AI-GEOSTATS: entering the fray
I'm hesitant to participate for fear of being jumped on, but I think there is another aspect to the original question that has not been addressed. From a practical standpoint, we have complete discretion in choosing a model. In theory, the properties of the sample do not inform the choice. That being said, I wonder why one would choose a random model with no sill (i.e., assume that the covariance function doesn't exist) rather than a non-stationary model? If the covariance function exists, then C(0) is your constant, and produces the odd result for a linear model that the covariance between more distant locations becomes increasingly negative. If the empirical semivariogram doesn't reach a sill near the overall variance of the sample=C(0), I would, consider choosing a different model that used covariates to remove drift, or vary the sill in different strata/regions if there is truly heteroscedascicity. I think part of the difficulty in the semivariogram vs. covariance war is that modeling is subjective, and the notion of covariance has become more intuitive for statisticians, while the notion of semivariance has become more intuitive for geologists. Yetta -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and unsubscribe ai-geostats followed by end on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: AI-GEOSTATS: entering the fray
Hi Yetta Jump in, the water is lovely! All contributions equally valid in my e-mail box ;-) I have to confess that I have rarely used an unbounded semi-variogram model. In mining applications, in my experience (which is limited to 30 years in economic mineralisations) semi-variograms which shoot off into the wild blue yonder are usually caused by trend, strong anisotropy or violation of the 'homogeneity' assumptions (stuff like faults etc or skewed distributions). However, the de Wijsian model is extremely popular in Southern Africa and widely used by some major mining houses along with simple kriging. Not my bag, but who am I to judge? There is an interesting paper by Cressie (not got reference to hand, but it must be in his book somewhere) where he treats the Wolfcamp data as an anisotropic generalised linear model. I use a quadratic trend surface and a spherical model for the residuals. The final estimates are almost identical, but the standard errors differ by an order of magnitude. Actually, I used that as an example in a talk in Ireland about 10 days ago. Noel is an archetypical ivory tower academic (and all round good guy), so I guess we did a bit of role reversal there ;-) I agree that the semi-variogram approach is easier for the non-statistician to grasp. Difference in value is a simpler concept to grasp than cross-product, especially when your boss wants to know the likely difference between what you tell him and what really happens! Keep it coming. It is your voices we want to hear, not us border line pensioners Isobel Clark http://uk.geocities.com/drisobelclark Do You Yahoo!? Get your free @yahoo.co.uk address at http://mail.yahoo.co.uk or your free @yahoo.ie address at http://mail.yahoo.ie -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and unsubscribe ai-geostats followed by end on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
Re: FW: AI-GEOSTATS: entering the fray
Hi guys,
I promised myself I would not waste more time
on this futile discussion about covariance and variogram,
but it seems that the discussion has drifted far away from
the initial comment by Isobel or that most people don't
remember what was the initial question.
Isobel's comment originated from my sideline remark
(it was not even part of Celia's initial question)
that the SIMPLE kriging system can not be written in terms of
semivariograms, which Isobel qualified of pure non sense..
It seems that my reference to the excellent
book by Chiles and Delfiner did not convince Isobel.
Let's then use Gslib book by Deutsch and Journel since
it is probably more widely used by members of this discussion
list and Isobel pointed out that anecdote with Andre.
On page 65 of Gslib user manual, 2nd paragraph, I quote:
In the sytem (IV.4) (SIMPLE kriging system!), the covariance
values C(h) cannot be replaced by semivariogram values
g(h)=C(O)-C(h) unless sum_lambda = 1, which is the ordinary
kriging constraint. I guess it's clear enough, and that is
nothing to do with whether we should solve an ordinary
kriging system in terms of covariances or semivariograms
(Everybody knows that you get the same results!), or
whether we should teach students in one way or another...
Given that SIMPLE kriging is rarely used, we might even
argue that all this discussion is pointless...
Again, the reason for that e-mail is to clarify the matter
for students or practitioners who might have been
confused by this exchange of e-mails... I don't have
a book, a software or a consulting company to advertise!
Cheers,
Pierre
|\/|Pierre Goovaerts
|_\ /_|Assistant professor
__|\/|__Dept of Civil Environmental Engineering
|| The University of Michigan
| M I C H I G A N| EWRE Building, Room 117
|| Ann Arbor, Michigan, 48109-2125, U.S.A
_||_\/_||_
||\ /||E-mail: [EMAIL PROTECTED]
|| \/ ||Phone: (734) 936-0141
Fax: (734) 763-2275
http://www-personal.engin.umich.edu/~goovaert/
On Wed, 23 May 2001, Steve Zoraster wrote:
1)What manager in the mining or petroleum industry who has graduated
from college hasn't taken a serious statistics course, including covariances
and correlations?
2)Surely when starting from scratch, educating someone about
geostatistics is more intuitive using covariances? (Just my opinion so far,
speaking as a mathematician who remembers teaching basic college level
statistics to nursing majors, education majors, sociology majors, etc. And
even succeeding occasionally.)
3)I have taken two multi-day courses in geostatistics from well known
industry experts. In each class they included significant material and time
on the first day explaining/justifying variograms by showing their
mathematical relationship to spatial covariance functions. It seems that
those instructors did not trust the variogram to be more intuitive than
spatial covariance functions.
4)The two basic level introductions to geostatistics I have on my
bookshelf replicate the experience at those two classes
Steven Zoraster
-Original Message-
From: Yetta Jager [SMTP:[EMAIL PROTECTED]] mailto:[SMTP:[EMAIL PROTECTED]]
I think part of the difficulty in the semivariogram vs. covariance war is
that modeling is subjective, and the notion of covariance has become more
intuitive for statisticians, while the notion of semivariance has become
more intuitive for geologists.
From: Isobel Clark [[EMAIL PROTECTED]]
I agree that the semi-variogram approach is easier for the non-statistician
to grasp. Difference in value is a simpler concept to grasp than
cross-product, especially when your boss wants to know the likely difference
between what you tell him and what really happens!
--
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Re: FW: AI-GEOSTATS: entering the fray
Hi all, A few more references to the Covariance Vs. Semi- variograme discussion: To support Semi-variograme: Cressie N.A.C. (1993) Statistics for spatial data. New York Wiley. ( Page 70- 73) I believe that the original discussion appears in: Cressie A.C. Noel. and Grondona O. Martin (1992); A comparison of Variogram Estimation with Covariogram Estimation, In The art of statistical Science Edited by K.V. Mardia Jhon pp:191-208, Wiley Sons Ltd. Cressie proves that semi- variogram estimation is to be preferred over covariogram estimation; the main reasons for that are: 1.In the Kriging process where we estimate the mean of the process and then predict the random process both the variogram estimator and covariogram estimator are biased. However the variogram bias is of smaller order. 2. If our data has trend contamination then it has disastrous effect on attempts to estimate the covariogram while on the variogram it has a small upward shift. There is more to that; check on the book.. To support Covariance: Barry and Pace (1997) Kriging with large data sets using sparse matrix techniques Communications in statistics simulation and computation Vol 26 (2) pp 619-629 exploit the sparseness of covariance matrix - with stationary models we have zeros for points outside the range - and they were able to dramatically lower the time and storage cost of kriging. Since the covariance matrix is a symmetric positive definite matrix, we can use the Cholesky factorization for its inversion. If A is n-by-n, the computational complexity of Cholesky(A) is O(n^3), but the complexity of the subsequent inversion solutions is only O(n^2). With Marco's suggestion of Matheron equations it seems that one can use Cholesky factorization even with semi- variogram matrix. Thanks for interesting discussions. Yaron Felus The Ohio-State University http://felus.mps.ohio-state.edu/yf/ -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and unsubscribe ai-geostats followed by end on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
