- Hello,

By multigaussian, I meant a combination of distributions. I used to assist with reserve reports with data that

displayed a close to three parameter lognormal distribution. Although the histogram displayed mixed distributions,

three parameter lognormal was the closest so I used sichel tables as discussed in Ch. 1 Geostatistical Ore Reserve Estimation M. David. Yes I could see data would be lost but the total reserve would not be skewed. This may be

useful for global reserve statements of an operating mine, while for grass roots operations alternates should be used

to avoid loss of real data.

Ruben,>Regarding lognormal data, if you are only interested in the mean of the

What does MLE stand for?

>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

>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.

>and the theory and tables in Land (1975, Tables of

What is asymmetric? Are you saying to develop a confidence interval as compared to

>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.

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.

> This solution might not be popular among members of the list, however, since it

Yes, I am concerned the model maybe be highly smoothed, and the affect this will

>involves abandoning the spatial analysis.

have on the global estimate.

Regards Digby Millikan.

[Non-text portions of this message have been removed] - Digby

> By multigaussian, I meant a combination of

You have to be a little careful with terminology ;-)

> distributions.

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

Better to use lognormal kriging. This is a local

> estimate the grade in every

> block? i.e. Moving Lognormal Average.

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

lognormal kriging avoids the over-smoothing and gives

> smoothed, and the affect this will

> have on the global estimate.

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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* Support to the list is provided at http://www.ai-geostats.org >===== Original Message From "Digby Millikan" <digbym@...> =====

Digby:

Isobel has responded to your questions about my comments, but i may add

something of value

>Ruben,

not

>>Regarding lognormal data, if you are only interested in the mean of the

>>regionalized variable and its confidence interval whithin the region, and

>>in the spatial mapping itself, you can just use the MLE estimator of the

Maximum likelihood estimators. The least square estimators, such as those

>>lognormal mean, the (Finney-)Sichel estimator that you mentioned

>

>What does MLE stand for?

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,

every

>

>Are you saying to use sichel mean of local data to estimate the grade in

>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

compared >to

>>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

>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

The confidence interval of the lognormal mean is asymmetric because the

>Applications" SAIMM.

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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