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

of your interpolation grid.

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 -

again please let me know.

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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PLEASE NOTE NEW EMAIL ADDRESS : simon.brewer@...

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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