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AI-GEOSTATS: Standard deviation, Variance

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  • Digby Millikan
    Hello, I was wondering if someone can tell me about statistical parameters, why standard deviation and variance is used as opposed to mean absolute deviation
    Message 1 of 9 , Nov 28, 2002
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      Hello,
      I was wondering if someone can tell me about statistical parameters,
      why standard deviation and variance is used as opposed to mean absolute
      deviation from the mean. It rings a bell that intergral calculus has
      something
      to do with it e.g. related to formulea for a normal distribution .
      M.David states the variogram uses the squared term as it makes calculations
      easier, as it would being related to statistical parameters such as
      variance,
      covariance similarly, A.Journel informed me, as Donald exaplained Kriging
      is Least Squared Error.

      Thanks in advance,

      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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    • Digby Millikan
      Hello, My inital enquiry about why variance is used as a basis for geostatistics appears to be the tip of the iceberg of many reasons and a large amount of
      Message 2 of 9 , Dec 4, 2002
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        Hello,
        My inital enquiry about why variance is used as a basis for geostatistics appears
        to be the tip of the iceberg of many reasons and a large amount of complex
        mathematical theory. I have included a summary of further postings I have received concerning this matter and will later write a short summary of these reasons based
        on the emails received.

        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
        //=======================================================
        Digby Millikan wrote:

        Hello,
        I was wondering if someone can tell me about statistical parameters,
        why standard deviation and variance is used as opposed to mean absolute
        deviation from the mean. It rings a bell that intergral calculus has
        something
        to do with it e.g. related to formulea for a normal distribution .
        M.David states the variogram uses the squared term as it makes calculations
        easier, as it would being related to statistical parameters such as
        variance,
        covariance similarly, A.Journel informed me, as Donald exaplained Kriging
        is Least Squared Error.

        Thanks in advance,

        Regards Digby Millikan
        //======================================================
        This question comes up from time to time in statistics and it is likely
        that the answer pertains to optimization. The variance is a second
        moment, i.e., it is related to a sum of squares. Problems pertaining to
        sums of squares arise in a number of places (e.g., moment of inertia,
        PCA, energy) but part of the reason for the emphasis on squares as
        opposed to absolute values probably has to do with differentiation. The
        absolute value function is not differentiable at zero whereas the sum of
        squares is differentiable. Moreove when optimizing a sum of squares one
        obtains a system of linear equations, to optimize a function involving
        the absolute value does not lead to a nice analytic solution. Note that
        Newton used squares in his landmark study on errors.

        The absolute value is not exactly a first moment but it certainly is not
        a second moment. Consequently if one constructs an objective function
        using absolute values as opposed to squares it will behave differently.

        The absolute deviation probably more naturally relates to the median
        (than to the mean).

        In summary I don't think there is an absolute answer to your question
        and you may get different answers/explanations from different people but
        I think all will include some of the ideas above.

        Donald E. Myers
        //=======================================================
        Virgil wrote;

        Partly because way back in the days when calculators and computers
        were people, there were nicely developed shortcuts for calculating
        means and variances which were not available for medians and mean
        absolute deviations (MADs).

        Secondly, the theoretical analysis of Gaussian distributions was
        easier to develop in terms of means and variances than in terms of
        medians and MADs, and, originally, Gaussian were, by far, the most
        studied of the continuous distributions in the early days of
        statistics. Then Gossett developed the Student distributions, again
        strongly dependent on means and variances.
        //=======================================================
        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.

        WDA

        end
        //=======================================================


        [Non-text portions of this message have been removed]
      • 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 3 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 4 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 5 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 6 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 7 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 8 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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