Please find below the replys I received on my questions regarding regression

slope and kriging

Dear ai-geostat following the original questions below.

Thanks to Donald Myers and Isoble Clark

I have some questions regarding kriging.

1.Kriging is often cited as least squares regression method - this I

understand for liner regression but how does this actually occur in the

kriging matrix ? Are the covariances the square values that are being

minimised ?

2.I read in seveal papers that it is possible to calculate the slope of

regression from parameters of the krigin system. Specifically:

(Block variance-kriging variance + abs(lagrangian))/(block varaince -

kriging variance + 2 x abs(lagrangian))

I can follow the derivation of the kriging variance and I can see the

purpose of the lagrangian, where does the block variance come from and how

(in conceptual terms) does this equation give the regression slope of true

versus actual block grades ?

Replys

Donald E. Myers

A couple of observations about kriging and regression

You might want to look at a paper by S. Goldberg in the J. American

Statistical Assn, 1962 where derives what turns out to be the kriging

estimator (albeit without any acknowledgement of the geostatistical

literature which was pretty sparse at that time. He does it entirely in

terms of regression.

Several important distinctions or contrasts between kriging and regression

(partly theoretical and partly practical)

1. For regression the response variable does not have to be the same as the

regressor/control variable(s)

2. The regression model includes an error term (there are different

possibilities for the assumptions on this error term, e.g., not

intercorrelated, constant variance,. Usually the regression approach focuses

on "removing" the error term.

3. At least sometimes in regression, the regressor /control variable(s) are

deterministic. To that degree, universal kriging is the analogue of

generalized regression (see a paper by M. David et al in Math Geology a few

years ago comparing universal kriging with a nugget effect model vs trend

surface analysis)

4. This one is probably most important, in the statistical form of

regression, the covariance values are estimates not computed (i.e., computed

from a model for the covariance function or variogram) .

5. From a least squares perspective, one can fit a "regression" model to

data without any statistical assumptions at all (of course then strictly

speaking you can't do any statistical inference).

6. Finally, and I am sure that this goes back much further (presumably it is

part of the motivation for Krige's and Matheron's work), if you have

jointly distributed random variables Z0, Z1, ...., Zn each with finite

variances then the "optimal" estimator of Z0 given the data Z1,..., Zn is

the conditional expectation of Z0 given Z1,...,Zm ("optimal" meaning

unbiased and minimal variance of the error of estimation). Moreover if in

addition the random variables are multivariate gaussian then the conditional

expectation is linear in Z1,...,Zn. That is,

E[Z0 | Z1,...,Zn] = mu0 + Sum (i=1,..,n)ai [ Zi- mui]

mo is the expected value of Z0, the mui's are the expected values of the

Zi's. This of course looks like the Simple Kriging estimator except that

usually we would assume that all the mu's are the same. This connection is

exploited by Journel in his 1980 paper in Math Geology discussing the bias

correction for lognormal kriging.

7. In kriging the form of the estimator is assumed, i.e., Simple kriging vs

Ordinary/Universal kriging.. IN PARTICULAR IT IS ASSUMED TO BE A LINEAR

FUNCTION OF THE DATA. There is no distributional assumption (although there

are authors that from time to time that keep saying there is a multivariate

gaussian assumption). There is some form of stationarity assumption although

this is primarily used to justify estimating and modeling the

covariance/variogram from the data. Two conditions are imposed on the

coefficients in the estimator, unbiaseness and minimal estimation variance.

These together with the lineaity assumption are sufficent to derive the

kriging equations, they are analogous to the regression equations but not

exactly the same. The kriging variance is the minimized estimation variance

(obtained from the specified covariance/variogram model).

Now to your questions, "slope" is only going to really make sense if there

is only one regressor variance, i.e., the regression equation is not only

linear but has only one variable. While the kriging equations will work with

only one data point one would usually not restrict it to that.

Kriging can be used in two general forms, point estimation and "block"

estimation. The form of the estimator is the same but for "block" estimation

there is a modification that accounts for the change in spatial correlation

resulting from a change in support. There really isn't an analogue of this

for regression (as an example however of an attempt to do this see

1984, DeVerle Harris and D.E. Myers, World Oil Resources/A Statistical

Perspective. in Advances in Energy Systems and Technology, Vol 4, Academic

Press).

It is also possible to use "non-point" grades, to that extent you might

think of the :"slope" as relating "true" to "estimated" grades but unless

you use a unique search neighborhood the "slope" will change from estimate

to estimate.

All of the above is probably a bit long winded but my point is that the

connection between kriging and regression is not a simple one, it has

multiple facets.

Donald E. Myers

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

Isobel Clark

>1.Kriging is often cited as least squares regression

Simple kriging as invented by Danie Krige in the

1950's was exactly a linear regression method. The

'kriging' system has a 'left-hand-side' consisting of

the variance/co-variance matrix between sample pairs

and a 'right-hand-side' consisting of the co-variances

between each sample and the unknown value. Krige

derived the variances and co-variances empirically

from 50 years of historical data.

In the early 1960s, Matheron's work put this on a

modelling (theoretical) footing by suggesting that the

co-cariances could be modelled by a function -- the

semi-variogram reversed. Thus the l.h.s became

co-variances or semi-variograms depending on your

personal preference and likewise the r.h.s.

However, Matheron also introduced the notion that the

weights should add up to one and invented 'ordinary'

kriging which is not (strictly) classical linear

regression.

>2.I read in seveal papers that it is possible to

All explained in my 1983 paper 'regression revisited'

>calculate the slope of regression from parameters of the krigin system.

in Mathematical Geology. I can send you a copy if you

can't find it or you can download it from:

http://uk.geocities.com/drisobelclark/resume/Publications.html

(note capital P)

Isobel Clark

http://geoecosse.bizland.com/whatsnew.htm

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