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

[Edzer Pebesma]

On 12/05/2017 04:35 PM, Joelle k. Akram wrote:
> hi Prof. Pebesma,
> 
> 
> just to follow up on my question about adding the nugget term to both V
> and v from your lecture 7 on github.
> 
> I do not want an exact interpolator. Instead I want to do a smoothing
> interpolator using UnivKrig. Would you 
> 
> recommend only adding the nugget to V only ? and set the nugget=0 for
> defining v (whilst retaining the same psill and range used for defining V).

Yes.

> 
> 
> thanks
> Chris Akram
> 
> ------------------------------------------------------------------------
> *From:* Edzer Pebesma <edzer.pebesma at uni-muenster.de>
> *Sent:* November 22, 2017 2:55 PM
> *To:* Joelle k. Akram; 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 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>.
> <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
> 
> 
> 
>> 
>> edzer/mstp <https://github.com/edzer/mstp/blob/master/lec7.Rmd>
> <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
> 
> 
> 
>> 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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> geographical, data.
> 
> 
> 
>>> 
>> 
>> -- 
>> Edzer Pebesma
>> Institute for Geoinformatics
>> Heisenbergstrasse 2, 48151 Muenster, Germany
>> Phone: +49 251 8333081
>> 
>> _______________________________________________
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>> R-sig-Geo at r-project.org
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> R-sig-Geo Info Page - SfS – Seminar for Statistics | ETH ...
> <https://stat.ethz.ch/mailman/listinfo/r-sig-geo>
> stat.ethz.ch
> A mailing list for discussing the development and use of R functions and
> packages for handling and analysis of spatial, and particularly
> geographical, data.
> 
> 
> 
> 
> -- 
> Edzer Pebesma
> Institute for Geoinformatics
> Heisenbergstrasse 2, 48151 Muenster, Germany
> Phone: +49 251 8333081

-- 
Edzer Pebesma
Institute for Geoinformatics
Heisenbergstrasse 2, 48151 Muenster, Germany
Phone: +49 251 8333081