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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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