Least absolute deviations

Summary

Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute residuals (LAR), or least absolute values (LAV), is a statistical optimality criterion and a statistical optimization technique based on minimizing the sum of absolute deviations (also sum of absolute residuals or sum of absolute errors) or the L1 norm of such values. It is analogous to the least squares technique, except that it is based on absolute values instead of squared values. It attempts to find a function which closely approximates a set of data by minimizing residuals between points generated by the function and corresponding data points. The LAD estimate also arises as the maximum likelihood estimate if the errors have a Laplace distribution. It was introduced in 1757 by Roger Joseph Boscovich. Formulation Suppose that the data set consists of the points (xi, yi) with i = 1, 2, ..., n. We want to find a function f such that f(x_i)\approx y_i. To attain this goal

Official source

https://en.wikipedia.org/wiki/Least_absolute_deviations

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In this paper we enunciate and rigorously demonstrate a new lemma that, based on a previously proposed theorem, proves the identifiability of leverage points in state estimation with specific reference to the Least Absolute Value (LAV) estimator. In this context, we also propose an algorithm for the leverage point identification in LAV estimators whose performance is validated by means of extensive numerical simulations and compared against the well-known approach of Projection Statistics (PS). The obtained results confirm that the proposed method outperforms PS and represents a significant enhancement for LAV-based state estimators as it correctly identifies all the leverage points in the measurement set.

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