Statistical inference of the eigenspace components of a two-dimensional, symmetric rank-two random tensor

Abstract

The eigenspace components (i.e. principal components and principal directions) play a key role in the validation of a symmetric rank-two random tensor, e.g. for strain and stress. They classify deformation and stress patterns in earthquake regions, plate tectonics and glacial isostatic adjustment (postglacial rebound). It is assumed that the strain or stress tensor has been directly observed or indirectly determined by other measurements. According to the Measurement Axiom, such a symmetric rank-two tensor is considered random. For its statistical inference, the random tensor is assumed to be tensor-valued Gauss–Laplace normally distributed. The eigenspace synthesis relates the eigenspace elements to the observations by means of a nonlinear vector-valued function, thus establishing a special nonlinear multivariate Gauss–Markov model. For its linearized form, the best linear uniformly unbiased estimation (BLUUE) of the eigenspace elements and the best invariant quadratic uniformly unbiased estimate (BIQUUE) of its variance–covariance matrix have been successfully constructed. The related linear hypothesis test has documented large confidence regions for both eigenvalues and eigendirections based upon real measurement configurations. They lead to a statement of caution when dealing with data concerning extension and contraction, as well as the orientation of principal stresses.