Re: AI-GEOSTATS: Kriging Error vs variance
Donald, Thanks for your reply, my comments; Digby I don't understand some of your comments: 1. Stationarity is not a property of data, it is a property of the underlying random function (no matter which form of stationarity you are considering). + +++ + + + ++ + + + + +++ + + + + + Michelle David describes this data as having local drift but overall stationarity, so the mean changes with location. 2. The actual kriging error is unknown so how can you compare the kriging error with the kriging variance? My apologies I mistyped the email, I intended to say estimation variance vs. sample variance. 3. In the absence of spatial correlation, i.e., pure nugget effect variogram, the kriged value at each location will be the sample mean. Is it possible you could have some type of drift as above, yet have a pure nugget effect variogram? 4. In what sense might the sample mean be better than the kriged estimate? If one uses the sample mean for all estimates then you will not have an exact interpolator hence at the data locations the sample mean could not be better than the kriged estimate. Except at data locations you will not know the true value and hence you can't compare which is better, the sample mean or the kriged estimate. I was meaning to say that if you have a drift in the mean is it possible you may have a kriging variance equal to the sample variance, but a large difference between the local mean and the sample mean (I should think about this), so your local kriged estimate would be far more accurate than the sample mean. Hence you should not adopt your sample mean if you have drift. I was really mixing terms of what Pierre described as trend which may be appropriate for universal kriging, but I meant trends on a small scale which I described as drift, where such methods may be too complex, or inconvenient to apply. (I should have read the book, Michelle does incidentally recommend universal kriging for the above sketch example, but assummed that in cases software would not be available to carry out universal kriging the drift may be ignored). If one argues that the sample mean is better than kriged estimates then you are really arguing that there is no spatial correlation (and hence that your variogram estimation and modeling step was not adequate). The sample mean is in fact a special case of the kriging estimator, i.e., where all the weights are the same. The kriging equatiions are derived by minimizing the estimation variance, hence an equal weight estimator has to be one of the possibilities. Of course this is all predicated on several assumptions that are not really testable a. The underlying random function satisfies an appropriate form of stationarity b. The variogram has been adequately estimated and modeled (the kriging variance does not capture any uncertainty related to inadequate estimation and modeling of the variogram) c. You are restricted to linear estimators (the conditional expectation would be the optimal estimator in general, it coincides with the simple kriging estimator in the case of multivariate normality) Ultimately however the question of whether using the sample mean (at all locations) or using kriging (assuming a non-pure nugget variogram) is better may depend on what you going to use the results for, that is not exactly a statistical question. Donald E. Myers http://www.u.arizona.edu/~donaldm Thankyou all for your patience, Best Regards Digby Millikan -- * 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: Kriging Error vs variance
Dear List members, I am looking for a reference on interpretation of the Kriging error versus the sample variance. Am I correct in assumung that in any kriged interpolation where the Kriging error is greater than the sample varience then the sample mean would be a better estimate at that location? Thanks for your help Russell Barbour Ph.D. Research Associate in Applied Mathematics Vector Ecology Laboratory Yale School of Medicine 60 College St. Rm 600 New Haven CT. 06520 TEL: 203 785 3223 FAX 203 785 3604 email: [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
Re: AI-GEOSTATS: Kriging Error vs variance
Russell Absolutely on the spot. We call this the 'ygiagam' criterion (your guess is as good as mine) ;-) Isobel Clark http://geoecosse.bizland.com/news.html __ 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
Re: AI-GEOSTATS: Kriging Error vs variance
Hi Russell, I am assuming you refer to kriging error variance. If your semivariogram is bounded and has a sill close to the sample variance, then the simple kriging estimate will automatically be the global mean when the kriging variance is the sample variance (that is when all observations are beyond the range of spatial correlation). Note that you might want to decluster your sample mean before using it as global mean. For ordinary kriging, the kriging variance would actually be greater than the sample variance because of the Lagrangian parameter. I don't think I would adopt a global/sample mean instead of the local mean provided by ordinary kriging even if the variance of the estimator is smaller. However, for kriging with a trend or universal kriging, I wouldn't trust too much the estimate obtained for large kriging variance since the extrapolated trend can be very unrealistic (e.g. negative concentration estimates). 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 Mon, 27 Jan 2003, Russell Barbour wrote: Dear List members, I am looking for a reference on interpretation of the Kriging error versus the sample variance. Am I correct in assumung that in any kriged interpolation where the Kriging error is greater than the sample varience then the sample mean would be a better estimate at that location? Thanks for your help Russell Barbour Ph.D. Research Associate in Applied Mathematics Vector Ecology Laboratory Yale School of Medicine 60 College St. Rm 600 New Haven CT. 06520 TEL: 203 785 3223 FAX 203 785 3604 email: [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 -- * 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: Kriging Error vs variance
You may have to consider the stationarity of your data, i.e. theoretically your sample mean is better than your kriged estimate if your kriginng error is greater than your kriging variance, but you did make the assumption that your data is stationary when you kriged it, i.e. constant mean and variance, if this is not the case then you should consider this when adopting your sample mean. Regards Digby Millikan B.Eng Geolite Mining Systems U4/16 First Ave., Payneham South SA 5070 Australia. Ph: +61 8 84312974 [EMAIL PROTECTED] http://www.users.on.net/digbym -- * 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: Kriging Error vs variance
Russell, If you have time to get to your library there is a book Geostatistical Ore Reserve Estimation 1977 M.David although related to geostatistics for mining this book is written by a renowned geostatistician and has an excellent diagrammatic representation of trend, drift and stationarity Fig. 188, 189 pp267 which relates to your question. Hope this is of some help. Thanks, Regards Digby Millikan B.Eng Geolite Mining Systems U4/16 First Ave., Payneham South SA 5070 Australia. Ph: +61 8 84312974 [EMAIL PROTECTED] http://www.users.on.net/digbym -- * 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
