[R-sig-Geo] Why is the covariance in Universal Kriging modeled this way in lectures by Prof. Edzer Pebesma?

[Edzer Pebesma]

On 11/22/2017 10:23 PM, Joelle k. Akram wrote:
> thank you for clarifying Prof. Pebesma. I have a couple more question
> for you regarding the inclusion of a
> 
> nugget to the diagonals of V. As we know,there are 2 covariances, V and
> v; one for the existing coordinates (i.e., V) and the other for the
> distances between these existing coordinates and other new locations
> (i.e., v).
> 
> 
> I) Assuming unscaled coordinates in latitude/longitude; should the
> Nugget theoretically be a small value (lets say typically less than <1)
> or does it depend on other the dataset's spatial distribution,etc?
> 

I would say it should depend on the data.

> 
> 2) When computing Beta coefficients as in you lecture 7 in github, do we
> have to add the nugget term to both V and v or only one of them?

For a nugget, by definition to each of them; if you'd only add it to V
you no longer obtain an exact interpolator (i.e., you no longer predict
the data value at data locations); if your measured process is subject
to a measurement error, this may however be preferred.

> 
> 
> thank you,
> 
> Chris Akram
> 
> 
> 
> 
> ------------------------------------------------------------------------
> *From:* R-sig-Geo <r-sig-geo-bounces at r-project.org> on behalf of Edzer
> Pebesma <edzer.pebesma at uni-muenster.de>
> *Sent:* November 22, 2017 6:16 AM
> *To:* r-sig-geo at r-project.org
> *Subject:* Re: [R-sig-Geo] Why is the covariance in Universal Kriging
> modeled this way in lectures by Prof. Edzer Pebesma?
>  
> 
> 
> On 11/22/2017 12:09 AM, Joelle k. Akram wrote:
>> Greetings,
>> 
>> There is code for Universal Kriging from Prof. Edzer Pebesma in GitHub<https://github.com/edzer/mstp/blob/master/lec7.Rmd>.
> 
> edzer/mstp <https://github.com/edzer/mstp/blob/master/lec7.Rmd>
> github.com
> mstp - Course slides: modelling spatio-temporal processes
> 
> 
> 
> 
> https://edzer.github.io/mstp/lec7.html
> 
> gives you the rendered version.
> 
>> 
>> The covariance function is defined as follows:
>> 
>> cov = function(h) exp(-h)
>> 
>> And defined without any variogram modeling/generation to produce partial sill, range or nugget parameters for defining the covariance matrix.
> 
> Well, it implies nugget=0, sill=1 and range parameter=1, it was the
> shortest covariance function I could think of.
> 
>> 
>> If  I want to include a regularization term to account for singularity effects caused due to close spatial points, how do I modify the matrix computation for computing the 'beta' coefficients ?
> 
> Add a nugget (i.e. add a constant to the diagonal of V)?
> 
>> 
>> I know there are standard formulae for different models (e.g. Matern, Exp,etc). But I would like to retain the simple cov function defined above and possibly use a regularizer (like ridge regression) to account for a nugget-like effect.
>> 
>> thanks,
>> 
>> Chris
>> 
>>        [[alternative HTML version deleted]]
>> 
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> 
> -- 
> Edzer Pebesma
> Institute for Geoinformatics
> Heisenbergstrasse 2, 48151 Muenster, Germany
> Phone: +49 251 8333081
> 
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-- 
Edzer Pebesma
Institute for Geoinformatics
Heisenbergstrasse 2, 48151 Muenster, Germany
Phone: +49 251 8333081