Thanks to Gerald for the replies to my latest questions
about the derivation of the estimation variance for the
kriging equations and the assumptions made. Below is the
Geralds email;
Dear Mr. Millikan,
> Hello,
>
> I was wondering about the technique of simulation. Is the basic method of
> simulation to
>
> generate random estimates using the kriging variance distributions, or is
> it more complex
>
> than this.
>
>
>
> Regards Digby Millikan
It more complex than that.
Honestly, when asked you to maybe read some more theoretical books, this
might
be seen as a very indirect hint to my feeling that asking unreflected
questions in a mailing list is not the pricinple way of learning a field of
science.
E.g. your question on the relation of mean squares and variance or why we
use
the terms variogram and variance for a quite similar thing can only be
answered either by understanding an arbitrary basic introductory chapter in
basic statistics or geostatistics book or by a stupid "pseudo answer" like:
a variance is a kind of mean square, i.e. the mean square difference of a
random variable from its mean.
or
a variogram is a function giving variances of differences of observations in
a
given distance.
However probably you well know both of these facts and would be able to give
this answer yourself.
Here is a list of answers to your last questions:
> Variograms represent only mean square differences, is this a major
> shortcoming of geostatistics?
>
A honest answer to this would need to discusse on the squared error as a
measure of predictivity. In case of the decission that this is a good
measure
we would have to discusse that is the genius thing about the variogram to
identifiable from the data and containing exactly the information to give
the
linear unbiased estimator with the least expected mean square. We could
question each part: linearity, unbiasedness, squared errors and would in all
cases get the hint to some extended geostatistical theory like Disjunktive
Kriging, IRFk, Conditional Expectation/Nonlinear Kriging, Baysian Prediction
and so on which e.g. discussed in Cressie 1993 and which are also based on
variograms or on more complex functions and typically not used because the
more complex functions can not be properly estimated. Also the shortcoming
of
the mean squares in the variogram have to dicussed in the case of
multivariate kriging and the variogram has to be replaced here. Again the
complexity of the problem would take several chapters.
> Is unbiasedness a fas? E(Z*-Z) = 0
>
> E(sumwZ(x) - Z) = 0
>
> sumwu - u =0 says all Z(x) =u this is
> not true?
Maybe a full understanding of the underlying algebra would have helped. Your
problem with the formula seamed to arise from the short writting in an email
sumwZ(x)-Z
instead of
E[ \sum_i w_i Z(x_i) - Z(x_0) ] =0
as it can be found in the books.
> I am still trying to gain a deeper understanding of the derivation
> of kriging variance, which I have not fully grasped, which is based
> on using mean variogram values, and was curious as to weather other
> values could be used or only the mean was appropriate.
>
I can not imagine how anyone could help get a better understanding of that,
by
answering an e-mail, than our most famouse people can help by the book
chapters they wrote on exactly that topic. The derivation of the kriging
variance is really explained in each an every geostatistics book.
> Sorry I meant I am trying to gain, a better understanding
> of the derivation of the estimation variance used for the kriging
> equations. I understand the derivation is based on a number of
> assumptions, which I am trying to understand the effect of these
> assumptions. e.g. using mean values, unbiassedness etc. I am
> reading Chapter 10 of Isobels book, which seems to be the greatest
> help.
>
So if Isobels book is a great help, why are you asking.
However I can give you an expert hint, where the answers are. However I do
expect any help from them:
The assumption of unbiasedness has a lot of implications.
-> You can write the formulae in terms of variances (This is understanding
the
underlying calculus on moments)
-> You can use variograms rather than covariance functions (this is
IRFk-theory)
-> Typically there is no way of given a best predictor without asking for
unbiasedness if the conditional expectation is not computable (this is the
Stein-Theory)
The idea of using mean values is not the idea of using mean values but the
idea of using moments, i.e. expected values. The definition of an expected
value is a basic notion of probability theory and well explained in all
books on probability or measure theory. The idea that moments can estimated
by means is called U-type estimate and basic notion of mathematical
statistics and is explained in detail in each and every book on statistics.
The idea using distance dependent moment functions is a basic notion of
geostatistics and explained in detail there.
The assumption of stationarity, is necessary to be able, to define the
distance dependent variogram, to estimate it from the observations and
distinguish the effect of mean and variance. However this is explained in
detail in the introctory chapters of geostatistical books.
> That's what I was wondering unbiasedness is based on E(Z*) = the mean,
> But we know Z* is not the mean because we are estimating it. I'm sure
> it's true, otherwise E(Z*-Z) = 0 would not be published so many books,
> however there is not a lot of explanation of this assumption.
>
Maybe an understanding of the expected value would help: Take a fair dice
showing the numbers 1,2,3,4,5,6 and than the expected value of i-th gambled
number X_i would be mu=3.5=1/6*1+1/6*2+1/6*3+1/6*4+1/6*5+1/6*6. And
obviously E[X_i - 3.5] = 0 although X_i is not (never) 3.5 and .nobody
expects it to be. This the basic of understanding moments. Moments are means
over the randomness.
>
>
> pp252 Practical Geostatistics 2000;
>
>
>
> " ue = E{g1} - E{T} = u - u = 0 "
>
>
> My question was how is does E{g1} = u, when we know it's value is g1.
This is the idea of Expectation and random variables. You need to read a
book
on probabily. The expected value of a random variable is a constant, not
equal to the true value.
>
> Is this unbiasedness an assumption or a reality? An assumption so that
It is a reality, because it is a restriction of the predictor to those
having
the property. However it is based on the assumption of stationarity.
However, what it is exactly can be found from understanding the derivations
in
a book on geostatistics and not by getting short answers to a mail.
>
> we only have to minimise, the variance, assuming the error distribution
>
> has a zero mean.
>
>
Not assumping but restricting to estimators with a given property.
However understanding this prerequires understanding of moments.
> I was wondering if anyone knows any accessible good published papers on
> nugget and proportional effects in mining applications?
>
We are at the level that this is discussed in textbooks published one 1-3
decades ago. I would expect these things to be discussed in a high level
journal level in the early (i.e. 1960s and 1970s) publications of Krige and
Matheron.
Best regards,
Gerald v.d. Boogaart
Am Mittwoch, 27. September 2006 04:27 schrieb Digby Millikan:
--
-------------------------------------------------
Prof. Dr. K. Gerald v.d. Boogaart
Professor als Juniorprofessor fuer Statistik
http://www.math-inf.uni-greifswald.de/statistik/
Bro: 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 fr Mathematik und Informatik
Jahnstr. 15a
17487 Greifswald
Germany
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