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AI-GEOSTATS: re CLT and mixtures

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  • Brian R Gray
    someone recently commented on the central limit theorem not being applicable to mixture distributions. this seems sensible but, given that that s all I ve
    Message 1 of 7 , Dec 30, 2002
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      someone recently commented on the central limit theorem not being
      applicable to mixture distributions. this seems sensible but, given that
      that's all I've ever read on the topic, I wonder if any more amplification
      might be provided. thanks, brian

      ****************************************************************
      Brian Gray
      USGS Upper Midwest Environmental Sciences Center
      575 Lester Avenue, Onalaska, WI 54650
      ph 608-783-7550 ext 19, FAX 608-783-8058
      brgray@...
      *****************************************************************



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    • Isobel Clark
      Brian There is an immense amount of literature on mixtures of distributions, generally Normal or lognormal. Unfortunately, I don t have my 1974 paper up on the
      Message 2 of 7 , Dec 30, 2002
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        Brian

        There is an immense amount of literature on mixtures
        of distributions, generally Normal or lognormal.
        Unfortunately, I don't have my 1974 paper up on the
        Web yet, but you can see examples of mixtures in my
        1993 paper at the conference in Perth WA.

        The CLT will not necessarily apply unless the separate
        populations are completely mixed geographically and
        the proportions between the components are reasonably
        constant. Then you might (only might) get some sort of
        convergence.

        There was a paper in Math Geol in the late 70s by M W
        Clark which reviewed mixture papers to that date. PDM
        MacDonald at MacMaster (Canada) has done a lot of work
        in the mixture field also. The ability to separate
        mixed distributions by statistical means in included
        in our free teaching software, but I am not supposed
        to mention that.

        Isobel Clark
        http://geoecosse.bizland.com/softwares

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      • Digby Millikan
        Brian, I am not familiar with the derivation of kriging equations, but of note I know the lognormal population does not follow the central limit theorem, and
        Message 3 of 7 , Dec 30, 2002
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          Brian,

          I am not familiar with the derivation of kriging equations, but of note
          I know the lognormal population does not follow the central limit
          theorem, and it is necessary to transform lognormal populations to normal
          populations before kriging using lognormal kriging. I know the same also
          applies to mixed populations and that mixed populations must be transformed
          to normal populations using hermite polynomials with disjunctive kriging,
          maybe this is for the same reason.

          Best wishes for the New Year everyone.

          Regards Digby Millikan.
          Geolite Mining Systems
          digbym@...


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        • Isobel Clark
          Digby The original derivation of the kriging equations does not demand any specific distribution. However, it only makes sense if the increments - difference
          Message 4 of 7 , Dec 31, 2002
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            Digby

            The original derivation of the kriging equations does
            not demand any specific distribution. However, it only
            makes sense if the 'increments' - difference in value
            - have a fairly stable variance.

            Kriging works much better IN PRACTICE if those
            differences are Normal, hence the various
            transformations offerred such as lognormal kriging,
            hermitian polynomials, Normal scores, rank transforms
            and so on.

            Normalising a mixture of distributions will result in
            specious answers unless the distributions are mixed in
            the same proportions all over the study area.

            Usually a mixed distribution is due to a mixture of
            real populations. For example, in geology oxide and
            sulphide samples may show different behaviour. There
            may be separate phases of deposition.

            In short, a mixture of distributions generally
            indicates a violation of the assumption of homogeneity
            (or, if your prefer, stationarity) which is needed for
            the proper application of any geostatistics. If
            possible the populations should be separated and the
            analysis carried out. If not, you may be able to cope
            with the data using indicator kriging such as
            suggested in my 1993 paper. There is also an example
            in my Cardiff paper of 2000, I think.

            Isobel
            http://geocities.com/drisobelclark/resume/Publications.html

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          • Digby Millikan
            Francis, Have you looked at indirect lognormal corrections and affine corrections for sample sizes. Many kriging softwares use diffrent sample sizes for their
            Message 5 of 7 , Jan 1, 2003
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              Francis,
              Have you looked at indirect lognormal corrections and affine corrections
              for sample
              sizes. Many kriging softwares use diffrent sample sizes for their algorithm
              e.g. a single
              line of zero diameter representing all core sizes, with each block
              discretized to match
              the sample sizes, probably sufficient. Where poor lognormal semivariograms
              are available
              in the past I have used a sichel estimate of the mean of the population then
              cut the grades
              until the arithmetic mean equals the sichel mean and used inverse distance
              modelling
              (White Devil, N.T. Australia). Populations where actually mixed (lognormal
              with a small
              high grade portion) it may have been more useful to provide a mixed
              distribution mean rather
              than a sichel mean which I think a paper has been published on. Unforunately
              it can be
              rare at some undergrounds and such operating environments where several
              mines target
              one mill. However this underground grade modelling was only for financial
              planning,
              as stope design was done entirely from geological interpretation. However at
              open pit
              operations with a single mine and mill operating and more sophisticated
              grade control
              samples are available it is possible to reconcile a resource model to a
              grade control model
              and mill production to within a 0.5% by adjusting variograms and selective
              mining unit
              cut off grades, though I personally have no experience with nugget effect
              adjustments.

              Regards Digby Millikan
              Geolite Mining Systems
              digbym@...




              ----- Original Message -----
              From: "Francis T. Manns" <artesian1@...>
              To: "Isobel Clark" <drisobelclark@...>; "Digby Millikan"
              <digbym@...>
              Cc: <ai-geostats@...>
              Sent: Wednesday, January 01, 2003 8:07 AM
              Subject: RE: AI-GEOSTATS: re CLT and mixtures


              > Dear all,
              >
              > Please excuse my jumping into the dialogue without a proper introduction.
              I
              > just enrolled in the forum in order to search for a collaborator to assist
              > me in understanding what I've done with grade distributions for gold and
              > copper. My recent manuscript on Sadiola was rejected by Canadian
              Institute
              > of Mining Metallurgy and Petroleum [CIM] on the basis that my empirical
              > nugget effect solution needed a mathematical basis. Therefore I would
              like
              > to interest someone, perhaps a grad student somewhere, in pursuing the
              math
              > that eludes me. Would that mathematical proofs be required to have
              > empirical support in earth sciences? How many new mines struggle because
              > the expected grades fall short of predictions, estimates, and
              'calculations'
              > from the geostatistical side?
              >
              > My hypothesis is that in far too many instances, the sample size (weight)
              is
              > too small to represent the end product - the estimate (assay) of the metal
              > content of a tonne of rock or a block of ore. Moreover, the industry
              > standard aliquot of 30 grams is practical for a lab is not practical for
              at
              > least half of the gold deposits in the universe. We're asking 30 grams to
              > represent 1 million grams and then require four or so assays to represent
              > 300 tonnes (e.g., 120 grams to represent to estimate 300 million grams of
              > rock).
              >
              > I use a uniform probability plot to describe the assay distribution. A
              > straight segment on a uniform plot indicates a random number sequence. In
              > most cases, the distributions shows two or more random runs of numbers at
              > the high end of the distribution (these separate runs represent geological
              > loci of crystallization). Though the sample size is small, we at least
              get
              > a collection of random numbers or samples from the deposit. Treating
              these
              > empirically it is possible to correct the arithmetic average of the sample
              > distribution for the sample size limitation.
              >
              > As for conventional geostatistics, it's GIGO. A systematic error or
              errors
              > appear if the sample size is too small for the distribution of metal in
              the
              > rocks. I believe that if we were able to twin all the holes of the garden
              > variety exploration level deposit, we would find that very few holes would
              > actually be twinned, but the mean, standard deviation and standard error
              > would be indistinguishable.
              >
              > I have found that the slope of the straight segment of the uniform
              > probability plot is proportional to the error of the arithmetic average.
              > When one deducts the intercept of the slope at the fiftieth percentile
              from
              > the arithmetic average, one gets a very good estimator of the plant grade
              of
              > the deposit. I would be very happy to correspond with someone who has
              > insight into this phenomenon. It works for a copper distribution that I
              > once had in my possession. The raw data are gone now. I believe the 50th
              > percentile is important on a semi-log plot as it represents the square
              root
              > of the slope "factor", for want of any clearer understanding. I'll add my
              > website, and hope someone could have a glance at what's happening here.
              > It's geologically based, as you can see from my Borneo photograph in the
              > site.
              >
              > http://www.geocities.com/fmanns_artesian/index.html
              >
              > Thank you,
              >
              > Francis Manns, PhD
              > Artesian Geological Research
              > Toronto Ontario
              >
              >
              > -----Original Message-----
              > From: ai-geostats-list@... [mailto:ai-geostats-list@...]On
              > Behalf Of Isobel Clark
              > Sent: December 31, 2002 5:30 AM
              > To: Digby Millikan
              > Cc: ai-geostats@...
              > Subject: Re: AI-GEOSTATS: re CLT and mixtures
              >
              >
              > Digby
              >
              > The original derivation of the kriging equations does
              > not demand any specific distribution. However, it only
              > makes sense if the 'increments' - difference in value
              > - have a fairly stable variance.
              >
              > Kriging works much better IN PRACTICE if those
              > differences are Normal, hence the various
              > transformations offerred such as lognormal kriging,
              > hermitian polynomials, Normal scores, rank transforms
              > and so on.
              >
              > Normalising a mixture of distributions will result in
              > specious answers unless the distributions are mixed in
              > the same proportions all over the study area.
              >
              > Usually a mixed distribution is due to a mixture of
              > real populations. For example, in geology oxide and
              > sulphide samples may show different behaviour. There
              > may be separate phases of deposition.
              >
              > In short, a mixture of distributions generally
              > indicates a violation of the assumption of homogeneity
              > (or, if your prefer, stationarity) which is needed for
              > the proper application of any geostatistics. If
              > possible the populations should be separated and the
              > analysis carried out. If not, you may be able to cope
              > with the data using indicator kriging such as
              > suggested in my 1993 paper. There is also an example
              > in my Cardiff paper of 2000, I think.
              >
              > Isobel
              > http://geocities.com/drisobelclark/resume/Publications.html
              >
              > __________________________________________________
              > Do You Yahoo!?
              > Everything you'll ever need on one web page
              > from News and Sport to Email and Music Charts
              > http://uk.my.yahoo.com
              >
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              > any useful responses to your questions.
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              >


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            • Digby Millikan
              Francis, On topic in Lognormal-de Wijsian Geostatistics for Ore Evaluation D.G.Krige pp5. points out that the slope of the log-probability plot provides an
              Message 6 of 7 , Jan 2, 2003
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                Francis,
                On topic in "Lognormal-de Wijsian Geostatistics for Ore Evaluation"
                D.G.Krige pp5. points out that the slope of the log-probability plot
                provides an estimate of the variance which Rendu 78 and Storrar 77
                also published information on. The graph paper in the book actually
                has a variance slope scale on it from which you can estimate the
                variance. There is also a graph which gives the realtionship between
                mean value, pay limit, pay value and percentage pay for lognormal
                distributions. I will take a look and see how this relates to the 50th
                percentile. It is interesting that your mill grade was related to the
                50th percentile as this would depend on the pay limit or cutoff grade
                which we call it in Australia which is entirely based on economic
                factors and hence should show no relationship unless at operations
                of similar commodity type and operating costs and revenues.
                Overestimating gold grade used to be due to a reluctance to cut grades
                so operators would hear the figures they wanted to, in regards to
                nugget effect, I am a mining engineer, on what basis do geologists
                select core size? is it e.g. hole depth, mineralization type etc.

                Regards Digby Millikan
                Geolite Mining Systems
                digbym@...
                http://www.users.on.net/digbym


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              • Digby Millikan
                Francis, Having looked at your website further comments regarding the Australian gold mining industry that only 2 out of 35 operations acheived feasibility
                Message 7 of 7 , Jan 2, 2003
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                  Francis,
                  Having looked at your website further comments regarding the Australian
                  gold mining industry
                  that only 2 out of 35 operations acheived feasibility head grades, that in
                  1990 it was not common
                  knowledge among the mining industry, though common knowledge to
                  geostatisticians, that gold
                  deposits are typically overestimated due to lognormality. This demonstrates
                  geostatistics has its
                  place in educational institutions and the Australian mining industry.
                  Hopefully with "due diligence"
                  (1995 Newmont Mining Australia appointed a technical due diligence officer,
                  and
                  "Basic Linear Geostatistics" 1998 Margaret Armstrong discusses due
                  diligence) on the increase
                  this is being addressed.
                  Is your Sadiola paper available online?

                  Regards Digby Millikan
                  Geolite Mining Systems
                  digbym@...
                  http://www.users.on.net/digbym


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