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Re: GEOSTATS: (Another) Question on log-transforms

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  • Pierre Goovaerts
    Hi Bill, The remark that was made earlier on the list regarding the exponentiation of lognormal kriging applies to both point and block estimates. Taking
    Message 1 of 2 , Feb 19, 1999
      Hi Bill,

      The remark that was made earlier on the list regarding the
      exponentiation of lognormal kriging applies to both point
      and block estimates. Taking simply the antilog
      of your kriged estimates leads to a biased estimate.
      As you rigthly guess, computing a semivariogram on
      the original values or the logtransforms usually leads
      to different results. Logarithmic transformation is typically
      performed to reduce the influence of high values on the estimation
      of the semivariogram (e.g. positively skewed histograms of
      heavy metal concentrations) and so the resulting histogram
      is expected to be less
      erratic than the original semivariogram.

      My advice is that if you have to deal with highly skewed
      histograms and want to avoid the problems associated with the
      logtransform and backtransform of your estimates, use an
      indicator approach to model the local cdf and use the ccdf mean
      as an estimate and the ccdf variance as a measure of uncertainty.
      Another option is to use a normal score transform (see Deutsch
      and Journel, p. 76 or my book) that ensures that the
      histogram of transformed data is perfectly symmetric
      regardless the shape of the original histogram.
      Use simple kriging to derive the mean and variance of the
      ccdf that is Gaussian in that case, and backtransform the
      results. Note that if you have a lot of data below the detection
      limits, you better go with an indicator approach or consider
      splitting your data into different subsets (use of different
      populations and corresponding RFs).


      ________ ________
      | \ / | Pierre Goovaerts
      |_ \ / _| Assistant professor
      __|________\/________|__ Dept of Civil & Environmental Engineering
      | | The University of Michigan
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      On Thu, 18 Feb 1999, Bill Thayer wrote:

      > Hello,
      > In the past I have asked the list for their opinion on the use of
      > log-transformed data in developing variograms and then using the
      > variograms to generate block estimates and estimation variances of the
      > log-transformed data. I am now aware of the problems this creates when
      > one tries to back transform the results (both the block estimates and
      > their estimation variances).
      > What is the opinion of the list on using log-transformed data to develop
      > variograms and using the variograms to generate a grid of point
      > estimates. I am not interested in the associated estimation variances.
      > I plan on back-transforming the point estimates (simply taking the
      > exponents of the point estimates) and then use bootstrapping to estimate
      > confidence intervals for the mean concentration (i.e. mean of the
      > back-transformed point estimates). My data is lead concentration
      > measured from (grab) soil samples. It seems to me that simply taking
      > the exponent of the kriged point estimates is OK because the kriged
      > estimates do not represent mean concentrations over a block. Is this
      > sound reasoning? I would also like feedback on the use of the
      > logtransformed data to develop variograms. In other words, it is not
      > clear to me that the difference between the squares of log-tranformed
      > data is analogous to the the same calculation using untransformed data.
      > Thanks in advance for any feedback. I will post a summary of the
      > responses I receive,
      > Bill

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