AI-GEOSTATS: Regression Slope and kriging - the answers
- Dear ai-geostats readers
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
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 ?
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
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
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
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
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
Donald E. Myers
>1.Kriging is often cited as least squares regressionSimple 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
>2.I read in seveal papers that it is possible toAll 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:
(note capital P)
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