Regression and Classification by Zonal Kriging
This work addresses regression and classification tasks for data analysts, but it appears incremental as it builds on existing kriging methods with a zonality hypothesis.
The paper tackles the problem of predicting scalar outputs from vector inputs using a zonal kriging approach, resulting in an unbiased estimator with minimal variance and a closed-form solution for weights without matrix inversion.
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 $y^{*}(\boldsymbol{x_{0}})=\sum_{i}λ^{i}z(\boldsymbol{x_{i}})$ 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 $λ^{i}$ without having calculated the inverse of the matrices.