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AI-GEOSTATS: Extreme bias in autocoviariance function estimator

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  • Nicholas Lewin-Koh
    Hi, I hope this is not a dumb question. I have set of sites which are discrete and connected by a graph. On each site (vertex in graph theory parlance) lives a
    Message 1 of 3 , Dec 18, 2001
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      Hi,
      I hope this is not a dumb question. I have set of sites which are discrete
      and connected by a graph. On each site (vertex in graph theory
      parlance) lives a population which is modeled as a discrete time ricker
      model, n(t+1)=n(t)exp[r(1-n(t)/k) + e] - E + M, E is a proportion of the
      population migrating out only along the edges of the graph and M is the
      sum of migrants coming in for other vertices and e is a random error
      term correlated in space but independent in time. Now I am interested in
      the correlation cor(v_i, v_j) as a function of distance. I set the
      migration to zero, so that after a sufficient amount of time all
      populations hover around their equilibrium K. I have the correlation in
      the random errors e specified as p^||s_i-s_j||, an isotropic covariance
      that is a function of distance alone.

      Now the question, if I calculate all pairwise correlations between
      vertices and fit a smoother, I recover the specified corelation function
      with very little bias (provided t is large enough), and this seems to work
      for a relatively small sample of vertices. However if I caculate the
      auto-correlation function using the standard estimators and average them
      over time, or calculate the average of the standardized populations and
      fit the auto-correlation function, the result is severely biased (under
      estimates) the correlation as a function of distance, even for large v. I
      have some hunches about this, but does anyone know where I can find
      some more information on this?

      Nicholas

      CH3
      |
      N Nicholas Lewin-Koh
      / \ Dept of Statistics
      N----C C==O Program in Ecology and Evolutionary Biology
      || || | Iowa State University
      || || | Ames, IA 50011
      CH C N--CH3 http://www.public.iastate.edu/~nlewin
      \ / \ / nlewin@...
      N C
      | || Currently
      CH3 O Graphics Lab
      School of Computing
      National University of Singapore
      The Real Part of Coffee kohnicho@...


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    • Isobel Clark
      Julhendra That is what your semi-variogram is for. Determine maximum distance and anisotropy (change with direction) from your experimental semi-variograms.
      Message 2 of 3 , Dec 19, 2001
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        Julhendra

        That is what your semi-variogram is for. Determine
        maximum distance and anisotropy (change with
        direction) from your experimental semi-variograms.

        Your search strategy should also change with the shape
        of your blast layout. Single blast patterns are
        usually 'long and thin', meaning that a circular
        search would be less than optimal. I don't know of any
        papers which discuss this, but you can see our
        strategy in the kriging game. This is freely
        downloadable at:

        http://uk.geocities.com/drisobelclark/briefcase.html

        and allows you to experiment with search patterns and
        changes in semi-variogram model. It also allows you to
        see the difference between kriging as a 'point'
        estimation method (for mapping) and kriging an average
        over a blast area. Unfortunately, this free package is
        only 2d, but you may find it useful.

        One other point you may find useful. In my experience,
        working in 3d reduces to using the previous bench
        blastholes. A full 3d approach is only useful if you
        have good diamond or percussion drilling within the
        search volume. If you are going to combine sampling
        types, you need to determine whether the samples are
        compatible or to use a co-kriging approach.

        Isobel Clark



        --- Julhendra Solin <Julhendra_Solin@...> wrote:
        > Dear All,
        >
        > I am working on blasthole interpolation in open pit
        > mine. Interpolation
        > using ordinary kriging and grades interpolated also
        > from above bench.
        > Blasthole spacing about 10 m and bench height 15 m.
        > Anybody could help me
        > how to determine search distance and min/max of
        > number of samples to krige
        > the grade. May be some technical paper related to
        > it.
        >
        > Thanks.
        >
        > Jul
        >
        >
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