- Dec 11, 1905Dear all,

here is a summary of the answers I got to my question about the

impact of local search on kriging variance.

I got many replies showing me that my question was still

unclear or could be understood in different ways.

My point really was that the kriging variance, which is

dependent on the covariance model and the data configuration,

can not properly reflect the variance of the estimates if one uses

a local search neighborhood since the experimental semivariogram

is usually modeled for the whole data set.

When I was asking about the impact of a search neighborhood

on the kriging variance, I was more thinking in terms of the

interpretation of the kriging variance.

I can only recommend to read pages 178-180 of Pierre Goovaerts'

book "Geostatistics for Natural Resources Evaluation" for what

concerns the practice of using only the data closest to the

location where an estimation is made.

Other references suggested by Victor De Oliveira and

Mustafa Touati are given hereafter. The first two references

propose a statistically sound likelihood-based method to choose

the "neighborhoods" for doing estimation and prediction.

Vecchi (1992), A new method of prediction for spatial regression models

with correlated errors, J. of the Royal Statistical Society B, 813-830

Vecchia (1988), Estimation and model identification for continuous

spatial processes, J. of the Royal Statistical Society B, 297-312.

Poids de Krigeage ( A Ph.D. Thesis from the Paris School of Mines).

(this was done early in the 80th) by Jaques Rivoirard from the

Centre de Geostatistique

You will find here under other replies that could be interesting for

the readers:

From Edzer J. Pebesma:

Obviously, when using kriging in a small neighbourhood,

you only use (and need) the variogram function for the smaller

distances. The good thing is that for smaller distances the

variogram is easier to estimate than for longer distances,

given a single replicate (realization) of a RF. OTOH, even

for kriging with a global neighbourhood the variogram at the

smaller distances is more important than at larger distances.

For ordinary or universal kriging, all data in the neigbourhood

are used for prediction, also values beyond the correlation

distance (range): one of the things they are used for is

(re-)estimating the trend value or parameters at the kriging location.

A larger neighbourhood will therefor usually lead to smaller

kriging variances, but the effect may be small after

taking, say, the 20 or 30 nearest observations. For a pure nugget

effect things are easy to write out analytically, because

data configurations don't matter here. For other cases, I would

just try a couple of neighbourhood sizes, perhaps do some cross

validations.

Robert C Reynolds & Yetta Jager remind that the kriging variance

is affected by the number, distance, position and orientation of data

used. A limited number of nearest neighbors is frequently used so as

to keep the kriging system of equations small. Kriging weights

become small for sample points as one move away from the location

estimated and have a negligible influence, especially of there is a

strong spatial correlation. Points beyond the sill contribute in no way

to the kriged estimate and so can be ignored. Ideally, points should

reside fairly evenly around (interpolation) the point to be estimated

rather than tending off to one side (extrapolation) for best results.

Mats Soderstrom reminds that it might be more efficient in some cases

to use local variograms instead of variograms made on the whole data set

and that the VESPER software (See the Software list of AI-GEOSTATS or

directly http://www.usyd.edu.au/su/agric/acpa/vesper/vesper.html) proposes

a local approach.

Last but not least, from Dan Cornford:

"The answer depends on the range of the process and the

distance beyond which the sites are and the sites locations - i.e.

it is not particularly easy to answer. In general a small number

of surrounding sites is used in kriging for several reasons:

Computational (matrix inversion scales as n^3, where n is the

number of sites. Stability (conditioning) of matrix inversion is

also likely to be improved. Stationarity becomes less of a problem

(only local needed) - but of course this is conditional on knowing

the variogram which was estimated locally thus this is a bit of a fix!

Nowadays it is possible to invert matrices of up to about 1000

observations in one go - that is there is no need to do the local

thing - but this is not always a good idea for the reasons given

above.

Adding more observations should (I think - it is a while since I have

looked at this) alweays decrease the kriging variance, but in practice

the decrease will be tiny".

Many other replies suggested me to test it by increasing the numbers

of points in the neighbourhood and to check the kriging variance.

Thanks a lot also to Nicla Giglioli and Andrew (Nefia ?)

for their kind replies.

Have a nice week !

Gregoire

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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