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GEOSTATS: Summary of Linear Model of Coregionalization

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  • gerhard
    Thanks for the overwhelming response to my question concerning the Linear Model of Coregionalization!!! Here is a short summary of the responses I got from
    Message 1 of 5 , Jun 3, 1999
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      Thanks for the overwhelming response to my question
      concerning the Linear Model of Coregionalization!!!

      Here is a short summary of the responses I got from several
      people.
      Special thanks go to Pierre Goovaerts, Phaedon C. Kyriakidis,
      Carme Hervada i Sala, Gilles Bourgault, John Kern, Kirsten
      de Beurs, John Kern, Syed Abdul Rahman, and Denis Allard.


      Linear Model of Coregionalization:
      "The key point in modeling coregionalization
      is to ensure that the matrix of auto and cross covariance
      models is positive semi-definite for any possible lag
      distance and direction" (P. Goovaerts).

      >Question 1:
      >if I use e.g. a spherical model for the cross-variogram of
      >variables A-B, I have to use a spherical model for both, the
      >variogram of variable A and the variogram of variable B?

      Yes!

      >Question 2:
      >The ranges of both variogram models and the range of the cross-
      >variogram model have to be identical?

      Yes!

      >Question 3:
      >What are the consequences of violating the Linear Model of
      >Coregionalization?

      Cokriging matrices might not be invertible.
      Cokriging variances might be negative.


      Alternatives to the Linear Model of Coregionalization
      exist:

      1) Fast Fourier Transformation (FFT)
      was mentioned by Phaedon C. Kyriakidis, Carme Hervada i Sala,
      John Kern, and Pierre Goovaerts. The references are:

      - Yao, T. and A.G. Journel. 1998. Automatic modeling of (cross)
      covariance tables using Fast Fourier Transform.
      Mathematical Geology 30 (6): 589-616.
      - Borgman and Kern, 1996. pg.155. Proceedings of Spatial Accuracy
      Assessment in Natural Resources and Environmental Sciences,
      Second International Symposium, May 21-23 1996. Mowrer,
      Czaplewski and Hamre, USDA, Forest Service, Rocky Mountain
      Forest and Range Experiment Station, Fort Collins, CO.
      General Technical Report: RM-GTR-277.
      - John Kern. PhD dissertation. University of Wyoming.

      2) combinatorial optimization schemes
      was mentioned by Syed Abdul Rahman

      3) hope (!?)
      was mentioned by Pierre Goovaerts:

      "Model independently the 3 variograms and hope that in the
      subsequent cokriging all matrices will be positive definite.
      Many people who modeled independently their variograms
      reported negative kriging variances and numerical instabilities.
      I remember that some authors proposed to perturb (slightly!)
      any troublesome cokriging matrix to make it positive definite,
      but I wouldn't recommend this practice.

      In summary, my advice is to stay with the well proven and yet
      reasonably flexible linear model of coregionalization."
      (P. Goovaerts)


      Literature recommended by Pierre Goovaerts:
      1. Morisette J. 1997. Examples using SAS to fit the model of linear
      coregionalization. Computer and Geosciences, 23:(3) 317-323.
      2. http://www.agro.ucl.ac.be/biom/recherche/projets/agromet/
      for public-domain programs to perform coregionalization analysis
      and cokriging.
      3. Goovaerts, P., 1994. On a controversial method for modeling a
      coregionalization. Math. Geol. 26: 197-204.
      4. Goovaerts, P., 1998. Ordinary cokriging revisited.
      Math. Geol. 30: 21-42.

      Literature recommended by Denis Allard:
      1. Wackernagel, H. (1999) Multivariate Geostatistics (2nd ed.),
      Springer Verlag.
      2. Goulard M. and Voltz M. (1992) Linear coregionalization model:
      tools for estimation and choice of multivariate variograms,
      Math. Geol. 24: 269-286.


      With best regards!

      Gerhard

      --




      ******************************************************************
      | Gerhard Hunner |
      |----------------------------------------------------------------|
      | Ph.D. candidate |
      | Geostatistics, GIS, Remote Sensing |
      |----------------------------------------------------------------|
      | GIS and Remote Sensing Program |
      | Department of Forest Sciences |
      | Colorado State University |
      | Fort Collins, CO 80523 |
      | USA |
      | Tel.: (970) 221-1826 |
      | Fax: (970) 491-6754 |
      | Email: gerhard@... |
      ******************************************************************


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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 2 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 3 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 4 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.
            > *To unsubscribe, send email to majordomo@... with no subject and
            > "unsubscribe ai-geostats" in the message body.
            > DO NOT SEND Subscribe/Unsubscribe requests to the list!
            >

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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 5 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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