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

Antonio Manuel Moreno Ródenas argantonio65 at gmail.com
Fri Feb 5 11:48:40 CET 2016

```Hello Ruben,

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 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
> >
> >       [[alternative HTML version deleted]]
> >
> > _______________________________________________
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> > R-sig-Geo at r-project.org
> > https://stat.ethz.ch/mailman/listinfo/r-sig-geo
> >
>
>
>         [[alternative HTML version deleted]]
>
>
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