Hi Mark, This technique is also known as uniform score transform and has been introduced in 1984 by Sullivan as part of probability kriging (e.g. see my book page 301). Like the normal score transform, it amounts at replacing original observations by the corresponding quantiles of a target distribution (e.g. uniform or standard normal distribution). You may want to check the following references that discuss the applications of the rank-order transform and its back-transform. Journel, A.G. and Deutsch, C.V., "Rank Order Geostatistics: A Proposal for a Unique Coding and Common Processing of Diverse Data," Geostatistics Wollongong '96, Vol 1, Baafi and Schofield, editors, Kluwer Academic Publishers, September 1996, pp 174-187. Bourgault, G. et al. 1997. Geostatistical Analysis of a soil salinity data set. Adv. Agronomy 58, 241-292. http://www.ars.usda.gov/SP2UserFiles/Place/53102000/pdf_pubs/P1410.pdf <http://www.ars.usda.gov/SP2UserFiles/Place/53102000/pdf_pubs/P1410.pdf> If you do not want the standardized ranks to be valued 0 or 1, just use the transform rank/n-0.5/n instead of rank/n+1.. For large n, it shouldn't make a lot of differences.. The back-transform is always the critical point. You have the same problem when conducting multiGaussian kriging: the normal score back-transform of the kriging estimate obtained in the normal space will lead to a biased estimation in the original space... You can back-transform any quantile of the distribution though, that is the normal score back-transform of the median will still be the median in the original space. I have 2 comments: 1. The equation [21] given in Juang et al. paper applies only to simple lognormal kriging. For ordinary lognormal kriging, the Lagrange parameter must also be part of the back-transform, see Eq. (6) in Saito, H. and P. Goovaerts. 2000. Geostatistical interpolation of positively skewed and censored data in a dioxin contaminated site. Environmental Science & Technology, vol.34, No.19: 4228-4235. 2. As mentioned in Journel & Deutsch's paper, the estimates back-transformed according to the middle-point model are only median-unbiased. The idea is that the kriged rank is an estimate of the rank of the unknown original value, hence you simply compute the value with the same rank in the original distribution. Personally I would use the method described for the normal score back-transform in Saito and Goovaerts (2000): you compute the 100 percentiles of the local uniform distribution of probability in the transformed space, then back-transform those 100 percentiles and use their arithmetical average as an estimate of the mean of the local distributions in the original space. Cheers, Pierre Pierre Goovaerts Chief Scientist at BioMedware Inc. Courtesy Associate Professor, University of Florida President of PGeostat LLC Office address: 516 North State Street Ann Arbor, MI 48104 Voice: (734) 913-1098 (ext. 8) Fax: (734) 913-2201 http://home.comcast.net/~goovaerts/
________________________________ From: [EMAIL PROTECTED] on behalf of Mark Dowdall Sent: Mon 2/12/2007 5:13 AM To: ai-geostats@jrc.it Subject: AI-GEOSTATS: Rank-order geostats Hello I have a VERY skewed data set that fails tests for normality and log normality. Variograms are OK for th elower percentiles of the set but as one goes above the median the variograms get quite poor. And that is causing me a bit of a headache for Indicator kriging. I came across a paper by Juang et al (J. Environ. Qual. 2001, 30:894-903) that discussed the use of Rank-order geostats for highly skewed data and had my interest peked. I transformed the data by ranking them, then dividing each rank transformed data point by the total number of data points. And the variogram (omni directional, all data) looked exceptionally well. Enthused, I began reading and searching the archive on ai-geostats but have some questions. 1. Is rank order (as in rank/number of samples) geostatistics known by some other name as there doesnt seem to be too much out there bar a couple of papers? 2. Is n-score geostatistics the same thing? 3. Some people seem to say the rank should be divided by N+1 and others N. Which should it be or have I misunderstood? 4. Juang discusses back transforming the data using a "middle point model". I cannot understand how he has acheived this. Has anyone any experience in back transforming the estimates to concentrations? I remember problems I had before with log transformed estimates and whether or not to add half the kriging variance to the back transformation value and would rather not fall into the same kind of problem. If any one has any info on rank order geostatistics and particularly back transforming, I would be very grateful. Thanks in advance M dowdall + + To post a message to the list, send it to ai-geostats@jrc.it + To unsubscribe, send email to majordomo@ jrc.it with no subject and "unsubscribe ai-geostats" in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list + As a general service to list users, please remember to post a summary of any useful responses to your questions. + Support to the forum can be found at http://www.ai-geostats.org/ + + To post a message to the list, send it to ai-geostats@jrc.it + To unsubscribe, send email to majordomo@ jrc.it with no subject and "unsubscribe ai-geostats" in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list + As a general service to list users, please remember to post a summary of any useful responses to your questions. + Support to the forum can be found at http://www.ai-geostats.org/
