## GEOSTATS: Re: Normality

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• 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
Message 1 of 6 , Feb 14, 2000
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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• 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.
Message 2 of 6 , Feb 14, 2000
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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• 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
Message 3 of 6 , Feb 15, 2000
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
Oxford OX1 3RB
UK
Tel. 01865 275000 Fax. 01865 275074

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• 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
Message 4 of 6 , Feb 15, 2000
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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>*As a general service to list users, please remember to post a summary
>of any useful responses to your questions.
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>"unsubscribe ai-geostats" in the message body.
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>

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