Bounded isotonic regression (Q1684147)

Let \(\{(x_1,y_1),\ldots,(x_n,y_n)\}\) be a set of observations each consisting of a vector of covariates \(x\in\mathcal{X}\) and a response \(y\in\mathbb{R}\). It is supposed that the covariate space is equipped with a partial order \(\preceq\). The class of isotonic models \(\mathcal{G}\) is defined as the collection of models \(g:\mathcal{X}\rightarrow \mathbb{R}\) having the following property \[ x,z\in\mathcal{X}, x\preceq z \;\Rightarrow\;g(x)\leq g(z). \] The authors of the paper resent the bounded isotonic regression (BIR) model which can be presented by the formula \[ \hat{g}=\text{argmin}_{g\in\mathcal{G}, \operatorname{range}(g)\leq s}\;\sum\limits_{i=1}^n\big(y_i-g(x_i)\big)^2, \] where \[ \operatorname{range}(g)=\sup\limits_{x\in\mathcal{X}} g(x)-\inf\limits_{x\in\mathcal{X}}g(x). \] The properties of the BIR model are presented and the algorithm to find this model is described. It is shown that the BIR algorithm can be easily generalized to other loss functions. The favorable empirical performance of the presented approach compared to various relevant alternatives is demonstrated.