Mechanical Representations

Abstract

We describe a set of mechanical models that may be used to represent the various $L_1$-norm, $L_2$-norm and $L_{\infty }$-norm fitting procedures: the $L_1$-norm estimation problems may be represented by the positioning of a ring or a rigid rod under the influence of a frictionless system of strings and pulleys; the $L_2$-norm estimation problems may be represented by the positioning of a ring or a rigid rod under the influence of a frictionless system of stretched springs; and, by combining disparate aspects from these two mechanical models, we find that $L_{\infty }$-norm estimation problems may be represented by the positioning of a ring or a rigid rod under the influence of a system of strings and blocks. In the first two cases the optimal position of the ring or rigid rod is determined by a minimisation of the total potential energy of the system. In the third case we only have to determine the physical limitations imposed by the lengths of string attached to the ring or rod. Moreover, the mechanical model for the $L_1$-norm problem may be generalised to cover Oja’s bivariate median and the $L_{\infty }$-norm model may be generalised to cover Rousseeuw’s least median of squares problem.