- G'day all,

I reckon we need to quantify the reliability of the kriging variance

map. Because sometimes its going to be an accurate map, and other times

its going to be way off the mark.

Imagine the situation when there are two maps with similar kriging

variances. However when we look at the semivariagram fit one of them

closely follow the line of fit while the other has a much larger

scatter. This means that one of the maps is actually much more accurate

then the other.

But as maps are currently presented we would never know!!

Could this be a big problem? I think it could. Particularly when the

estimation is quite bad, meaning that the variances have been

underestimated and should likely be much larger.

One solution could be to make the kriging variances proportional to the

model fit. Maybe the error between the kriging variance (as estimated

using the semivariagram) and the estimation variance (using real data

points) could be used to do this?

Does anyone know if this has been discussed before? Has it ever been

considered. Or am I totally off the trail and should activate my GIS

beacon?

For those that are interested I'll explain how I got to the above

conclusion:

Kriging can be summarised by the following:

Var(est) = f(weights and semivariance between all points that have a

positive weight), and we obtain the Var(krig) by minimising Var(est)

with respect to the weights. This is how we get the weights.

But in order to do this we need to know what the semivariance between

the points is. However if we're estimating a point we don't have then we

can't calculate the semi-variance, so we can't find the appropriate

weights. However, if we have a model for the semi-variance then we can

predict what the semi-variance should be using this model and we can

then calculate the appropriate weights. Which is why we require a

semivariagram model. So the semivariagram fit is vital in generating not

only the estimates, but their reliability also.

What this all boils down to is that the most important thing when

kriging is the ASSUMPTION that the points used to generate the

semi-variagram are capable of representing the semivariance for all

points. As well as the ASSUMPTION that the correct model has been fit,

and that its a good fit.

If either of these assumption fails then the kriging variance is

incorrect.

More to the point if the model is a poor fit then the kriging variance

is less likely to be accurate.

This brought me to my question. Should we have some statitistic that

quantifies how reliable our kriging variances are?

Christopher G Howden

Statistical Ecologist

Department of Land and Water Conservation

(Work) 02 9895 7130

(Fax) 02 9895 7867

(Mob) 0410 689 945

--

* To post a message to the list, send it to ai-geostats@...

* As a general service to the users, please remember to post a summary of any useful responses to your questions.

* To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list

* Support to the list is provided at http://www.ai-geostats.org - Dear Chris

Bayesian kriging is what you should use, if you want to include estimation

uncertainty into the kriging variances. Some useful references are:

Le, N.D. and Zidek, J.V. (1992).

Interpolation with uncertain covariances: a Bayesian alternative to

Kriging.

Journal of Multivariate Analysis, 43, p. 351-74.

Handcock, M.S. and Stein, M.L. (1993).

A Bayesian analysis of kriging.

Technometrics, 35, p. 403-10.

Kitanidis, P.K. (1986).

Parameter uncertainty in estimation of spatial functions: Bayesian

analysis.

Water Resources Research, 22, p. 499-507.

Best regards / Venlig hilsen

Søren Lophaven

******************************************************************************

Master of Science in Engineering | Ph.D. student

Informatics and Mathematical Modelling | Building 321, Room 011

Technical University of Denmark | 2800 kgs. Lyngby, Denmark

E-mail: snl@... | http://www.imm.dtu.dk/~snl

Telephone: +45 45253419 |

******************************************************************************

On Wed, 19 Feb 2003, Chris Howden wrote:

> G'day all,

>

> I reckon we need to quantify the reliability of the kriging variance

> map. Because sometimes its going to be an accurate map, and other times

> its going to be way off the mark.

>

> Imagine the situation when there are two maps with similar kriging

> variances. However when we look at the semivariagram fit one of them

> closely follow the line of fit while the other has a much larger

> scatter. This means that one of the maps is actually much more accurate

> then the other.

>

> But as maps are currently presented we would never know!!

>

> Could this be a big problem? I think it could. Particularly when the

> estimation is quite bad, meaning that the variances have been

> underestimated and should likely be much larger.

>

> One solution could be to make the kriging variances proportional to the

> model fit. Maybe the error between the kriging variance (as estimated

> using the semivariagram) and the estimation variance (using real data

> points) could be used to do this?

>

> Does anyone know if this has been discussed before? Has it ever been

> considered. Or am I totally off the trail and should activate my GIS

> beacon?

>

>

>

>

> For those that are interested I'll explain how I got to the above

> conclusion:

>

> Kriging can be summarised by the following:

> Var(est) = f(weights and semivariance between all points that have a

> positive weight), and we obtain the Var(krig) by minimising Var(est)

> with respect to the weights. This is how we get the weights.

>

> But in order to do this we need to know what the semivariance between

> the points is. However if we're estimating a point we don't have then we

> can't calculate the semi-variance, so we can't find the appropriate

> weights. However, if we have a model for the semi-variance then we can

> predict what the semi-variance should be using this model and we can

> then calculate the appropriate weights. Which is why we require a

> semivariagram model. So the semivariagram fit is vital in generating not

> only the estimates, but their reliability also.

>

> What this all boils down to is that the most important thing when

> kriging is the ASSUMPTION that the points used to generate the

> semi-variagram are capable of representing the semivariance for all

> points. As well as the ASSUMPTION that the correct model has been fit,

> and that its a good fit.

>

> If either of these assumption fails then the kriging variance is

> incorrect.

>

> More to the point if the model is a poor fit then the kriging variance

> is less likely to be accurate.

>

> This brought me to my question. Should we have some statitistic that

> quantifies how reliable our kriging variances are?

>

>

>

>

>

> Christopher G Howden

> Statistical Ecologist

> Department of Land and Water Conservation

> (Work) 02 9895 7130

> (Fax) 02 9895 7867

> (Mob) 0410 689 945

>

>

> --

> * To post a message to the list, send it to ai-geostats@...

> * As a general service to the users, please remember to post a summary of any useful responses to your questions.

> * To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list

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

>

--

* To post a message to the list, send it to ai-geostats@...

* As a general service to the users, please remember to post a summary of any useful responses to your questions.

* To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list

* Support to the list is provided at http://www.ai-geostats.org - Hi Christopher,

I believe you forgot a key assumption, homoscedasticity.

In most situations this assumption is not realistic

and we would like the kriging variance to somehow

depend on the local variability of data.

Rescaling globally the kriging variance to

account for uncertainty about variogram model

won't solve this problem.

Your map might be "globally" more accurate

but locally it will still fail to indicate

where prediction errors might be larger.

Regarding statistics to account for reliability of kriging variance,

the key question is what do you want to do with that variance.

If it's used to derive local probability distributions

under the multiGaussian model, you can assess

precision and accuracy of uncertainty models

using cross-validation. I addressed this issue

in the following paper:

Goovaerts, P. 2001.

Geostatistical modelling of uncertainty in soil science.

Geoderma, 103: 3-26.

and would be glad to send you a PDF copy of the paper if needed.

Regards,

Pierre

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

Dr. Pierre Goovaerts

President of PGeostat, LLC

Chief Scientist with Biomedware Inc.

710 Ridgemont Lane

Ann Arbor, Michigan, 48103-1535, U.S.A.

E-mail: goovaert@...

Phone: (734) 668-9900

Fax: (734) 668-7788

http://alumni.engin.umich.edu/~goovaert/

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

On Wed, 19 Feb 2003, Chris Howden wrote:

> G'day all,

>

> I reckon we need to quantify the reliability of the kriging variance

> map. Because sometimes its going to be an accurate map, and other times

> its going to be way off the mark.

>

> Imagine the situation when there are two maps with similar kriging

> variances. However when we look at the semivariagram fit one of them

> closely follow the line of fit while the other has a much larger

> scatter. This means that one of the maps is actually much more accurate

> then the other.

>

> But as maps are currently presented we would never know!!

>

> Could this be a big problem? I think it could. Particularly when the

> estimation is quite bad, meaning that the variances have been

> underestimated and should likely be much larger.

>

> One solution could be to make the kriging variances proportional to the

> model fit. Maybe the error between the kriging variance (as estimated

> using the semivariagram) and the estimation variance (using real data

> points) could be used to do this?

>

> Does anyone know if this has been discussed before? Has it ever been

> considered. Or am I totally off the trail and should activate my GIS

> beacon?

>

>

>

>

> For those that are interested I'll explain how I got to the above

> conclusion:

>

> Kriging can be summarised by the following:

> Var(est) = f(weights and semivariance between all points that have a

> positive weight), and we obtain the Var(krig) by minimising Var(est)

> with respect to the weights. This is how we get the weights.

>

> But in order to do this we need to know what the semivariance between

> the points is. However if we're estimating a point we don't have then we

> can't calculate the semi-variance, so we can't find the appropriate

> weights. However, if we have a model for the semi-variance then we can

> predict what the semi-variance should be using this model and we can

> then calculate the appropriate weights. Which is why we require a

> semivariagram model. So the semivariagram fit is vital in generating not

> only the estimates, but their reliability also.

>

> What this all boils down to is that the most important thing when

> kriging is the ASSUMPTION that the points used to generate the

> semi-variagram are capable of representing the semivariance for all

> points. As well as the ASSUMPTION that the correct model has been fit,

> and that its a good fit.

>

> If either of these assumption fails then the kriging variance is

> incorrect.

>

> More to the point if the model is a poor fit then the kriging variance

> is less likely to be accurate.

>

> This brought me to my question. Should we have some statitistic that

> quantifies how reliable our kriging variances are?

>

>

>

>

>

> Christopher G Howden

> Statistical Ecologist

> Department of Land and Water Conservation

> (Work) 02 9895 7130

> (Fax) 02 9895 7867

> (Mob) 0410 689 945

>

>

> --

> * To post a message to the list, send it to ai-geostats@...

> * As a general service to the users, please remember to post a summary of any useful responses to your questions.

> * To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list

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

>

--

* To post a message to the list, send it to ai-geostats@...

* As a general service to the users, please remember to post a summary of any useful responses to your questions.

* To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list

* Support to the list is provided at http://www.ai-geostats.org >G'day all,

Because sometimes its going to be an accurate map, and other times its

>

>I reckon we need to quantify the reliability of the kriging variance map.

going to be way off the mark.>

variances. However when we look at the semivariagram fit one of them

>Imagine the situation when there are two maps with similar kriging

closely follow the line of fit while the other has a much larger scatter.

This means that one of the maps is actually much more accurate then the

other.

Statistically, your question is related to the fact that the covariance

matrix from the fit of the variogram model is usually disregarded in

subsequent analyses. Some time ago i asked about this discarding of the

variogram parameters covariance matrix and geostatisticians replied that

this source of uncertainty was usually less important than other problems

and should not be of much importance.

Rubén

http://webmail.udec.cl

--

* To post a message to the list, send it to ai-geostats@...

* As a general service to the users, please remember to post a summary of any useful responses to your questions.

* To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list

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

they are completely and perfectly "reliable"

>A couple of observations on your problem

>

>1. The kriging variances are computed, not estimated, hence in that sense

IOW, the kriging variances are the objective functions to be optimized not

the parameters to be estimated (the latter ones are the values of the

spatial variable at non observed locations). It seems to me they are

perfectly "reliable" (or perhaps 'exact') in linear kriging but not so in

non linear kriging, in which they should be approximate.

>2. However, the kriging variances are computed using (1) the variogram or

covariance model, (2) the coordinates of the data locations used to

generate the kriged value, (2) the coordinates of the location being

kriged, (3) the degree of the polynomial used to represent the drift (if

Universal Kriging is used). (4) Decisions made about the search

neighborhood. Therein lies the problem, while it may be that the

coordinates are known (nearly perfectly), the variogram or covariance must

be estimated from the data and that is not a "perfect" process.

Under standard geostatistical procedures the variogram or covariance is

taken as known during kriging. I once reviewed a ms. sent to a statistical

journal in which the authors assessed by simulation the degree to which

this assumption affects OK (IIRW). They found that for fairly low sample

sizes, the effect of ignoring the covariance matrix of parameters in the

model variogram was very minor.

>While changing the search neighborhood may not change the estimates much,

it can sometimes have a big effect on the kriging variances, an easy way to

see this is to do a little experimenting in cross-validation.>

bigger or smaller than one) you will NOT change the kriging weights and

>3. Note that if you multiply the variogram by a positive number (either

hence not change the kriged values but you will change the kriging

variance.

That is why the use of standardized experimental variograms to fit the

model variogram do not change the kriging weights as compared with the

standard experimental variogram.

Your other points (snipped) are very well taken.

Note that i did not posit the original question as may appear by the way

Donald's message showed up on the list.

Rubén

http://webmail.udec.cl

--

* To post a message to the list, send it to ai-geostats@...

* As a general service to the users, please remember to post a summary of any useful responses to your questions.

* To unsubscribe, send an email to majordomo@... with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list

* Support to the list is provided at http://www.ai-geostats.org