Kriging

Remarks

The constraint $\sum_i \lambda_i = 1$ makes the predictor unbiased, $E[\hat Z(s_0)] = \mu = E[Z(s_0)]$, whatever the value of $\mu$. The minimal mean squared error, the kriging variance, is $\sum_{i=1}^{n} \lambda_i \gamma_{i0} + m$. Simple kriging assumes $\mu$ known and drops the constraint; universal kriging replaces $\mu$ by a linear combination of known functions of $s$.

Example: with one observation, $n = 1$, the constraint forces $\lambda_1 = 1$, so $\hat Z(s_0) = Z(s_1)$. The first equation then gives $m = \gamma_{10} - \gamma_{11} = \gamma(s_1 - s_0)$, since $\gamma(0) = 0$, so the kriging variance is $2\gamma(s_1 - s_0)$, which is indeed $E[(Z(s_0) - Z(s_1))^2]$.

Non-example: inverse distance weighting also predicts by a weighted average with weights summing to $1$. Its weights are fixed by the distances alone and do not minimise the mean squared error for the given variogram, so it is not kriging.