Dear Ai_GeoStatistians,
Regarding my query on 'LogNormal Kriging', I got responses from four
people.
Scroll down to read them.
[My questions are given with Dr. Oliveira's reply]
Thanks for all.
-/ramani.
======================================================
1. Dr. Victor De Oliveira
--------------------------
On Fri, 13 Sep 2002 [EMAIL PROTECTED] wrote:
> Dear all,
>
> Even after learning geostats for couple of years, I am not sure I got
> the concepts right. (My lame excuse, "Since I am a statistician, I
have
> to understand a lot" does not help) Hence, pardon me if/since my
> questions are very fundamental to this group. Whenever I have
> geostatistical concept and implementation issues, I explore the
> ai-geostats (old & new) archives at yahoo!groups to clear clouds over
my
> head. However, I haven't found any relevant threads deeply addressing
on
> the following questions. Thus, I am posting here.
>
> 1. Having normality in data always bothers me; I see the text books
and
> papers comment on this issue, mostly saying, "Normality is not
necessary
> for kriging, but having normal data distribution makes the prediction
> better." I am yet to understand this statement.
The second part of the statement is vague, but what I pressume it means
is that kriging predictors are optimal, in the mean square sense, among
the class of all possible predictors when random field is Gaussian.
>
> 2. On Lognormal kriging, I have two questions.
>
> 2.1. If my (original) data shows a second order polynomial trend,
while
> fitting a lognormal distribution, how to proceed further along
variogram
> fitting & kriging?
>
> Removing the trend from the original data and fitting the variogram to
> the 'residual'? or, can this second order polynomial trend be directly
> incorporated into the lognormal OK or lognormal KT?
Estimating the variogram by some form of least squares for random fields
with non-constant mean function is problematic even when no
transformation
is entertained (the problems are described in Cressie's book).
For lognormal kriging these problems become more severe.
On the other hand, estimation of the variogram by likelihood methods
(ML, REML, Bayesian) do not require "removing the trend" and behave
better, I believe, although they are not free from other drawbacks.
>
> [In the archives, I saw a post by Prof. Donald Myers touched this
issue;
> 'Dioxin contaminated site' paper by Dr. Goovaerts has touched the
> comparative studies among different types of kriging]
>
> Prof. Cressie's book ("Statistics for Spatial Data", pp. 135-137)
> discusses the lognormal kriging to an extent. However, I do not
> understand well how the 'Mean' discussed in the lognormal kriging is
> connected to the trend in the original data.
>
When using transformed Gaussian random fields there is always an issue
about the interpretation of the parameters. What is usually done is to
model the mean and covariance functions in the transformed scale where
the process is (approximatelly) Gaussian. For the case of the
exponential
transformation there are closed formulas that express the mean and
covariance functions of the original process in terms of those of the
transformed process (this is not the case for general monotone
transformations).
If you use a quadratic polynomial as the mean of the transformed
process,
then the mean of the origianl process is not a quadratic polynomial.
Although undesirable, this fact is not as bad as it may seem at first
because assuming a quadratic polynomial is a rough approximation in the
first place and the mean of the original process will end up having
similar global features since the exponential transformation is
monotone.
In addition, for many data sets and transformations (including the
exponential) holds that the correlation functions of the transformed
and original processes are very close to each other.
> 2.2. How to do the lognormal cokriging? (as a matter of fact, is it
> anything called 'lognormal cokriging'?) I haven't seen any lognormal
> cokriging models in the books.
>
> I have to cokrige in two ways. In the first case, both cokriging data
> sets are lognormal. In the second case, while one is lognormal, the
> other is normally distributed. I would appreciate if anyone can give a
> short statistical methodology on how these problems can be approched.
> Even giving citations may be helpful for my reference. I am also
> searching for papers that talk about principal component kriging on
> non-normally distributed data & handling anisotropy in lognormal
> kriging.
>
> 3. As the result of the complexity wrapped with the lognormal & the
> transgaussian transformations, is it alright "getting away" with the
> traditional OK/KT keeping a "blind eye" on the normality? [I do not
know
> how scientific this question is. However, I could not resist the
> temptation
> of asking it]
>
> Thank you very much.
>
> Sincerely,
> -/ramani.
+++++++++++++++++++++++++
Since, some of Dr. I. Clark's papers & her book 'Practical Geostatistics
2000' had discussed about lognormal kriging, I also add the link that
connects to those papers.
http://uk.geocities.com/drisobelclark/resume/Publications.html
+++++++++++++++++++++++++
3. Denis ALLARD
---------------
Hi,
your questions are indeed fundamenal and by no means trivial.
1. A correct version of this statement is "Normality is not necessary
or kriging, but since kriging is linear, kriging is equivalent to the
conditinial expectation for normality. As one departs from normality,
the difference between kriging and conditional expectation increases".
This statement is very similar to the statements made for linear
regression. Of course, normality cannot be checked, nor tested, and
strictly speaking never holds.
2.1. The theory of non stationary, non linear geostatistics is yet to be
developped. This could be an interesting part of your dissertation.
2.2 Same for multivariate non linear geostastitics (to my knowledge).
3. Only an experimental study can answer to this question. For some data
sets it is sort of OK; for some, introducing some non linear geostat
really
improves the performances. Experimental studies can be conducted by
cross-validation, either in a leave-one-out fashion, or by subdiving
your data set into a calibration set and a validation set.
Hope this helps
Denis
+++++++++++++++++++++++++
4. Marta Rufino
---------------
Dear Ramani,
I can not answer you to all questions, but I may consider some
suggestions:
>1. Having normality in data always bothers me; I see the text books and
>papers comment on this issue, mostly saying, "Normality is not
necessary
>for kriging, but having normal data distribution makes the prediction
>better." I am yet to understand this statement.
Your objective when doing kriging is to make the best possible
estimator,
which means, that what you want is that the estimations of the predicted
grid are the most close to reality as possible. So when you have normal
data yours estimations should be better, or better sid: more reliable.
>2. On Lognormal kriging, I have two questions.
>2.1. If my (original) data shows a second order polynomial trend, while
>fitting a lognormal distribution, how to proceed further along
variogram
>fitting & kriging?
I suggest you have a look on the package geoR for R
(http://www.r-project.org/).
This is the most complete program and better I found till now (and
besides
it is free- not only you will get a great statistical package).
With it you can do lognormal kriging without any problem.
I would say you have to steps where you can introduce the
trasformation...
First when you produce the variogram (you can consider a value for
lambda
from box cox transformation) Second in the kriging.
Another thing I suggest is that instead of co-kriging, you can do
kriging
with external trend. When you do the krging in geor, in the final you
simply insert an object with your covariate predictions, and this may
have
been produced any way you want and as tbetter is the precision (more
points) the better will be your predictions.
For example, I used depth as a external trend variable, and I used the
coast points from a CD room to make the predictoins, not only my sample
points.
Another important thing about the external trend is that you can
introduce
a quadractic relationship with the covariable.
Also, you can correct for anysotropy,
All this you may do simultaneoulsy within the same model (with geor).
_However I may advert you that often the differences in practice are
very
little, although you may complicate your model very much.
Another possibility is backtransform the model you have produced and do
kriging with it.
I hope this help,
Marta Rufino
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