Introducing covariances of observations in the minimum L1-norm, is it needed?

Abstract

Abstract The most common approaches for assigning weights to observations in minimum L1-norm (ML1) is to introduce weights of p or p \sqrt{p} , p being the weights vector of observations given by the inverse of variances. Hence, they do not take covariances into consideration, being appropriated only to independent observations. To work around this limitation, methods for decorrelation/unit-weight reduction of observations originally developed in the context of least squares (LS) have been applied for ML1, although this adaptation still requires further investigations. In this article, we presented a deeper investigation into the mentioned adaptation and proposed the new ML1 expressions that introduce weights for both independent and correlated observations; and compared their results with the usual approaches that ignore covariances. Experiments were performed in a leveling network geometry by means of Monte Carlo simulations considering three different scenarios: independent observations, observations with “weak” correlations, and observations with “strong” correlations. The main conclusions are: (1) in ML1 adjustment of independent observations, adaptation of LS techniques introduces weights proportional to p \sqrt{p} (but not p); (2) proposed formulations allowed covariances to influence parameters estimation, which is unfeasible with usual ML1 formulations; (3) introducing weighs of p provided the closest ML1 parameters estimation compared to that of LS in networks free of outliers; (4) weighs of p \sqrt{p} provided the highest successful rate in outlier identification with ML1. Conclusions (3) and (4) imply that introducing covariances in ML1 may adversely affect its performance in these two practical applications.