Loading ...
Sorry, an error occurred while loading the content.

Lognormal data. Re: AI-GEOSTATS: Variogram behaviour

Expand Messages
  • Digby Millikan
    Hello, By multigaussian, I meant a combination of distributions. I used to assist with reserve reports with data that displayed a close to three parameter
    Message 1 of 3 , Mar 18 2:30 AM
    • 0 Attachment
      Hello,
      By multigaussian, I meant a combination of distributions. I used to assist with reserve reports with data that
      displayed a close to three parameter lognormal distribution. Although the histogram displayed mixed distributions,
      three parameter lognormal was the closest so I used sichel tables as discussed in Ch. 1 Geostatistical Ore Reserve Estimation M. David. Yes I could see data would be lost but the total reserve would not be skewed. This may be
      useful for global reserve statements of an operating mine, while for grass roots operations alternates should be used
      to avoid loss of real data.

      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?

      >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.

      >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.

      > This solution might not be popular among members of the list, however, since it
      >involves abandoning the spatial analysis.

      Yes, I am concerned the model maybe be highly smoothed, and the affect this will
      have on the global estimate.

      Regards Digby Millikan.





      [Non-text portions of this message have been removed]
    • 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 2 of 3 , Mar 18 8:18 AM
      • 0 Attachment
        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

        ____________________________________________________________
        Do You Yahoo!?
        Get your free @... address at http://mail.yahoo.co.uk
        or your free @... address at http://mail.yahoo.ie

        --
        * To post a message to the list, send it to ai-geostats@...
        * As a general service to the users, please remember to post a summary of any useful responses to your questions.
        * To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list
        * Support to the list is provided at http://www.ai-geostats.org
      • 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 3 of 3 , Mar 18 5:17 PM
        • 0 Attachment
          >===== 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


          --
          * To post a message to the list, send it to ai-geostats@...
          * As a general service to the users, please remember to post a summary of any useful responses to your questions.
          * To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list
          * Support to the list is provided at http://www.ai-geostats.org
        Your message has been successfully submitted and would be delivered to recipients shortly.