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Re: Lognormal data. Re: AI-GEOSTATS: Variogram behaviour

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  • Isobel Clark
    Digby ... You have to be a little careful with terminology ;-) To a geostatistician multigaussian means multivariate gaussian. Most people call a combination
    Message 1 of 3 , Mar 18, 2001
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      Digby

      > By multigaussian, I meant a combination of
      > distributions.
      You have to be a little careful with terminology ;-)

      To a geostatistician "multigaussian" means
      multivariate gaussian. Most people call a combination
      of populations a mixture. I have published quite a lot
      of 'statistical' work on mixtures of Normals and
      lognormals, the earliest being my 1974 paper in the
      Transactions of the Institution of Mining and
      Metallurgy.

      Depending on how heavily your populations overlap, you
      may be able to use a combination of indicator and more
      'traditional' ordinary kriging to produce estimates
      from mixtures. You might want to look at my keynote
      address in the MRE symposium for Alwyn Annels last
      March, which is on exactly this topic. Those papers
      are up on the Web, although I don't have the address
      to hand. If you do a Yahoo! search using my name and
      MRE 21, you should be able to track it down.

      The three parameter lognormal is a good substitute,
      particularly if the proportion of one population is
      very low.

      > What does MLE stand for?
      Maximum likelihood estimator. Most geostatistical (and
      classical statistical) methods are based on a "least
      squares" or closest fit criterion. MLE is based on the
      "solution that the data is most likely to fit".
      Different philosophy, much harder mathematics. The
      examples in my 1974 paper were tackled with MLE.

      > Are you saying to use sichel mean of local data to
      > estimate the grade in every
      > block? i.e. Moving Lognormal Average.
      Better to use lognormal kriging. This is a local
      sichel-type estimator and you can use Sichel theory
      for local confidence bounds.

      > What is asymmetric?
      A 'central 90% confidence' -- that is 5% at bottom and
      5% at top -- will be assymetric around the estimators
      because the original distribution is assymetrical
      around the mean. The lower confidence level is closer
      to the estimate than the upper one, if you keep the %
      risk the same.

      All explained in my 1987 paper in SAIMM. Not on the
      Web, unfortunately. Also updated Sichel's tables. The
      paper was refereed by Sichel, so it must be OK ;-)

      > Yes, I am concerned the model maybe be highly
      > smoothed, and the affect this will
      > have on the global estimate.
      lognormal kriging avoids the over-smoothing and gives
      an unbiassed estimator with the narrowest confidence
      intervals. This is, of course, provided your
      distribution is lognormal or three parameter
      lognormal.

      If you need any clarification on this, please do not
      hesitate to contact me direct. Complete references can
      be found at
      http://uk.geocities.com/drisobelclark/resume/Publications.html

      Isobel Clark

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    • Ruben Roa
      ... Digby: Isobel has responded to your questions about my comments, but i may add something of value ... not ... Maximum likelihood estimators. The least
      Message 2 of 3 , Mar 18, 2001
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        >===== Original Message From "Digby Millikan" <digbym@...> =====
        Digby:

        Isobel has responded to your questions about my comments, but i may add
        something of value

        >Ruben,
        >>Regarding lognormal data, if you are only interested in the mean of the
        >>regionalized variable and its confidence interval whithin the region, and
        not
        >>in the spatial mapping itself, you can just use the MLE estimator of the
        >>lognormal mean, the (Finney-)Sichel estimator that you mentioned
        >
        >What does MLE stand for?

        Maximum likelihood estimators. The least square estimators, such as those
        normally used in goestatistics, are MLE when the distribution of errors is
        Gaussian, so that least square estimators are a particular case of MLE. The
        (Finney-)Sichel lognormal mean is a case of MLE, though only approximate.

        >>to obtain the point estimate,
        >
        >Are you saying to use sichel mean of local data to estimate the grade in
        every
        >block? i.e. Moving Lognormal Average.

        I'm not familiar with mining or geology. What i say is that if in a region,
        the variable of interest measured at several points distribute lognormally,
        then the Finney-Sichel lognormal mean is the MLE of the mean, and that
        estimate can be used as the mean of the regionalized variable without regards
        to its spatial distribution. In the geostatistical paradigm the equivalent
        value would be the kriged mean. Insofar as the kriged mean is a better
        estimate (more unbiased and with less variance) than the lognormal mean of
        lognormally distributed spatial data, you have done something of value by
        performing the spatial analysis. But even when the kriged mean and the
        lognormal mean are similar in terms of the central estimate and the measure of
        precision, the spatial analysis gives additional products which are not
        obtained from a purely distributional analysis of the data.

        >>and the theory and tables in Land (1975, Tables of
        >>confidence limits for linear functions of the normal mean and variance,
        >>Selected Tables in Mathematical Statistics, vol. III, Am. Math. Soc.
        >>Providence, pages: 385-419) to obtain the asymmetric confidence interval.
        >
        >What is asymmetric? Are you saying to develop a confidence interval as
        compared >to
        >kriging which reports the 95% confidence interval, or the estimation
        variance.
        >I note that sichel provides 90% confidence interval tables in his original
        paper on
        >the t-estimator found in "Symposium On Mathematical Statistics and Computer
        >Applications" SAIMM.

        The confidence interval of the lognormal mean is asymmetric because the
        distribution is asymmetric. I am not familiar with Sichel's tables but my
        previous reference to Land's (1975) tables refer to tables of the quantiles of
        the lognormal distribution which are needed to build confidence interval
        around the lognormal mean.

        Ruben


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