L1-norm pre-analysis measures for geodetic networks

Abstract.

 Several pre-analysis measures which help to expose the behavior of L ₁ -norm minimization solutions are described. The pre-analysis measures are primarily based on familiar elements of the linear programming solution to L ₁-norm minimization, such as slack variables and the reduced-cost vector. By examining certain elements of the linear programming solution in a probabilistic light, it is possible to derive the cumulative distribution function (CDF) associated with univariate L ₁-norm residuals. Unlike traditional least squares (LS) residual CDFs, it is found that L ₁-norm residual CDFs fail to follow the normal distribution in general, and instead are characterized by both discrete and continuous (i.e. piecewise) segments. It is also found that an L ₁ equivalent to LS redundancy numbers exists and that these L ₁ equivalents are a byproduct of the univariate L ₁ univariate residual CDF. Probing deeper into the linear programming solution, it is found that certain combinations of observations which are capable of tolerating large-magnitude gross errors can be predicted by comprehensively tabulating the signs of slack variables associated with the L ₁ residuals. The developed techniques are illustrated on a two-dimensional trilateration network.