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Re: AI-GEOSTATS: Standard deviation, Variance

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  • Isobel Clark
    ... Things have obviously changed since I was a lad. I was taught that the Central Limit Theorem was a theorem NOT a law. There are distributions which do not
    Message 1 of 9 , Dec 5, 2002
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      > The reason is simple and comprehensive....
      >
      > Assume a population with ANY distribution of
      > elements. Then randomly select
      > a number of sample elements from the population to
      > characterize the
      > underlying population. That distribution of sample
      > elements ALWAYS tends
      > toward a normal [Gaussian] distribution. And the
      > mean and standard deviation
      > of the sample distribution are unbiased
      > representations of the mean and
      > standard deviation of the underlying population.
      Things have obviously changed since I was a lad. I was
      taught that the Central Limit Theorem was a theorem
      NOT a law. There are distributions which do not
      conform to this behaviour and (alas for us) the
      lognormal is one of them.

      The Central Limit theorem also does not apply to mixed
      distributions or in cases of non-stationarity. Mind
      you, neither does geostatistics................

      Isobel Clark
      http://geoecosse.bizland.com/news.html

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    • Digby Millikan
      Isobel, ... Is this the reason for transforming the data (only upto page 14). At the moment I am thinking kriging minimizes the variance of the sampling
      Message 2 of 9 , Dec 5, 2002
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        Isobel,

        > NOT a law. There are distributions which do not
        > conform to this behaviour and (alas for us) the
        > lognormal is one of them.
        >

        Is this the reason for transforming the data (only upto page 14).
        At the moment I am thinking kriging minimizes the variance of the
        sampling distribution as I am also reading a book on classical
        statistics.

        Is this distribution common in elements other than gold and uranium.

        >
        > The Central Limit theorem also does not apply to mixed
        > distributions or in cases of non-stationarity. Mind
        > you, neither does geostatistics................
        >

        John Sturgul was my lecturer in mine evaluation, I think he mentioned your
        1979 book in that course, but I did use it as a reference for a project I
        did on geostatistics.

        Thanks again,


        Regards Digby Millikan B.Eng

        Geolite Mining Systems
        U4/16 First Ave.,
        Payneham South SA 5070
        Australia.
        Ph: +61 8 84312974

        digbym@...
        http://www.users.on.net/digbym


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      • Isobel Clark
        I find this fascinating. Apparently what I said is almost entirely wrong. What I said was I was taught that....... I do not recollect Don Myers being in my
        Message 3 of 9 , Dec 5, 2002
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          I find this fascinating.

          Apparently what I said is almost entirely wrong.

          What I said was 'I was taught that.......' I do not
          recollect Don Myers being in my classrooms as an
          undergraduate (or during my MSC for that matter).

          You know, I welcome criticism, especially when I get
          things wrong. I have a big problem with people who do
          not actually read what I write but react at some
          visceral level to what they think I said.

          Also, I must be really stupid, because the comments
          given by Don include the statement

          " If any of the conditions in the theorem are not
          satisfied then the theorem may not apply. "

          Which, I am fairly sure, is what I was trying to say.
          Isobel Clark
          http://uk.geocities.com/drisobelclark/resume


          --- "Donald E. Myers" <myers@...> wrote:
          > Regrettably the following statement by I. Clark is
          > almost entirely wrong
          > See below for a correct statement of the CLT, the
          > problem in part is
          > simply carelessness in terminology and replacing
          > correct
          > statements/formulations by sort of heuristic ones
          > (which are not correct)
          > Donald E. Myers
          > http://www.u.arizona.edu/~donaldm
          >
          ***********************************************************************
          > Isobel Clark wrote:
          >
          > >>The reason is simple and comprehensive....
          > >>
          > >>Assume a population with ANY distribution of
          > >>elements. Then randomly select
          > >>a number of sample elements from the population to
          > >>characterize the
          > >>underlying population. That distribution of sample
          > >>elements ALWAYS tends
          > >>toward a normal [Gaussian] distribution. And the
          > >>mean and standard deviation
          > >>of the sample distribution are unbiased
          > >>representations of the mean and
          > >>standard deviation of the underlying population.
          > >>
          > >
          > >
          >
          ***************************************************************************
          >
          > CLT
          > Let X_1,...., X_n be a sequence of independent,
          > identically distributed
          > random variables with common mean m and common
          > standard deviation
          > sigma. Let Z_n be defined as a normalized sum
          >
          > Z_n = [S_n - m]/ (sigma/sqt root of n),
          > S_n = [Z_1
          > +.....+ X_n]/n
          >
          > S_n is the sample mean
          >
          > Let F_n(z) be the cumulative probability
          > distribution function for Z_n
          > and let G(z) be the cumulative probability
          > distribution function for the
          > standard Normal,. Then F_n(z) --> G(z) as n
          > increases.
          >
          > Note two things about this statement, (1) the
          > theorem does not say how
          > "fast" the cdf for Z_n approaches the standard
          > Normal, (2) the speed of
          > convergence depends on z. Also the speed of
          > convergence depends on the
          > distribution type of the X_i's
          >
          > If any of the conditions in the theorem are not
          > satisfied then the
          > theorem may not apply. The convergence in this
          > theorem is what is called
          > "convergence in distribution", this is one of the
          > weakest forms of
          > convergence for a sequence of random variables.
          > There are theorems that
          > will give estimates or bounds on the speed of
          > convergence. There are
          > also special cases of this theorem that are somewhat
          > simpler such as the
          > the Normal approximation to the Binomial
          >
          > The simplest proof of the theorem above uses
          > characteristic functions
          > (Fourier Transforms of the densities).
          >

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        • Digby Millikan
          Apologise, The original email, ... was not written by Isobel, it came from W.D. Allen on sci.stat.math which I posted in the summary of my replies. Thankyou
          Message 4 of 9 , Dec 5, 2002
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            Apologise,
            The original email,

            > > >>The reason is simple and comprehensive....
            > > >>
            > > >>Assume a population with ANY distribution of
            > > >>elements. Then randomly select
            > > >>a number of sample elements from the population to
            > > >>characterize the
            > > >>underlying population. That distribution of sample
            > > >>elements ALWAYS tends
            > > >>toward a normal [Gaussian] distribution. And the
            > > >>mean and standard deviation
            > > >>of the sample distribution are unbiased
            > > >>representations of the mean and
            > > >>standard deviation of the underlying population.

            was not written by Isobel, it came from W.D. Allen on sci.stat.math
            which I posted in the summary of my replies.
            Thankyou both for your help in this matter, I am currently reading
            Practical Geostatistics 2000 and have ordered the statistics books
            as recommended by Donald.


            Regards Digby Millikan B.Eng

            Geolite Mining Systems
            U4/16 First Ave.,
            Payneham South SA 5070
            Australia.
            Ph: +61 8 84312974

            digbym@...
            http://www.users.on.net/digbym


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          • Ruben Roa
            Isobel did not write the careless paragraph about the central limit theorem (CLT) Don replied to, as pointed out by Digby. I wish to add something to what Don
            Message 5 of 9 , Dec 5, 2002
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              Isobel did not write the careless paragraph about the central limit theorem
              (CLT) Don replied to, as pointed out by Digby. I wish to add something to
              what Don said about the conditions under which the CLT applies, and that
              people usually miss in considering the universality of the CLT. See below.

              >> Let X_1,...., X_n be a sequence of independent,
              >> identically distributed
              >> random variables with common mean m and common
              >> standard deviation
              >> sigma. Let Z_n be defined as a normalized sum
              >>
              >> Z_n = [S_n - m]/ (sigma/sqt root of n),
              >> S_n = [Z_1
              >> +.....+ X_n]/n
              >>
              >> S_n is the sample mean
              >>
              >> Let F_n(z) be the cumulative probability
              >> distribution function for Z_n
              >> and let G(z) be the cumulative probability
              >> distribution function for the
              >> standard Normal,. Then F_n(z) --> G(z) as n
              >> increases.
              >>
              >> Note two things about this statement, (1) the
              >> theorem does not say how
              >> "fast" the cdf for Z_n approaches the standard
              >> Normal, (2) the speed of
              >> convergence depends on z. Also the speed of
              >> convergence depends on the
              >> distribution type of the X_i's

              Note also the sum operation. The CLT, more precisely called the Additive
              CLT, applies to sums of pairwise independent random variables as n tends to
              infinity. But if the operation is multiplication with equal-signed r.v.,
              then convergence in distribution is towards the lognormal, not the normal.
              It might well be that when considering natural phenomena, multiplicative
              processes be more or equally common than additive ones, as we oftenly
              observed skewed continuous data.
              Rubén
              http://webmail.udec.cl

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            • Isobel Clark
              Thanks to Rubén and Digby for pointing out what I had misunderstood about Don Myers email. It had not occurred to me (duh) that the lines starting would
              Message 6 of 9 , Dec 6, 2002
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                Thanks to Rubén and Digby for pointing out what I had
                misunderstood about Don Myers' email.

                It had not occurred to me (duh) that the lines
                starting '>' would be read as being from me rather
                than part of a forwarded email.

                Another score on the dumb side. Apologies for the
                strong reaction to Don's email if (on this occasion)
                he was not criticising my contribution.

                Isobel

                http://uk.geocities.com/drisobelclark


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