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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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    • Syed Abdul Rahman Shibli
      ... There are in fact situations where you might want to use the mean absolute deviation from the mean (madogram) since any squared term would amplify the
      Message 2 of 9 , Nov 28, 2002
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        On 29/11/02 5:32 AM, "Digby Millikan" <digbym@...> 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

        There are in fact situations where you might want to use the mean absolute
        deviation from the mean (madogram) since any squared term would amplify the
        effect of outliers or other extreme values on the spatial correlation
        measure. For example, a range might be better defined from an inspection of
        the madogram instead of the variogram.


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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 3 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 4 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 5 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 6 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 7 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 8 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 9 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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