Evaluation of the estimation variance

The estimation variance is defined as the variance of the error:

$$\epsilon = Y^\ast_0- Y_0$$

Hence:

   var$$[Y^\ast_0- Y_0] = E[(Y^\ast_0- Y_0)^2] - E[(Y^\ast_0- Y_0)]^2$$

but:

$$\begin{multline*} E[(Y^\ast_0- Y_0)] = E[Y^\ast_0] - E[Y_0]= \sum_i \lambda_i E[Y_i] - E[Y_0] = 0 \\ (E[Y]=m=0) \end{multline*}$$

and since

$$\sum_j \lambda_j C(x_i-x_j) = C(x_i-x_0)$$

we finally obtain:

$$[Y^\ast_0- Y_0] = E[(Y^\ast_0- Y_0)^2]$$

$$= \sum_i \sum_j \lambda_i\lambda_j C(x_i-x_j) - 2 \sum_i \lambda_i C(x_i-x_0) + C(0)$$

$$= - \sum_i \lambda_i C(x_i-x_0) + C(0)$$

$$= [Y] - \sum_i \lambda_i C(x_i-x_0)$$

which, since $\sum_i \lambda_i C(x_i-x_0) > 0$, shows that the estimation variance of $Y_0$ is smaller than the dispersion variance of $Y$ (the real variance of the RF). In statistical terms, we can interpret this result by saying that since we have observed Y at some points $x_i$ then the uncertainty on $Y$ decreases.

It is important to remark the difference between estimation and dispertion variance. The latter is representative of the variation interval of the RF $Y$ within the interpolation domain, while the estimation variance represents the residual uncertainty in the estimation of the realization $Y^\ast_0$ of $Z$ when $n$ observations are available. The dispersion variance is a constant, while the estimation variance varies from point to point and is zero at the observation points.

Our original variable was $Z=Y+m$ and its estimate is thus:

$$Z^\ast_0 = m + \sum_i \lambda_i (Z_i-m)$$

$$[Z^\ast_0 - Z_0] = [Z_0] - \sum_i \lambda_i C(x_i-x_0)$$