Sebastiano, I am struggling to understand why you are interested in doing trend + residual separation? There can be no unique decomposition of a data set into 'trend' and 'residual', it is a judgement about what model you feel is most appropriate given your prior beliefs and observations (evidence). The only thing you can say about the model is to validate it on out of sample data (even as a Bayesian I say this!). So in a sense there is no correct decomposition, and any decomposition is valid (so long as it is correctly implemented - maybe that is your question?). Are some decompositions better than others? Well yes they are likely to be, but this largely depends on your data (and the completeness of the overall model).
In terms of your original question about the shape of the kernel there is no overall theory that I am aware of - different kernels will have different properties in terms of the function classes that they represent (e.g. differentiability, frequency response / characteristic length scales). Kernel families will have different null spaces which might or might not be important for your specific application and what you want to find out. I'm not sure if this is terribly helpful ... but I think it is the reality - everything depends on your data and your judgement (prior). Conditional on those you get a model and you need to validate this model carefully ... then you are OK. cheers Dan ------------------------------------------- Dr Dan Cornford Senior Lecturer, Computer Science and NCRG Aston University, Birmingham B4 7ET www: http://wiki.aston.ac.uk/DanCornford/ tel: +44 (0)121 204 3451 mob: 07766344953 ------------------------------------------- ________________________________ From: [email protected] [mailto:[email protected]] On Behalf Of seba Sent: 02 February 2010 08:39 To: Pierre Goovaerts Cc: [email protected] Subject: Re: AI-GEOSTATS: moving averages and trend Hi Pierre I think that for my task factorial kriging is a little bit too much sophisticated (nevertheless, is there any open source or free implementation of it ??? I remember that it is implemented in Isatis.....). I have an exhaustive and regularly spaced data set (i.e. a grid) and I need to calculate locally the spatial variability of the residual surface or better I would like to calculate the spatial variability of the high frequency component. Here I'm lucky because I know exactly what I want to see and what I need to filter out. In theory, using (overlapping) moving window averages (but here it seems better to use some more complex kernel) one should be able to filter out the short range variability (characterized by an eventual variogram range within the window size???). Seeing the problem from another perspective, in the case of a perfect sine wave behavior, I should be able to filter out spatial variability components with wave lengths up to the window size. But maybe there is something flawed in my reasoning....so feedback is appreciated! Bye Sebastiano At 16.27 01/02/2010, you wrote: well Factorial Kriging Analysis allows you to tailor the filtering weights to the spatial patterns in your data. You can use the same filter size but different kriging weights depending on whether you want to estimate the local or regional scales of variability. Pierre 2010/2/1 seba < [email protected]<mailto:[email protected]>> Hi José Thank you for the interesting references. I'm going to give a look! Bye Sebastiano At 15.46 01/02/2010, José M. Blanco Moreno wrote: Hello again, I am not a mathematician, so I never worried too much on the theoretical reasons. You may be able to find some discussion on this subject in Eubank, R.L. 1999. Nonparametric Regression and Spline Smoothing, 2a ed. M. Dekker, New York. You may be also interested on searching information in and related to (perhaps citing) this work: Altman, N. 1990. Kernel smoothing of data with correlated errors. Journal of the American Statistical Association, 85: 749-759. En/na seba ha escrit: Hi José Thank you for your reply. Effectively I'm trying to figure out the theoretical reasons for their use. Bye Sebas -- Pierre Goovaerts Chief Scientist at BioMedware Inc. 3526 W Liberty, Suite 100 Ann Arbor, MI 48103 Voice: (734) 913-1098 (ext. 202) Fax: (734) 913-2201 Courtesy Associate Professor, University of Florida Associate Editor, Mathematical Geosciences Geostatistician, Computer Sciences Corporation President, PGeostat LLC 710 Ridgemont Lane Ann Arbor, MI 48103 Voice: (734) 668-9900 Fax: (734) 668-7788 http://goovaerts.pierre.googlepages.com/
