The Fisher-Bingham Distribution on the Sphere

Abstract

The Fisher distribution is the analogue on the sphere of the isotropic bivariate normal distribution in the plane. The purpose of this paper is to propose and analyse a spherical analogue of the general bivariate normal distribution. Estimation, hypothesis testing and confidence regions are also discussed.with respect to sin e de dfjJ. For plotting purposes Lambert's equal area projection, Z2 = Pcos fjJ, Z3 = Psin fjJ, (1.5) where p = 2 sin (0/2), 0~p~2, is useful because dz1dZ 2 = sin 0 dO dfjJ (see, for example, Mardia, 1972, p. 215). Note that the Lambert projection maps 0 3 onto the disc {(ZI,Z2): (zI +z~)t~2}.

The axes of random infinitesimal rotations and the propagation of orientation uncertainty

“…The object was represented by a mesh and for each k-th vertex of the mesh, a corresponding pair of elevation and azimuth angles (# k , u k ) was determined as in Eq. (7). Finally, the same angular uncertainty data r(# k , u k ) shown in Fig.…”

“…Parameters b and r (or b and j) can be calculated from the zero centered covariance matrix for a set of vectors {w j } as in [7]. This probability measure is used to process the noisy orientation data acquired in the experiment described in Section 3.…”

“…If the Cartesian coordinates of a point w j (x,y,z) have a Gaussian distribution with the constraint that ||w j || = 1, then the probability of finding a vector w j on the unit sphere is described by the Fisher-Bingham-Kent (FBK) distribution [7]. FBK distributions are the core of directional statistics [8][9][10] and were first used in paleomagnetism to study the spatial distribution of magnetic properties in rocks [11].…”

“…For every pair of elevation and azimuth angles (# m , u n ) on the unit sphere, a vector u(# m , u n ) was built using (7). Then, for each acquired dataset of rotations {R j }, a set of N rotated vectors {w j } was created using (1) from which the corresponding angular uncertainty r(# m , u n ) was calculated as in [7].…”

Is the Fisher Distribution Additive?

“…Due to the folding, the normals to the strata, initially vertical, are randomly distributed and their distribution is combined with the assumed Fisher distribution of the direction of primary magnetization before the folding. Unfortunately, experts say, the bedding-normals have never the Fisher distribution, but rather a more general Kent distribution (Kent, 1982). That is why the model of combination of two Fisher-distributed rotations can be hardly applied to paleomagnetic directions.…”