Cornford, Dan wrote:
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).
An article by Tomislav Hengl goes into separately estimating the
trend and then interpolating the residuals [1]. He calls it
regression kriging. Might be interesting to have a look at it in
light of this discussion.
cheers,
Paul
[1] http://dx.doi.org/10.1016/j.cageo.2007.05.001
@ARTICLE{Hengl2007,
author = {Hengl, T. and Heuvelink, G.B.M. and Rossiter, D.G.},
title = {About regression-kriging: From equations to case studies},
journal = {Computers \& Geosciences},
year = {2007},
volume = {33},
pages = {1301--1315},
number = {10},
}
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:* owner-ai-geost...@jrc.ec.europa.eu
[mailto:owner-ai-geost...@jrc.ec.europa.eu] *On Behalf Of *seba
*Sent:* 02 February 2010 08:39
*To:* Pierre Goovaerts
*Cc:* ai-geostats@jrc.it
*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 < sebastiano.trevis...@libero.it
<mailto:sebastiano.trevis...@libero.it>>
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
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