- Sep 6, 1905Dear all,

here is a first summary of the replies I received to my first question (on the

use of resampling techniques & cross-validation techniques applied to spatial

dependent data) and that have not been sent to the list.

Thanks a lot to all who contributed.

1st question:

Denis Marcotte ( Denis.Marcotte@... )write

"It is still an open question. Some have suggested to work with

orthogonal residuals. The problem is that the orthogonalization calls

for the covariance function which is precisely what we want to estimate with

these techniques. Journel (1994) suggested to produce conditional realizations

and then to resample from these realizations. This calls

for a knowledge of the covariance function (to use in the conditional

simulation) and of the sampling strategy used to get the original

sample that will be applied to the various realizations

(But, of course not at the same points, as these are perfectly

reproduced by the conditional simulations). Naraghi and Marcotte (1996) used

the fact that increments computed for variogram computation are rather weakly

correlated in average for small lags and when the same

data point is not used for the construction of more than one increment

for a given lag distance. They then resampled the increments within

each lag distance class. A bad idea is certainly to resample directly

the data points, as this biased strongly the covariance function toward

a pure nugget effect."

Two other references are given by Edzer Pebesma

A. Solow, bootstrapping correlated data,

Math. Geol. early eighties.

P. Kitanidis, something with `Orthogonal residuals' in the title, also

Math. Geol., year unknown.

For what concerns the use of cross-validation applied to geostatistical

data,Donald Myers writes

There have also been a couple of papers by Bruce Davis, A. Solow and

Kathryn Campbell (all in Math Geology) with warnings about the use of

cross-validation.

The cross-validation statistics are not equally sensitive to changes in the

variogram model and or the parameters of the variogram. Some of them are

sensitive to changes in the search neighborhood. In the case of simple

kriging and cross-validation, the theoretical maximum value of the

correlation between the observed and estimated values is 1. However if

ordinary kriging is used then the theoretical maximum value is less because of

the LaGrange multiplier. Similarly if simple kriging is used the theoretical

minimum for the correlation between estimated and the

estimation error is zero, this is not true if ordinary kriging is used.

Again the LaGrange multiplier has an effect.

References:

1992, Myers, D.E., Selection of a radial basis function for data

interpolation. in

Advances in Computer Methods for Partial Differential Eq. VII, R.

Vichnevetsky, D. Knight and G.Richter (eds), IMACS, 553-558

1991, Myers,D.E., On Variogram Estimation. in Proceedings of the First

Inter. Conf. Stat. Comp., Cesme, Turkey, 30 Mar.-2 April 1987, Vol II,

American Sciences Press, 261-281

1991, Myers,D.E., Interpolation and Estimation with Spatially

Located Data, Chemometrics and Intelligent Laboratory Systems 11, 209-228

Gregoire Dubois

Section of Earth Sciences

Institute of Mineralogy and Petrography

University of Lausanne

Switzerland

Currently detached in Italy

http://curie.ei.jrc.it/ai-geostats.htm

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