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The uncentered empirical second-moment matrix averages the outer products of the observations. Its expected value is $\mathrm{Cov}(X)+\mathbb{E}[X]\mathbb{E}[X{]}^{\top }$. It is an unbiased estimator of the covariance matrix when the population expected value is known to be zero. It differs from a sample covariance matrix centered at the sample mean.

For unknown drift $X\beta$, a full-column-rank design matrix $X$, a known positive-definite matrix observation covariance matrix $\Sigma$, and target drift row $x_0^T$, universal kriging minimizes prediction variance subject to $X^{T}w=x_0$. Equivalently, with $\hat{\beta }=(X^T{\Sigma }^{-1}X{)}^{-1}{X}^T{\Sigma }^{-1}z$, predict $x_0^T\hat{\beta }+c^T{\Sigma }^{-1}(z-X\hat{\beta })$. The first term is the estimated trend alone; the second is a correlated residual prediction.