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Re: AI-GEOSTATS: Normal distributions

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  • Ruben Roa
    Chaosheng Zhang wrote: [snip] Hi Chaosheng: A few points about log transforms. See below. ... There shouldn t be any loss of information since the log
    Message 1 of 4 , May 20, 2002
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      Chaosheng Zhang wrote:

      [snip]

      Hi Chaosheng:

      A few points about log transforms. See below.

      > The reason why I care about this issue is that there are at least two
      > problems related to data transformation (in order to follow the normal
      > distribution):
      >
      > (1) The measurement scale is reduced. The orignal ratio/interval scale may
      > be reduced to the lower level of ordinal, even close to nominal, which
      > results in loss of raw information.

      There shouldn't be any loss of information since the log transformation
      is a one-to-one mapping. The sample variance is much smaller in the log
      scale but the log transform is often used precisely for that purpose.

      > (2) Artificial relationship is introduced. We know that the lognormal
      > distribution is widely accepted. In correlation analysis, if the
      > log-transformed data are used, the correlation becomes the "log-log"
      > relationship, not the oginal linear relationship. In bivariate regression
      > analysis, the original function is:
      > y = a x + b
      > However, for the log-transformed data, the function becomes:
      > log(y) = a log(x) + b
      > or y = exp (a log(x) + b)
      > In many cases, it is not clear if the relationship should be linear or
      > "log-linear". However, the artificially introduced "log-linear" relationship
      > need to be proved.

      Rather, when you apply the log transform you a-priori assume the
      existence of what you call the 'artificial relation', and this
      assumption refer to the algebraic form of the error term rather than to
      the relation between y and x. Say E(y)=f(x) is the model for y versus x,
      where E is the expectation operator. If you assume an additive error
      structure y_i=f(x_i)+e_i, where i indexes observation, and you consider
      the e_i's as iid normal random variates, then there is no reason to
      apply the log tranform. On the other hand if you assume a multiplicative
      error structure such as y_i=f(x_i)*e_i, and you assume that the e_i are
      iid lognormal random variates, then the log transform yields
      ln(y_i)=lnf(x_i)+ln(e_i) and now the ln(e_i) are iid normal random
      variates (with a much smaller variance than the e_i). The theory of the
      lognormal is well developed so that there isn't actually much need to
      transform the data to make it normal (e.g. Crow and Shimizu, 1988,
      Lognormal distributions, Dekker Inc, NY).

      > The most difficult situation is that if scientifically the relationship
      > between x and y is linear, should the data transformation still be carried
      > out (just to satisfy the statistical requirement)?

      When the relation between x and y is linear such as in E(y)=a*x+b, the
      errors are assumed to be additive (people usually do not believe that
      y_i=(a*x_i+b)*e_i but rather y_i=(a*x_i+b)+e_i), so applying a log
      transform to such case does not satisfy statistical requirements. On the
      contrary, it goes against statistical advice.
      The multiplicative error structure arises in models of the form
      E(y)=a*x^b or E(y)=a*exp(b*x), or in general, in all multiplicative
      processes.
      As there is a central limit theoren for additive processes leading to
      the normal, there is a central limit theorem for multiplicative
      processes leading to the lognormal.

      In the case of geostatistics, as pointed out by other people, the
      kriging equations do not require distributional assumptions (though the
      fitting of the model variogram to the moment-based Matheron variogram
      does). If the frequency distribution of the regionalised variable looks
      lognormal, it means that there is un underlying mechanism which is
      multiplicative, but still i don't see why the variable should be
      transformed for geostatistical analysis, except perhaps for fitting the
      variogram.

      Cheers
      Ruben

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