Remarks about Kriging

1. Kriging is a BLUE (Best Linear Unbiased Estimator) interpolator. In other word it is a Linear estimator that matches the correct expected value of the population (Unbiased) and that minimizes the variance of the observations (Best).

2. Kriging is an exact interpolator if no errors are present. In fact, if we set $x_0=x_i$ in (2.5) we obtain immediately $\lambda_i=1, \lambda_j=0, j=1,\ldots,n,j\ne i$.

3. If we assume that the the error $\epsilon$ is Gaussian, then we can associate to the estimate $Y^\ast(x_0)$ a confidence interval. For example, the 95% confidence interval is $\pm 2 \sigma_0$ where:
   $$\sigma_0 = \sqrt{\mbox{var}[Y^*(x_0)-Y(X_0)]}$$
   Then the krigin estimator (2.4 becomes:
   $$Y^\ast (x_0) = \sum_{i=1}^n \lambda_i Y_i$$

4. The solution of the linear system does not depend on the observed value but only on $x_i$ and $x_0$.

5. A map of the estimated regionalized variable, and possibly its confidence intervals, can be obtained be defining a grid and solving the linear system for each point in the grid.