Isotone optimization in R: . . . Set methods

ing case antitonic regression. The corresponding umbrella term for both cases is monotonic regression (for a compact description, see de Leeuw 2005). Suppose that Z is the finite set {z1, z2,..., zn} of the ordered predictors with no ties, i.e., z1 < z2 < · · · < zn. Let y be again the observed response vector and x = (x1,..., xi,... xn) the unknown response values to be fitted. The least squares problem in monotonic regression can be stated as f(x) = n∑ i=1 wi (yi − xi)2 → min! (2) which has to be minimized over x under the inequality restrictions x1 ≤ x2 ≤ · · · ≤ xn for isotonic regression and x1 ≥ x2 ≥ · · · ≥ xn for the antitonic case. These restrictions avoid that we always get perfect fit. The basic theorem the isotonic solution of (2) is based on, is that if yi ≥ yi+1, then the fitted values x̂i+1: = x̂i. Correspondingly, the antitonic solution requires yi ≤ yi+1