale) | ||
| Support | [math]\displaystyle{ \boldsymbol \theta \in [0, \pi]^{n - 2} \times [0, 2 \pi) }[/math] | ||
| complicated, see text | |||
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.
Definition and properties
Given a random variable [math]\displaystyle{ \boldsymbol X \in \R^n }[/math] that follows a multivariate normal distribution [math]\displaystyle{ \mathcal{N}_n(\boldsymbol\mu,\, \boldsymbol\Sigma) }[/math], the projected normal distribution [math]\displaystyle{ \mathcal{PN}_n(\boldsymbol\mu, \boldsymbol\Sigma) }[/math] represents the distribution of the random variable [math]\displaystyle{ \boldsymbol Y = \frac{\boldsymbol X}{\lVert \boldsymbol X \rVert} }[/math] obtained projecting [math]\displaystyle{ \boldsymbol X }[/math] over the unit sphere. In the general case, the projected normal distribution can be asymmetric and multimodal. In case [math]\displaystyle{ \boldsymbol \mu }[/math] is orthogonal to an eigenvector of [math]\displaystyle{ \boldsymbol \Sigma }[/math], the distribution is symmetric.[2]
Density function
The density of the projected normal distribution [math]\displaystyle{ \mathcal{P N}_n(\boldsymbol\mu, \boldsymbol\Sigma) }[/math] can be constructed from the density of its generator n-variate normal distribution [math]\displaystyle{ \mathcal{N}_n(\boldsymbol\mu, \boldsymbol\Sigma) }[/math] by re-parametrising to n-dimensional spherical coordinates and then integrating over the radial coordinate.
In spherical coordinates with radial component [math]\displaystyle{ r \in [0, \infty) }[/math] and angles [math]\displaystyle{ \boldsymbol \theta = (\theta_1, \dots, \theta_{n-1}) \in [0, \pi]^{n - 2} \times [0, 2 \pi) }[/math], a point [math]\displaystyle{ \boldsymbol x = (x_1, \dots, x_n) \in \R^n }[/math] can be written as [math]\displaystyle{ \boldsymbol x = r \boldsymbol v }[/math], with [math]\displaystyle{ \lVert \boldsymbol v \rVert = 1 }[/math]. The joint density becomes
- [math]\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)} }[/math]
and the density of [math]\displaystyle{ \mathcal{P N}_n(\boldsymbol\mu, \boldsymbol\Sigma) }[/math] can then be obtained as[3]
- [math]\displaystyle{ p(\boldsymbol \theta | \boldsymbol \mu, \boldsymbol \Sigma) = \int_0^\infty p(r, \boldsymbol \theta | \boldsymbol \mu, \boldsymbol \Sigma) dr . }[/math]
Circular distribution
Parametrising the position on the unit circle in polar coordinates as [math]\displaystyle{ \boldsymbol v = (\cos\theta, \sin\theta) }[/math], the density function can be written with respect to the parameters [math]\displaystyle{ \boldsymbol\mu }[/math] and [math]\displaystyle{ \boldsymbol\Sigma }[/math] of the initial normal distribution as
- [math]\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) }[/math]
where [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ \Phi }[/math] are the density and cumulative distribution of a standard normal distribution, [math]\displaystyle{ T(\theta) = \frac{\boldsymbol v^\top \boldsymbol \Sigma^{-1} \boldsymbol \mu}{\sqrt{\boldsymbol v^\top \boldsymbol \Sigma^{-1} \boldsymbol v}} }[/math], and [math]\displaystyle{ I }[/math] is the indicator function.[2]
In the circular case, if the mean vector [math]\displaystyle{ \boldsymbol \mu }[/math] is parallel to the eigenvector associated to the largest eigenvalue of the covariance, the distribution is symmetric and has a mode at [math]\displaystyle{ \theta = \alpha }[/math] and either a mode or an antimode at [math]\displaystyle{ \theta = \alpha + \pi }[/math], where [math]\displaystyle{ \alpha }[/math] is the polar angle of [math]\displaystyle{ \boldsymbol \mu = (r \cos\alpha, r \sin\alpha) }[/math]. 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 [math]\displaystyle{ \theta = \alpha }[/math] and an antimode at [math]\displaystyle{ \theta = \alpha + \pi }[/math].[4]
Spherical distribution
Parametrising the position on the unit sphere in spherical coordinates as [math]\displaystyle{ \boldsymbol v = (\cos\theta_1 \sin\theta_2, \sin\theta_1 \sin\theta_2, \cos\theta_2) }[/math] where [math]\displaystyle{ \boldsymbol \theta = (\theta_1, \theta_2) }[/math] are the azimuth [math]\displaystyle{ \theta_1 \in [0, 2\pi) }[/math] and inclination [math]\displaystyle{ \theta_2 \in [0, \pi] }[/math] angles respectively, the density function becomes
- [math]\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) }[/math]
where [math]\displaystyle{ \phi }[/math], [math]\displaystyle{ \Phi }[/math], [math]\displaystyle{ T }[/math], and [math]\displaystyle{ I }[/math] have the same meaning as the circular case.[5]
See also
References
- ↑ Wang & Gelfand 2013.
- ↑ 2.0 2.1 Hernandez-Stumpfhauser & Breidt 2017, p. 115.
- ↑ Hernandez-Stumpfhauser & Breidt 2017, p. 117.
- ↑ Hernandez-Stumpfhauser & Breidt 2017, Supplementary material, p. 1.
- ↑ Hernandez-Stumpfhauser & Breidt 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 (Elsevier) 10 (1): 113–127.
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