A note about Gaussian statistics on a sphere
Chave, Alan D.
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KeywordNumerical approximations and analysis; Probability distributions; Marine magnetics and palaeomagnetics
The statistics of directional data on a sphere can be modelled either using the Fisher distribution that is conditioned on the magnitude being unity, in which case the sample space is confined to the unit sphere, or using the latitude–longitude marginal distribution derived from a trivariate Gaussian model that places no constraint on the magnitude. These two distributions are derived from first principles and compared. The Fisher distribution more closely approximates the uniform distribution on a sphere for a given small value of the concentration parameter, while the latitude–longitude marginal distribution is always slightly larger than the Fisher distribution at small off-axis angles for large values of the concentration parameter. Asymptotic analysis shows that the two distributions only become equivalent in the limit of large concentration parameter and very small off-axis angle.
Author Posting. © The Author, 2015. This article is posted here by permission of The Royal Astronomical Society for personal use, not for redistribution. The definitive version was published in Geophysical Journal International 203 (2015): 893-895, doi:10.1093/gji/ggv324.