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Re: [SPAM] RE: [ai-geostats] Kriging Variance

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  • Gerald van den Boogaart
    Dear List, A comment for discussion: In the worlds leading mailing list on Geostatistics, we should be more precise in wording, especially in the answers given
    Message 1 of 3 , Nov 7, 2005
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      Dear List,

      A comment for discussion:
      In the worlds leading mailing list on Geostatistics, we should be more precise
      in wording, especially in the answers given by the experts and experienced.

      I will use a current example for explanation. However this is a critic remark
      on a general tendency in the list, and not aimed to an individual:

      Somebody wrote to the list:
      > I think if your data is normally distributed, your kriging variance is
      > normally distributed.

      This sentence is a loose explanation of a simple fact:
      Universal/ordinary/simple kriging errors on Gaussian random fields are
      normally distributed.

      Hower taken in literal sense, the sentence is pure nonsense:

      -> kriging variance is not distributed at all, since it is a moment (a
      variance) and not a random variable.

      Footnote: {Kriging variance might be seen as a random variable, when we
      consider the randomness of the estimation of the variogram. However this is
      almost never considered and probably not the intended meaning. Especially is
      that distribution of these positive number not a Gaussian one and has never
      been calculated analytically.}

      -> It is the kriging error , which is the random and unkown difference
      \hat{Z(x)}-Z(x) of the predicted and the true value, which is normally
      distributed under some circumstances.

      -> It is not the data to be normally distributed, but the random field. The
      dataset can easily be not normally distributed (e.g. multimodal), even when
      the underlying random field is Gaussian.

      Footnote: {When loosely speaking about applied statistics, this loose wording
      identifing the random variable and the data is quite common, since in the end
      both follow more or less the same distribution in case of independent
      sampling. However things simply get wrong in case of geostats.)}

      I would like to encourage the list on discussion on that and ask for opinions.

      Dialectic Discussion:
      Clearly this is a one sided view of a mathematician. And one could say from a
      newbee perspective: It is to hard to understand all these complicated exact
      language. I would ask: Is it really more easy to understand objectively
      wrong sentences in the right way, than to understand true but complicated
      information. I would suspect that it only feels more simple, as long as you
      do not realize the problem or as long you know the truth anyway.

      Futhermore some of our list members will use the answers, not only those to
      their own questions, in their seminar and thesis works and need literally
      correct wording there.

      Best regards,
      Gerald v.d. Boogaart

      PS:
      I will try to contribute to the actual problem in my way:

      Am Montag, 7. November 2005 06:49 wrote Digby Millikan:
      > I think if your data is normally distributed, your kriging variance is
      > normally distributed.

      If the field is jointly normally distributed (it is a Gaussian random field),
      the kriging error (the difference between prediction and true value) is
      normally distributed.

      The kriging variance has a very complicated distribution, which is never
      considered.
      >
      > You will have higher kriging variances for samples, which have higher
      > variogram values

      This assertation does not hold in general.
      The kriging variance has a complex dependence on the local geometry of
      observations, the shape of the variogram and its sill.
      Clearly, when all else is equal larger the scaled variograms lead to a scaled
      kriging variance.

      Very loose wording again: Samples i.e. observations have no variogram value at
      all. Probably the sentence is speaking on geostatistical datasets, however
      it does not mention the necessary conditions.

      >If you had a set of blocks is it possible to add their kriging variances
      >together,
      >to get a standard error for the mine head grade, is this something that is
      >done in practice or has been done in the past to estimate possible deviations
      > from akriged head grade?

      No, it is not correct to just add kriging variances to get the kriging
      variance of the sum of the blocks, since the kriging errors are not
      independent. The correct procedure would be to add all kriging variances and
      all kriging covariances:
      with kriging errors:
      k_i=\hat{Z}_i- Z_i
      it holds:
      \var(\sum_i k_i) = \sum_i \sum_j \cov(k_i,k_j)

      > From: Feng Liu [mailto:asherliu@...]
      > Sent: Wednesday, 2 November 2005 4:32 AM
      >
      > Got two questions about Kriging variance, If i have an area of interest,
      > many known points and I want to use kriging to estimate grid points all
      > over the area.
      >
      > 1. Suppose stationarity, should kriging variance following a normal
      > distribution? or should it be random?

      It is not random at all and has no distribution.

      The kriging variance itself has no probabilistic distribution, since it
      depends functionally on the locations of the observations and on the
      variogram. Asking for the distribution of the kriging variance is the same as
      asking for the distribution of values of \sin(x+y) on some grid.


      >
      >
      > 2. Besides configuration of known samples, what are the other factors that
      > may affect the magnitude of kriging variance at a give location?

      The kriging variance depends functionally on the configeration of known
      samples and the variogram.

      However the kriging error can depend on more. E.g. will the kriging variance
      underestimate the true conditional variance of error in regions of high local
      variability e.g. in regions of high ore grades.

      >
      > Thank you very much
      >
      >
      > Feng Liu

      Best regards,
      Gerald v.d. Boogaart

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