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Re: [RE: AI-GEOSTATS: SUMMARY: log-kriging & back transformation]

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  • Gregoire Dubois
    Thanks Daniel, I will forward it to the list since it may interest other readers. Best regards, Gregoire ... they ... is ... Evan ... AI-GEOSTATS, ... in ...
    Message 1 of 1 , May 15, 2003
      Thanks Daniel,

      I will forward it to the list since it may interest other readers.

      Best regards,

      Gregoire

      daniel guibal <dguibal@...> wrote:
      > Gregoire,
      >
      > In addition to the references given by Denis Marcotte, I would add the
      > following a bit more recent, and which summarises the various options very
      > neatly:
      >
      > Rivoirard, J, 1990 A review of Lognormal estimators for in situ reserves
      > Math. Geology, v. 22, no. 2, p. 213 and sq.
      >
      > Regards
      >
      > Daniel
      >
      > Daniel Guibal, FAusIMM(CP), MMICA, MGAA, Min.Eng.
      > Technical Director
      > SRK Consulting
      > SRK Consulting are Australasia's
      > leading integrated science and engineering consultancy.
      > visit our web site===> www.srk.com.au
      > 1064 Hay St
      > West Perth 6005
      > ph: +61 (0) 8 9288 2000
      > fax: +61 (0) 8 9288 2001
      > mob: +61 (0) 409 382 427
      > PO Box 943 West Perth
      > Western Australia 6872
      >
      >
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      >
      > -----Original Message-----
      > From: Gregoire Dubois [mailto:gregoire.dubois@...]
      > Sent: Thursday, 15 May 2003 5:45 PM
      > To: ai-geostats@...
      > Subject: AI-GEOSTATS: SUMMARY: log-kriging & back transformation
      >
      > Dear all,
      >
      > here is a (long) summary to the replies I got to my question on back
      > transformation after log-normal kriging. I imagine there will be a few
      > reactions to the possible approaches.
      >
      > Open questions still remains, e.g.see the 6 (!) back transformations in
      Evan
      > Englund's paper, which one to use, or what is the relative impact
      > of the Lagrange parameter, ...
      >
      > The replies summarised hereafter are from Denis Marcotte, Evan Englund,
      > Edzer
      > Pebesma, Paulo Ribeiro, Hirotaka Saito, Pierre Goovaerts and Isobel Clark.
      >
      > Thanks a lot to everyone.
      >
      >
      > A. Block kriging & back-transformation of logarithms
      > ++++++++++++++++++++++++++++++++++++++++++++++
      >
      > 1.1.) Pierre reminded me that back transformation of log transformed block
      > kriging estimates does not make sense.
      >
      > See Goovaerts' book (page 156)
      >
      > 1.2) Denis Marcotte suggested the following:
      >
      > "I think a possible way to diminish the risk of bias is to do point
      > conditional simulation. The point values are back-transformed and then
      > the block spatial average is done for each realization. The average of
      > many realizations gives you an estimate of the block mean value. I
      > expect the results with this approach to be less sensitive to the choice
      > of the variogram model than the usual bias correcting formulas but I did
      > not check this conjecture. In any case, whatever the method you choose,
      > you must conduct a careful cross-validation study to ensure there is no
      > bias (at least at the global level).
      >
      > If you are only interested in the mean value over blocks, I think it is
      > safer to use ordinary kriging on the raw data rather than lognormal
      > kriging or conditional simulation. You can estimate the variogram on the
      > log values and then transform it to the raw space (under a bivariate
      > lognormal hypothesis). Written in terms of covariances, the relation is
      > simply C(h)=m^2(exp(Cl(h))-1) where m is the population raw mean, C(h)
      > is the raw covariance and Cl(h) is the log-covariance. Note that because
      > of the non-linearity of the relation, the shape of the variogram also
      > changes. Thus a spherical variogram in the log-space does not map to a
      > spherical variogram in the raw-space.
      >
      > Two alternatives are possible:
      >
      > 1- compute numerically all the covariances required for kriging or
      >
      > 2- transform numerically the log-covariances and fit a combination of
      > standard models to the transformed covariances."
      >
      >
      >
      >
      > B. Point kriging & back-transformation of logarithms
      > ++++++++++++++++++++++++++++++++++++++++++++++
      >
      >
      > 2.1) From Denis again, I got the following references which discuss the
      > various aspects and hypothesis involved in point and block lognormal
      > kriging:
      >
      > Dowd, P. A., 1982, Lognormal kriging-The general case; Math. Geology, v.
      > 14, no. 5, p. 475-499.
      >
      > Journel, A.G., 1980, The lognormal approach to predicting local
      > distributions of selective mining unit grades: Math. Geology, v. 12,
      > no. 4, p. 285-304.
      >
      > Rendu, J.M., 1979, Normal and lognormal estimation: Math. Geology, v.
      > 11, no. 4, p. 407-422.
      >
      > 2.2) Evan provided me with 2 papers (will be available today on
      AI-GEOSTATS,
      > see papers section)
      >
      > In his paper "Evaluation and comparison of spatial interpolators" (Math.
      > geol.
      > 24:4, pp. 381-391, 1992), 6 different log kriging back-transforms are
      > compared
      > with other estimators.
      >
      > His "conclusion about log kriging is that the variance and Lagrange terms
      in
      > the back transform aren't enough - you still need the mean
      > correction.... but if you do the mean correction, you don't need the
      > others. That means you don't need to worry about extracting the
      > Lagrange parameter from the kriging run. My conclusion about
      > interpolation in general is that if I can estimate the ordinary
      > variogram, I stick with ordinary kriging, even with log normal data.
      > All the transforming, back-transforming and bias correcting aren't worth
      > the trouble."
      >
      >
      > 2.3) Paulo suggested to use simulations to compute the necessary
      quantities,
      > including the confidence intervals.
      >
      > In summary the idea is to:
      >
      > - work in the transformed scale
      > - simulate from the predictive distribution
      > - back-transform simulation
      > - use simulations to compute estimations (mean, median, quantiles,
      intervals
      > etc.)
      >
      > This avoids the Lagrangian parameyet and provides accurate results
      regarding
      > the back-transformation. Furthermore this is valid for a wider group of
      > transformation, not only for the log.
      >
      > 2.4) The approach used by Edzer in Journal of Hydrology 200, p. 364-386 was
      >
      > 1. estimate block mean concentrations on the log scale, and std.errors
      > 2. calculate approximate 95% predictions intervals by est +/- 2 * std.err
      > 3. back-transform both sides of the interval by taking the exponent.
      >
      > What results is not an interval estimate of the block mean value (which
      > may be outside this interval!) but an estimate of the block geometric
      > mean value. When, on the log scale block mean and block median coincide
      > (e.g. when log-concentrations within a block are symmetrically distributed)
      > this value coincides with the block median value.
      >
      >
      > C. geostats packages & back-transformation of logarithms
      > ++++++++++++++++++++++++++++++++++++++++++++++
      >
      > 3.1) Hirotaka added the unbiased transformation in GSLIB. By adding the
      > following line into kt3d.f ,that kriging estimates are automatically
      > backtransformed
      >
      > c
      > c back transform for log transform
      > c
      > est=exp(est+estv/2-s(na+1))
      >
      > The file will be available online when received.
      >
      > 3.2) Edzer and Paulo reffered to GeoR that is doing the back
      > transformation.geoR implements kriging and bayesian kriging including the
      > family of Box-Cox transformation for which the log. See point 2.3. above.
      >
      > 3.3) Isobel mentioned that her software does such transformation. The
      > software
      > is a commercial package. http://geoecosse.bizland.com
      >
      >
      > Gregoire
      >
      >
      >
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