Parametrized Approximation Estimators for Mixed Noise Distributions

Consider approximating a set of discretely defined values f1, f2,…, fm say at x = x1, x2,…, xm, with a chosen approximating form. Given prior knowledge that noise is present and that some might be outliers, a standard least squares approach based on an l2 norm of the approximation error є may well provide poor estimates. We instead consider a least squares approach based on a modified measure taking the form , where c is a constant to be fixed. Given a prior estimate of the likely standard deviation of the noise in the data, it is possible to determine a value of c such that the estimator behaves like a robust estimator when outliers are present but like a least squares estimator otherwise. We describe algorithms for computing the parameter estimates based on an iteratively weighted linear least squares scheme, the Gauss-Newton algorithm for nonlinear least squares problems and the Newton algorithm for function minimization. We illustrate their behaviour on approximation with polynomial and radial basis functions and in an application in co-ordinate metrology.