L1-norm pre-analysis measures for geodetic networks |
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| Authors: | J Marshall |
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| Institution: | (1) National Geodetic Survey, NOS/NGS6, 1315 East–West Highway, Room 8138, Silver Spring, MD 20910, USA e-mail: john.marshall@noaa.gov; Tel.: +1-301-713-2850; Fax: +1-301-713-4475, US |
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| 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.
Received: 6 July 2001 / Accepted: 21 February 2002 |
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| Keywords: | : Estimation – Statistics – Probability – Linear Programming – Redundancy Numbers – Gross Error Detection |
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