Dear list 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: ******************************************************************** 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 S�ren Lophaven ****************************************************************************** Master of Science in Engineering | Ph.D. student Informatics and Mathematical Modelling | Building 321, Room 011 Technical University of Denmark | 2800 kgs. Lyngby, Denmark E-mail: [EMAIL PROTECTED] | http://www.imm.dtu.dk/~snl Telephone: +45 45253419 | ****************************************************************************** -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
