- Hi Pat

I asked this question a while ago, as I have a very similar problem with my data - too many zeros!!

Here's the summary of the responses I received. I also made a copy of other emails I found on this topic. I will send these on to you as well.

hth

Simon

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

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.

"Patrick J. Doran" a écrit :

> Greetings!

--

>

> I am a graduate student at Dartmouth College and a new user to AI-GEOSTATS. I am hoping to use geostatistical techniques to help describe the distribution and abundance of bird species in a forested ecosystem, with the goal of understanding the mechanisms that influence spatial variation in abundance. I have censused bird species at approx. 350 locations in a 3000ha forested valley and am beginning to analyze the data. Additionally, I have access to a wide variety of other environmental and ecological variables from each census plot.

>

> As a first cut I hope to use variogram analysis and kriging to provide a thorough description of the spatial distribution of the bird census data. However, I am concerned with the nature of the census data - as with most bird census data, the range is generally from 0-3 individuals/point (in intervals of 0.33) and the data is highly skewed to the left with many zero values. For example, for one species at 373 plots the mean per plot is 0.55, the range is 0-2.67 and I have 110 values of "0", 91 values of "0.33", 81 of "0.66", 38 of "1", 30 of "1.33", 14 of "1.66", 5 of "2", 3 of "2.33" and 1 of "2.67"

>

> Due to the highly skewed nature of this data, do I have to transform it before I attempt the variogram analysis and kriging? What transformation would work best? I have attempted to search the ai-geostats archive and the intro texts and have not been able to answer these simple questions. Are there any other references that may help me with this analysis?

>

> Any help would be very much appreciated and will be summarized for the list.

>

> Thanks,

>

> Pat Doran

>

> Patrick J. Doran

> Department of Biological Sciences

> Dartmouth College

> Hanover, NH 03755

> 603-646-3688

> Patrick.J.Doran@...

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-------------------------------------------------------

Simon Brewer email: 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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DO NOT SEND Subscribe/Unsubscribe requests to the list! - Just a word of caution about tests for normality, i.e., your question about

whether normality is required for geostatistical tools.

In geostatistics the data is considered to be a non-random sample from one

realization of the random function (otherwise the underlying concepts of

geostatistics would not apply). There are at least two different notions of

normality to consider:

1. If one considered the data obtained from all the locations in the region

of interest, would the distribution of this data be normal? The data is a

sample from this distribution but is not a random sample (random location

selection is not quite the same thing as random sampling).

Note that the region of interest is often defined after the fact, i.e.,

after data has been collected.

2. Assuming that the random function is strongly stationary, is the

univariate distribution normal ? Note that the data is not a sample from

this distribution.

The standard tests for normality are based on random sampling FROM the

distribution, it is difficult to modify the tests to allow for the spatial

correlation (especially without knowing the "true" variogram/covariance).

Hence one should be cautious in treating the results of a test of

hypothesis (for normality) as really definitive.

Although not often mentioned, we use a form of egodicity in geostatistics

but the simplest form of this pertains to moments not distributions. For

example, the weak law of Large numbers implies that the sample proportion

(for a random sample) will converge (in a certain sense) to the population

proportion. Similarly, we expect the sample mean to converge to the

population mean (as the sample size increases) but note that the

distribution of the sample mean tends to the standard normal not to the

original distribution.

We do know that if the random function is multivariate normal that the

simple kriging estimator is the conditional expectation and hence is THE

minimum variance, unbiased estimator/predictor. In general however it is

only the minimum variance, unbiased LINEAR estimator/predictor.

The bottom line however is probably not a statistical question, do the

geostatistical tools produce useful results? "Useful" has to be decided by

the user, not the statistician. Across a wide spectrum of applications the

answer seems to be yes but in specific instances the answer may be no

because of a lack of data, difficulty in estimating/modeling the variogram,

sensitivity of any linear estimator to unusual data values, etc.

Donald E. Myers

Department of Mathematics

University of Arizona

Tucson, AZ 85721

http://www.u.arizona.edu/~donaldm

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DO NOT SEND Subscribe/Unsubscribe requests to the list! - Assume that we have two sets of geostatistical data. Is there any

statistical test to determine whether variograms on those two sets are the

same?

Thanks,

A. Lazarevic

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DO NOT SEND Subscribe/Unsubscribe requests to the list! - I asked a similar question a while back, and was sent the following

reference by Andrew Lister

Kabrick J. M., Clayton M. K. & McSweeney K. 1997. Spatial patterns of carbon

texture on

drumlins in northeastern Wisconsin. Soil Sci. Soc. Am. J. 61(2):541-548

This contains a method for estimating the errors on a semivariogram value,

and testing for differences between them. If anyone has any comments on this

methodology I would be interested to hear them.

Dan

_____________________________________

Mr. Daniel P. Bebber

Department of Plant Sciences

University of Oxford

South Parks Road

Oxford OX1 3RB

UK

Tel. 01865 275000 Fax. 01865 275074

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DO NOT SEND Subscribe/Unsubscribe requests to the list! - One should be a little careful about accepting the validity of a test for

the "equality" of two variograms. If one uses an estimator such as the

sample variogram, one only obtains estimates of the values of the variogram

for a finite number of lags (note that dealing with a possible anisotropy

makes it even more complicated). Moreover the reliability of these

estimates varies, in part because the numbers of pairs will vary. If one is

using the variogram for kriging or simulation then one is most interested

in the behavior of the variogram, i.e., the values for short lags and

unfortunately the short lags usually have the smallest numbers of pairs. If

one uses least squares or maximum likelihood then one must first choose a

model (or models in the case of a nested model) and then one of these is

used to estimate the parameters.

There is an old paper by Davis and Borgman in Mathematical Geology (circa

1980) on the distribution of the sample variogram, they give two results:

(1) beginning with an assumption of multivariate normality (which is not

testable) and an assumed model type then they obtain numerical results for

the distribution , (2) they obtain asymptotic results which are

theoretically interesting but probably not much help in practice.

There is also a paper in Mathematical Geology, circa 1990, on the "true"

numbers of pairs. The problem as is well known is that there is an

interdependence between the pairs used to estimate for one lag and those

used to estimate for another. The author has to assume multivariate

normality to derive the results.

It is known that the kriging estimator is relatively robust with respect to

the variogram, i.e., slight changes in the variogram will result in only

slight changes in the kriging weight vector and hence in general only

slight changes in the kriged values. There are at least two different ways

to quantify the "distance" between two variograms, these correspond to a

notion of continuity. A third one corresponds to differentiability, none of

the three implies the others.

In practice one often uses a search neighborhood in kriging hence it is

only of interest whether the variograms match or are at least close for up

to some maximum lag. One will have very little information about the

variogram for longer lags anyway.

In general statistical tests will require some distributional assumptions

and these are hard to obtain for variograms/variogram estimators. It is an

interesting question to ask, i.e., are the variograms for two different

variables or the same variable for two different regions the same but one

that will be hard to test without making very strong assumptions

(non-testable assumptions).

Finally one might want to consider the question of sample location pattern

design relative to testing the equality of two variograms. I have an old

paper with A.W. Warrick on the design of sampling plans in order to control

the numbers of pairs for each lag. If one assumes isotropy (it is even more

complicated in the case of anisotropy) then the pattern that generates an

equal number of pairs is a spiral, not a very practical result.

Note also that if one assumes normality then the distribution of the

half-squared differences will be Chi-Squared (one can see this effect in

most sample variograms, the VARIO component of GEOEAS will provide

histograms for these distributions). Not a particularly nice distribution

for testing because of the "fat" tails.

Donald E. Myers

Department of Mathematics

University of Arizona

Tucson, AZ 85721

http://www.u.arizona.edu/~donaldm

At 05:02 PM 2/14/00 -0800, you wrote:>Assume that we have two sets of geostatistical data. Is there any

--

>statistical test to determine whether variograms on those two sets are the

>same?

>

>Thanks,

>

>A. Lazarevic

>

>

>

>

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