Least absolute deviations

Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute value (LAV), least absolute residual (LAR), sum of absolute deviations, or the L₁ norm condition, is a statistical optimality criterion and the statistical optimization technique that relies on it. Similar to the popular least squares technique, it attempts to find a function which closely approximates a set of data. In the simple case of a set of (x,y) data, the approximation function is a simple "trend line" in two-dimensional Cartesian coordinates. The method minimizes the sum of absolute errors (SAE) (the sum of the absolute values of the vertical "residuals" between points generated by the function and corresponding points in the data). The least absolute deviations estimate also arises as the maximum likelihood estimate if the errors have a Laplace distribution.

Suppose that the data set consists of the points (x_i, y_i) with i = 1, 2, ..., n. We want to find a function f such that ${\displaystyle f(x_{i})\approx y_{i}.}$

To attain this goal, we suppose that the function f is of a particular form containing some parameters which need to be determined. For instance, the simplest form would be linear: f(x) = bx + c, where b and c are parameters whose values are not known but which we would like to estimate. Less simply, suppose that f(x) is quadratic, meaning that f(x) = ax² + bx + c, where a, b and c are not yet known. (More generally, there could be not just one explanator x, but rather multiple explanators, all appearing as arguments of the function f.)

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