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GEOSTATS: Summary 2: decomposition of nugget effect

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  • Gregoire Dubois
    Hello again, here is a summary of the replies I received to my second question (on the decomposition of the nugget effect). Thanks again a lot to all who
    Message 1 of 1 , Sep 6, 1905
      Hello again,

      here is a summary of the replies I received to my second question (on the
      decomposition of the nugget effect). Thanks again a lot to all who
      contributed.

      2nd question:

      Andy Long (aelon@...), Donald Myers (myers@...) and
      Denis Marcotte (Denis.Marcotte@...) have underlined the
      differences between the theoretical definition of Kriging and its
      implementation in most software. Kriging is an exact interpolator since it
      always will return the value of the site if evaluated at a site. Kriging makes
      no allowance for how the nugget is modelled. Kriging is a JUMP interpolator
      in the case of a nugget, which means that the surface will not be continuous
      but still exact.

      This is clear from the simple kriging equations:

      |v(1,1) v(1,2) ... v(1,N)| |lambda_1| = |v(1,x)|
      |v(2,1) v(2,2) ... v(2,N)| |lambda_2| = |v(2,x)|
      | | | | = | |
      |v(N,1) v(N,2) ... v(N,N)| |lambda_N| = |v(N,x)|

      Now if x happens to be one of the points 1,...,N, say i, then the solution of
      this system is simply lambda_i=1, all other lambda=0, which means that the
      data location i will be given all the weight, and will return thus z_i - exact
      interpolation

      Donald Myers (myers@...) also writes that "The usual versions of
      the ordinary, universal kriging equations do not separate the nugget into two
      parts and hence if the nugget term is non-zero there will be a jump
      discontinuity at the data locations (it will still be an exact interpolator).
      However one can modify the kriging equations slightly, if written in variogram
      form this means that the diagonal entries are non-zero (the values are the
      value of the error variance), the remainder of the nugget term appears in the
      variogram values off-diagonal. You will not see this modified version in much
      of the geostatistical literature. Cressie discusses it in his book and also in
      a paper that appeared in the American Statistician.

      You will also find the equations in the following papers

      1994, Myers,D.E., Statistical Methods for Interpolation of Spatial Data.
      J. Applied Science and Computations 1, 283-318

      1994, Myers,D.E., Spatial Interpolation: An Overview. Geoderma 62, 17-28"

      Denis Marcotte (Denis.Marcotte@...) writes


      The difference between both form of estimates appear only at a
      sample point. For every other point (even an epsilon away), you will
      get exactly the same estimates. This is possibly one of the reason why
      the distinction between EV and MV contributions to the nugget was not
      explicitely taken into account in the kriging equations in geostatistical
      textbooks before Cressie's book. At a sample point (and of course using this
      sample point for the estimation), splitting the nugget in EV and MV will
      produce different estimates if programmed properly (it is then nothing else
      than a special case of factorial kriging). The reason why the estimates differ
      is that with Cressies'equation you are estimating Y(xi), not
      Z(xi)=Y(xi)+e(xi) like with other kriging programs.


      Another reference given by E. Pebesma is

      Ronald Christensen Linear Models for Multivariate, Time Series and Spatial
      Data (Springer Texts inStatistics)
      Hardcover (January 1991)
      Springer Verlag; ISBN: 038797413X

      Jeff Myers (jeff_myers@...) also underlines the following:
      "The micro variance MV should be highly correlated with the Fundamental
      Error (FE) of sampling, and to some degree the grouping and segregation
      error (GE). For particulate materials (soils, etc.), the FE can be
      extremely high (> 1000% relative error). This is true for environmental
      contaminants such as explosives and PCBs or mining variables such as
      precious metals. the United States Environmental Protection Agency has
      recognized this and now has guidance documents (SW-846 Chapter Nine, FFFI)
      to assist in the evaluation of the FE, based on the work of Pierre Gy.
      This problem is discussed in Pitard (1993), Myers (1997), and Gy (1998?).
      FE is highly dependent on the sample and subsample support (mass) and
      occurs at each sampling step (which means it is additive), so it can can
      have a great influence on the nugget effect.

      In contrast, typical laboratory measurement errors in the environment are
      on the order of 20 to 30%. If the FE is less than 20-30%, it will be hard
      to distinguish whether the nugget is a result of sampling or measurement
      error. An interesting website which deals with this problem is

      http://pubs.acs.org/hotartcl/ac/99/aug/settle.html

      "

      It is 1 AM here, I'm off to bed and will dream of worlds without
      nuggets...

      Gregoire


      Gregoire Dubois
      Section of Earth Sciences
      Institute of Mineralogy and Petrography
      University of Lausanne
      Switzerland

      Currently detached in Italy

      http://curie.ei.jrc.it/ai-geostats.htm

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