Universal kriging

Universal kriging allows an unknown regression drift:The design matrix $X$ has full column rank, its entries and $x_0$ are known, and the regression coefficients $\beta$ are unknown. For example, rows $(1,s_i)$ and target row $(1,s_0)$ allow a linear spatial trend. Requiring $w^{T}Z$ to have zero bias for every $\beta$ gives $X^{T}w=x_0$. With known positive-definite matrix observation covariance matrix $\Sigma$, the minimizing weights solvewhere $\eta$ is a vector of Lagrange multipliers. Equivalently, the generalized least squares estimate and prediction areThe first term is the estimated trend alone; the second predicts the correlated regression residuals. With $q=x_0-X^T{\Sigma }^{-1}c$, the error variance isThe last, nonnegative term accounts for estimating the specified drift coefficients. Merely inserting a fitted drift into simple kriging and keeping its known-mean error variance would omit this term. For a Gaussian random field, the same expressions arise by integrating over drift coefficients in a diffuse prior distribution limit; this is an additional inferential interpretation, not an assumption required by the linear optimization.^[ref]