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GEOSTATS: Anamophosis

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  • GillesG@aol.com
    ... 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
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      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 ?
      >
      > All advice, comments, or references are wellcomed.
      >
      > 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,
      address="Dordrecht",
      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,
      address="Fontainebleau"}

      @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",
      address="Oxford",
      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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