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Re: GEOSTATS: Summary of Linear Model of Coregionalization

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  • Steven I . Citron-Pousty
    Dear Gerhard: I have also started reading about coregionalization and cokriging and my take on question 2 from the literature doesn t match... ... Yes! ... In
    Message 1 of 5 , Jun 4, 1999
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      Dear Gerhard:
      I have also started reading about coregionalization and cokriging and my
      take on question 2 from the literature doesn't match...
      ---------
      >Question 2:
      >The ranges of both variogram models and the range of the cross-
      >variogram model have to be identical?

      Yes!
      ----------
      In Isaaks and Srivastava and in Goovaerts (1998) in Biol. Fert. of Soils
      they have models with different ranges. Actually in the Goovaerts he has
      a nested variogram model of which one model shares a common range with
      the other models variograms but the short scale model has a different
      range from the other semivariogram models. He gives the same type of
      example in his book on pg 211. Does this mean that the
      coregionalization is only occuring at the larger spatial scale.
      In Issaks and Srivastava I may also be confusing their use of symbols.
      Any feedback on this issue would be appreciated...
      Steve

      *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^
      Steven I. Citron-Pousty
      Dept. of Ecology and Evol. Biology
      University of Connecticut
      75 N. Eagleville Rd, U-43
      Storrs, CT 06269
      P: 860-486-4157 F: 860-486-6364
      NOTE NEW E-MAIL ADDRESS!!
      E-mail: Steven.I.Citron-Pousty@...
      *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^


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    • Colin Daly
      Steven et al. I think it is incontestable that in the context of a linear model of co-regionalisation that the answer to question 2 is YES !!! Denis
      Message 2 of 5 , Jun 4, 1999
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        Steven et al.

         I think it is incontestable that in the context of a linear model of co-regionalisation that the answer to question 2 is 'YES' !!!

         Denis' suggestion about looking at Hans Wackernagel book deserves upgrading. Hans worked quite a bit on these issues and presents a good discussion of these topics (including a generalisation of the linear model due to Michel Grzebyk)

         Two side issues are (possibly) worth mentioning

         (1) Pierre Goovaerts mentions that variables that are cross correlated will tend to show similar patterns of variability. In this he is quite correct when in comes to scalar variables. It is worth noting (for the very few who may care) that for vector random variables - the cross correlation between components of the vector will not look anything like the variogram of the components but can not be ignored. There are a class of variogram/cross variogram models for this situation which does not follow the linear model of co-regionalisation.

         (2) The FFT algorithm of Yao and Journel is a fairly heurestic one. It is very interesting and may offer good models in many cases. However it is an iterative algorithm and there is no proof offered that the iterations (fixing the amplitudes but varying the phase) will converge. My experience (in the rather nasty case of the vector fields mentioned above) is that it did not give a good solution. This may be a problem with the algorithm or simply my bad programming! Too early to tell!

         Have a good weekend
         

        Colin Daly
         
        Steven I . Citron-Pousty wrote:

        Dear Gerhard:
        I have also started reading about coregionalization and cokriging and my
        take on question 2 from the literature doesn't match...
        ---------
        >Question 2:
        >The ranges of both variogram models and the range of the cross-
        >variogram model have to be identical?

        Yes!
        ----------
        In Isaaks and Srivastava and in Goovaerts (1998) in Biol. Fert. of Soils
        they have models with different ranges. Actually in the Goovaerts he has
        a nested variogram model of which one model shares a common range with
        the other models variograms but the short scale model has a different
        range from the other semivariogram models. He gives the same type of
        example in his book on pg 211.  Does this mean that the
        coregionalization is only occuring at the larger spatial scale.
        In Issaks and Srivastava I may also be confusing their use of symbols.
        Any feedback on this issue would be appreciated...
        Steve

        *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^
        Steven I. Citron-Pousty
        Dept. of Ecology and Evol. Biology
        University of Connecticut
        75 N. Eagleville Rd, U-43
        Storrs, CT  06269
        P: 860-486-4157  F: 860-486-6364
        NOTE NEW E-MAIL ADDRESS!!
        E-mail: Steven.I.Citron-Pousty@...
        *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^

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      • Pierre Goovaerts
        Hello, As I explained in the Biology and Fertility paper, the linear model of coregionalization is built by considering a common set of basic variogram models
        Message 3 of 5 , Jun 4, 1999
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          Hello,

          As I explained in the Biology and Fertility paper, the
          linear model of coregionalization is built by considering
          a common set of basic variogram models for all direct and
          cross variograms. The condition for the model to be permissible
          is that, for each basic structure g_l(h), the matrix of coefficients
          b^l_ij (sill for bounded models such as spherical and exponential,
          or slope for the linear model) must be semi-definite positive.
          It means that some of the coefficients can actually be equal
          to zero! In case of two variables, if one structure is missing from
          a direct variogram (say b^l_ii=0) it should also be missing for
          the cross variogram to keep the matrix semi-definite positive,
          e.g. b^l_ij=0 to ensure that |b^l_ij| <= sqrt(b^l_ii * b^l_jj)
          (Note that the absolute value is not mentioned in my book
          but it's fixed in forthcoming printing).

          As I mentioned before, if you consider several structures besides
          the nugget effect (i.e. nested model), you have some flexibility to model
          a set of variograms with different ranges. As Steven rightly interpreted,
          a missing basic structure in a cross variogram means that the two
          variables are not correlated at that scale, although we have to be
          cautious because of the underlying assumptions of the Linear
          model of coregionalization, such as independence of different
          spatial processes (phenomena are rarely additive).

          Since Gerhard considered the case of a single structure
          (besides the nugget effect) and that I assumed that the
          cross variogram was not a pure nugget effect, the answer
          was yes: direct and cross variograms must include the
          same structures with the same range. But, it is not
          a general answer.

          Hope now it is clear

          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 Fri, 4 Jun 1999, Steven I . Citron-Pousty wrote:

          > Dear Gerhard:
          > I have also started reading about coregionalization and cokriging and my
          > take on question 2 from the literature doesn't match...
          > ---------
          > >Question 2:
          > >The ranges of both variogram models and the range of the cross-
          > >variogram model have to be identical?
          >
          > Yes!
          > ----------
          > In Isaaks and Srivastava and in Goovaerts (1998) in Biol. Fert. of Soils
          > they have models with different ranges. Actually in the Goovaerts he has
          > a nested variogram model of which one model shares a common range with
          > the other models variograms but the short scale model has a different
          > range from the other semivariogram models. He gives the same type of
          > example in his book on pg 211. Does this mean that the
          > coregionalization is only occuring at the larger spatial scale.
          > In Issaks and Srivastava I may also be confusing their use of symbols.
          > Any feedback on this issue would be appreciated...
          > Steve
          >
          > *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^
          > Steven I. Citron-Pousty
          > Dept. of Ecology and Evol. Biology
          > University of Connecticut
          > 75 N. Eagleville Rd, U-43
          > Storrs, CT 06269
          > P: 860-486-4157 F: 860-486-6364
          > NOTE NEW E-MAIL ADDRESS!!
          > E-mail: Steven.I.Citron-Pousty@...
          > *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^
          >
          >
          > --
          > *To post a message to the list, send it to ai-geostats@....
          > *As a general service to list users, please remember to post a summary
          > of any useful responses to your questions.
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          > "unsubscribe ai-geostats" in the message body.
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          >

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        • Steven I . Citron-Pousty
          Dear Geostats folks: After your responses and talking to a more geostats conversant friend, I now have a better understanding of whats going on. Thanks for
          Message 4 of 5 , Jun 4, 1999
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            Dear Geostats folks:
            After your responses and talking to a more geostats conversant friend, I
            now have a better understanding of whats going on. Thanks for being
            tolerant of a neophyte to the world of geostats...
            Steve

            *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^
            Steven I. Citron-Pousty
            Dept. of Ecology and Evol. Biology
            University of Connecticut
            75 N. Eagleville Rd, U-43
            Storrs, CT 06269
            P: 860-486-4157 F: 860-486-6364
            NOTE NEW E-MAIL ADDRESS!!
            E-mail: Steven.I.Citron-Pousty@...
            *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^


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