Directional Statistics
Directional statistics is concerned mainly with
observations which are unit vectors (possibly with sign unknown) in the plane
or in three-dimensional space. Thus the sample space is typically a circle or a
sphere, so that standard methods for analysing univariate
or multivariate measurement data cannot be used. Special directional methods
are required which take into account the structure of these sample spaces.
Extensions include rotational data for example.
TRAPS IN USING
LINEAR STATISTICS !!
Image © K.V.
Mardia
See also:
http://en.wikipedia.org/wiki/Directional_statistics
Some key contributions: Bingham-Mardia model for small circle, line find procedure, circular regression model, Mardia’s bivariate von Mises distributions, circular rank tests.
Books:
1972 Statistics of Directional Data. Academic Press,
2000 Directional Statistics,
2nd edition (with P. E. Jupp). J. Wiley,
Read Paper:
1975 Statistics of directional data. J. Roy. Statist. Soc. B 37, 349-393.
Until 1975, the field of directional statistics was the “Cinderella” of statistical science. This paper coupled with the 1972 book can be said to have had some visible impact in changing this otherwise neglected field by bringing it into its deserving limelight.
This paper introduced new models such as the bivariate von Mises distribution and the famous characterization that it is not possible to have both conditional von Mises as well as both marginal von Mises distributions from any bivariate circular density. It also introduced uniform scores on a circle for circular non-parametrics originating the notion of key circular correlation. The Mardia-Wheeler-Watson two-sample test turns out to be a particular case. The recent surge of research activities in Shape Analysis also owes a great deal to this paper. For example, we may cite the complex Watson shape model and Mardia-Dryden offset shape distribution.
Presenting
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Papers in Journals:
1967 A non-parametric test for the bivariate two-sample location problem. J. Roy. Statist. Soc. Ser. B 29, 320-342.
1969 On Wheeler-Watson’s two-sample test on a circle. Sankhya Ser. A 31, 177-190.
1972 A multisample uniform scores test on a circle and its parametric competitor. J. Roy. Statist. Soc. Ser. B 34, 102-113.
1975 On the modes of a mixture of two von Mises distributions (with T. W. Sutton). Biometrika 62, 699-701.
1975 Spherical statistics (with P. J. Zemroch). J. Roy. Statist. Soc. Ser. C 24, 144-146.
1975 Circular statistics (with P. J. Zemroch). J. Roy. Statist. Soc. Ser. C 24, 147-150.
1975 von Mises distribution function (with P. J. Zemroch). J. Roy. Statist. Soc. Ser. C 24, 262-265.
1976 Bayesian inference for the von Mises-Fisher distribution (with S. A. M. El-Atoum). Biometrika 63, 302-306.
1976 Linear-circular correlation coefficients and rhythmometry. Biometrika 63, 403-405.
1977 The uniform distribution on Stiefel manifold (with C. G. Khatri). J. Multivar. Anal 7, 468-473.
1977 Analysis of central place theory (with R. Edwards and M. L. Puri). Proc. 41st Conf. Int. Statist. Inst. 47, 93-110.
1977 A circle of best-fit for spherical data and areas of vulcanism (with R. J. Gadsen). J. Roy. Statist. Soc. Ser. C 16, 238-245.
1977 The von Mises-Fisher matrix distribution (with C.G. Khatri). J. Roy. Statist. Soc. Ser. B 39, 95-106.
1978 A small circle distribution on a sphere (with C. Bingham). Biometrika 65, 379-389.
1978 A spherical correlation coefficient robust against scale (with M. L. Puri). Biometrika 65, 391-396.
1979 Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions (with P. E. Jupp). Ann. Statist. 7, 599-606.
1979 Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions (with P. E. Jupp). Ann. Statist. 7, 599-606.
1979 A characterization of the uniform distribution on the circle (with J. T. Kent and J. S. Rao). Ann. Statist. 7, 882-889.
1980 A general correlation coefficient for directions data (with P. E. Jupp). Biometrika 67, 163-174.
1980 A statistical analysis of megalithic data under elliptic pattern (with D. Holmes). J. Roy. Statist. Soc. Ser. A 143, 293-302.
1982 Weighted distributions and rotating caps (with R. Edwards). Biometrika 69, 323-330.
1984 A goodness-of-fit test for the von Mises-Fisher distribution (with J. T. Kent and D. Holmes).
1989 Unified view of the theory of directional statistics, 1975-1985 (with P. E. Jupp). Int. Statist. Review 57, 261-294.
2002 A bootstrap approach to Pluto's origin (with V. Patrangenaru) J. Appl. Statist. 29 (6) 935-943.
2002 Circular regression (with T. D. Downs). Biometrika 89, 683-697.
2004 Statistics of shape, direction and cylindrical variables (with F. L. Bookstein, and J. Kirkbride). J. Appl. Statist. 31 465-479.
2007 Markov chain