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Axial Data Analysis

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

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Axial Data Analysis

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  1. Axial Data Analysis

  2. Random Vector

  3. Axial Data

  4. 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 .

  5. Random axis Maps to a Projection

  6. Distance .

  7. Uniform distribtuion .

  8. Density for the Uniform .

  9. Finding the Mean .

  10. Intrinsic Mean .

  11. Distance between axes .

  12. .

  13. The minimum of expected distance squared .

  14. Finding the Sample Mean .

  15. Central Limit Theorem .

  16. 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

  17. Bingham distribution . Where the integration is with respect to the uniform distribution on

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