Re: AI-GEOSTATS: How reliable are your kriging variances?
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: [EMAIL PROTECTED] | 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 [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] 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 [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] 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
Re: AI-GEOSTATS: How reliable are your kriging variances?
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: [EMAIL PROTECTED] 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 [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] 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 [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] 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
Re: AI-GEOSTATS: How reliable are your kriging variances?
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. 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 [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] 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
AI-GEOSTATS: Post-Doctoral position
POST-DOCTORAL POSITION - Quantitative Ecology/Landscape Ecology/Spatial Statistics BACKGROUND: A postdoctoral position will be available shortly at the Forest Landscape Ecology Laboratory, Ontario Forest Research Institute, Ministry of Natural Resources, Sault Ste. Marie, Ontario, Canada. It will be for 2 years initially, with the possibility of a 2-yr. extension. This is for a collaborative project between Ontario Ministry of Natural Resources, several forest companies in Ontario, University of Waterloo, and the Joint Fire Science Program (University of Toledo and US Forest Service - Grand Rapids, MN). GOAL: The goal of the research project is to test and calibrate a spatial forest cover transition matrix component of a boreal forest landscape dynamics model (see ecol. modelling 150:189-209, 2002, and For. Chronicle 79 (1)1-15, 2003 ). The main responsibilities of the post doctoral researcher include developing the experimental design, formulating spatially explicit null hypotheses using the semi-Markov transition matrices, testing the model against the already complied extensive spatial database of historical forest cover (based on aerial photographs) in boreal Ontario, Canada. Based on the success of the testing stage, it will be possible to secure funding to revise the model. QUALIFICATIONS: A recent Ph.D. in Quantitative Ecology, Landscape Ecology, Spatial Statistics or a related field. Just-about-done is acceptable. Excellent quantitative, statistical, spatial modeling, and computing skills are essential. Good working knowledge of ARCGIS/ArcView and VB/C++ is necessary. The salary will be competitive based on the qualifications and experience. Send a letter of intent listing your qualifications, including the Ph.D. dissertation title (and expected completion date if JAD), and the earliest possible start date by e-mail to [EMAIL PROTECTED] You may also mail your CV and relevant reprints to: Dr. Ajith H. Perera, Ontario Forest Research Institute, 1235 Queen St. East, Sault Ste. Marie, Ontario P6A 2E5, Canada. === Ajith H. Perera Research Scientist Program Leader Forest Landscape Ecology Program Ontario Forest Research Institute 1235 Queen St. East Sault Ste. Marie ON P6A 2E5 CANADA Ph: (705) 946-7426 Fax: (705) 946-2030 [EMAIL PROTECTED] -- * To post a message to the list, send it to [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] 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
AI-GEOSTATS: SUM: weighted cross-validations?
Dear all, to my question on weighted cross-validations, I got also a reply from Donald Myers that has not appeared on the list and which is here forwarded. To summarize in two words Pierre and Donald's replies : weighted cross-validations might be an interesting step but the problem of interpretation of the results remains unexplored... I also managed to get a copy of Peter Rogerson's paper, a reading suggested by Ana Militino. The abstract and reference are given here: Optimal Spatial Sampling for Variables With Varying Spatial Importance Peter A. Rogerson, Eric Delmelle, Rajan Batta, Mohan Akella, Alan Blatt, and Glenn Wilson Paper presented at the annual meeting of the North American Regional Science Association, San Juan, Puerto Rico, November, 2002. Abstract It is often desirable to sample in those locations where uncertainty associated with a variable is highest. However, the importance of knowing the variable+IBk-s value may vary across space. We are interested in the spatial distribution of Received Signal Strength Indicator (RSSI), a measure of the signal strength from a cell tower received at a particular location. Estimation of RSSI is important to estimate the effectiveness of mayday systems designed for rapid emergency notification following vehicle crashes. RSSI estimation is less important for locations where the probability of a crash is low and where the likelihood of call completion is either close to zero or one. We develop a method for augmenting a spatial sample of RSSI values to achieve a high-precision estimate of the probability of call completion following a crash. We illustrate the approach using data on RSSI and vehicle crashes in Erie County, NY. Thanks again to all for the kind feedback. Best regards Gregoire ---BeginMessage--- Gregoire There is no theoretical reason why you can't weight the estimation errors but the key question then is how to interpret the results. At least some geostatistics packages compute a mean square normalized error (in the cross-validation stage), in this case "normalized" means that each "error" is divided by the kriging standard deviation. Then it is easy to show that the expected value of this is one, i.e.., yoiu have a theoretical value to compare against. The second way to use the normalized errors is to look at the empirical distribution. Under an assumption of normality, 95+ACU- of the values should be between -2.5 and +ACs-2.5. Even without the normality assumption you can use Chebycheff's inequality to get almost the same result. The locations for which the normalized error is too large (in absolute value) could then be considered as "unusual" locations and these would merit further investigation. If you normalize the errors in some other way you lose these theoretical results but you might still find it useful to look at the empirical distribution as well as at a coded plot of the locations +AKA-(each data location coded by the value of the normalized error. The question probably is whether you are using the cross-validation to evaluate the fit of the variogram or whether you are using it to identify unusual locations. For the former I suggest that using the krigng standard deviation is better once you have fitted the variogram satisfactorily then the other may also be useful. I use the word "satisfactorily" because you can never be sure you have gotten the best possible fit. Cross validation is useful to compare one variogram fit against another but does not work to "optimize" the variogram fit in an abolute sense. This is because forcing one cross-validation statistic to be best may cause another one to be sub-optimal. All of the cross-validation statistics are somewhat sensitive to the choice of the search neighborhood and its parameters. See 1991, Myers,D.E., On Variogram Estimation. in Proceedings of the First Inter. Conf. Stat. Comp., Cesme, Turkey, 30 Mar.-2 April 1987, Vol II, American Sciences Press, 261-281 A caution about "robust" variogram estimators, one should ask "robust" with respect to what? +AKA-Note that the kriging estimator is already moderately robust with respect to small changes in the variogram model. When estimating and fitting the variogram model, what really matters is how the fitted variogram affect the interpolation? When looking at "robustness" in estimating the variogram this may not relate to how the variogram affects the interpolation. Unfortunately, given only a finite data set both the problem of estimating and modeling the variogram and the subsequent interpolation step are ill-posed problems. That means that they do not have unique solutions. We like to believe (or at least we act like we do) that there is only one true variogram model for our problem and hence we want to find the best way to find that true variogram. One way to make it look more unique is to put more assumptions in such as multivariate normality which may or may not be reasonable. Donald Myers
AI-GEOSTATS: Practical questions about Universal Kriging.
I pretend to perform Universal Kriging with GSLIB, the problem is how to get the vertical drill in bore hole data set, for use de residual variogram. If exist other solution like use Pair-wise Relative Variogram for example.? Other question is how to detect the best drift. -- * To post a message to the list, send it to [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] 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
AI-GEOSTATS: Variograms models
Hi all, Does anyboby can explain to me the origin of the variogram models: spherical and exponential ? Why the names spherical and exponential ? Sincerely Charles -- * To post a message to the list, send it to [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] 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
Re: AI-GEOSTATS: Variograms models
Matheron used the term spherical to describe the semi-variogram model which represents the concept of two overlapping 'spheres of influence'. The formula is actually the geometric calculation of the amount by which two spheres of diameter 'a' (range of influence) do NOT overlap when their centres are separated by a given distance. The exponential model contains an exponential term and is exactly equivalent to the 'exponential decay' beloved of economists and other predicters. BTW: the Gaussian is so-called simply because it is the same shape as a Normal cumulative frequency plot (ogive). Isobel Clark http://uk.geocities.com/drisobelclark/resume/Publications.html --- Serele, Charles [EMAIL PROTECTED] wrote: Hi all, Does anyboby can explain to me the origin of the variogram models: spherical and exponential ? Why the names spherical and exponential ? Sincerely Charles __ Do You Yahoo!? Everything you'll ever need on one web page from News and Sport to Email and Music Charts http://uk.my.yahoo.com -- * To post a message to the list, send it to [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] 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
