The estimation of second-order tensors, with related tests and designs
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二阶张量的估计及相关检验与实验设计
- 研究问题
- 如何由物理向量之间关系的测量估计对称二阶张量及其主轴，并对轴长、方向、各向同性和不同位置的张量进行推断？
- 方法与思路
- 作者用最小二乘估计张量分量，在测量误差较小且正态的假设下构造区间与球面置信域，并研究可旋转实验设计。
- 材料陈述的贡献
- 摘要称给出主轴和张量比较的检验、七种可旋转设计及岩石磁学示例，并将结果扩展到更一般的对称矩阵。
- 阅读关注点
- 应核对小误差近似、主轴相等情形的处理、球面方向置信域的构造，以及可旋转设计所称最优性质的具体标准。
- 阅读边界
- 仅据版本未注明的 OpenAlex 摘要；未列出近似误差与最优性条件，不能将物理张量的估计直接视为经济结构参数估计。
原文摘要
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.