A question about simulation

I understand in the simulation methods SGS that each node the data a kriging

is performed than an value is drawn from a normal distribution with the

local mean (from data and previously simulated nodes captured in a search

neigbourhood) and variance defined by the kriging variance.

My understanding of kriging variance is that this variance is really an

index of data configuration independent of the data values. This leads to

my question.

How a set of multiple simulations capture information about the variability

of conditioning data values when the kriging variance is only a data

location index. Specifically, it is possible to have the same conditioning

data configuration but the variability of the values attached to the data

can be quite different. How is this recognised in the simulation process.

Thanks

Joe

Hi Joe,

After the backtransform, the distribution of simulated values

at each node is not Gaussian anymore and its variance can be

used as a local index of uncertainty, which accounts for both

the range of surrounding values and their closeness in

terms of data configuration.

Regards,

Pierre Goovaerts

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

Dr. Pierre Goovaerts

President of PGeostat, LLC

Chief Scientist with Biomedware Inc.

710 Ridgemont Lane

Ann Arbor, Michigan, 48103-1535, U.S.A.

E-mail: goovaert@...

Phone: (734) 668-9900

Fax: (734) 668-7788

http://alumni.engin.umich.edu/~goovaert/

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Answer Donald E. Myers

A couple of additional observations about SGS. This algorithm is based on

properties of the multivariate normal distribution. Let Z0, Z1,....., Zn be

jointly distributed with all having finite variances then the best estimator

of Z0 given Z1,...., Zn is the conditional expectation of Z0 given Z1,...,

Zn. "Best" in this case means unbiased and with minimal estimation variance.

IF in addition the joint distribution is multivariate normal then the

conditional expectation has a particularly simple form, i.e. it is the same

as the simple kriging estimator. MOREOVER the conditional distribution of Z0

given the Z1,..., Zn is normal AND the mean of this distribution is the

conditional mean and its variance is the estimation variance. All of the

above is true without any reference to spatial problems (except in drawing

the analogy to simple kriging).

One critical practical problem then that arises is how does one know that

the assumption of multivariate normality is satisfied? In practice one does

not, it is simply an assumption. Note that doing a statistical test on the

data does not answer the question, the data is not the right kind to test

for multivariate distributional properties.

In the SGS algorithm we sort of turn things around. The simple kriging

equations are derived without any distributional assumptions and we know how

to compute the kriging variance and the kriged estimate without the

distributional assumptions. IF the multivariate normality assumption were

true then the simple kriging value would be the conditional mean, i.e., the

mean of the conditional distribution and the kriging variance would be the

variance of the conditional distribution. As you have noted the kriging

variance is computed without using the data values and is completely

determined by (1) the covariance model, (2) the pattern of the data

locations. Under the multivariate normality assumption however this is the

right quantity. There is no way to get away from that important assumption.

A histogram and other descriptive statistics are useful in deciding whether

that assumption is "reasonable" but you can't absolutely answer the

question.

Donald E. Myers

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

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