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, New York, 1972. Translated in Russian.

2000    Directional Statistics, 2nd edition (with P. E. Jupp). J. Wiley, Chichester, 2000 [for Errata, see the link on Peter Jupp’s website].

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 this paper at the RSS London 1975. Image © K.V. Mardia

 

 

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 Monte Carlo implementation of rock fracture modeling (with V. B. Nyirongo, A. N. Walder, C. Xu, P. A. Dowd, R. J. Fowell, and J. T. Kent). Mathematical Geology, 39, 355-381.

2009    Probabilistic Model of RNA Conformational Space. (with J. Frellsen, I. Moltke, M. Thiim, J. Ferkinghoff-Borg and T.A. Hamelryck). PLoS Computational Biology, 5(6), 1-11.

2011    Hierarchical Bayesian Modelling of Pharmacophores in Bioinformatics. (with V.B. Nyirongo, C.J. Fallaize, S. Barber and R.M. Jackson). Biometrics, 67(2), 611-619.

2011    Log-Linear Modelling of Protein Dipeptide Structure Reveals Interesting Patterns of Side- Chain-Backbone Interactions. (with K. Hommola, and W.R. Gilks). Statistical Applications in Genetics and Molecular Biology, 10(1), Article 8, 1-29.