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Quaternions, Haar Measure and the Estimation of a Palaeomagnetic Rotation

Published online by Cambridge University Press:  05 September 2017

P. A. P. Moran
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Abstract

Representing a rotation in three dimensions by a unit tensor quaternion and supposing that the errors in the measurements of a number of vector directions follow Fisher's distribution, the maximum likelihood estimator of a rotation is obtained. It is shown that the natural Haar measure for three-dimensional rotations is mapped 1:2 onto the natural Haar measure for random directions in 4-space. The natural Haar measure for rotations in 4-space is also mappable onto the product measure of the Haar measures for two separate random directions in 4-space.

Type
Part VII — Probability Models in the Physical Sciences
Information
Journal of Applied Probability , Volume 12 , Issue S1 , 1975 , pp. 295 - 301
Copyright
Copyright © 1975 Applied Probability Trust 

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References

Fisher, R. A. (1953) Dispersion on a sphere. Proc. Roy. Soc. A 217, 295–305.CrossRefGoogle Scholar
Hargraves, R. B. and Duncan, R. A. (1973) Does the mantle roll? Nature 245, 361–363.Google Scholar
James, A. T. (1954) Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. 25, 40–75.Google Scholar
Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
Klein, F. (1932) Elementary Mathematics from an Advanced Standpoint. Vol. I. Arithmetic, Algebra, Analysis. (translated by Hedrick, E. R. and Noble, C. A.). Macmillan, London.Google Scholar
Littlewood, D. E. (1948) Invariant theory under orthogonal groups. Proc. London Math. Soc. (2) 50, 349–379.Google Scholar
Mackenzie, J. K. (1957) The estimation of an orientation relationship. Acta Cryst. 10, 61–62.Google Scholar
Miles, R. E. (1965) On random rotations in R3. Biometrika 52, 636–639.Google Scholar
Whittaker, E. T. (1937) A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press.Google Scholar