Last week I asked the following question regarding the gaussian

semivariogram model:

I have experienced that the gaussian semivariogram model sometimes lead

to a covariance matrix which is not positive definite. I am aware that

the parabolic behavior of the function near the origin could give these

kinds of problems, but I dont think this is the whole story. Do you about

this phenomenon, and where to read more about it ??

and got some nice and helpful answers. Thanks to Pierre Goovaerts, Donald

Myers, Sean McKenna and Benjamin Warr for providing these answers, which

are given below:

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Pierre Goovaerts wrote:

Problems with the Gaussian semivariogram typically

arise when no nugget effect is specified and

some observations are very close to each other,

leading to covariances matrice with very similar rows.

You can read more about this "pathological" model in Hans

Wackernagel's book "multivariate geostatistics"

or the recent book by Chiles and Delfiner.

*******************************************************************

Donald Myers wrote:

Theoretically this can not happen (because the gaussian variogram is a

valid model) BUT:

The problem is that the graph of the gaussian model is almost horizontal

for some distance near the origin and if there is no nugget term then

the computed values (for multiple pairs of locations) is either zero or

almost zero. When you have a lot of zeros or entries that are almost

zero in the covariance matrix, i.e., in the coefficient matrix for the

kriging equations, that matrix will not be invertible. The solution is

to incorporate a small nugget term into the variogram. What I said above

is at least related to your observation that the gaussian variogram is

nearly parabolic in shape near the origin.

Note that even though the gaussian covariance is positive definite and

will result in positive definite matrices, if you are using the

variogram form in ordinary or universal kriging then the coefficient

matrix is NOT postive definite although the coefficient matrix is

invertible. See a paper by D. Posa and A. Journel in Math. Geology ,

early 1990's. This distinction is not related to the point I made above.

*******************************************************************

Sean McKenna wrote:

Soren, try Ababou et al., 1994, On the Condition Number of Covariance

Matrices in Kriging, Estimation and Simulation of Random Fields,

Mathematical Geology, 26 (1), pp. 99-133.

******************************************************************

Benjamin Warr wrote:

the addition of a miniscule nugget variance to a variogram model that

inclludes a Gaussian model can rectify this problem, by introducing a

discontinuity at the origin,

Best regards / Venlig hilsen

SÃ¸ren Lophaven

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Informatics and Mathematical Modelling | Building 321, Room 011

Technical University of Denmark | 2800 kgs. Lyngby, Denmark

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