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Re: GEOSTATS: Linear Model of Coregionalization

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  • Pierre Goovaerts
    Hi Gerhard, You are perfectly right that there has been a lot of confusion about the modeling of direct and cross variograms in the literature. As explained in
    Message 1 of 3 , Jun 2, 1999
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      Hi Gerhard,

      You are perfectly right that there has been a lot of
      confusion about the modeling of direct and cross variograms
      in the literature.
      As explained in my book (p. 123) and illustrated in
      Goovaerts (1994), 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.
      You have essentially 2 options:
      1. use the Linear model of coregionalization that allows
      an easy check of the permissibility of your model.
      The price to pay is the requirement that all direct
      and cross variograms share the same set of basic structures.
      Thus if you use your program that doesn't allow nested structures,
      you have to fit the same model with the same range to the three
      variograms.
      Note that: a) Variables that are well cross-correlated are likely
      to show similar patterns of spatial variability.
      What's the point of modeling the coregionalization
      of variables that are weakly correlated since
      it won't bring any benefit in cokriging (goovaerts, 1998).
      b) If you don't restrict yourself to single-structure
      models, you get much more flexibility because
      there is no need for the direct and cross semivariograms to
      include all the basic structures, see Fig. 1.16 in my book.

      2. 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. You may want to check the
      following sources for programs that fit nested models of
      coregionalization:
      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.


      References
      ---------
      Goovaerts, P., 1994. On a controversial method for modeling a
      coregionalization. Math. Geol. 26, 197--204.

      Goovaerts, P., 1998. Ordinary cokriging revisited.
      Math. Geol. 30, 21--42.

      Cheers,

      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 Tue, 1 Jun 1999, gerhard wrote:

      > Dear all,
      >
      > I am looking for the "ultimate" truth about the Linear Model of
      > Coregionalization!
      >
      > I want to model the variograms of variable A and B, and the
      > cross-variogram of variables A-B. The variogram and cross-
      > variogram models cannot be built independently from one
      > another (Issaks & Srivastava, 1989; Goovaerts, 1997).
      > This is where the Linear Model of Coregionalization comes
      > into play: any basic model that is included in the cross-
      > variogram model must be included in all the variogram
      > models (ibid.).
      > I have a program for calculating variogram and cross-
      > variogram models that can handle only one basic model for
      > each (cross-) variogram, i.e. no nested structures are allowed.
      > From the above description of the Linear Model of
      > Coregionalization does it now follow that
      >
      > 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?
      > And further:
      >
      > Question 2:
      > The ranges of both variogram models and the range of the cross-
      > variogram model have to be identical?
      >
      > Question 3:
      > What are the consequences of violating the Linear Model of
      > Coregionalization? Some people seem to think that it is o.k.
      > to ignore the Linear Model of Coregionalization as long as
      > they appear to get reasonable results from their analysis.
      >
      > The reason for my confusion comes from the fact that I have
      > seen both cases in the literature:
      > people using the same variogram models and the same ranges
      > for their variograms and cross-variograms, and
      > people (non-geostatisticans?) using different models (e.g.
      > an exponential model for variogram A, a spherical model
      > for variogram B, and a Gaussian model for the cross-
      > variogram A-B) with different ranges.
      >
      > I would really appreciate any comments that could
      > shed some light on this (for me) confusing topic!
      >
      > Thanks for your help and have a nice day!
      >
      > 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@... |
      > ******************************************************************
      >
      >
      > --
      > *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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    • Syed Abdul Rahman/SINGPROD1/Landmark
      There is also the well trodden path of combinatorial optimization schemes (annealing, GAs) whereby one can flexibly model the structure of variates and
      Message 2 of 3 , Jun 2, 1999
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        There is also the well trodden path of combinatorial
        optimization schemes (annealing, GAs) whereby one
        can flexibly model the structure of variates and co-variates
        without having to worry about LMC. At the cost of CPU time,
        of course.

        Syed





        Pierre Goovaerts <goovaert@...> on 06/03/99 01:03:41 AM
        To: gerhard <gerhard@...>
        cc: ai-geostats@...
        Subject: Re: GEOSTATS: Linear Model of Coregionalization

        Hi Gerhard,

        You are perfectly right that there has been a lot of
        confusion about the modeling of direct and cross variograms
        in the literature.
        As explained in my book (p. 123) and illustrated in
        Goovaerts (1994), 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.
        You have essentially 2 options:
        1. use the Linear model of coregionalization that allows
        an easy check of the permissibility of your model.
        The price to pay is the requirement that all direct
        and cross variograms share the same set of basic structures.
        Thus if you use your program that doesn't allow nested structures,
        you have to fit the same model with the same range to the three
        variograms.
        Note that: a) Variables that are well cross-correlated are likely
        to show similar patterns of spatial variability.
        What's the point of modeling the coregionalization
        of variables that are weakly correlated since
        it won't bring any benefit in cokriging (goovaerts, 1998).
        b) If you don't restrict yourself to single-structure
        models, you get much more flexibility because
        there is no need for the direct and cross semivariograms to
        include all the basic structures, see Fig. 1.16 in my book.

        2. 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. You may want to check the
        following sources for programs that fit nested models of
        coregionalization:
        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.


        References
        ---------
        Goovaerts, P., 1994. On a controversial method for modeling a
        coregionalization. Math. Geol. 26, 197--204.

        Goovaerts, P., 1998. Ordinary cokriging revisited.
        Math. Geol. 30, 21--42.

        Cheers,

        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 Tue, 1 Jun 1999, gerhard wrote:

        > Dear all,
        >
        > I am looking for the "ultimate" truth about the Linear Model of
        > Coregionalization!
        >
        > I want to model the variograms of variable A and B, and the
        > cross-variogram of variables A-B. The variogram and cross-
        > variogram models cannot be built independently from one
        > another (Issaks & Srivastava, 1989; Goovaerts, 1997).
        > This is where the Linear Model of Coregionalization comes
        > into play: any basic model that is included in the cross-
        > variogram model must be included in all the variogram
        > models (ibid.).
        > I have a program for calculating variogram and cross-
        > variogram models that can handle only one basic model for
        > each (cross-) variogram, i.e. no nested structures are allowed.
        > From the above description of the Linear Model of
        > Coregionalization does it now follow that
        >
        > 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?
        > And further:
        >
        > Question 2:
        > The ranges of both variogram models and the range of the cross-
        > variogram model have to be identical?
        >
        > Question 3:
        > What are the consequences of violating the Linear Model of
        > Coregionalization? Some people seem to think that it is o.k.
        > to ignore the Linear Model of Coregionalization as long as
        > they appear to get reasonable results from their analysis.
        >
        > The reason for my confusion comes from the fact that I have
        > seen both cases in the literature:
        > people using the same variogram models and the same ranges
        > for their variograms and cross-variograms, and
        > people (non-geostatisticans?) using different models (e.g.
        > an exponential model for variogram A, a spherical model
        > for variogram B, and a Gaussian model for the cross-
        > variogram A-B) with different ranges.
        >
        > I would really appreciate any comments that could
        > shed some light on this (for me) confusing topic!
        >
        > Thanks for your help and have a nice day!
        >
        > 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@... |
        > ******************************************************************
        >
        >
        > --
        > *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!
        >

        --
        *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.
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