On the solution of the errors in variables problem using the l1 norm

A fundamental problem in data analysis is that of fitting a given model to observed data. It is commonly assumed that only the dependent variable values are in error, and the least squares criterion is often used to fit the model. When significant errors occur in all the variables, then an alternative approach which is frequently suggested for this errors in variables problem is to minimize the sum of squared orthogonal distances between each data point and the curve described by the model equation. It has long been recognized that the use of least squares is not always satisfactory, and the l1 criterion is often superior when estimating the true form of data which contain some very inaccurate observations. In this paper the measure of goodness of fit is taken to be the l1 norm of the errors. A Levenberg-Marquardt method is proposed, and the main objective is to take full advantage of the structure of the subproblems so that they can be solved efficiently.