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This paper enhances and develops bridges between statistics, mechanics, andgeometry. For a given system of points in $\mathbb R^k$ representing a sampleof a full rank, we construct an explicit pencil of confocal quadrics with thefollowing properties: (i) All the hyperplanes for which the hyperplanar momentsof inertia for the given system of points are equal, are tangent to one of thequadrics from the pencil of quadrics. As an application, we developregularization procedures for the orthogonal least square method, analogues oflasso and ridge methods from linear regression. (ii) For any given point $P$among all the hyperplanes that contain it, the best fit is the tangenthyperplane to the quadric from the confocal pencil corresponding to the maximalJacobi coordinate of the point $P$; the worst fit among the hyperplanescontaining $P$ is the tangent hyperplane to the ellipsoid from the confocalpencil that contains $P$. The confocal pencil of quadrics provides a universaltool to solve the restricted principal component analysis restricted at anygiven point. Both results (i) and (ii) can be seen as generalizations of theclassical result of Pearson on orthogonal regression. They have natural andimportant applications in the statistics of the measurement error models (EIV).For the classical linear regressions we provide a geometric characterization ofhyperplanes of least squares in a given direction among all hyperplanes whichcontain a given point. The obtained results have applications in restrictedregressions, both ordinary and orthogonal ones. For the latter, a new formulafor test statistic is derived. The developed methods and results areillustrated in natural statistics examples.

Orthogonal and Linear Regressions and Pencils of Confocal Quadrics