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[ai-geostats] Traditional OCK or Standardize OCK?

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  • Behrang Kushavand
    Dear All, Is it true that estimation variance of standardize Ordinary Co-Kriging (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging (TOCK)?
    Message 1 of 5 , Jan 4, 2006

      Dear All,

       

      Is it true that estimation variance of standardize Ordinary Co-Kriging (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging (TOCK)?

      What is the advantage of TOCK to SOCK (I think it is about negative weights) and are there any criteria to choice TOCK or SOCK?

       

      Thanks

      Behrang

       
    • Pierre Goovaerts
      Hi, The main difference between SOCK and TOCK is that, in the standardized form, only one unbiasedness constraint is imposed, i.e. the sum of all primary and
      Message 2 of 5 , Jan 4, 2006
        Hi,

        The main difference between SOCK and TOCK is that, in the standardized
        form, only one unbiasedness constraint is imposed, i.e. the sum of all
        primary and secondary data weights is one, while in the traditional
        version a separate constraint is applied for each variable, i.e.
        sum of primary data weights is one and the sum of secondary data
        weights is zero for each secondary variable. The traditional
        constraints lead to larger and more frequent negative weights
        for the secondary variables. The difference between SOCK and
        TOCK estimates is expected to increase as differences between
        the variance of primary and secondary variables increases.
        The different types of cokriging are described and compared in the
        following paper:
        Goovaerts, P. 1998. Ordinary cokriging revisited.
        Mathematical Geology, 30(1): 21-42.

        Cheers,

        Pierre

        Pierre Goovaerts
        Chief Scientist at BioMedware
        516 North State Street
        Ann Arbor, MI 48104
        Voice: (734) 913-1098 (ext. 8)
        Fax: (734) 913-2201
        http://home.comcast.net/~goovaerts/



        -----Original Message-----
        From: Adrián Martínez Vargas [mailto:amvargas@...]
        Sent: Wed 1/4/2006 12:53 PM
        To: Behrang Kushavand; ai-geostats@...
        Cc:
        Subject: Re: [ai-geostats] Traditional OCK or Standardize OCK?
        In the definition of the cross variogram you can see that it is not
        adimentional (depend of units >> Km, %, ppm, etc.), you can avoid this
        effect using standardize Ordinary Co-Kriging.

        Adrian

        -----Original Message-----
        From: "Behrang Kushavand" <Kushavand@...>
        To: <ai-geostats@...>
        Date: Wed, 4 Jan 2006 19:55:01 +0330
        Subject: [ai-geostats] Traditional OCK or Standardize OCK?

        > Dear All,
        >
        >
        >
        > Is it true that estimation variance of standardize Ordinary Co-Kriging
        > (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging
        > (TOCK)?
        >
        > What is the advantage of TOCK to SOCK (I think it is about negative
        > weights) and are there any criteria to choice TOCK or SOCK?
        >
        >
        >
        > Thanks
        >
        > Behrang
        >
        >


        ____________________________________________________________________________________________
        Participe en el V Congreso Internacional de Educación Superior
        "Universidad 2006". La Habana, Cuba, del 13 al 17 de Febrero del 2006
        http://www.universidad2006.cu
        _____________________________________
        Instituto Superior Minero Metalúrgico de Moa
        Dr. Antonio Núñez Jiménez
        http://www.ismm.edu.cu
      • Heuvelink, Gerard
        The downside of SOCK (often not mentioned) is that as a minimum requirement one must know the difference(s) between the population means (i.e., the means of
        Message 3 of 5 , Jan 5, 2006
          The downside of SOCK (often not mentioned) is that as a minimum requirement one must know the difference(s) between the population means (i.e., the means of the random functions) of the primary and secondary variables. In practice, one rarely knows these and uses the differences between the sample means instead, which is incorrect, unless one takes the associated estimation errors into account. However, when the BLUE of the differences between population means is used and the associated estimation errors are taken into account, then I suspect that SOCK boils down to something very close or identical to TOCK. Along similar lines, recall that substituting the BLUE of the population mean in the simple kriging equations yields a predictor that is identical to the ordinary kriging predictor (I think it is in Cressie's book, but in fact it is not that difficult to establish this result).

          The main (only?) purpose of using ordinary kriging instead of simple kriging is that one often does not know the population mean and cannot simply assume that it is equal to the sample mean or some other combination of the sample data. That is why ordinary kriging is used much more often than simple kriging. It puzzles me why so many geostatisticians so easily replace TOCK by SOCK and ignore the problem above. It is not the right method to avoid large and many negative weights, there are much better ways for that (see discussion of one month ago).

          Gerard

          Gerard B.M. Heuvelink
          Soil Science Centre
          Wageningen University and Research Centre
          P.O. Box 47
          6700 AA Wageningen
          The Netherlands

          tel +31 317 474628 / 482420
          email gerard.heuvelink@...
          http://www.sil.wur.nl/UK/


          -----Original Message-----
          From: Pierre Goovaerts [mailto:Goovaerts@...]
          Sent: donderdag 5 januari 2006 0:20
          To: Adrián Martínez Vargas; Behrang Kushavand; ai-geostats@...
          Subject: RE: [ai-geostats] Traditional OCK or Standardize OCK?



          Hi,

          The main difference between SOCK and TOCK is that, in the standardized
          form, only one unbiasedness constraint is imposed, i.e. the sum of all
          primary and secondary data weights is one, while in the traditional
          version a separate constraint is applied for each variable, i.e.
          sum of primary data weights is one and the sum of secondary data
          weights is zero for each secondary variable. The traditional
          constraints lead to larger and more frequent negative weights
          for the secondary variables. The difference between SOCK and
          TOCK estimates is expected to increase as differences between
          the variance of primary and secondary variables increases.
          The different types of cokriging are described and compared in the
          following paper:
          Goovaerts, P. 1998. Ordinary cokriging revisited.
          Mathematical Geology, 30(1): 21-42.

          Cheers,

          Pierre

          Pierre Goovaerts
          Chief Scientist at BioMedware
          516 North State Street
          Ann Arbor, MI 48104
          Voice: (734) 913-1098 (ext. 8)
          Fax: (734) 913-2201
          http://home.comcast.net/~goovaerts/



          -----Original Message-----
          From: Adrián Martínez Vargas [mailto:amvargas@...]
          Sent: Wed 1/4/2006 12:53 PM
          To: Behrang Kushavand; ai-geostats@...
          Cc:
          Subject: Re: [ai-geostats] Traditional OCK or Standardize OCK?
          In the definition of the cross variogram you can see that it is not
          adimentional (depend of units >> Km, %, ppm, etc.), you can avoid this
          effect using standardize Ordinary Co-Kriging.

          Adrian

          -----Original Message-----
          From: "Behrang Kushavand" <Kushavand@...>
          To: <ai-geostats@...>
          Date: Wed, 4 Jan 2006 19:55:01 +0330
          Subject: [ai-geostats] Traditional OCK or Standardize OCK?

          > Dear All,
          >
          >
          >
          > Is it true that estimation variance of standardize Ordinary Co-Kriging
          > (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging
          > (TOCK)?
          >
          > What is the advantage of TOCK to SOCK (I think it is about negative
          > weights) and are there any criteria to choice TOCK or SOCK?
          >
          >
          >
          > Thanks
          >
          > Behrang
          >
          >


          ____________________________________________________________________________________________
          Participe en el V Congreso Internacional de Educación Superior
          "Universidad 2006". La Habana, Cuba, del 13 al 17 de Febrero del 2006
          http://www.universidad2006.cu
          _____________________________________
          Instituto Superior Minero Metalúrgico de Moa
          Dr. Antonio Núñez Jiménez
          http://www.ismm.edu.cu
        • Pierre Goovaerts
          Hello, It is indeed correct that as for simple cokriging, the standardized OCK requires knowledge of the population means for both primary and secondary
          Message 4 of 5 , Jan 5, 2006
            Hello,

            It is indeed correct that as for simple cokriging, the standardized OCK
            requires knowledge of the population means for both primary and
            secondary variables, and as I mentioned in my book p. 232 "Provided the
            data are representative of the study area, these means can be estimated
            from the sample means". Of course, we could also account for the uncertainty
            attached to those samples means.. but the same can be said regarding the
            uncertainty attached to the parameters of the semivariogram model...

            The main reason ordinary kriging is used instead of simple kriging is
            its ability to accommodate changes in the mean across the study area
            (what I called global trend in my book) through the use of local
            search windows. The interesting fact for standardized OCK is that,
            even if a global mean is used in the standardization, local means
            are still re-estimated within each search window thanks to the
            unbiasedness constraint. The main assumption however is that after
            rescaling by their global means both primary and secondary variables
            have the same local mean, see Goovaerts (1997, 1998). For me, this
            might be the main weakness/limitation of the approach. As always, cross-validation is a good way to compare the prediction performances
            of the different estimators.

            Pierre

            Pierre Goovaerts
            Chief Scientist at BioMedware
            516 North State Street
            Ann Arbor, MI 48104
            Voice: (734) 913-1098 (ext. 8)
            Fax: (734) 913-2201
            http://home.comcast.net/~goovaerts/

            -----Original Message-----
            From: Heuvelink, Gerard [mailto:Gerard.Heuvelink@...]
            Sent: Thu 1/5/2006 4:31 AM
            To: Pierre Goovaerts; Adrián Martínez Vargas; Behrang Kushavand; ai-geostats@...
            Cc:
            Subject: RE: [ai-geostats] Traditional OCK or Standardize OCK?
            The downside of SOCK (often not mentioned) is that as a minimum requirement one must know the difference(s) between the population means (i.e., the means of the random functions) of the primary and secondary variables. In practice, one rarely knows these and uses the differences between the sample means instead, which is incorrect, unless one takes the associated estimation errors into account. However, when the BLUE of the differences between population means is used and the associated estimation errors are taken into account, then I suspect that SOCK boils down to something very close or identical to TOCK. Along similar lines, recall that substituting the BLUE of the population mean in the simple kriging equations yields a predictor that is identical to the ordinary kriging predictor (I think it is in Cressie's book, but in fact it is not that difficult to establish this result).

            The main (only?) purpose of using ordinary kriging instead of simple kriging is that one often does not know the population mean and cannot simply assume that it is equal to the sample mean or some other combination of the sample data. That is why ordinary kriging is used much more often than simple kriging. It puzzles me why so many geostatisticians so easily replace TOCK by SOCK and ignore the problem above. It is not the right method to avoid large and many negative weights, there are much better ways for that (see discussion of one month ago).

            Gerard

            Gerard B.M. Heuvelink
            Soil Science Centre
            Wageningen University and Research Centre
            P.O. Box 47
            6700 AA Wageningen
            The Netherlands

            tel +31 317 474628 / 482420
            email gerard.heuvelink@...
            http://www.sil.wur.nl/UK/


            -----Original Message-----
            From: Pierre Goovaerts [mailto:Goovaerts@...]
            Sent: donderdag 5 januari 2006 0:20
            To: Adrián Martínez Vargas; Behrang Kushavand; ai-geostats@...
            Subject: RE: [ai-geostats] Traditional OCK or Standardize OCK?



            Hi,

            The main difference between SOCK and TOCK is that, in the standardized
            form, only one unbiasedness constraint is imposed, i.e. the sum of all
            primary and secondary data weights is one, while in the traditional
            version a separate constraint is applied for each variable, i.e.
            sum of primary data weights is one and the sum of secondary data
            weights is zero for each secondary variable. The traditional
            constraints lead to larger and more frequent negative weights
            for the secondary variables. The difference between SOCK and
            TOCK estimates is expected to increase as differences between
            the variance of primary and secondary variables increases.
            The different types of cokriging are described and compared in the
            following paper:
            Goovaerts, P. 1998. Ordinary cokriging revisited.
            Mathematical Geology, 30(1): 21-42.

            Cheers,

            Pierre

            Pierre Goovaerts
            Chief Scientist at BioMedware
            516 North State Street
            Ann Arbor, MI 48104
            Voice: (734) 913-1098 (ext. 8)
            Fax: (734) 913-2201
            http://home.comcast.net/~goovaerts/



            -----Original Message-----
            From: Adrián Martínez Vargas [mailto:amvargas@...]
            Sent: Wed 1/4/2006 12:53 PM
            To: Behrang Kushavand; ai-geostats@...
            Cc:
            Subject: Re: [ai-geostats] Traditional OCK or Standardize OCK?
            In the definition of the cross variogram you can see that it is not
            adimentional (depend of units >> Km, %, ppm, etc.), you can avoid this
            effect using standardize Ordinary Co-Kriging.

            Adrian

            -----Original Message-----
            From: "Behrang Kushavand" <Kushavand@...>
            To: <ai-geostats@...>
            Date: Wed, 4 Jan 2006 19:55:01 +0330
            Subject: [ai-geostats] Traditional OCK or Standardize OCK?

            > Dear All,
            >
            >
            >
            > Is it true that estimation variance of standardize Ordinary Co-Kriging
            > (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging
            > (TOCK)?
            >
            > What is the advantage of TOCK to SOCK (I think it is about negative
            > weights) and are there any criteria to choice TOCK or SOCK?
            >
            >
            >
            > Thanks
            >
            > Behrang
            >
            >


            ____________________________________________________________________________________________
            Participe en el V Congreso Internacional de Educación Superior
            "Universidad 2006". La Habana, Cuba, del 13 al 17 de Febrero del 2006
            http://www.universidad2006.cu
            _____________________________________
            Instituto Superior Minero Metalúrgico de Moa
            Dr. Antonio Núñez Jiménez
            http://www.ismm.edu.cu
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