Regression and Classification by Zonal Kriging

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.