This paper enhances and develops bridges between statistics, mechanics, and geometry. For a given system of points in $\mathbb R^k$ representing a sample of full rank, we construct an explicit pencil of confocal quadrics with the following properties: (i) All the hyperplanes for which the hyperplanar moments of inertia for the given system of points are equal, are tangent to the same quadrics from the pencil of quadrics. As an application, we develop regularization procedures for the orthogonal least square method, analogues of lasso and ridge methods from linear regression. (ii) For any given point $P$ among all the hyperplanes that contain it, the best fit is the tangent hyperplane to the quadric from the confocal pencil corresponding to the maximal Jacobi coordinate of the point $P$; the worst fit among the hyperplanes containing $P$ is the tangent hyperplane to the ellipsoid from the confocal pencil that contains $P$. The confocal pencil of quadrics provides a universal tool to solve the restricted principal component analysis restricted at any given point. Both results (i) and (ii) can be seen as generalizations of the classical result of Pearson on orthogonal regression. They have natural and important applications in the statistics of the errors-in-variables models (EIV). For the classical linear regressions we provide a geometric characterization of hyperplanes of least squares in a given direction among all hyperplanes which contain a given point. The obtained results have applications in restricted regressions, both ordinary and orthogonal ones. For the latter, a new formula for test statistic is derived. The developed methods and results are illustrated in natural statistics examples.

翻译：本文强化并发展了统计学、力学与几何学之间的桥梁。针对表示满秩样本的 $\mathbb R^k$ 空间中的给定点系，我们构造了一个显式的共焦二次曲面束，其具有以下性质：(i) 所有使得给定点系的平面惯性矩相等的超平面，均与来自该二次曲面束的同一二次曲面相切。作为应用，我们为正交最小二乘法开发了正则化程序，类似于线性回归中的 lasso 和 ridge 方法。(ii) 对于任意给定点 $P$，在所有包含该点的超平面中，最佳拟合超平面是与共焦束中对应于点 $P$ 最大雅可比坐标的二次曲面相切的超平面；而包含点 $P$ 的超平面中的最差拟合，则是与包含点 $P$ 的共焦束中的椭球面相切的超平面。该共焦二次曲面束为在任意给定点处进行限制主成分分析提供了通用工具。结果 (i) 和 (ii) 均可视为皮尔逊正交回归经典结论的推广。它们在变量误差模型 (EIV) 的统计学中具有自然且重要的应用。对于经典线性回归，我们给出了在包含给定点的所有超平面中，沿给定方向的最小二乘超平面的几何特征描述。所得结果可应用于限制回归，包括普通回归与正交回归。对于后者，我们推导了一种新的检验统计量公式。所发展的方法与结论通过自然统计实例进行了阐释。