- 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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>of any useful responses to your questions.

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>

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