## Re: Lognormal data. Re: AI-GEOSTATS: Variogram behaviour

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• Digby ... You have to be a little careful with terminology ;-) To a geostatistician multigaussian means multivariate gaussian. Most people call a combination
Message 1 of 3 , Mar 18, 2001
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Digby

> By multigaussian, I meant a combination of
> distributions.
You have to be a little careful with terminology ;-)

To a geostatistician "multigaussian" means
multivariate gaussian. Most people call a combination
of populations a mixture. I have published quite a lot
of 'statistical' work on mixtures of Normals and
lognormals, the earliest being my 1974 paper in the
Transactions of the Institution of Mining and
Metallurgy.

Depending on how heavily your populations overlap, you
may be able to use a combination of indicator and more
'traditional' ordinary kriging to produce estimates
from mixtures. You might want to look at my keynote
address in the MRE symposium for Alwyn Annels last
March, which is on exactly this topic. Those papers
are up on the Web, although I don't have the address
to hand. If you do a Yahoo! search using my name and
MRE 21, you should be able to track it down.

The three parameter lognormal is a good substitute,
particularly if the proportion of one population is
very low.

> What does MLE stand for?
Maximum likelihood estimator. Most geostatistical (and
classical statistical) methods are based on a "least
squares" or closest fit criterion. MLE is based on the
"solution that the data is most likely to fit".
Different philosophy, much harder mathematics. The
examples in my 1974 paper were tackled with MLE.

> Are you saying to use sichel mean of local data to
> estimate the grade in every
> block? i.e. Moving Lognormal Average.
Better to use lognormal kriging. This is a local
sichel-type estimator and you can use Sichel theory
for local confidence bounds.

> What is asymmetric?
A 'central 90% confidence' -- that is 5% at bottom and
5% at top -- will be assymetric around the estimators
because the original distribution is assymetrical
around the mean. The lower confidence level is closer
to the estimate than the upper one, if you keep the %
risk the same.

All explained in my 1987 paper in SAIMM. Not on the
Web, unfortunately. Also updated Sichel's tables. The
paper was refereed by Sichel, so it must be OK ;-)

> Yes, I am concerned the model maybe be highly
> smoothed, and the affect this will
> have on the global estimate.
lognormal kriging avoids the over-smoothing and gives
an unbiassed estimator with the narrowest confidence
intervals. This is, of course, provided your
distribution is lognormal or three parameter
lognormal.

If you need any clarification on this, please do not
hesitate to contact me direct. Complete references can
be found at
http://uk.geocities.com/drisobelclark/resume/Publications.html

Isobel Clark

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• ... Digby: Isobel has responded to your questions about my comments, but i may add something of value ... not ... Maximum likelihood estimators. The least
Message 2 of 3 , Mar 18, 2001
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>===== Original Message From "Digby Millikan" <digbym@...> =====
Digby:

something of value

>Ruben,
>>Regarding lognormal data, if you are only interested in the mean of the
>>regionalized variable and its confidence interval whithin the region, and
not
>>in the spatial mapping itself, you can just use the MLE estimator of the
>>lognormal mean, the (Finney-)Sichel estimator that you mentioned
>
>What does MLE stand for?

Maximum likelihood estimators. The least square estimators, such as those
normally used in goestatistics, are MLE when the distribution of errors is
Gaussian, so that least square estimators are a particular case of MLE. The
(Finney-)Sichel lognormal mean is a case of MLE, though only approximate.

>>to obtain the point estimate,
>
>Are you saying to use sichel mean of local data to estimate the grade in
every
>block? i.e. Moving Lognormal Average.

I'm not familiar with mining or geology. What i say is that if in a region,
the variable of interest measured at several points distribute lognormally,
then the Finney-Sichel lognormal mean is the MLE of the mean, and that
estimate can be used as the mean of the regionalized variable without regards
to its spatial distribution. In the geostatistical paradigm the equivalent
value would be the kriged mean. Insofar as the kriged mean is a better
estimate (more unbiased and with less variance) than the lognormal mean of
lognormally distributed spatial data, you have done something of value by
performing the spatial analysis. But even when the kriged mean and the
lognormal mean are similar in terms of the central estimate and the measure of
precision, the spatial analysis gives additional products which are not
obtained from a purely distributional analysis of the data.

>>and the theory and tables in Land (1975, Tables of
>>confidence limits for linear functions of the normal mean and variance,
>>Selected Tables in Mathematical Statistics, vol. III, Am. Math. Soc.
>>Providence, pages: 385-419) to obtain the asymmetric confidence interval.
>
>What is asymmetric? Are you saying to develop a confidence interval as
compared >to
>kriging which reports the 95% confidence interval, or the estimation
variance.
>I note that sichel provides 90% confidence interval tables in his original
paper on
>the t-estimator found in "Symposium On Mathematical Statistics and Computer
>Applications" SAIMM.

The confidence interval of the lognormal mean is asymmetric because the
distribution is asymmetric. I am not familiar with Sichel's tables but my
previous reference to Land's (1975) tables refer to tables of the quantiles of
the lognormal distribution which are needed to build confidence interval
around the lognormal mean.

Ruben

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