Distribution of directional data and fabric tensors

Distribution of directional data is characterized by what is termed fabric tensors. A formal least square approximation is applied, and three kinds of fabric tensors are defined in connection with the choice of a basis of the space of functions on a unit sphere or a unit circle. All the resulting equations are Cartesian tensor equations, and they are interpreted in terms of the representation theory of the rotation group and the potential theory in electrodynamics. It is also shown how this characterization is related to the spherical harmonics expansion or the Fourier series expansion. Finally, a method of statistical test is presented in the Cartesian tensor form to check the true form of the distribution. A physical example is also given to illustrate the proposed technique.