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Re: [ai-geostats] Re: descriptive statistics or inference?

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  • Digby Millikan
    Meng-Ying, Even if your population variance and sill do not match identically, the sample sill should still be a better estimate than the sample variance, when
    Message 1 of 9 , Dec 8, 2004
      Meng-Ying,

      Even if your population variance and sill do not match identically,
      the sample sill should still be a better estimate than the sample variance,
      when you consider the amount of clustering which occurs in sampling.

      Digby
    • Meng-Ying Li
      ... variance, ... All right, if you think the clustering of data values (I m not talking about clustering of locations) are not be part of the representation
      Message 2 of 9 , Dec 8, 2004
        On Thu, 9 Dec 2004, Digby Millikan wrote:

        > Meng-Ying,
        >
        > Even if your population variance and sill do not match identically,
        > the sample sill should still be a better estimate than the sample
        variance,
        > when you consider the amount of clustering which occurs in sampling.
        >
        > Digby
        >

        All right, if you think the clustering of data values (I'm not talking
        about clustering of locations) are not be part of the representation of
        population.

        I just found an example that I can use as population, with 2500 points
        and it's 2-D (in the GSLIB manual, second realization of SGSIM.OUT-- if
        you happen to have this data set) and found the sill and variance of this
        population not matching (sill~20, variance=18.63).

        I intended to use a smaller sample so everyone can have fun playing the
        data (even if you use M$-Excel to calculate the variogram), which also
        speaks out more what I'd like to say. But seems like people are more
        interested in discussing the size of population. . . I'll leave it here
        then, if nobody found any problem estimating population variance using the
        sill value. Maybe I'm just psychologically not comfortable estimating
        variance like that. . . (I'll probably follow you people if I found no
        theoretic derivation for my thinking.)

        It's fun discussing with you people though, and I'm happy to have this
        much discussion for my debut.


        Meng-ying
      • Digby Millikan
        Meng-ying, Sorry I don t have time to process the data yet, but a summary so far, is that geostatisitcs was originally developed in the mining field, and for
        Message 3 of 9 , Dec 12, 2004
          Meng-ying,

          Sorry I don't have time to process the data yet, but a summary so far,
          is that geostatisitcs was originally developed in the mining field, and
          for the application of mineral resource assesment with which I am involved
          where sampling patterns are generally clustered and the use of the sill of
          the
          variogram can be used as an estimate of the variance at your discression.
          As Isobel says this is where the half originates in the equation for the
          experimental variogram, as the sill will often approximate the variance.
          However with many applications of geostatisitics much more detailed and
          regular sampling patterns are used (even in mining e.g. soil geochemistry),
          in which case the sill is not an exact estimator (I'm not sure of the
          correct
          terminology here) of the variance and should be treated as such, though
          Isobel
          seems to be more familiar with the mathematics of this, and it might be
          interesting a further study of this.

          Digby
        • Isobel Clark
          Digby The variance/sill relationship is theoretical and does not depend on the layout of the samples, regular or clustered. Since the sill only uses pairs
          Message 4 of 9 , Dec 12, 2004
            Digby

            The variance/sill relationship is theoretical and does
            not depend on the layout of the samples, regular or
            clustered. Since the sill only uses pairs where
            samples are uncorrelated from one another, the
            clustering is irrelevant.

            It does depend on the distribution of the samples
            values being 'stationary', that is having constant
            mean and variance over the study area. It also depends
            on that distribution having a valid variance. For
            example, the variance of samples from a lognormal
            distribution depends on the average of those samples -
            hence the proportional effect.

            All of this is explained in any basic geostatistics
            book, including Matheron's original Theory of
            Regionalised Variables and my Practical Geostatistics
            (Chapter 3) which cn be freely downloaded from
            http://geoecosse.bizland.com/practica.htm

            Isobel
            http://uk.geocities.com/drisobelclark/seasonsgreetings.htm
          • Meng-Ying Li
            Like Isobel mentioned, the sill only uses pairs where samples are uncorrelated from one another, and in this case the clustering is irrelevant. And I totally
            Message 5 of 9 , Dec 12, 2004
              Like Isobel mentioned, the sill only uses pairs where samples are
              uncorrelated from one another, and in this case the clustering is
              irrelevant.

              And I totally agree with that. The crucial thing, Digby, is that you want
              to make sure the variance estimated reflects the characteristic of what
              you actually wanted, regardless of the terminology that may or may not be
              stated for different purposes.


              Meng-ying

              On Sun, 12 Dec 2004, Isobel Clark wrote:

              > Digby
              >
              > The variance/sill relationship is theoretical and does
              > not depend on the layout of the samples, regular or
              > clustered. Since the sill only uses pairs where
              > samples are uncorrelated from one another, the
              > clustering is irrelevant.
              >
              > It does depend on the distribution of the samples
              > values being 'stationary', that is having constant
              > mean and variance over the study area. It also depends
              > on that distribution having a valid variance. For
              > example, the variance of samples from a lognormal
              > distribution depends on the average of those samples -
              > hence the proportional effect.
              >
              > All of this is explained in any basic geostatistics
              > book, including Matheron's original Theory of
              > Regionalised Variables and my Practical Geostatistics
              > (Chapter 3) which cn be freely downloaded from
              > http://geoecosse.bizland.com/practica.htm
              >
              > Isobel
              > http://uk.geocities.com/drisobelclark/seasonsgreetings.htm
              >
              >
            • Digby Millikan
              Isobel, Thankyou, in relation to Colins work then the variance may be estimated from the sill of the variograms for the two orebodies and if the two orebodies
              Message 6 of 9 , Dec 12, 2004
                Isobel,

                Thankyou, in relation to Colins work then the variance
                may be estimated from the sill of the variograms for
                the two orebodies and if the two orebodies had lognormal
                distributions, they may have a different mean and variance,
                but may still display the proportional effect, i.e. similar
                coefficients of variation in which case Geostatistical
                Ore Reserve Estimation, pp172 M. David points out
                that lognormal kriging may be avoided, from what I
                understand as relative variograms may be used instead
                of lognormal variograms.

                Digby
              • Meng-Ying Li
                Hi people, Finally I got the point of argues for the estimation of population variance. What I had in mind as an overall variance is the variance of all
                Message 7 of 9 , Dec 14, 2004
                  Hi people,

                  Finally I got the point of argues for the estimation of population
                  variance.

                  What I had in mind as an "overall" variance is the variance of all
                  possible locations in any realization of the random field, while Isobel
                  and some other people are trying to explain to me is the variance of all
                  possible realization at any location of the random field.

                  I realized, by noticing this, why along the discussion the stationarity
                  and existence of variance has been emphasized. If the random field is not
                  stationary then we'll have no consistant population variance as Isobel
                  explained. I also learned that my understanding of the population variance
                  has a name called "areal variance."

                  Now, it should be clear why I emphasized on the expected variance of the
                  "future samples." This would be the variance of the any possible sample
                  taken from the current realization (which I called "population"
                  previously) by some planned sampling scheme. And this variance will have
                  to do with the clustering or non-clustering of the future sampling
                  scheme. I'm aware, of course, that in practice "future" samples may no
                  longer be taken from the current realization since in the real case
                  the study site would be changing. Calling it a "future" sample is
                  just a convenient saying for the expected variance based on possible
                  sampling schemes.


                  Hope I'm not getting things more confused.


                  Mng-yng
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