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Axial Data Analysis. Random Vector. Axial Data. Properties of Axial data.

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properties of axial data
Properties of Axial data

Sometime the observations are not direction but axes, that is, the unit vector and – are indistinguishable, so that it is which is observed. In this context it is appropriate to consider probability density functions for onwhich are anitpodally symmetric (diametrically opposite <an antipodal point on a sphere>)

  • i.e.
  • in such cases the observations can be regrarded as being on the projective space , which is obtained by identifying opposite points on the sphere .
random axis
Random axis

Maps to a Projection

watson distribution
Watson Distribution
  • One of the simplest models for axial data is the Dimroth-Scheidegger-Watson model, which has densities
  • Where

Note: the density is rotationally symmetric about

bingham distribution
Bingham distribution

.

Where the integration is with respect to the uniform distribution on