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AI-GEOSTATS: Gaussian semivariogram model

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  • Soeren Nymand Lophaven
    Dear list I have experienced that the gaussian semivariogram model sometimes leads to a covariance matrix which is not positive definite. I am aware that the
    Message 1 of 3 , May 1, 2002
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      Dear list

      I have experienced that the gaussian semivariogram model sometimes leads
      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 ??

      Best regards / Venlig hilsen

      Søren 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: snl@... | http://www.imm.dtu.dk/~snl
      Telephone: +45 45253419 |
      ******************************************************************************


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    • Pierre Goovaerts
      Hello, Problems with the Gaussian semivariogram typically arise when no nugget effect is specified and some observations are very close to each other, leading
      Message 2 of 3 , May 1, 2002
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        Hello,

        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.

        Pierre
        <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

        ________ ________
        | \ / | Pierre Goovaerts
        |_ \ / _| Assistant professor
        __|________\/________|__ Dept of Civil & Environmental Engineering
        | | The University of Michigan
        | M I C H I G A N | EWRE Building, Room 117
        |________________________| Ann Arbor, Michigan, 48109-2125, U.S.A
        _| |_\ /_| |_
        | |\ /| | E-mail: goovaert@...
        |________| \/ |________| Phone: (734) 936-0141
        Fax: (734) 763-2275
        http://www-personal.engin.umich.edu/~goovaert/

        <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>


        On Wed, 1 May 2002, Soeren Nymand Lophaven wrote:

        > Dear list
        >
        > I have experienced that the gaussian semivariogram model sometimes leads
        > 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 ??
        >
        > Best regards / Venlig hilsen
        >
        > Søren 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: snl@... | http://www.imm.dtu.dk/~snl
        > Telephone: +45 45253419 |
        > ******************************************************************************
        >
        >
        > --
        > * To post a message to the list, send it to ai-geostats@...
        > * 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 majordomo@... 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
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        >


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      • Soeren Nymand Lophaven
        Dear list Last week I asked the following question regarding the gaussian semivariogram model: I have experienced that the gaussian semivariogram model
        Message 3 of 3 , May 7, 2002
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          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

          Søren 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: snl@... | http://www.imm.dtu.dk/~snl
          Telephone: +45 45253419 |
          ******************************************************************************


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