Regression and Classification by Zonal Kriging

Abstract: Consider a family $Z=\{\boldsymbol{x_{i}},y_{i}$,$1\leq i\leq N\}$ of $N$ pairs of vectors $\boldsymbol{x_{i}} \in \mathbb{R}^d$ and scalars $y_{i}$ that we aim to predict for a new sample vector $\mathbf{x}_0$. Kriging models $y$ as a sum of a deterministic function $m$, a drift which depends on the point $\boldsymbol{x}$, and a random function $z$ with zero mean. The zonality hypothesis interprets $y$ as a weighted sum of $d$ random functions of a single independent variables, each of which is a kriging, with a quadratic form for the variograms drift. We can therefore construct an unbiased estimator $$y^{*}(\boldsymbol{x_{0}})=\sum_{i}\lambda^{i}z(\boldsymbol{x_{i}})$$ de $y(\boldsymbol{x_{0}})$ with minimal variance $$E[y^{*}(\boldsymbol{x_{0}})-y(\boldsymbol{x_{0}})]^{2}$$, with the help of the known training set points. We give the explicitly closed form for $\lambda^{i}$ without having calculated the inverse of the matrices.