Ordinary kriging variance of the prediction with error structure
Hello Ruben, Thanks for your answer, I tried to check your reference, but still I'm having the same issue, Cressie 1993, he says in 3.2.27: var(S|Y) = Sum(w_i)*V_io + m - c_me Being: [w_i, vector of weights] [V_io, Semivariogram value from the sampled point to the predictied] [m, lagrange multiplier] [c_me, nugget (due to measurement error)] This is equivalent to the notation I posted previously as C(h>0)= partial variance + nugget - V(h>0) [C, Covariance function] [V, Semivariogram function] Substituting V_io and knowing that Sum(w_i)= 1, this gives: Var(S| Y) = partial variance - Sum(w_i) * C_oi - m [C_oi, vector Covariance between predicted and sampled coordinates] This is the expression that I'm working with, in which the nugget should not appear (unless x_o=x_i) isn't it? I'm working with the covariance matrix as the nugget will only affect C(h=0) and not C(h>0) which is a property I need. And then the layout of the kriging system should be the same yet adding a nugget term in the diagonal of the covariance matrix. Am I misunderstanding something? Thanks and kind regards Antonio
On 5 February 2016 at 10:56, rubenfcasal <rubenfcasal at gmail.com> wrote:
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 R?denas escribi?:
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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