Mario, you use some tricky reasoning here:
Mario Rossi wrote:
Edzer,
the kriging variance is not data-dependent, which means that, no
matter how perfectly the second order assumption applies to your
domain, on a point by point or block by block basis, the kriging
variance cannot be used to give you confidence intervals on individual
estimated points.
Here you make a statement which is not true. (First, the primary
question was whether the kriging variance is possibly a valid measures
of uncertainty, which is true; second you introduce CI, which indeed can
be obtained from kriging prediction and variance if the the multivariate
normal model assumption holds, as you acknowledge below. Kriging
variances and kriging prediction intervals are two different things)
The reason is that, for most variables, the estimation error is
dependent on sample values, a property called heteroscedasticity; the
only exception is if the errors are Gaussian.
Here you told us why your first statement is false.
Using the KV to provide local uncertainties (point-by-point) can only
be done if the error distribution is homoscedastic,a property that
only the Gaussian dist. has; in practice, error distributions tend to
be far from Gaussian, even if the original variable
is quasi-Gaussian at the univariate (histogram) level.
I strongly doubt whether no non-Gaussian second order stationary
functions exist.
In cases where you averaging up your points to large (very large)
blocks, then the use of the KV becomes more acceptable (realistic)
because the Central Limit Theorem will help in making those errors
"more" Gaussian-like.
I agree with this, for the case of non-Gaussian variables.
Hope this helps,
Yes and no, if I'm concerned. It feels like you want to forbid any
multi-Gaussian assumption; I believe that people should realise what
they assume, but that they should decide for themselves.
Best regards,
--
Edzer
Regards,
Mario
*/"Edzer J. Pebesma" <[EMAIL PROTECTED]>/* wrote:
Dear Kerry and Gustavo,
The kriging variance is a perfect measure for estimation
uncertainty as
long as a second order stationary model is a good representation
of the
data under study. Obviously, if variability and/or spatial
correlation
varies over the field of interest and you have sufficient data to
characterize this, or e.g. do a non-linear transform such as a
log-transform to correct for a proportional effect, than you can and
will do better when taking this into account.
In my opinion papers such as those by Journel and Rossi have
over-shouted their point, and have ignored that for many cases a
second
order stationary random field is a suitable model, if not the only
possible.
The argument that after rejecting the kriging variance, conditional
simulation is suddenly needed as the solution get some measure of
uncertainty is invalid: if you create a large enough set of
conditional
Gaussian simulations, their mean value equals the kriging mean and
their
variance equals the kriging variance. Nothing is gained, only an
expensive approximation of something rather cheap is obtained.
You will not find many papers that make this point, as the only
point is
that someone else is wrong. Not many people like to write such stuff.
Below is a reference that may be hard to get (but you can google
for the
first author). I for instance didn't enjoy writing this email.
Best regards,
--
Edzer
Heuvelink, G.B.M. and E.J. Pebesma, 2002, Is the ordinary kriging
variance a proper measure of interpolation error? In: Proceedings
of the
fifth International Symposium on Spatial Accuracy Assessment in
Natural
Resources and Environmental Sciences (eds. G. Hunter and K. Lowell).
Melbourne: RMIT University, 179-186.
Gustavo G. Pilger wrote:
> Hi,
>
> Indeed the kriging variance is only semi-variogram and spatial data
> configuration dependent. The kriging variance is calculated taking
> into account only the geometry of the samples, i.e. their spatial
> arrangement and the semi-variogram. Basically, kriging variance
do not
> take into account the value of the samples, but only their location
> (and the semi-variogram), consequently ignoring the local
variability.
> Therefore this parameter is not appropriate to measure uncertainty.
> For this purpose you should consider the use of conditional
simulation
> methods.
>
> I wrote some papers about this subject some years ago. For exemple:
>
> PILGER, Gustavo G.; COSTA, Joao Felipe Coimbra Leite; KOPPE, Jair
> Carlos, 2001. Additional Samples: Where they Should be Located?.
> Natural Resources Research, New York, v. 10, n. 3, p. 197-207.
>
> I can send you a copy if you wish.
>
> I hope this helps you.
>
> Cheers.
> <><><><><><><><><><><><><><><><><><><><><><><>
> Gustavo G. Pilger, Mining Engineer, MSc, PhD
> Senior Geostatistician
> MBR - Brazil
> <><><><><><><><><><><><><><><><><><><><><><><>
>
>
>
>> Hi. I just read through Journel and Rossi's 1999 paper, "When do we
>> need a trend model in Kriging". In the appendix they say "A kriging
>> variance is but a variogram-model dependent ranking of data
>> configurations; being data-value independent, it is generally not a
>> measure of local accuracy...This fact is unfortunately not yet
fully
>> appreciated by some practitioners". Can someone explain the
>> implications of this in terms of determining cost-efficiency
analysis
>> for sample designs? Specifically, can we use kriging variance
>> estimates across potential sampling grids, (from modeled variograms
>> estimated from say a pilot study) to estimate the variability
>> associated with different sampling densities/configurations. In
>> addition, can someone provide some references that address this
topic.
>>
>> Thanks,
>> Kerry
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