Hi Dan
Thank you for your reply, I was forgetting that I had to send
another mail about this issue..... so your mail
is not too late!!!
I understand (or I think to!) your points and also the Nicholas Nagel's ones.
Following your considerations and my own experiences with
different data sets related to different physical-chemical processes
I'm oriented to draw the following practical considerations.
I think that it could be useful to make a distinction between
"poor" (for example few sparse data of water levels) data sets and
"almost exhaustive" data sets (for example topographic
LiDAR data). Clearly it is not easy to say where the limit among the
two cases lies: we should know the real spatial
variability to state the effective informative content of our data.
From my point of view, in the case of a "poor" data set the choice
of the trend model should be based on our expert knowledge
(knowledge about the processes). In this case, if there is no
physical reason to do something different, the trend model should be
fitted globally (better if by means of generalized least squares) and
consequently the universal kriging should be performed using all data
and not by means of a local search. For sure also a Bayesan approach
could be followed.
Differently, in case of a "almost exhaustive" data set, above all
when related to a non stationary spatial distribution, things are a
little bit different. Theoretically I would like to have a method
which is able to define
the smallest moving search windows in which is possible to choose
and automatically fit the most "simple" combination of trend and,
eventually non isotropic, covariance function (or
variogram). Practically, considering also the computational cost, I
guess that good results could be achieved detrending data with local
polynomials and performing ordinary kriging on the residuals, with a
globally or, better, a locally fitted covariance function.
The choice of the search window, the degree of polynomials, the
bounding values for covariance function fitting should be based on
cross validation and the relative diagnostics (but with massive data
this could be quite costly from the computational point of view).
Finally, in the case of "almost exhaustive" data set it is also
possible to think to perform some preprocessing of data before
interpolation. For
example data could be classified considering their spatial
variability. And for sure, before interpolation, zones of
discontinuity, or abrupt changes, could be delineated.
I would like to know if you agree with this considerations or if you
have other operational suggestions.
Sebastiano
At 22.47 13/07/2008, you wrote:
Sebastiano,
I have just got the end of my mailbox and I noticed your email
(meant to reply a long time back). I also read Nicholas Nagel's
reply. I think he does a good job of summarising where we stand at
the moment, but I would add a comment (or two).
In my PhD many years ago I spent a lot (LOT) of time worrying about
this trend versus residual interpretation. This was before I saw the
Bayesian light :-). Now I tend to worry less - it is clear that
there is no unique interpretation of this issue.
I have worked a lot in machine learning over the years, and in this
field many researchers tend to think about generating 'kernel
functions' (covariance functions / variograms) often based on
'trend-like models' and using mixture of such covariances to create
models which have complex additive covariance structures. To
generalise crudely, machine learning people tend to assume zero mean
trend and have complex priors over functions in the covariance part.
I also work a lot with statisticians - they tend to put lots of
things into the trend part, and model the covariance as simply as possible.
Quite often the two approaches are theoretically equivalent (in the
infinite data limit!). I quite like the rationale for trend + simple
covariance - in a linear in parameters model, it is generally easier
to marginalise (integrate out) the effect of uncertainty on the
trend parameters in a computationally efficient manner.
For my part, I believe that the effort in the modelling should,
where possible, produce the 'correct' covariance structure from the
known process based dynamics (i.e. the scientific knowledge of the
system) that actually generated the data. In this case one can put
rather more belief into what is trend and what is not. Of course
this is hugely difficult and often not possible in many
applications, but I am pretty confident that it will give the most
honest (with respect to reality) answers, if done.
If you can't do this, then it is a question of belief (or
judgement). The model is purely a representation of the data, and
your beliefs about the data should inform this judgement on the
model structure (i.e. I'm being Bayesian here). I'd like to suggest
that an appropriate model structure could be determined using some
sort of Bayesian model comparison (e.g. Bayes factors) but I
recently attended a very good talk where is was suggested that
estimating these reliably is very tricky, even in quite simple models.
So if you don't apriori know about what the trend might look like
what can you do?: well to be honest I would use careful cross
validation! And appropriate diagnostics (checking in particular on
my predictive DISTRIBUTIONS) - Malhalanobis distance, test set likelihood.
Anyway hope that is not too late to be of relevance,
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
-------------------------------------------
> -----Original Message-----
> From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf
> Of seba
> Sent: 16 May 2008 08:22
> To: ai-geostats@jrc.it
> Subject: AI-GEOSTATS: The trend ...again
>
> There was something wrong with the last message (the message for
> some reason appear cutted!?)
> ...I send it again...
>
> Dear list members
>
> Another time, I have some questions about the decomposition between
> the trend and residuals.
> I have some philosophical as well as technical questions for you.
>
> First I would like to start with philosophy: given that
> the trend is a deterministic component, why I have to evaluate
> its uncertainty? Can I consider it determined and consider the stochastic
> part as the exclusive source of uncertainty?
>
>
> Then I come back with technical stuff:
> Following an iterative approach I was able to
> calculate, considering the whole spatial domain of interest, a trend model
> by means of generalized least squares (GLS) and consequently the
> residuals.
> Then I followed two approaches to make predictions (in reality three
> because I tried also ordinary kriging that could
> be useful if I should use a search window smaller than the whole
> spatial domain) :
>
> 1) Universal kriging
> 2) Simple kriging on the residuals and composition with the estimated
> trend.
>
> I compared the estimation variances of UK predictions with the
> estimation variances
> of the second approach (variance of simple kriging + variance of GLS
> trend model, supposing
> that there is not correlation between the two components).
> I have seen that the results are a little bit different, differently
> from the predictions that are practically the same.
> I would like to know if the difference in variances between the two
> approaches
> are related to some my misunderstanding or if it is justified
> theoretically.
>
>
> Then a short consideration: for sure a GSL approach should give
> better results than a ordinary least square approach (OLS) above all
> when considering clustered data.
> But maybe other choices, regarding for example the characteristics
> of the trend (the degree of polynomial or the characteristics of
> auxiliary regressors in case of Kriging with
> external drift or regression kriging) and above the search windows,
> play a more important role.
> In particular I'm interested in the last point. The decomposition
> between trend an residuals is a matter of scale, given that we are
> trying to separate low frequency variability from high frequency
> variability. For example, considering a 2D spatial domain, if
> I consider a small search window a planar trend could be good, but
> when I enlarge the windows a quadratic trend could be better.
> I have the feeling that (having many data compared to the variability
> of the phenomenon)
> there should be a way to choose locally a search window giving a good
> balance
> between "complexity" of the trend model and "complexity" of the residuals.
>
> Thank you in advance for you time.
> Sebastiano Trevisani
>
>
>
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