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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
      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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    • 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 2 of 7 , Dec 30, 2002
        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 3 of 7 , Dec 31, 2002
          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 4 of 7 , Jan 1 11:26 PM
            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 5 of 7 , Jan 2 7:41 PM
              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 6 of 7 , Jan 2 11:12 PM
                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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