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AI-GEOSTATS: Geostatistics and likelihood - List of replies

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  • Ruben Roa Ureta
    Hi: Regarding my recent question, i received the replies listed below. Thanks to all those who responded giving me useful hints. Rubén My question: Hi people:
    Message 1 of 2 , Nov 5, 2003
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      Hi:
      Regarding my recent question, i received the replies listed below. Thanks
      to all those who responded giving me useful hints.
      Rubén

      My question:
      Hi people:
      I am looking for information (journal papers, reports, etc) about the
      maximum likelihood estimation of variogram parameters and the total of a
      single variable over the spatial field. My final purpose is to produce
      profile likelihood plots of fish biomass. I know there is PhD thesis by
      Matern which seems to have been published as a lecture note in
      biomathematics, and that work is not available to me in my location.
      There is a theoretical variogram called the Matern variogram, which is
      presented as a theoretically sound version of the variogram, as compared
      to 'curve-fitting' variograms (spherical, Gaussian, etc) fitted by
      various forms of least-squares. I am specially interested in the
      derivation of Matern variogram from first principles. I have some papers
      by Diggle, Ribeiro and colleagues, in Conference proceedings and applied
      statistics journals but i have not found anything in
      theoretically-inclined statistical journals or in the web (apart from
      the material on GeoR. I am planning to implement this approach in GeoR).
      Thanks for any help on this.
      Rubén
      --------------
      Replies:

      Hi Ruben

      Have you had a look at M. Stein book?
      Interpolation of Spatial Data: Some Theory for Kriging (Springer Series in
      Statistics)
      It is more teoretically inclined.

      P.J.
      ---------------
      Hi,

      Look up work done by Peter Kitinidis and Bob Hoeksema.

      Yetta
      ---------------
      based on a lecture of his, you might also try searching Peter Guttorp's
      work at the stat dept at the univ of washington. brian

      Brian Gray
      ---------------
      Hello ruben,

      I have been applying Maximum likelihood in geoR. there is a function in
      the program where you can estimate the covariance model parameters by ML
      and REML. You can also estimate the profiles of the parameters calculated.
      Also it acepts 'box-cox' tranformations.
      Papers on ML and REML... in my data base I have these:

      Akaike, H., 1973. Information theory and an extension of the maximum
      likelihood principle. In Second International Symposium on Information
      Theory, pp. 267-281. Akadémiai Kiadó, Budapest.
      Dolan, D. M., A. H. El-Shaarawi & T. B. Reynoldson, 2000. Predicting
      benthic counts in Lake Huron using spatial statistics and
      quasi-likelihood. Environmetrics 11: 284-304.
      Fuentes, M., 2001. A high frequency kriging approach for non-stationary
      environmental processes. Environmetrics 12(5): 469-483.
      Lai, H.-L. & D. K. Kimura, 2002. Analyzing survey experiments having
      spatial variability with an application to a sea scallop fishing
      experiment. Fish. Res. 56(3): 239-259.
      Anselin, L., 2001. Rao's score test in spatial econometrics. Journal of
      Statistical Planning and Inference 97(1): 113-139.
      Berberoglu, S., C. D. Lloyd, P. M. Atkinson & P. J. Curran, 2000. The
      integration of spectral and textural information using neural networks for
      land cover mapping in the Mediterranean. Computers & Geosciences 26(4):
      385-396.
      Diggle, P. J., 1988. An approach to the analysis of repeated measurements.
      Biometrics 44(4): 959-971.
      Hollenbeck, K. J. & K. H. Jensen, 1998. Maximum-likelihood estimation of
      unsaturated hydraulic parameters. Journal of Hydrology 210(1-4): 192-205.
      Houwing-Duistermaat, J. J., H. C. Van Houwelingen & A. Terhell, 1998.
      Modelling the cause of dependency with application to filaria infection.
      Statistics in Medicine 17(24): 2939-2954.
      Jarvis, C. H., 2001. GEO_BUG: a geographical modelling environment for
      assessing the likelihood of pest development. Environmental Modelling &
      Software 16(8): 753-765.
      Jones, W. L., V. J. Cardone, W. J. Pierson, J. Zec, L. P. Rice, A. Cox &
      W. B. Sylvester, 1999. NSCAT high-resolution surface wind measurements in
      Typhoon Violet. Journal of Geophysical Research. C. Oceans [J. Geophys.
      Res. (C Oceans)] 104(C5): 11247-11259.
      Lark, R. M., 2002. Optimized spatial sampling of soil for estimation of
      the variogram by maximum likelihood. Geoderma 105(1-2): 49-80.
      Lele, S. & M. L. Taper, 2002. A composite likelihood approach to
      (co)variance components estimation. Journal of Statistical Planning and
      Inference 103(1-2): 117-135.
      Mowrer, H. T., 2000. Uncertainty in natural resource decision support
      systems: sources, interpretation, and importance. Computers and
      Electronics in Agriculture 27(1-3): 139-154.
      Nanos, N. & G. Montero, 2002. Spatial prediction of diameter distribution
      models. Forest Ecology and Management 161(1-3): 147-158.
      Pardo-Iguzquiza, E., 1998. MLREML4: A program for the inference of the
      power variogram model by maximum likelihood and restricted maximum
      likelihood. Computers & Geosciences 24(6): 537-543.
      Pardo-Iguzquiza, E., 1998. Inference of spatial indicator covariance
      parameters by maximum likelihood using MLREML. Computers & Geosciences
      24(5): 453-464.
      Pardo-Iguzquiza, E. & P. A. Dowd, 1997. AMLE3D: A computer program for the
      inference of spatial covariance parameters by approximate maximum
      likelihood estimation. Computers & Geosciences 23(7): 793-805.
      Park, J.-S. & J. Baek, 2001. Efficient computation of maximum likelihood
      estimators in a spatial linear model with power exponential covariogram.
      Computers & Geosciences 27(1): 1-7.
      Porter, D. W., B. P. Gibbs, W. F. Jones, P. S. Huyakorn, L. L. Hamm & G.
      P. Flach, 2000. Data fusion modeling for groundwater systems. Journal of
      Contaminant Hydrology 42(2-4): 303-335.
      Skene, A. M. & S. A. White, 1992. A latent class model for repeated
      measurements experiments. Statistics in Medicine 11(16): 2111-2122.
      Todini, E. & M. Ferraresi, 1996. Influence of parameter estimation
      uncertainty in Kriging. Journal of Hydrology 175(1-4): 555-566.
      Watkins, A. J., 1992. On models of spatial covariance. Computational
      Statistics & Data Analysis 13(4): 473-481.
      Yokozawa, M., Y. Kubota & T. Hara, 1999. Effects of competition mode on
      the spatial pattern dynamics of wave regeneration in subalpine tree
      stands. Ecological Modelling 118(1): 73-86.

      Now I dont knw which would be interesting or not. (I can send you the file
      in endnote format, or data base whatever...).

      I hope this helps,
      Marta
      ---------------

      I think the SAS procedure "proc mixed" can do all of these.
      Din Chen



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    • Ruben Roa Ureta
      ... Thanks, i noticed the publication in a biostatistics series and it s not available right now at Amazon. ... I read the statement in one of Diggle, Ribeiro,
      Message 2 of 2 , Nov 5, 2003
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        > A couple of observations
        >
        > 1. Matern's thesis originally was published in Swedish but reprinted in
        > English in 1986 by Springer-Verlag, the title is "Spatial Variation"
        > (this appears in a statistics series)

        Thanks, i noticed the publication in a biostatistics series and it's not
        available right now at Amazon.

        > 2. Matern introduced a class of isotropic positive definite functions,
        > i.e., isotropic covariances. Each covariance function then determines a
        > variogram. See a discussion of these in M. Stein's book (Interpolation
        > of Spatial Data, Springer-Verlag)
        >
        > 3. Your comment
        >
        > "There is a theoretical variogram called the Matern variogram, which is
        > presented as a theoretically sound version of the variogram, as compared
        > to 'curve-fitting' variograms (spherical, Gaussian, etc) fitted by
        > various forms of least-squares."
        >
        > is puzzling.

        I read the statement in one of Diggle, Ribeiro, and Christensen papers on
        GeoR and what they call model-based geostatistics. The point they were
        making was that Matern covariance function is mean-square continuous and
        mean-square differentiable up to high orders, while the popular spherical
        variogram say, is not, and then the spherical variogram would produce
        serious problems for likelihood estimation (especially at the range
        parameter dimension). When i wrote the paragraph you quote i was thinking
        that the Matern covariance function was obtained from first principles, by
        some sound mechanistic reasoning, but eventually i realized that the
        authors were just referring to the mathematical convenience of Matern
        covariance function, because of differentiability, which allowed formal
        estimation methods like maximum likelihood.
        Diggle, Ribeiro and Christensen also consider their description of
        geostatistics theoretically sounder because they use the concepts of
        stochastic processes.

        > Valid variograms must satisfy two conditions, (i) they must be
        > conditionally negative definite (not just semidefinite) (ii) the growth
        > rate must be less than quadratic. Any positive definite function (again
        > not just semidefinite) will correspond to a variogram satisfying these two
        > conditions. However there are valid variograms that do not correspond to
        > covariances, e.g. the power model. While it true that practitioneers will
        > sometimes use least squares to fit the parameters in a variogram, e.g.,
        > spherical, gaussian, exponential, this has absolutely nothing to do with
        > the two properties listed above. For that matter neither does maximum
        > likelihood fitting

        Yes, it is the fact that Matern covariance function has certain
        mathematical properties in addition to conditional negative definiteness
        that make it suitable for maximum likelihood estimation. This was a bit
        dissapointing to me since i was looking for mechanistically derived
        covariance/variogram functions.

        > 4. In general variogram modeling and fitting involves two steps; (a)
        > determining/choosing model types (e.g., Matern, spherical, gaussian,
        > exponential, power, etc and if a nested model is to be used, the number
        > of terms) , (b) estimating the parameters for the model types. MLE is
        > not so useful for the first step.

        Except for the case mentioned above in which for mathematical convenience
        a model is selected so it can be differentiated up to a high term in all
        dimensions of the parameter space.

        > 5. The easiest way to deal with MLE for variogram estimation is based on
        > an assumption of multivariate normality (a very strong assumption and
        > one not really appropriate in many applications). See some discussion of
        > this in M. Stein's book (pages 171-
        >
        > 6. In common models such as the spherical, gaussian, exponential the
        > "shape" of the variogram doesn't really change when the parameter values
        > change. That could be one of the advantages of the Matern class (which
        > has more parameters and some of which change the "shape" of the model).
        >
        > 7. The kriging estimator is moderately robust with respect to the
        > variogram (and its parameters),i.e., slight changes in the variogram
        > model and/or its parameters will cause only slight changes in the
        > solution vector(s) hence only slight changes in the estimated values.
        >
        > 8. I think that your fish data should really be viewed as non-point
        > data, this has an effect on the variogram modeling that should not be
        > ignored.

        I am a little intrigued by this comment. I am imagining that the fish
        density field is continuous.

        > Donald E. Myers

        Thanks for your comments.
        Rubén Roa

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