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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
    • 0 Attachment

      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

       
    • Adrián Martínez Vargas
      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
      Message 2 of 5 , Jan 4, 2006
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        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 Educaci�n Superior
        "Universidad 2006". La Habana, Cuba, del 13 al 17 de Febrero del 2006
        http://www.universidad2006.cu
        _____________________________________
        Instituto Superior Minero Metal�rgico de Moa
        Dr. Antonio N��ez Jim�nez
        http://www.ismm.edu.cu
      • 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 3 of 5 , Jan 4, 2006
        • 0 Attachment
          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 4 of 5 , Jan 5, 2006
          • 0 Attachment
            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 5 of 5 , Jan 5, 2006
            • 0 Attachment
              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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