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RE: [ai-geostats] Traditional OCK or Standardize OCK?

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  • 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 1 of 5 , Jan 5, 2006
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      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
      >
      >


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