Least absolute deviations

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. 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 To attain this goal, we suppose that the function f is of a particular form containing some parameters that 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.) We now seek estimated values of the unknown parameters that minimize the sum of the absolute values of the residuals: Though the idea of least absolute deviations regression is just as straightforward as that of least squares regression, the least absolute deviations line is not as simple to compute efficiently. Unlike least squares regression, least absolute deviations regression does not have an analytical solving method. Therefore, an iterative approach is required. The following is an enumeration of some least absolute deviations solving methods.