Use of lp norms in fitting curves and surfaces to data

Abstract

Given a family of curves or surfaces in R s , an important problem is that of finding a member of the family which gives a "best" fit to m given data points. A criterion which is relevant to many application areas is orthogonal distance regression, where the sum of squares of the orthogonal distances from the data points to the surface is minimized. For example, this is important in metrology, where measured data from a manufactured part may have to be modelled. The least squares norm is not always suitable (for example, there may be wild points in the data, accept/reject decisions may be required, etc). So we use this to justify looking at the use of other l p norms. There are different ways to formulate the problem, and we examine methods which generalize in a natural way those available for least squares. The emphasis is on the efficient numerical treatment of the resulting problems.