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GEOSTATS: kriging in binary fields

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  • Ali_Alaa@seo.state.nm.us
    Dear Victor, You really don t need a stochastic interpolator, instead you need a stochastic simulator. There are two ways: 1) Gaussian Sequential Simulation,
    Message 1 of 3 , Apr 30, 1997
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      Dear Victor,
      You really don't need a stochastic interpolator, instead you need a
      stochastic simulator. There are two ways:
      1) Gaussian Sequential Simulation, or Kernel Sequential Simulation
      2) Boot strap.
      In (1), Gaussian Sequential Simulation is a Kriging based algorithm.
      Let's say if we have 1, we have an event occurrence, and if we have 0,
      we don't have such an occurrence. Given n data points, and m locations
      to be simulated, the simulation proceeds as follows:
      1) visit the m locations randomly (i.e., select a random path that
      visits each location only once).
      2) At location i on this random path, Apply Indicator Kriging using all
      n data points and the previously simulated data at i-1 locations, to
      find the probability "p" of an event occurrence. Note that 0=<p=<1. SO
      if we are a locale where data points are mostly 0, p will be very close
      to 0, and vice versa.
      3) Sample this location using p, and a random Uniform variate U[0,1]
      if p.gt.U then this location is 1, and if p.lt.0, this location is 0.
      The equal sign can be used with either without any significant bias.
      The question which will immediately arises is "do I need to define the
      variogram every time I increase the number of data points during the
      progress of the simulation". The answer is yes, but there is a smart
      way of doing it. Refer to GSLIB for a complete description and a
      readily available code for this method.

      In (1), also you will find Kernel Sequential simulation, it is similar
      to the one described above except that you use Kernel estimator rather
      than Indicator Kriging (refer to my paper Ali and Lall, Ground Water,
      34, 4, 647, 1996). Also visit my homepage to see images based on this
      methodology.

      In (2), bootstrap technique:
      This is a naturally resampling method without any assumption (such as
      stationarity), and much more simpler than the above methods.
      Define KNN (Kth Nearest Neighbor) value (sqrt of the data available as a
      rule of thumb). Given n data points, and m unsampled locations,
      simulation proceeds as follows:
      1) select a random path to visit all m points
      2) at location i, define KNN (KNN =sqrt(n+i-1)) points to location i
      3) Define associated weights to these points (Inverse distance, Kernel
      weights, ...etc). Definitely, if you want to be fancy, you may select
      these weights based on the local correlation structure (if you have
      enough data). Those weights have to be normalized to sum up to 1.
      4) Using such weights, and a generated U[0,1], resample location i,
      for example
      say you have 20 points with 20 weight: i.e., w1, w2, w3, ...w20
      then you have the following quantities:
      W1=w1
      W2=w1+w2
      W3=w1+w2+w3
      ..
      ..
      W19=w1+w2+......w19
      W20=1

      At location i, resample point 11 for example, if W10<u<W11, and so on.

      Now if you used any weights which are not based on the local correlation
      structure, you may need to compare the correlation structure with the
      correlation structure of the resulting realizations. (if your algorithm
      is correct, 100 to 1000 realizations should reproduce an average
      structure similar to that observed from data).

      Good luck..
      Alaa Ali

      > Dear ai-geostats members,
      >
      > This note is to request information about the application of
      > kriging methodology for prediction in binary random fields, i.e.
      > the observed response can take only two possible values, say 0 and 1,
      > and the purpose is to infer the values of the process at many
      > ungauged locations, so the predictions must be also 0 or 1.
      >
      > The problem and a description of its possible solution is described
      > in a paper by Solow (1986), Math. Geol., but there is no real
      application
      > where its soundness is tested. I have been unable to find any paper
      > where this (or some other variant) is applied to an actual data set.
      > Information on this regard is highly appreciated.
      >
      > The only related methodology I'm aware of is what is called Indicator
      > kriging, but it does not seem appropriate for these sort of data.
      > As far as I understand, this is used in the prediction of
      > continuous variables, which are clipped at various thresholds, and
      then
      > all the induced binary variables are used to predict the original
      > continuous variable (by cokriging).
      > In the case I'm interested in, the original response variable is
      > binary to start with.
      >
      > Thanks in advance.
      >
      > Victor De Oliveira
      > Department of Mathematics
      > University of Maryland
      > College Park, MD 20742
      >
      > (301)4055074
      > vdo@...
      >
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    • Ashley Francis
      -- *To post a message to the list, send it to ai-geostats@gis.psu.edu. *As a general service to list users, please remember to post a summary of any useful
      Message 2 of 3 , May 8, 1997
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