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Re: AI-GEOSTATS: non-ergotic covariance

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  • Marco Alfaro
    Dear Sara: Sorry, but the non ergodic variogram is an artifact! If you do not believe to me, see the comments about my paper in NACOG 1996 or in
    Message 1 of 3 , Mar 15, 2001
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      Dear Sara:

      Sorry, but the "non ergodic" variogram is an artifact!

      If you do not believe to me, see the comments about my paper in NACOG 1996 or in Geostatistics, Volume 8, No.2.

      The Mathematical proof is in my paper titled "Acerca del variograma no ergódico", in Spanish (difficult to get, if you wish I can
      send a copy for you).

      I think that is more easy to see an example (in my paper you have more examples):

      Let a line with data (the data is very regular and has a trend) sampled at regular intervals of 1. Tha data are (20 data):

      1, 1, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 12

      Compute by hand or by your software the non ergodic and the classical variogram (the classical variogram is also non
      ergodic!):

      Conclusions:

      The behavior of the non ergodic variogram, near the origin is linear, and, for the classical variogram is parabolic.

      The non ergodic variogram has a range and a sill, and, the classical variogram always grows.

      I attach a little Qbasic (or QuickBasic) program you can run.

      Best regards,

      Marco Alfaro.

      Listing of the program:

      n = 20
      DIM z(n)
      m = 0 ' mean
      v = 0 ' variance
      FOR i = 1 TO n
      READ z(i)
      m = m + z(i)
      v = v + z(i) * z(i)
      NEXT i
      m = m / n ' mean
      ' the variance is = the non ergodic variogram in the origin
      v = v / n - m * m
      CLS : SCREEN 12
      WINDOW (-10, -10)-(30, 80)
      LINE (0, 0)-(20, 65), 8, B
      LOCATE 3, 10: PRINT "In red, classical variogram, in green, non ergodic variogram"
      FOR k = 0 TO n - 1
      cov = 0 ' the non ergodic variogram.
      head = 0
      tail = 0
      gama = 0 ' the classical variogram.
      FOR i = 1 TO n - k
      hh = z(i)
      tt = z(i + k)
      head = head + hh
      tail = tail + tt
      cov = cov + hh * tt
      gama = gama + (hh - tt) * (hh - tt)
      NEXT i
      head = head / (n - k)
      tail = tail / (n - k)
      cov = cov / (n - k) - head * tail
      gama = .5 * gama / (n - k)
      CIRCLE (k, v - cov), .1, 2
      PAINT (k, v - cov), 2
      CIRCLE (k, gama), .1, 4
      PAINT (k, gama), 4
      NEXT k
      a$ = INPUT$(1)
      END
      DATA 1,1,2,3,3,3,4,4,5,5,6,6,7,7,8,9,10,10,11,12





      Sara Kustron wrote:

      I have been using Variowin for my variogram analysis. My
      covariograms are much better behaved compared to semivariograms, and
      I would like to use the covariance to estimate the semivariance.

      I understand that under the assumption of strict stationarity, which
      is harder to satisfy than the intrinsic hypothesis, semivariance and
      covariance are essentially equivalent. Since I have a non-stationary
      process I would have expected that my covariograms to be just as bad
      as my semivariograms, but this is not the case.

      I've read that Variowin uses "non-ergotic"
      covariance. Exactly what is this and how does it differ from regular
      old covariance? I just want to make sure I understand the
      assumptions of using non-ergotic covariance to estimate model
      parameters.

      Thanks!
      Sara Kustron
      Boston University Department of Geography

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      Sara Kustron wrote:

      > I have been using Variowin for my variogram analysis. My
      > covariograms are much better behaved compared to semivariograms, and
      > I would like to use the covariance to estimate the semivariance.
      >
      > I understand that under the assumption of strict stationarity, which
      > is harder to satisfy than the intrinsic hypothesis, semivariance and
      > covariance are essentially equivalent. Since I have a non-stationary
      > process I would have expected that my covariograms to be just as bad
      > as my semivariograms, but this is not the case.
      >
      > I've read that Variowin uses "non-ergotic"
      > covariance. Exactly what is this and how does it differ from regular
      > old covariance? I just want to make sure I understand the
      > assumptions of using non-ergotic covariance to estimate model
      > parameters.
      >
      > Thanks!
      > Sara Kustron
      > Boston University Department of Geography
      >
      > --
      > * To post a message to the list, send it to ai-geostats@...
      > * As a general service to the users, please remember to post a summary of any useful responses to your questions.
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