Fitting the Most Likely Curve through Noisy Data

摘要

At present the preferred method for fitting a general curve through scattered data points in the plane is orthogonal distance regression, i.e., by minimising the sum of squares of the distances from each data point to its nearest neighbor on the curve. While generally producing good fits, in theory orthogonal distance regression can be both biased and inconsistent: in practice this manifest itself in overfitting of convex curves or underfitting of corners. The paper postulates this occurs because orthogonal distance regression is based on an incomplete stochastic model of the problem. It therefore presents an extension of the standard model that takes into accounts both the noisy measurement of points on the curve and their underlying distribution along the curve. It then derives the likelihood function of a given curve being observed under this model. Although this cannot be evaluated exactly for anything other than the simplest curves, it lends itself naturally to asymptotic approximation. Orthogonal distance regression corresponds to a first order approximation to the maximum likelihood estimator in this model: the paper also derives a second order approximation, which turns out to be a simple modification of the least squares penalty that includes a contribution from the curvature at the closest point. Analytical and numerical examples are presented to demonstrate the improvement achieved using the higher order estimator.