> 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.

RubĂ©n Roa

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