GEOSTATS: Non-normal distribution

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• Non-normal distribution I posted a question to the mailing list on the 30 Nov, concerning the non-normal distribution of some data, that I am trying to map
Message 1 of 2 , Dec 4, 1998
Non-normal distribution

I posted a question to the mailing list on the 30 Nov, concerning the
non-normal distribution of some data, that I am trying to map using the
kriging process. To follow this up, I am posting a brief synopsis of the
(many) replies I received. So first, a big THANK YOU to everyone who
took the time to send me answers, suggestions and references. I have put
where possible the names of the people who have supplied the
information. If I've missed any one - I'm very sorry, please let me
know.

1) Should the data be normally distributed? The first point, stressed by
a number of people, was that there is NO requirement for data to be
normally distributed for use in the computation of semi-variograms, or
predictions made by kriging. (Gilles Guillot, Daniel Guibal, Pierre
Goovaerts). However, as it is a linear estimator, kriging is sensitive
to a few large samples, which may bias the results. Notably, the
variogram may become unstructed, close to a pure nugget effect. As Med
Bennett pointed out, normality is a rare find in mining and
environmental data! - On this point it also worth reading Don Myers'
message dated 25 June 1997, and the subsequent postings.

A number of tests of normality have been mentioned, and I list some of
them here:
Q-Q (quantile-quantile) plot
Kolgorov-Smirnov test
Data density vs. Theoretical density

2) What can I do with the data?
Three possibilities were suggested, as follows:

a) Transformations - Skewed data may be transformed to a normal
distribution by a non-linear transformation, e.g. logarithmic. If,
however, the data are then back-transformed to real values, then the
unbiased property of the kriging estimates is lost. Other
transformations have been suggested: the family of Box-Cox power
transformations, the square-root transformation (for count data), the
arcsine square-root transformation for percentage data, and the normal
score transform, of which, I'm afraid, I know nothing! (Joyce Witebsky,
Daniel Guibal, Vera Pawlowsky-Glahn)

However, for my data (which is the percentage fossil pollen of various
tree taxa in lake sediments), the distribution contains a large number
(approx 50%) of zero values (where no pollen was found). Any attempt to
transform this simply results in a large number of arbitrary values,
replacing the zeros. So, another method was needed...

b) Data partitioning. If the zero values were grouped in a way, that a
physically different domain could be identified, then it would be
possible to construct variograms for the different domains. I cannot do
this, except by arbitrary domaining, which would defeat the object of
the exercise! (Daniel Guibal, Vera Pawlowsky-Glahn),

c) Indicator kriging. This would allow an estimate at each prediction
location of whether or not the fossil pollen would have been present
(zero or non-zero value). This could then be combined with ordinary
kriging of the non-zero points. (Daniel Guibal).

Alternatively, the range could be discretised, using a number of
thresholds. This I have taken directly from Pierre Goovaerts message:
What I would suggest is to use an indicator approach,
that is:
1. discretize the range of variation of your data using
a given number of thresholds, say 5: the first threshold
would be 0% (which is close to the median of your sample
distribution) and 4 other thresholds corresponding to the
0.6, 0.7, 0.8 and 0.9 quantiles of your distribution.
2. for each threshold, code each observation into an indicator value
which is zero if the measured percentage is larger than the threshold
and one otherwise.
3. Compute and model the 5 corresponding indicator semivariograms,
that is the semivariograms of indicator values.
4. Use indicator kriging to interpolate the probabilities
to be no greater than each of the 5 thresholds at the nodes
5. At each location, you can now model the conditional cumulative
distribution function which provides you with the probability
that the unknown percentage value is no greater than any
given threshold. You could use the mean of that distribution
as your estimate and the variance as a measure of uncertainty.

The method of indicator kriging seems most appropriate to the data I
have, it appears to be a more robust method. So I will try this at the
weekend - wish me luck...

Well, it's been quite a crash course!! I don't believe I have understood
everything, so if anyone sees any blinding errors in this message -

If anyone wants a copy of all the replies received, and a small
collection of other messages related to the issue of non-normality,
please contect me. I did not include all this on this mail, to keep it's
size down. On the other hand, if it is generally felt that all replies
should be posted, I would be happy to do that.

Thanks again

Simon
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Simon Brewer

European Pollen Database
(Laboratoire de Botanique Historique et Palynologie)
(IMEP CNRS URA 1152)
(Faculte St Jerome - Aix Marseille III)

Centre Universitaire d'Arles
Place de la Republique
13200 Arles - France
Tel: (33)-(0)4 90 96 18 18 Fax: (33)-(0)4 90 93 98 03
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