# Why is the covariance in Universal Kriging modeled this way in lectures by Prof. Edzer Pebesma? Classic List Threaded 7 messages Open this post in threaded view
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## Why is the covariance in Universal Kriging modeled this way in lectures by Prof. Edzer Pebesma?

 Greetings, There is code for Universal Kriging from Prof. Edzer Pebesma in GitHub. 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. 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 ? 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]] _______________________________________________ R-sig-Geo mailing list [hidden email] https://stat.ethz.ch/mailman/listinfo/r-sig-geo
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## Re: 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://edzer.github.io/mstp/lec7.htmlgives 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]] > > _______________________________________________ > R-sig-Geo mailing list > [hidden email] > https://stat.ethz.ch/mailman/listinfo/r-sig-geo> -- Edzer Pebesma Institute for Geoinformatics Heisenbergstrasse 2, 48151 Muenster, Germany Phone: +49 251 8333081 _______________________________________________ R-sig-Geo mailing list [hidden email] https://stat.ethz.ch/mailman/listinfo/r-sig-geo
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## Re: Why is the covariance in Universal Kriging modeled this way in lectures by Prof. Edzer Pebesma?

 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? 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? thank you, Chris Akram ________________________________ From: R-sig-Geo <[hidden email]> on behalf of Edzer Pebesma <[hidden email]> Sent: November 22, 2017 6:16 AM To: [hidden email] 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. edzer/mstp github.com mstp - Course slides: modelling spatio-temporal processes https://edzer.github.io/mstp/lec7.htmlgives 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]] > > _______________________________________________ > R-sig-Geo mailing list > [hidden email] > https://stat.ethz.ch/mailman/listinfo/r-sig-geo> -- Edzer Pebesma Institute for Geoinformatics Heisenbergstrasse 2, 48151 Muenster, Germany Phone: +49 251 8333081 _______________________________________________ R-sig-Geo mailing list [hidden email] https://stat.ethz.ch/mailman/listinfo/r-sig-geo        [[alternative HTML version deleted]] _______________________________________________ R-sig-Geo mailing list [hidden email] https://stat.ethz.ch/mailman/listinfo/r-sig-geo
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## Re: 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 <[hidden email]> on behalf of Edzer > Pebesma <[hidden email]> > *Sent:* November 22, 2017 6:16 AM > *To:* [hidden email] > *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. > > edzer/mstp > 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]] >> >> _______________________________________________ >> R-sig-Geo mailing list >> [hidden email] >> https://stat.ethz.ch/mailman/listinfo/r-sig-geo>> > > -- > Edzer Pebesma > Institute for Geoinformatics > Heisenbergstrasse 2, 48151 Muenster, Germany > Phone: +49 251 8333081 > > _______________________________________________ > R-sig-Geo mailing list > [hidden email] > https://stat.ethz.ch/mailman/listinfo/r-sig-geo-- Edzer Pebesma Institute for Geoinformatics Heisenbergstrasse 2, 48151 Muenster, Germany Phone: +49 251 8333081 _______________________________________________ R-sig-Geo mailing list [hidden email] https://stat.ethz.ch/mailman/listinfo/r-sig-geo
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