Dear List, dear Mr. Merks,
As a mathematician, a statistician, and a person well familiar with
geostatistics I would like to hint that Matherons concepts of geostatistics
and especially of the kriging variance is a well founded mathematical theory,
well compatible with all concepts of statistics and probability theory.
Indeed it is a special case of Kolmogorov-Wieder-Predicition and of a Best
Linear Unbiased Prediction (BLUP). It just differes from the most popular
part of statistics by the missing assumption of stochastic independency
between observations.
As in all instances of applied math one could discusse on the validity of the
assumptions of the theory to field of application. And as in all instance of
statistics one could discusse the statististical population, which is in the
classical case a quite large one of including all possible landscapes of
which only one is realized. In in other model of nature it might be
different. E.g in sampling theory the statistical population is the set of
locations, which leads to a totally different ideas of what is going
mathematically.
And I personally think that discussing the validity of the assumptions for a
specific situation is for all good applied statistics. We should always be
aware if the fact that things can be seen from different sides.
However claiming again and again that kriging is not a well done mathematical
theory, by wrong things like
> What a pity that Krige, Matheron, and scores of first generation
> geostatisticians, were not aware that each distance-weighted average had
> its own variance long before it was reborn as an honorific but
> variance-deprived kriged estimate.
does not make it more true.
Matheron, as can be concluded from his publications, was well aware of that
fact and made an excellent mathematical theory on it. It is obviously someone
in this list who is not aware of the fact that a variance is a statistical
moment of a single random variable (The difference of true and predicted
value) and not a proporty of a population (like it might be understood in
some introductory courses for on an undergraduate level for
non-mathematician). And also someone seams not to be aware of the fact that
e(x)=\hat{Z}(x)-Z(x) might have a variance with the same right as Z(x) and
\hat{Z}(x) (the weighted mean) have variances.
Maybe we should propose that we organize a mini-Workshop for those interested
in discussing the foundations of geostatistics including Kriging, BME,
Sampling and alternatives. 1,5 days, 4 big (invited but not payed) talks,
each with 1-2 discussents, and plenty of time for discussion and some small
contributed talks on foundations. In the end an internet-publication with
papers, discussent papers and a review of discussion. That would increase
the level of this polemic discussion to a usefull review of foundations,
wouldn't it. However I wonder whether we would get more that say 5
participants on any continent which are convinced enought that there is a
need for discussion to come.
So this is my poll:
Please send me an e-mail if you would visit such a workshop in Europe next
spring.
Best regards,
Gerald v.d. Boogaart
Am Montag, 17. Juli 2006 18:20 schrieb JW:
> Hello Tomasz,
>
>
>
> What you should do with a set of N measured values, determined in samples
> selected at positions with different coordinates in a finite sample space
> is verify spatial dependence by comparing the calculated F-value between
> var(x), the variance of the set, and var1(x), the first variance of the
> ordered set, with F0.05;n-1;2(n-1) and F0.01;n-1;2(n-1), the tabulated
> values of F-distributions at 5% and 1% with the proper degrees of freedom.
> If the set does not display a significant degree of spatial dependence, its
> distance-weighted average-cum-kriged estimate is not necessarily an
> unbiased estimate for the central values of the set. However, the variance
> of a single distance-weighted average is a genuine variance irrespective of
> the degree of spatial dependence. In fact, it would be misleading to
> compute confidence limits for that central value.
>
>
>
> What you ought not to do is compute pseudo kriging variances of sets of
> kriged estimates because a set of N functionally dependent kriged estimates
> gives exactly zero degrees of freedom. In fact, a compelling case can be
> made that the concept of degrees of freedom evolved to ensure that infinite
> sets of kriged estimates become the equivalent of perpetual motion in data
> acquisition.
>
>
>
> What a pity that Krige, Matheron, and scores of first generation
> geostatisticians, were not aware that each distance-weighted average had
> its own variance long before it was reborn as an honorific but
> variance-deprived kriged estimate. Here's a link that may guide you into
> mathematical statistics http://ai-geostats.jrc.it/documents/JW_Merks/
>
>
>
> What you may want to do is print out Readme and do read it. Most high
> school graduates are able to deduct that the posted formula for the
> weighted average converges on the variance of the arithmetic mean when
> variable weighting factors converge on 1/n. What she or he may not know
> that this variance is called the Central Limit Theorem. If you really want
> to know more about sampling and statistics, you should visit my website.
>
>
>
> What you should not do is blame me if you become addicted to commonsensical
> sampling practices and scientifically sound statistical methods.
>
>
>
> Kind regards,
>
> Jan W Merks
> ----- Original Message -----
> From: tom andrews
> To: JW
> Cc: [email protected]
> Sent: Sunday, July 16, 2006 4:05 PM
> Subject: Re: AI-GEOSTATS: Re: generalize kriging variance to
> average-based estimators different than
>
>
> Dear Jan W Merks
> <!--[if !supportEmptyParas]--> <!--[endif]-->
> I have a simple question that should be a piece of cake for such great
> expert like You. For a set of N input samples I can do by kriging the only
> ONE estimate and compute kriging variance for this single estimate. So,
> why do You call the kriging variance the "variance of some set of kriged
> estimates"? or "variance of a set of distance-weighted averages" et al ?
> <!--[if !supportEmptyParas]--> <!--[endif]-->
> Best Regards
> Tomasz Suslo
>
>
> ---------------------------------------------------------------------------
>--- Do you Yahoo!?
> Get on board. You're invited to try the new Yahoo! Mail Beta.
--
-------------------------------------------------
Prof. Dr. K. Gerald v.d. Boogaart
Professor als Juniorprofessor fuer Statistik
http://www.math-inf.uni-greifswald.de/statistik/
B�ro: 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 f�r Mathematik und Informatik
Jahnstr. 15a
17487 Greifswald
Germany
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