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Re: GEOSTATS: Nugget Effect, Jacknife, CV & Bootstrap

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  • Pierre Goovaerts
    Hello, The main difference between the Error Variance (EV) and Micro Variance (MV) is that MV=0 for h=0 and not EV. Thus, whenever h 0 both components are
    Message 1 of 2 , Apr 3 2:55 PM
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      Hello,

      The main difference between the Error Variance (EV) and
      Micro Variance (MV) is that MV=0 for h=0 and not EV.
      Thus, whenever h>0 both components are incorporated
      into the computation of the nugget effect.
      I am not surprised that you get similar estimates
      (at unsampled locations) if you use the same NE
      since the repartition of the nugget variance between
      Error Variance and Micro Variance won't impact the
      computation of the covariance values in your kriging
      system except for the diagonal of the left-hand-side
      kriging matrix. At sampled locations though the two
      systems will yield different estimates.

      The relative importance of the two components can be quantified
      if you have collected information about measurement errors
      (e.g. through duplicates or triplicates): many examples can
      be found in the soil science literature. An alternative is to look
      at the nugget effect of the cross variogram between two
      variables which, under the assumption that measurement errors
      of two variables are uncorrelated, can help the interpretation of
      the nugget effect of the two direct variograms (non-zero nugget
      effect on the cross variogram is due to micro-scale variation
      common to both variables), see my book. p. 102-103
      and the reference:
      Goovaerts, P. and R. Webster. 1994.
      Scale-dependent correlation between topsoil copper and cobalt
      concentrations in Scotland.
      European Journal of Soil Science, 45(1):79--95.

      Cheers,

      Pierre


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      ________ ________
      | \ / | Pierre Goovaerts
      |_ \ / _| Assistant professor
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      On 2 Apr 2000, Gregoire Dubois wrote:

      > Dear all,
      >
      > I have two questions.
      >
      > The first one is about the use of cross-validation, jacknife and bootstrap in
      > geostatistics while developed for independent and identically distributed
      > observations. What are the consequences of the application of these techniques
      > to dependent data and what can be done to adapt these techniques to
      > geostatistics.
      >
      > My second question is about the nugget effect, which can be defined as
      >
      > Nugget Effect (NE) = Error Variance (EV) + Micro Variance (MV) ,
      >
      > where the error variance is the variance of the measurement errors, and the
      > micro variance is the variance of the small scale structure.
      >
      > This definition is the one implemented in Surfer and follows the
      > recommendation of Cressie
      >
      > CRESSIE N. (1993)
      > Statistics for spatial data
      > John Wiley & Sons Inc. (Revised Edition)
      > (see pages 127 - 130)
      >
      > When EV = 0 and NE > 0, the estimates will honor every observation, when MV =
      > 0 and NE > 0, the estimates do not honor every observation. Both situations
      > will have a smoothing effect.
      >
      > I have made a few tests to evaluate the relative impact of both components but
      > could not see differences between EV and MV. For a given value of the NE and
      > for different combinations of EV and MV, I always get the same estimates. Does
      > anyone have an explanation for that ?
      >
      > I couldn't find any references to publications where both components are
      > quantified in a case study and would welcome any suggestions.
      >
      > Thank you for any help
      >
      > 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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