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Re: GEOSTATS: Summary of Linear Model of Coregionalization

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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 1 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
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      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@...
      > *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^
      >
      >
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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 2 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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