Optimal designs for regression with spherical data
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Date
2017
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Abstract
In this paper optimal designs for regression problems with spherical predictors of
arbitrary dimension are considered. Our work is motivated by applications in material
sciences, where crystallographic textures such as the missorientation distribution
or the grain boundary distribution (depending on a four dimensional spherical predictor)
are represented by series of hyperspherical harmonics, which are estimated
from experimental or simulated data.
For this type of estimation problems we explicitly determine optimal designs with
respect to Kiefers op-criteria and a class of orthogonally invariant information criteria
recently introduced in the literature. In particular, we show that the uniform
distribution on the m-dimensional sphere is optimal and construct discrete and implementable
designs with the same information matrices as the continuous optimal
designs. Finally, we illustrate the advantages of the new designs for series estimation
by hyperspherical harmonics, which are symmetric with respect to the first and
second crystallographic point group.