Consider a family $Z={boldsymbol{x_{i}},y_{i}$,$1leq ileq N}$ 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.
Download