Distribution of directional data and fabric tensors. (English) Zbl 0586.73004

Kanatani, Ken-Ichi

Int. J. Eng. Sci. 22, 149-164 (1984).

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.

MSC:

74A40	Random materials and composite materials
74A99	Generalities, axiomatics, foundations of continuum mechanics of solids
62H15	Hypothesis testing in multivariate analysis
53A45	Differential geometric aspects in vector and tensor analysis

Keywords:

tensors of second kind; third kind; Laplace harmonics; distribution of directional data; Fisher information matrix; fabric tensors; least square approximation; Cartesian tensor equations; representation theory of the rotation group; potential theory in electrodynamics; spherical harmonics expansion; Fourier series expansion; method of statistical test

Citations:

Zbl 0244.62005

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