## GEOSTATS: Anamophosis

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• ... making ... Hi, if I understand well, what you need is an anamorphosis to transform your data into gaussian data. There are at least 3 questions : 1) Is it
Message 1 of 1 , Aug 11, 1998
Dans un courrier daté du 10/08/1998 20:26:24, vous avez écrit :

> Hello ,
>
> I want to transform data in gaussian data by using a gaussian anamorphosis
> based on Hermite Polynomials,
> The purpose of this transformation is to use turning bands method for
making
> simulations.
> I would like to have some comments on this kind of transformation : which
> criteria I have to take care, problems that may appear after this
> manipulation...
>
> I have an exhaustive population and a sample taken from it : can I apply a
> same anamorphosis on the two data sets ?
>
>
> Thank you.
>
> Isabelle Feunette
>

Hi,
if I understand well, what you need is an anamorphosis to transform your data
into
gaussian data.
There are at least 3 questions :
1) Is it appropriate to consider that your data are a transformation of
gaussian data
(is it consistent with the bi-variate , tri-variate ... distribution of your
sample ?)
2) How to estimate the marginal distribution of your data
3) How to estimate properly the covariance of your underlying gaussian random
function
in a consistent way

1) you can find in the following reference some statistical tests
@MISC{Fouquet93,
author="de Fouquet, C.",
title="Simulation conditionnelle de fonctions al\'eatoires~:
cas gaussien stationnaire et sch\'ema lin\'eaire",
year=1993,
month=nov,
howpublished="Cours du Centre de G\'eostatistique de
l'\'Ecole des Mines de Paris"}

@TECHREPORT{Matheron82,
author="Matheron, G.",
institution=EMP,
title="La destructuration des hautes teneurs et le krigeage des indicatrices",
year=1982,
month=jun,
number="N-761",
type="Note du centre de G\'eostatistique"}

2) There are a lot of parametric and non parametric
(and also semi parametric) methods to do that.
Hermite Polynomial expansion of the empirical c.d.f
is a non parametric method among others.
It is described in

@MISC{Lajaunie93,
author="Lajaunie, C.",
title="L'estimation g\'eostatistique non lin\'eaire",
year=1993,
howpublished="Cours du Centre de g\'eostatistique de l'\'Ecole
des Mines de Paris",
month=nov,
number="C-152"}

This method have some statisitcal optimality properties
but as all parametric method
it is not very flexible both from a theorical and a practical point of view.

3) The problem is that you can't perform the estimation of the covariance
and of the monovariate distribution of your data separately.
Basically, the choice of the distribution implies some limitations for the
covariance.
(e.g the periodic covariance h->cos(wh) is not compatible with a lognormal
distribution)

The mathematical questions behind this are really difficult and most
of the basic questions around this are still open.
The implementation of consistent estimation schemes are rather cumbersome.
I suggest you to have a look to some papers of Matheron to have a flavour of
this.
The book of Rivoirard is very clear and usefull from a practical point of
view.
Armstrong and Diamond give an illustration of some problems pointed by
Matheron.
My own papers tackle with the question of non continuous cumulative
distribution function
model for data and relate an implementation for rain data.

@ARTICLE{Armstrong92,
author="Armstrong , M.",
title="Positive definiteness is not enough",
journal=MG,
year=1992,
volume=24,
number=1,
pages="135-143"}

@ARTICLE{GG98a,
author="Guillot, G.",
title="Sahelian rainfall fields modelling with meta-gaussian random functions
1 : model definition and methodology",
journal=Stochastic Hydrology and Hydraulics ,
note="in print"
year=1998,
key= "a"}

@ARTICLE{GG98b,
author="Guillot, G. and Lebel, T.",
title="Sahelian rainfall fields modelling with meta-gaussian
random functions 2 : parameter estimation and comparison to data",
journal=SHH,
note="in print",
year=1998,
key= "b"}

@INPROCEEDINGS{Matheron89b,
editor="Dowd, P.A. and Armstrong, M.",
title="The internal consistency of models in geostatistics",
year=1989,
author="Matheron,G.",
booktitle="Geostatistics",
publisher=KAP,
volume=1,
pages="21-38"}

@INPROCEEDINGS{Matheron93,
title="Une conjecture sur la covariance d'un ensemble al\'eatoire",
year=1993,
author="Matheron, G.",
booktitle="Cahiers de G\'eostatistique",
volume=3,
pages="107-113",
type="Compte-rendus des journ\'ees de G\'eostatistique",
organization=CG,

@ARTICLE{Rivoirard85,
author="Rivoirard , J.",
title="Convergence des d\'eveloppements
en polyn\^omes d'{H}ermite",
journal=Sciences de la Terre serie Inf.,
year=1985,
volume=24}

@BOOK{Rivoirard94,
author="Rivoirard, J.",
title="Introduction to disjunctive kriging and non-linear geostatistics",
publisher="Oxford University Press",
year=1994}

Good luck

Gilles Guillot
Laboratoire d'étude des Transferts en Hydrologie et Environnement
CNRS-INPG-ORSTOM-UJF
BP 53 X
38041 Grenoble cedex 9
FRANCE
tel +33 4 76 82 52 85 (professional)
+33 4 78 72 67 71 (home)
fax +33 4 76 82 52 86
email Gilles.Guillot@...
URL http://soul.hmg.inpg.fr/users/guillot/home_page/

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