- 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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* Support to the list is provided at http://www.ai-geostats.org > A couple of observations

Thanks, i noticed the publication in a biostatistics series and it's not

>

> 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)

available right now at Amazon.

> 2. Matern introduced a class of isotropic positive definite functions,

I read the statement in one of Diggle, Ribeiro, and Christensen papers on

> 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.

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

Yes, it is the fact that Matern covariance function has certain

> 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

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)

Except for the case mentioned above in which for mathematical convenience

> 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.

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

I am a little intrigued by this comment. I am imagining that the fish

> 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.

density field is continuous.

> Donald E. Myers

Thanks for your comments.

Rubén Roa

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