The estimation of second-order tensors, with related tests and designs

Abstract

The tensors considered in this paper are the linear relationships between certain pairs of physical vector quantities which may be described by 3 × 3 symmetric matrices. Such a linear relationship may well vary from point to point in a non-homogeneous medium. From measurements at a given point in the medium, the least squares estimates of the components of the tensor at that point are obtained; from these the tensor's principal axes may be estimated. On the assumption that the errors of measurement are small and normally distributed, confidence intervals for the lengths of the principal axes are derived, together with confidence regions on the unit sphere for the directions of these axes. Tests for equality of pairs of principal axes, for isotropy, and for comparing the tensors at two or more points are given. In certain common experimental situations, a design for this work may be represented by a set of points on the unit sphere. If such a design is rotatable (Box & Hunter, 1957), the estimates have optimum properties, and the confidence intervals and regions take on particularly simple forms. Seven rotatable designs are given, which should cover most practical requirements. These results are illustrated with a numerical example and diagram, taken from recent work on rock magnetism. Finally, all the results are extended to more general symmetric matrices, and some further rotatable designs are given.