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Re: [SPAM] AI-GEOSTATS: Microgeostatistics

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  • Gerald Boogaart
    Dear Steven, A new term such as microgeostatistics is necessary only, when there is something (mathematically) special with it. Special features might be: -
    Message 1 of 5 , Mar 8, 2004
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      Dear Steven,

      A new term such as microgeostatistics is necessary only, when there is
      something (mathematically) special with it.

      Special features might be:

      - The measurement locations might have a relevant error.

      - It is no longer possible to measure a point or block, but you get weighted
      means or maximum measurements in of a little area (e.g. of the size of a
      measurement needle)

      - Stationarity might not be given any longer, and might not be necessary
      since we get the same covariance structure for a large set of individual data
      sets (you don't have many earths but many microorganisms and many
      experimenting bowls)

      - The location support might be the threedimensional mannifold-surface of e.g.
      a hair and not the in first order flat earth

      - interpolation might not be the final goal, but an intermediate one leading
      to the more complex question of local influences.

      What does the list think is special about geostatistics applied to microscale
      problems?

      Best regards,
      Gerald

      >I'm working in
      > microbial ecology and am interested in finding information about spatial
      > analysis or interpolation at extremely small scales (millimeters, microns,
      > etc.)., especially as related to survival and fitness of microorganisms.
      > I've coined the term, "microgeostatistics" to describe this subject. I
      > could be wrong, but it seems like there is comparatively little information
      > on this general area.Any suggestions are
      > greatly appreciated!
      >Steven Rogers
      >
      --
      -------------------------------------------------
      Prof. Dr. K. Gerald v.d. Boogaart
      Professor als Juniorprofessor für Statistik
      http://www.math-inf.uni-greifswald.de/statistik/

      office: Franz-Mehring-Str. 48, 1.Etage rechts
      e-mail: Gerald.Boogaart@...
      phone: 00+49 (0)3834/86-4621
      fax: 00+49 (0)89-1488-293932 (Faxmail)
      fax: 00+49 (0)3834/86-4615 (Institut)

      paper-mail:
      Ernst-Moritz-Arndt-Universität Greifswald
      Institut für Mathematik und Informatik
      Jahnstr. 15a
      17487 Greifswald
      Germany
      --------------------------------------------------


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    • Noemi Barabas
      Dear list, I am working on a kriging problem of log-PCB concentrations in river sediments (the coordinates have been straightened ), using GSLib. I have
      Message 2 of 5 , Mar 8, 2004
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        Dear list,

        I am working on a kriging problem of log-PCB concentrations in
        river sediments (the coordinates have been "straightened"), using GSLib.
        I have strong anisotropy with a ratio of about 1:6 (x:y). I have some
        clustered locations as well as some sparsely sampled areas, and several
        instances where the high and low concentrations are found very close to
        eachother. The distribution is lognormal and I am working with
        log-transformed values. The variograms are rather nice in both
        directions. Nevertheless, ordinary kriging gives a very peculiar-looking
        map (of log-concentrations). It would be too difficult to put into
        words, so I have included maps of estimates, variance and local mean as
        an attachment.

        Does anybody know what causes this "plaid" effect? Looking at the map
        of variances, it appears that an estimation location has low variance
        if it has a data point directly above and next to it, but intermediate
        variance if those same two data points are in a diagonal direction
        relative to the axes of anisotropy, even if the new position takes the
        estimation point closer to the data points. I would like to undestand the
        reason for this effect, as well as whether there is something that can be
        done about it.

        Could the fact that there are high values embedded in low value locations
        be partially responsible for these strange maps?

        (I did experiment with octant search, various maximum search radii,
        various min and max number of data points for estimation, and this effect
        persists. I even reversed the angles of anisotropy, tried different
        variogram ranges. The variogram ranges are about 20% of the width/length
        of the domain, and the relative nugget effect is about 6% in both
        directions)

        Thanks very much!

        Noemi




        [Non-text portions of this message have been removed]
      • Monica Palaseanu-Lovejoy
        Hi, I am working myself with pollution data in soils and i have very high values very close to very low values, and highly skewed distribution. I am more and
        Message 3 of 5 , Mar 9, 2004
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          Hi,

          I am working myself with pollution data in soils and i have very high
          values very close to very low values, and highly skewed
          distribution. I am more and more concerned with doing kriging on
          transformed data. This simply means we believe the data came
          from only one population. But what if it comes from 2 different
          populations representing 2 different polluting processes? Much
          more if we do believe there are no gross error measurements. The
          fact that high values are very close to low values would tell me that
          the spatial autocorrelation is violated locally. I would try first to see
          if the outliers (local and global) represent a different population, if
          these values cluster or not, how significant is the association high-
          low values, and if the global Moran's I increases if i eliminate the
          "outliers". Maybe the majority of the data which have a higher
          spatial autocorrelation belong to a "better expressed" diffusive
          process, (maybe an older one) while the rest of the data which
          were identified as outliers before, represent a more patch-y or point
          source pollution process which didn't have time to diffuse over the
          entire study area (a younger process, maybe?).

          Of course if you have proof that the data came from only one
          population then .... it is a different story.

          I will really appreciate to hear other opinions about these thoughts.

          Thanks,

          Monica

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        • Ruben Roa Ureta
          ... Exploratory analysis of the frequency distribution of the data (i.e. the aggregated, non-spatial, frequency) could reveal the existence of two (or more)
          Message 4 of 5 , Mar 9, 2004
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            > Hi,
            >
            > I am working myself with pollution data in soils and i have very high
            > values very close to very low values, and highly skewed
            > distribution. I am more and more concerned with doing kriging on
            > transformed data. This simply means we believe the data came
            > from only one population. But what if it comes from 2 different
            > populations representing 2 different polluting processes? Much
            > more if we do believe there are no gross error measurements. The
            > fact that high values are very close to low values would tell me that
            > the spatial autocorrelation is violated locally. I would try first to see
            > if the outliers (local and global) represent a different population, if
            > these values cluster or not, how significant is the association high-
            > low values, and if the global Moran's I increases if i eliminate the
            > "outliers". Maybe the majority of the data which have a higher
            > spatial autocorrelation belong to a "better expressed" diffusive
            > process, (maybe an older one) while the rest of the data which
            > were identified as outliers before, represent a more patch-y or point
            > source pollution process which didn't have time to diffuse over the
            > entire study area (a younger process, maybe?).

            Exploratory analysis of the frequency distribution of the data (i.e. the
            aggregated, non-spatial, frequency) could reveal the existence of two (or
            more) populations. To evaluate the evidence in favour of such an
            hypothesis, you could compare the hypothesis that the frequency
            distribution is formed by a mixture of two (or more) specified
            distributions versus the hypothesis that it is formed by only one. The
            general topic in statistics is called 'mixture distribution analysis' (not
            to be confused with 'mixture models'). Useful references are:

            Everitt & Hand, 1981, Mixture distribution analysis. Chapman & Hall
            Chen & Chen, 2001, Statistics and Probability Letters 52:125
            Hawkins et al., 2001, Computational Statistics & Data Analysis 38:15
            http://www.math.mcmaster.ca/peter/mix/mix.html

            Some robust regression methods, for example, are based on treating the
            data as coming from a mixture of two distributions, the main one, and a
            contaminating distribution.

            If you conclude that there are two (or more) distributions, then you can
            compute the maximum conditional probability that any given data point
            belong to any of the two (or more) distributions, and use this computation
            to classify data. After this exploratory analysis, you could treat the two
            (or more) populations differently, if there is evidence for a mixture, and
            maybe even perform separate geostatistical analyses on the separate
            populations.

            I used this general strategy in the analysis of a time series of an index
            of returns from investments in finantial markets. The strategy was
            proposed by Hamilton, 1994, Time Series Analysis, Ch. 22, Princeton U. P.

            Ruben

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          • Pierre Goovaerts
            Hello, I agree that in many environmental datasets we could question the assumption of existence of a single population. Although there are ways to split the
            Message 5 of 5 , Mar 9, 2004
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              Hello,

              I agree that in many environmental datasets we could question the
              assumption of existence of a single population. Although there are
              ways to split the data into several populations, the key issue is
              that the study area needs also to be stratified into several populations.
              In some fields, such as geology, geological maps could provide
              a stratification of the study area and helps delineating the boundaries
              between populations. This is far less obvious for environmental
              data sets.

              Looking at Noemi's maps, I would agree with Richard's comment that
              nothing seems to be out of the ordinary. Of course, when dealing with
              streams the data configuration is far from optimal and screening effects
              abound. Also, the strong anisotropy ratio means that we deal with
              a "zonal-like" anisotopy which might cause sudden changes of covariance
              for slight difference of angles. In particular, this covariance model
              could lead to very small correlations off the two main axes of anisotropy,
              which could explain the larger kriging variance observed along the
              diagonal directions.

              Pierre

              <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

              Dr. Pierre Goovaerts
              President of PGeostat, LLC
              Chief Scientist with Biomedware Inc.
              710 Ridgemont Lane
              Ann Arbor, Michigan, 48103-1535, U.S.A.

              E-mail: goovaert@...
              Phone: (734) 668-9900
              Fax: (734) 668-7788
              http://alumni.engin.umich.edu/~goovaert/

              <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

              On Tue, 9 Mar 2004, Monica Palaseanu-Lovejoy wrote:

              > Hi,
              >
              > I am working myself with pollution data in soils and i have very high
              > values very close to very low values, and highly skewed
              > distribution. I am more and more concerned with doing kriging on
              > transformed data. This simply means we believe the data came
              > from only one population. But what if it comes from 2 different
              > populations representing 2 different polluting processes? Much
              > more if we do believe there are no gross error measurements. The
              > fact that high values are very close to low values would tell me that
              > the spatial autocorrelation is violated locally. I would try first to see
              > if the outliers (local and global) represent a different population, if
              > these values cluster or not, how significant is the association high-
              > low values, and if the global Moran's I increases if i eliminate the
              > "outliers". Maybe the majority of the data which have a higher
              > spatial autocorrelation belong to a "better expressed" diffusive
              > process, (maybe an older one) while the rest of the data which
              > were identified as outliers before, represent a more patch-y or point
              > source pollution process which didn't have time to diffuse over the
              > entire study area (a younger process, maybe?).
              >
              > Of course if you have proof that the data came from only one
              > population then .... it is a different story.
              >
              > I will really appreciate to hear other opinions about these thoughts.
              >
              > Thanks,
              >
              > Monica
              >
              > --
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              >

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