## AI-GEOSTATS: curve fitting summary

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• Here is the summary to my question: I have used geoeas until now, which uses a limited number of models. However, I found a program, CurveExpert (AVAILABLE AT
Message 1 of 5 , Nov 20, 2002
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Here is the summary to my question:

I have used geoeas until now, which uses a limited number of models.
However, I found a program, CurveExpert (AVAILABLE AT
http://www.ebicom.net/~dhyams/cvxpt.htm), which finds the best fitting
model, which is
usually different from the options in geoeas. For example, for my data, I
foundthat the best fitting models were a 4th level polynomial model and a
"Hoerl" model.
Is there any program (downloadable if possible) that would krige with a
custom
model?
Thanks,
Carolina

Pierre Goovaerts, Donald E. Myers and Isobel Clark answered

You can not fit any type of curve to
your experimental variograms since the model
needs to be permissible, hence the practice
to fit only a limited number of models
that are known to be permissible.

Pierre

Our software, EcoSSe, is enormously flexible and could
easily be modified to take a generic model, but it
does cost US1,000
When you fit a model you might want to consider a
Cressie-like statistics which weights inversely by the
model value (or distance) and directly by the number
of pairs.

Isobel

A 4th level polynomial can not be a valid variogram/covariance or even
a generalized covariance, consequently you would not want a
geostatistics package that would allow the use of such a "model". The

problem is that in general the the kriging system would not have a
unique solution (in fact might not have a solution at all). While there
are more possible valid models (some of which can be obtained by
"nesting" the basic models in GeoEas, it is not entirely clear that
this
would improve the results.

To be a valid covariance function, it must be positive definite (as a
function). In particular this implies that the function is bounded
(hence no polynomials)

To be a valid variogram, it must be conditionally negative definite and
have a growth rate that is less than quadratic.

Donald E. Myers

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• ... I hate to sound ignorant here, but aren t most of the standard semi-variogram models polynomials of one kind or another? I remember seeing a paper a few
Message 2 of 5 , Nov 21, 2002
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> To be a valid covariance function, it must be
> positive definite (as a function). In particular
> this implies that the function is bounded
> (hence no polynomials)
I hate to sound ignorant here, but aren't most of the
standard semi-variogram models polynomials of one kind
or another?

I remember seeing a paper a few years ago by a coupl
eof blokes from Pretoria University on a generalised
polynomial fit which would be positive definite. I
don't have it to hand but can probably track it down
if given sufficient motivation ;-)

Isobel Clark
http://geoecosse.bizland.com/news.html

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• I think that Pierre and Don Myers gave the correct response. For beginners the lesson must be that you cannot use any old function (and in particular a
Message 3 of 5 , Nov 22, 2002
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I think that Pierre and Don Myers gave the correct response. For beginners
the lesson must be that you cannot use any old function (and in particular a
polynomial) as a covariance function. It will lead to singular kriging
matrices for some configurations of data points. That is why we stick to a
few tried and tested covariance functions in most computer packages. In fact
there are lots of 'esoteric' covariance functions known. You don't have to
just stick to the spherical, exponential and Gaussian ( it might be an idea
to have a list of covariance functions somewhere on the AI geostats site)

To get after the point that Isobel said about polynomials - the facts are
best stated in terms of the type of random function

If the random function is stationary - then a covariance function C(h)
exists and is bounded - C(h) is a positive definite function so absolutely
no polynomials allowed whatsoever!

If the data is from an intrinsic function of order 0 (IRF-0) then the
variogram must have growth less than h**2, so the only polynomial is of
the form "gamma(h) = a+b*h where b>=0" (Of course you can have "gamma(h)
= h**c where c<2 but the only polynomial is with c=1)

If the data has more general nonstationarity such as the IRF-k, then the
generalised covariance can be a polynomial. However
1) the polynomial is not arbitrary (there are constraints on the
coefficients)
2) it is definitely not found by fitting a curve to the experimental
variogram - you need special fitting techniques which are not found in many
packages

In other words, if your variogram appears to grow without bounds, you have
got non staionarity. You can either try a linear model (if appropriate) or
try to tackle the non-stationarity head on by removing the trend from your
data and then using ordinary variogram fitting techniques to the stationary
residuals (or by applying an IRF-k model if you have the software)

For more details, see the book by Chiles and Delfiner for example (or for
the real purist you can go to Matheron's original publication. "The
intrinsic random functions and their applications Adv. App. Prob., 5, pp
439-468" but beware the maths is not simple in Matheron's paper)

Regards

Colin Daly

----- Original Message -----
From: "Isobel Clark" <drisobelclark@...>
To: <ai-geostats@...>
Sent: Thursday, November 21, 2002 10:23 AM
Subject: Re: AI-GEOSTATS: curve fitting summary

> > To be a valid covariance function, it must be
> > positive definite (as a function). In particular
> > this implies that the function is bounded
> > (hence no polynomials)
> I hate to sound ignorant here, but aren't most of the
> standard semi-variogram models polynomials of one kind
> or another?
>
> I remember seeing a paper a few years ago by a coupl
> eof blokes from Pretoria University on a generalised
> polynomial fit which would be positive definite. I
> don't have it to hand but can probably track it down
> if given sufficient motivation ;-)
>
> Isobel Clark
> http://geoecosse.bizland.com/news.html
>
> __________________________________________________
> Do You Yahoo!?
> Everything you'll ever need on one web page
> from News and Sport to Email and Music Charts
> http://uk.my.yahoo.com
>
> --
> * To post a message to the list, send it to ai-geostats@...
> * 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 majordomo@... 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
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