[R-sig-Geo] Ordinary kriging variance of the prediction with error structure

rubenfcasal rubenfcasal at gmail.com
Fri Feb 5 10:56:40 CET 2016

Kriging the noiseless version of Y is not “the solution of the 
(standard) kriging system with a nugget effect in the Covariance 
structure” (the semivariances/variogram at lag 0 may not be zero).
See e.g. Cressie, 1993, p. 128 (for instance, eq. 3.2.27 shows the 
correct expression of the kriging variance).

Best regards, Ruben.

El 04/02/2016 a las 15:56, Antonio Manuel Moreno Ródenas escribió:
> Dear r-sig-geo community,
> I would like to bring a conceptual question on the implementation of
> ordinary kriging in gstat.
> I'm trying to account for my measurement error in a OK scheme. I assume
> that my sampled vector Y(x_i) is a noisy realisation of S(x_i) (the real
> variable), thus:
> Y(x_i) = S(x_i) + e_i, where e_i is ~N(0,tau^2).
> If that error is assumed to follow a certain set of conditions (unbiased,
> uncorrelated between itself/the variable and tau=constant), this is
> analogous to the solution of the kriging system with a nugget effect in the
> Covariance structure.
> I coded the kriging system and its solution. In order to assess if my
> implementation is correct I contrasted it with the krige function in gstat.
> The predicted value at each point is the same, meaning that I got correctly
> the weights of the system.
> However, I'm really confused when dealing with the variance in the
> prediction. Which should have this form:
> Var(S(x_o) | Y) = Var(S(x_o)) - w' * C_oi - mu
> w' weights vector
> C_oi vector Covariance between predicted and sampled coordinates
> mu lagrange multiplier
> If my objective would be to predict the signal of the variable S(x_o), the
> term of Var(S(x_o)) will correspond to the partial variance, (*the sill
> without the nugget*). This is what I understood by following the notation
> of Model-based Geostatistics from Peter J.Diggle, where it is explicitly
> mentioned in (pag 137 (6.8)).
> However, I only get agreement in my comparison with the krige (gstat)
> variance results if I use the total sill as Var(S(x_o)) that is (partial
> variance + nugget). So my question is:
> I'am right by thinking that still Var(S(x_o)) should not include the nugget?
> What is the outcome of krige in gstat when you consider a nugget? is it the
> prediction of the signal? or is it the prediction of what you would measure
> at that location?
> Kind regards,
> Antonio
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