Re: [RE: AI-GEOSTATS: SUMMARY: log-kriging & back transformation]
- Thanks Daniel,
I will forward it to the list since it may interest other readers.
daniel guibal <dguibal@...> wrote:
> 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
> Rivoirard, J, 1990 A review of Lognormal estimators for in situ reserves
> Math. Geology, v. 22, no. 2, p. 213 and sq.
> Daniel Guibal, FAusIMM(CP), MMICA, MGAA, Min.Eng.
> Technical Director
> SRK Consulting
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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
> Englund's paper, which one to use, or what is the relative impactAI-GEOSTATS,
> of the Lagrange parameter, ...
> The replies summarised hereafter are from Denis Marcotte, Evan Englund,
> 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
> 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
> see papers section)in
> In his paper "Evaluation and comparison of spatial interpolators" (Math.
> 24:4, pp. 381-391, 1992), 6 different log kriging back-transforms are
> with other estimators.
> His "conclusion about log kriging is that the variance and Lagrange terms
> the back transform aren't enough - you still need the meanquantities,
> 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
> including the confidence intervals.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,
> This avoids the Lagrangian parameyet and provides accurate results
> 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
> c back transform for log transform
> 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
> is a commercial package. http://geoecosse.bizland.com
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