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Example text

Let t = (tl ,.. ,t d I T be a cl-tfinensional furiction on tlie circle arid BASIC COKCEPTS AND MODELS 43 let. distribution the circle for which the ent~-opyis maximal subject to E[t(z)J = c. y of the function h ( z ) = zlogz that this rnaxirnurri entropy tfistribution is a InerIiber of the ( d ,d ) exponential model with canonical statistic t. (This result holds for any sample space. See Kagm, Linnik & Rao, 1973, p. 409. iaies d l e d 'the maximuni entropy principle'. ) = p. This characterisation is due to hlardia (1972a, pp.

Ion of one of the b;tuir: tfistributions in circular statistics consists of the fractiorial parts of atomic weights. Before the discovery of isotopes, it was hypothesised that measured atomic weighits are iIitegeru siibjec%to error. Von Misw (1918) proposed testing this by testing whether or not the corresponding distribution on the circle has a mode at . ) Axial data occur as directions of o3pticaJ. axes of crystals. Thc distribution of the resultant of a random sample of unit vectors arises in representations of sourid waves (Rayieigli, 1919), interference among oscillations with random phases (Beckmann, 1959) and in niolecillar links (Rayleigh, 1919: Kuhn & Grun, 1942).

1997). 7 Medicine The incidence of onsets of'a particular disease (or of deaths due to the disease) at. various timw of yosr provides cireuliu di5tta. 4 are such an exasnple. Spherical data occur in vector cardiology. ion about the electrical activity in a heart. during a heartbeat is described in t e r m of a near-planar orbit in three-dimensiond space. See Downs & Licbmsn (1969) imd Gould (1969). ion on distance is often not readily available for ;tutronomir:al objects, many astronomical observations ase of points on the celestial sphere, and so provide spherical data.

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