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Projected normal distribution

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Projected normal distribution
Notation
Parameters (location)
(scale)
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In directional statistics, the projected normal distribution (also known as offset normal distribution, angular normal distribution or angular Gaussian distribution)[1][2] 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

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Given a random variable that follows a multivariate normal distribution , the projected normal distribution represents the distribution of the random variable obtained projecting over the unit sphere. In the general case, the projected normal distribution can be asymmetric and multimodal. In case is orthogonal to an eigenvector of , the distribution is symmetric.[3] The first version of such distribution was introduced in Pukkila and Rao (1988).[4]

Density function

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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

and the density of can then be obtained as[5]

The same density had been previously obtained in Pukkila and Rao (1988, Eq. (2.4))[4] using a different notation.

Circular distribution

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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

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

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Parametrising the position on the unit sphere in spherical coordinates as where are the azimuth and inclination angles respectively, the density function becomes

where , , , and have the same meaning as the circular case.[7]

See also

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References

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Sources

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  • Pukkila, Tarmo M.; Rao, C. Radhakrishna (1988). "Pattern recognition based on scale invariant discriminant functions". Information Sciences. 45 (3): 379–389.
  • 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.