I completely agree with your review. My only justification is that the main
intention of my previous thought was strictly to follow the idea of kriging.
The primitive statistics can only cut flowers in garden (to remove spatial
dependence) measure (how many measured flowers do you count?) and
compute central value. Weighted average is strictly restricted to central value.
The idea of modern statistics is much more intelligent not to cut
but to consider spatially dependent mathematical model of garden with
predictable spread values (how many calculated flowers do you count?)
Weighted average is not strictly restricted to central value.
Since D^2{sum_i=1^n omega_i V_i} = D^2{V} sum_i=1^n omega_i^2 each
distance weighted average can have its own variance but we have to know
constant process variance associated to central value first. The process
variance is genuine variance and variance of single weighted average
is second hand variance.
Best Regards
Tomasz Suslo
Gerald van den Boogaart <[EMAIL PROTECTED]> wrote:
Dear Mr. Andrews,
> Kriged estimate can not have always its own variance since kriged
> estimate can not be always considered (thanks God !!!) as the central
> value.
I only partially agree with your clearification:
A variance is general concept for random variables.
And random variables might or might not have variances. This is not a
priviledge of central values.
So under some assumptions (i.e. second order stationarity of the field) it is
well true that kriging predictors have a variance even if this variance might
be unknown and even when the kriging predictor is not intended to be a
central value.
The problem is more complicated:
In classical unbiased estimation theory the variance of the estimator tells us
something about its precision of the estimation by the equation:
Var( Estimator ) = E[ ( Estimator - trueFixedValue )^2 ]
however for unbiased predictors a similar equality does not hold :
Var( Predictor(x) ) not= E[ ( Predictor(x) - trueRandomValue(x) )^2 ]
However the right hand side of this equation is the relevant description of
the precision of the prediciton and not the left hand side. And this right
hand side becomes the the left hand side only in the special case of
nonrandom values.
And clearly E[ ( Predictor(x) - trueRandomValue(x) )^2 ] and
Var( Predictor(x) ) depend on x which means that each predictor and each
prediction error ( Predictor(x) - trueRandomValue(x) ) have their own
variance. And nobody wants to claim something different!!!
However things become more complicated just because the Var( Predictor(x) )
does not need to exist, and since is not identifiable even if exists if the
mean of the field is unkown, und does definitly not exist for generalized
(e.g. intrinsic) random fields.
And this makes it so difficult to debate on that subject properly: The
existence problem is quite unrelated to kriging and emerges from the
mathematical model. However it implies that one can not just say: Yes, each
kriging predictor has its own variance, but that variance is irrelevant for
the description of the precession of the prediction.
Best regards,
Gerald v.d. Boogaart
Am Donnerstag, 27. Juli 2006 10:05 schrieb tom andrews:
> Dear Jan W Merks
>
>
> ABSTRACT (as You wish):
> Yes each distance-weighted average that is considered as the central
> value has its own variance but no each distance-weighted average has to
> have its own variance since no each distance-weighted average can be
> considered as the central value.
>
> BODY:
> If the distance-weighted average (mean) is taking into consideration as
> the central value then the kriged estimate (mean) has its own variance
> equal to the kriging variance multiplied by the sum of kriging weight
> squares since in such case the kriging variance is the constant variance of
> the process. Please, see http://arxiv.org/abs/cs.NA/0409033 for details.
>
> SUMMARY:
> Kriged estimate can not have always its own variance since kriged
> estimate can not be always considered (thanks God !!!) as the central
> value.
>
> Best Regards
> Tomasz Suslo
>
>
> ---------------------------------
> Talk is cheap. Use Yahoo! Messenger to make PC-to-Phone calls. Great rates
> starting at 1¢/min.
--
-------------------------------------------------
Prof. Dr. K. Gerald v.d. Boogaart
Professor als Juniorprofessor fuer Statistik
http://www.math-inf.uni-greifswald.de/statistik/
Büro: Franz-Mehring-Str. 48, 1.Etage rechts
e-mail: [EMAIL PROTECTED]
phone: 00+49 (0)3834/86-4621
fax: 00+49 (0)3834/86-4615 (Institut)
paper-mail:
Ernst-Moritz-Arndt-Universitaet Greifswald
Institut für Mathematik und Informatik
Jahnstr. 15a
17487 Greifswald
Germany
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